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1 Statistical mechanics, dynamical systems, and turbulence
Abstracts
We present here abstracts of some lecture series in the areas of statistical mechanics, dynamical systems, and turbulence together with reading lists in the hope that they will provide a useful guide to others who wish to learn these subjects.
2 A simple proof of C. Siegel's center theorem
Raphael De laLlave
We give an elementary proof of a particular case of C. Siegel's center theorem, based on a method of M. Herman. Even if the proof has less generality than the standard one, it is simpler and provides sharper bounds.
3 On copositive matrices and strong ellipticity for
isotropic elastic materials
H. Simpson and S. Spector
In this paper we establish necessary and sufficient conditions for the strong-ellipticity of the equations governing an isotropic (compressible) nonlinerly elastic material at equilibrium. Our work extends results of Knowles and Sternberg [5] who obtained such conditions for both ordinary and strong ellipticity in the special case when the underlying deformations are plane.
4 Vector fields in the vicinity of a compact invariant
manifold
George R. Sell
Let us consider two vector fields
(1) X' = F(X)
(2) Y' = F(Y)
defined on a give Euclidean space E where F and G are of class C^{N+1}.
Furthermore, assume that there is a smooth compact manifold M smoothly imbedded
in E and that M is invariant for both vector fields. Also that F and G agree
on M, i.e. F|M = G|M.
We wish to study the question of C^{S}-conjugacies between (1) and (2).
5 Non-linear stability of asymptotic suction
Milan Miklavcic
A flow over a plane y = 0 in R^{3} given by
U(x,y,z) = (1 - e^{-y}, -1/R, 0)
is called an asymptotic suction velocity profile [12]. R>0 is the Reynolds
number. U satisfies the Navier-Stokes equation
ðv / ðt + (v ·
) v = -
p_{0} + 1/R v
div v = 0
with p_{0} = 0. In the present paper it is proved that the stability
of U for small perturbations which initially decay exponentially in the y direction
and are periodic in the x and z direction is governed by the eigenvalues of
the classical Orr-Sommerfeld equation [1, 8, 12]. For precise statements see
Theorems 4, 5, 9, and 15.
6 A simple system with a continuum of stable inhomogeneous
steady states
Hans Weinberger
The system
u_{t} = {(1 + v)}_{xx}
+ (R_{1} - au - bv)u
v_{t} = (R_{2}
- bu - av)v
{(1 + u)}_{xx}
= 0 at x = 0 and x = 1
with
1/2 (a/b + b/a) < R_{1}/R_{2}
< a/b
and
> a(a^{2} - b^{2}) / 2abR_{1} - (a^{2} +
b^{2}) R_{2}
was considered by M. Mimura [2] as a model for the population densities of two
competing species, one of which increases its migration rate in response to
crowding by the other species. It is a special case of the model of N. Shigesada,
K. Kawasaki, and E. Teramoto [3].
7 Period 3 bifurcation for the logistic mapping
Bau-Sen Du
In the context of continuous mappings of the interval, one of the most striking features may be Sharkovsky's theorem [6] which, among other thing, shows that the existence of a period 3 point implies the existence of periodic points of every period (see also [2, 5]). Therefore, for a one-parameter family of interval mappings, the determination of period 3 bifurcation points may be interesting. In recent years, the logistic mapping f_{}(x) = 1 - x^{2} has been entensively studied ([1, 4]). By using computer simulation for this family f_{}(x), as the parameter is increased from 0, we can observe the Feigenbaum "cascades" [3]. That is, stable periodic points of double periods accumulate in a geometric and universal way. As the parameter is approximately equal to 1.7498 ([1, p.129]), there seems to be a period 3 bifurcation. In this note, we show that this family f_{}(x) does have a period 3 bifurcation exactly at = 7/4.
8 Optimal Numerical Approximation of a Linear Operator
Hans Weinberger
Many linear problems of numerical analysis can be formulated in the following way: One is given a set of n linear data Nu = and a bound for the norm ||u||_{B} of an otherwise unknown element of u of a hilbert space B. One wishes to find a best approximation to the element Su, where S is a bounded linear operator from B to another Hilbert space . For example, Su may be the solution of an ordinary or partial differential equation with right-hand side, initial data, or boundary data u.
9 Three component ionic microemulsions
L.R. Angel, D.F. Evans, and B. Ninham
Necessary design features of microemulsions formed from cationic surfactant without any requirement for cosurfactant are illustrated by a study of microemulsions formed from didodecyldimethylammonium bromide in various oils. Ease of purification, preparation and manipulation give this and related systems a considerable advantage over conventional systems in enhancing our understanding of microemulsions and emulsion behavior.
10 Surfactant diffusion; new results and interpretations
D.F. Evans, D. Mitchell, S. Mukherjee, and B. Ninham
Data for surfactant diffusion are reproted for sodium dodecylsulfate at 25° and tetradecyltrimethylammonium bromide at 25°, 90°, and 135°C, as measured by Taylor tube dispersion. These data are analyzed in terms of two limiting forms of theory, one appropriate to "slow" reaction rates, the other to "fast" rates. It is shown that the usual extrapolation to the critical micelle concentration to infer intrinsic diffusion constants is not permissible. The data is explicable if transport occurs by a process wherein ionic micelles disassociate, diffuse as monomers and reassemble into micelles. This is directly contrary to current ideas on diffusion of surfactants.
11 Remark about the final aperiodic regime for maps
on the interval
Leif Arkeryd
We consider families of maps on the interval with one maximum, and prove the geometric convergence of the bifurcation parameter for the case of superstable periodic orbits converging towards the final aperiodic regime.
12 Manifolds of global solutions of functional differential
equations
Luis T. Magalhães
This paper consider smooth invariant manifolds of global solutions of retarded Functional Differential Equations in R^{n}. The persistence, under small perturbations, of such manifolds where the flow is given by an Ordinary Differential Equation in R^{n} is studied. The novelty of the present approach lies on the use of the dynamics of the flow on the manifolds, instead of their attractivity properties.
13 Tori in resonance
Kenneth Meyer
This paper gives three examples of ordinary differential equations which depend on one or more parameters and which admit invariant tori for some values of the parameters. These examples illustrate how invariant tori evolve as the parameters are changed; in particular how they disappear, bifurcate and lose smoothness. The equations presented are choosen to be as simple as possible in order to clearly show the interesting phenomenon without unnecessary details. However, the theory of normal forms and unfoldings was used to select typical examples, but no attempt will be made to define precisely the universe of discourse where these examples are generic. The unfolding of invariant tori would consist of a mutitude of cases not all of which are that interesting.
14 Surface models with nonlocal potentials: Upper
bounds
C. Eugene Wayne
The behavior of fluctuations in a class of surface models with exponentially decaying nonlocal potentials is studied. Combining a Mayer expansion with a duality transformation we demonstrate the equivalence of these models to a class of two dimensional spin systems with nonlocal interactions. The expansions give sufficient control over the potentials to allow the fluctuations to be bounded from above by the means of complex translations in the spin representation of the model.
15 On stability and uniqueness of fluid flow through
a rigid porous medium
K.A. Pericak-Spector
We study a set of equations describing the flow of an incompressible viscous fluid through a rigid porous medium. Existence, uniqueness and stability results are established for the case of a region impregnated with fluid, and uniqueness for an unsaturated region.
16 Smooth linearization near a
fixed point
George R. Sell
In this paper we extend a theorem of Sternberg and Bileckii. We study a vector field, or a diffeomorphism, in the vicinity of a hyperbolic fixed point. We show that if the eigenvalues of the linear part (at the fixed point) satisfy 2N-algebraic conditions (where N > 1), then there is a C^{N}-linearization in the vicinity of this fixed point. If the fixed point is stable, then the C^{N}-linearization theorem follows when only (N + 1)-algebraic conditions are satisfied. Examples are given which show that the first of these results is sharp. An application to celestial mechanics is included.
17 A nonlinear stability analysis of a model equation
for alloy solidification
David Wollkind
Controlled plane front solidification of alloys and other binary substances under an imposed temperature gradient is used in practice to grow single crystals, refine materials (e.g., zone refining), and obtain uniform or non-uniform composition within the material grown [1]. The most important industrial applications of this type of solidification are for growth of crystals for metal oxide semiconductors (MOS's) [1]. Growth of oxide crystals for jewels is another, much older commercial application of single crystal growth [1]. Another important application is in growth of oxides for laser systems and other optical devices [1]. Further industrial applications arise in ingot casting and in the steel and glass industries [2]. For all of these solidification situations involving binary materials, quantitative predictions of interfacial cellular morphology, including information on cell size and intracellular solute distribution, prove to be extremely valuable and are of a particular aid to industrial researchers.
18 Local
conjugacy on the Julia set for some holomorphic perturbations of
Pierre Collet
We consider holomorphic perturbations f of f_{0}, f_{0}(z) = z^{2}, which are small in a neighborhood of the unit circle (the Julia set of f_{0}). We show that if the C^{1} conjugacy invariants of f and f_{0} are identical, then f and f_{0} are conjugate on their part of the Julia set which remains near the unit circle.
19 On the modified Bessel functions of the first
kind (1st paper); and On barrelling for a material in finite elasticity
(2nd paper)
Henry C. Simpson, Scott J. Spector
A. On the modified Bessel functions of the first kind: We consider
the functions v_{} (t)
t I_{} (t) / I_{
+ 1} (t) where I_{} are
the modified Bessel functions of the first kind of order
0. We prove that v_{}
is strictly monotone and strictly convex on R^{+}. These results have
application in finite elasticity.
B. On barrelling for a material in finite elasticity: In this paper we
investigate the question of stability for a solid circular cylinder, composed
of a particular homogeneous isotropic (compressible) nonlinearly elastic material,
that is subjected to compressive end forces in the direction of its axis (so
as to give fixed axial displacements at the ends).
20 Linearization and global dynamics
George R. Sell
In this paper we show how the spectral theory of linear skew-product flows
may be used to study the following three questions in the qualitative theory
of dynamical systems: (1) when is an -limit
set or an attractor a manifold?
(2) Under which conditions will a dynamical system undergo a Hopf-Landau bifurcation
from a k-dimensional torus to a (k + 1)-dimensional torus?
(3) When is a vector field i the vicinity of a compact invariant manifold smoothly
conjugate to the linearized vector field and how smooth is the conjugacy?
21 Global Lyapunov exponents, Kaplan-Yorke formulas
and the dimension of the attractors for 2D Navier-Stokes equations
P. Constantin and C. Foias
We study the fractal and Hausforff dimensions of the universal attractor for the Navier-Stokes equations in two space dimensions. The finite dimensionality of the attractors for the Navier-Stokes equation was first implicitly proven in [16] and explicitely in [10]. The subject has been investigated recently by several authors.
22 Stability for semilinear parabolic equations
with noninvertible linear operator
Milan Miklavcic
Suppose that
x'(t) + Ax(t) = f(t, x(t)),
t ≥ 0
is a semilinear parabolic equation, e^{-At} is bounded and f satisfies
the usual continuity condition. If for some 0 <
≤ 1, 0 < < 1,
p > 1, > 1
||t^{}
Ae^{-At}|| ≤ C, t ≥ 1
||f(t, x)|| ≤ C(||A^{
}x|| ^{p} + (1 + t)^{-}),
t ≥ 0
whenever ||A^{ }x|| + ||x|| is
small enough, then for small initial data there exist stable global solutions.
Moreover, if the space is reflexive then their limit states exist. Some theorems
that are useful for obtaining the above bounds and some examples are also presented.
23 Perturbations of geodesic flows on surfaces of
constant negative curvature and their mixing properties
P. Collet, H. Epstein, and G. Gallavotti
We consider one parameter analytic Hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones. One condition says that the "Hamilton-Jacobi" (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic Hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, R) group. We also prove that the analytic functions on the phase space for the eodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time ("up to logarithms").
24 On the thermodynamics of interstitial working
J.E. Dunn and J. Serrin
In order to model fluid capillarity effects, the Dutch physicist Korteweg
formulated in 1901 a constitutive equation for the Cauchy stress that included
density gradients. Specifically, Koretweg proposed for study a compressible
fluid model in which the "elastic" or "equilibrium" portion of the Cauchy stress
tensor T is given by
T = $hat T$ ( ,
, grad ,
grad^{2} )
= (-p +
+ |grad |^{2})
1 + grad
grad
+ grad^{2} ,
where = (x,
t) is the density of the fluid at the place x at time t, where grad
and grad^{2} are, respectively,
the first and second (spatial) gradients of
with respect to x (with = tr (grad^{2} )
= the Laplacian of ), and where p, ,
, ,
and are material functions of
and the temperature . To model viscous
effects in the dynamic response of his fluids, Korteweg added to the right hand
side of (1.1) the classic form of Cauchy and Poisson, i.e., (tr
D) 1 + 2 D, where D is the usual stretching
tensor of hydrodynamics, and where
and are the usual viscosity coefficients
and may depend on and .
25 On the absence of bifurcation for elastic bars
in uniaxial tension
Scott J. Spector
We prove that an elastic bar undergoing uniaxial tension will not neck before the axial load on the bar attains a (local) maximum. Further, if the bar is in a hard loading device we show that necking is delayed until after maximum load is achieved. The key ingredient in the latter result is a generalized Korn inequality.
26 Maps
of an interval
W. A. Coppel
(There are three expository lectures in this preprint)
27 Phase transitions in the Ising model with traverse
field
James Kirkwood
The Ising model perturbed by a small transverse field is shown to have a phase transition by two methods. With the first method, using a Peierls' contour argument, we are only able to show that spontaneous magnetization occurs with the transverse field goint to 0 as ^{-1/3}. With the second method, which used reflection positivity, long range order is shown to occur for a small transverse field independent of temperature.
28 The asymptotics of solutions of singularly perturbed
functional differential equations: distributed and concentrated delays are different
Luis Magalhães
This paper illustrated the differences between systems with distributed delays and systems having only concentrated delays in what concerns the asymptotic rates of solutions of singularly perturbed linear retarded functional differential equations. An example of a system with distributed delays shows that the introduction of a ``slow" variable coupled with the ``fast" variable may decrease the asymptotic rates of solutions observed when the perturbation parameter is close to zero. Such a situation cannot happen for ordinary differential equations, or even for differential-difference equations.
29 Homoclinic orbits for flow in R^{3}
Charles Tresser
We propose a rough classification for volume contracting flows in R^{3} with chaotic behavior. In the simplest cases, one looks at the nature of a homoclinic loop for the flow. Most configurations have been studied at length in the literature; here we examine briefly the ``forgotten" case.
30 About some theorems by L.P. Sil'nikov
Charles Tresser
Some theorems by L.P. Sil'nikov, which describe the dynamics in the neighborhood of homoclinic orbits, bi-asymptotic to a saddle focus, and initially proved for real analytic vector fields are collected here. Recent results in dynamical systems theory allow us to precise some of the conclusions and to generalize these theorems to the C^{1,1} class. Certain heteroclinic loops involving a saddle focus are also considered.
31 On the renormalized coupling constant and the
susceptibility in _{4}^{4}
field theory and the Ising model in four dimensions
Michael Aizenmann
We discuss the Euclidean _{4}^{4} field theory, and the critical behavior in ferromagnetic systems in four dimensions. It is rigorously shown that there are at most logarithmic corrections to the mean field law in the behavior of the magnetic susceptibility = _{4}^{4} S_{2} (0, x). Furthermore, if any such corrections are present in a continuum limit which is used to construct a _{4}^{4} field theory the limiting theory would be non-interacting. Our analysis extends to ferromagnetic systems of variables which belong to the Simon-Griffiths class.
32 The KAM theory of systems with short range interactions
I
C. Eugene Wayne
The Kolmogorov, Arnol'd, Moser (KAM) theory [15, 1, 16] proves that ``small" perturbations of integrable Hamiltonian systems possess ``large" sets of initial conditions for which the trajectories remain quasiperiodic. In this paper we discuss how the ``strength" of the allowed perturbation varies with the number of degrees of freedom, N, in the system.
33 Temporal and spatial chaos in a Van der Waals
fluid due to periodic thermal fluctuations
M. Slemrod and J.E. Marsden
This paper applies the Mel'nikov technique to prove the existence of deterministic chaos in two problems for a Van der Waals fluid. The first problem shows that temporal chaos results as a result of small time periodic fluctuations about a subcritical temperature when the fluid is initially quenched in the unstable spinoidal region. The second problem shows that spatial chaos arises from small spatially periodic flunctions in an infinite tube of fluid if the ambient pressure is appropriately chosen.
34 Percolation in continuous systems
James R. Kirkwood and C. Eugene Wayne
A rigorous proof of the existence of a percolation phase transition in a system of noninteracting discs in the plane is presented. In addition, bounds on the critical density and critical area fraction are derived. The lower bound makes use of Halperin's idea of a self-avoiding walk of discs. The upper bound is proved by relating the continuum model to the site percolation problem on a triangular lattice, whose critical probability is exactly known.
35 Invariant manifolds for Functional Differential
Equations close to ordinary differential equations
Luis T. Magalhães
This paper considers invariant manifolds of global trajectories of retarded Functional Differential Equations in R^{n}. The persistence, smoothness and stability of such manifolds where the flow is given by an Ordinary Differential Equation (ODE) in R^{n} is studied for small perturbations of ODEs. The novelty of the present approach lies in the use of the dynamics of the flow on the manifolds, instead of their attractivity properties.
36 The KAM theory of systems with short range interactions,
II
C. Eugene Wayne
The proof of the results on the KAM theory of systems with short range interactions, stated in [4] is completed. Estimates on the decay of the interactions generated by the iterative procedure in the KAM theorem are proved, as well as the modification of the theorems of [1-3] needed for our results.
37 Passive quasi-free states of the noninteracting
Fermi gas
Jean De Cannière
The passive quasi-free states of the noninteracting Fermi gas with continuous one-particle Hamiltonian H are computed. They turn out to be the well known Fermi-Dirac states, or limits thereof. This still holds true if the spectrum of H has both a continuous and a discrete part, except for the appearance of a class of "ground state-like" states showing a local random excitation of the point spectrum in a neighborhood of the Fermi energy. When H has only pure point spectrum, the requirement that a state be passive and quasi-free is no longer sufficient to characterize the Fermi-Dirac distributions.
38 Maxwell and van der Waals
revisited
Elias C. Aifantis
We utilize a modern continuum mechanic framework to reconsider an old problem for fluid interfaces, also addressed by Maxwell and van der Waals. We prove that their results need not be valid necessarily. This conclusion is arrived at as a consequence of questioning the existence of thermodynamic potentials and the validity of usual thermodynamic relations within unstable (spinodal) regions. One central result is that Maxwell's equal area rule needs not be valid and certain statistical models are shown to be internally inconsistent. Prescise conditions for the validity of Maxwell's rule and the variational theory of van der Waals established in terms of the coefficients defining the interfacial stress.
39 On the mechanics of modulated structures
Elias C. Aifantis
The purpose of this lecture is to illustrate the appropriateness and potential of the methods of continuum mechanics in modeling modulated structures. Modulations are viewed, in general, as occurrences which may involve one or more properties of a system and extend from a submicroscopic to a macroscopic scale. They are also viewed as capable of possessing wave lengths and amplitudes which may vary from very small to very large values.
40 The strong
topology on symmetric sequence spaces
William H. Ruckle
Let S be a linear space of real sequences written in functional notation s=(s(j))=(s(1),
s(2),...). There is a natural duality between S and the space
of sequences which are eventually 0 given by the equation
<s,t> = _{j}
s(j) t(j) s S, t
.
41 A characterization of Borda's rule via optimization
Charles R. Johnson
It is shown that Borda's social welfare rule coincides with a social welfare function resulting from a well-defined optimization principle applied to a collection of individual binary preferences.
42
The spatial homogeneity of stable equilibria of some reaction-diffusion systems
on convex domains
Kazuo Kishimoto and Hans F. Weinberger
43 On work and constraints in mixtures
K.A. Pericak-Spector and W.O. Williams
In recent years workers in mixture theory have become aware of the central role that volume-fraction, the parameter describing the relative proportion of the volume occupied by a constituent, must play in that theory. In particular, the rate of change of volume-fraction, which we here call the chority, must appear in a working term as contributing to the energy, in order to avoid various inconsistencies. This is true both in theories in which volume-fraction appears as a parameter of microstructure and in complete mixture theories.
44 Some remarks on deformations of minimal surfaces
Harold Rosenberg and Eric Toubiana
We consider complete minimal surfaces (c.m.s.'s) in R^{3} and their deformations. M_{1} is an deformation of M_{0} if M_{1} is a graph over M_{0} in an tubular neighborhood of M_{1} and M_{1} is - C^{1} close to M_{0}. A c.m.s. M_{0} is isolated if all minimal surfaces M_{1}, which are sufficiently small deformations of M_{0}, are congruent to M_{0}. Many of the classical minimal surfaces in R^{3} are known to be isolated [2]; however, no example was known of a nonisolated minimal surface.
45 The duration of transients
S. Pelikan
Imagine a particle moving in a box and making elastic collisions with the sides. Suppose there is a small hole in one side of the box. For many initial conditions the particle will bounce around for a long time and then leave the box. These trajectories are examples of transients. In this paper we investigate the average duration of transients for a certain class of transformations T.
46 Random fluctuations of the duration of harvest
V. Capasso, K.L. Cooke, and M. Witten
In this report, we wish to discuss models of harvesting of a population when the duration of the harvest interval is subject to random fluctuations. This kind of situation arises, for example, if the harvestor or predator can harvest only when weather conditions are favorable. Clearly, the length of the favorable period will be subject to random variations.
47 The L^{p}-intergrability of Green's functions
and fundamental solutions for elliptic and parabolic equations
E. Fabes and D.W. Stroock
Given d ≥ 1 and in (0,1) denote by A_{d}() the class of smooth, symmetric, d X d matrix-valued functions a (a^{ij} (x)) on R^{d} which satisfy
48 Semilinear equations in R^{N} without
conditions at infinity
Haïm Brezis
The purpose of this paper is to point out that some nonlinear elliptic (and parabolic) problems are well-posed in all of R^{N} without conditions at infinity.
49 Lax-Friedrichs and the viscosity-capillarity
criterion
M. Slemrod
It has been shown by Lax some time ago that for hyperbolic conversation laws solutions obtained as limits of the Lax-Friedrichs finite difference scheme will actually satisfy an "entropy" admissibility criterion. The goal of this paper is to attempt to extend Lax's idea to a form which is amenable to mixed problems as well, e.g. the dynamics of a van der Waals fluid. Specifically, we compare shocks obtained by the Lax-Friedrichs scheme with those permitted by the viscosity-capillarity criterion of [2, 3]. We show that for isothermal motion it is expected that shocks produced by Lax-Friedrichs will satisfy the viscosity-capillarity criterion.
50 Spanning tree extensions of the Hadamard-Fischer
inequalities
Charles R. Johnson and Wayne W. Barrett
All possible graph theoretic generalizations of a certain sort for the Hadamard-Fischer determinantal inequalities are determined. These involve ratios of products of principal minors which dominate the determinant. Furthermore, the cases of equality in these inequalities are characterized, and equality is possible for every set of values which can occur for the relevant minors. This relates recent work of the authors on positive definite completions and determinantal identities. When applied to the same collections of principal minors, earlier generalizations give poorer, more difficult to compute bounds than the present inequalities. Thus, this work extends, and in certain sense completes, a series of generalizations of Hadamard-Fischer begun in the 1960's.
51 Revelation and implementation under differential
information
Andrew Postlewaite and David Schmeidler
Our goal in this pape is to merge several central ideas in economic theory; strategic behavior (incentive compatibility), differential (or incomplete) information, and the Arrow-Debreu model of general equilibrium. By strategic behavior we refer to the literature which models economic institutions as games in strategic form and uses Nash equilibrium as the solution concept. This literature, motivated by informational decentralization questions, deals not with a single economic environment and a single game, but rather considers a class of environments and a strategic outcome function (game form) which is applied uniformly to this class. The concept of differential information is that of Bayesian equilibrium as it has been applied in the literature on implicit contracts, principal-agent problems and bidding and auction models.
52 Complex analytic dynamics on the Riemann sphere
Paul Blanchard
Holomorphic, non-invertible dynamical systems of the Riemann sphere are surprisingly intricate and beautiful. Often the indecomposable, completely invariant sets are fractals (a la Mandelbrot [M1]) because, in fact, they are quasi-self-similar (see Sullivan [S3] and (8.5)). Sometimes they are nowhere differentiable Jordan curves whose Hausdorff dimension is greater than one (Sullivan [S4] and Ruelle [R]). Yet these sets are determined by a single analytic function z_{n+1} = R(z_{n}).
53 Topology and differentiability of labyrinths
in the disc and annulus
Gilbert Levitt and Harold Rosenberg
The study of differential equations in the plane which are locally of the form ðy / ðx =F(x,y), gives rise to labyrinths. They are limit sets of bounded solutions to this equation. This is made precise in [Ro], where the singularities considered are thorns and tripods. In part I of this paper, we shall extend the results of [Ro] to differential equations with n-prong singularities, in the disc and annulus. For the disc, the story is not essentially different from the previous case. However, for the annulus, the study is quite different and more complicated. In both cases, we obtain a topological structure theorem for solutions of the equation.
54 Symmetry of constant mean curvature hypersurfaces
in hyperbolic space
Gilbert Levitt and Harold Rosenberg
In a recent paper, M. Do Carmo and B. Lawson studied hypersurfaces M of constant
mean curvature in hyperbolic space [2]. They use the Alexandrov reflection technique
to study M given the asymptotic boundary ð_{}M.
For example, one of their theorems says M is a horoshpere when ð_{}M
reduces to a point. They also prove a Bernstein type theorem for minimal graphs.
In this paper we shall extend their results to other boundary conditions. We
prove an embedded M, of constant mean curvature, with ð_{}M
a subset of a codimension one sphere S, either is invariant by reflection in
the hyperbolic hyperplane containg S or is a hypersphere. In the former case
M is a "bigraph" over H: it meets any geodesic orthogonal to H either not at
all or transversaly in two points (one on each side of H) or tangentially on
H.
55 Analysis of a dynamic, decentralized exchange
economy
Ennio Stacchetti
A dynamic exchange economy model is presented. Similarly to the Walrasian equilibrium problem, each consumer is characterized by a feasible set and by an instantaneous demand function, that depends on the price vector, time, and the commodity holding. The commodity holding of each consumer varies according to his instantaneous demand function at each moment. We show that the market can choose prices that will lead the commodity holding of each consumer to remain in his consumption set while the aggregate commodity holding satisfies the scarcity constraints of the market.
56 On failure of the complementing condition and
nonuniqueness in linear elastostatics
Henry Simpson and Scott J. Spector
Consider a homogeneous, isotropic body composed of a compressible linearly
elastic material and assume that the body is at equilibrium in a state of plane
strain. The traction problem for such a body (in the absence of body forces
and surface tractions) consists of finding a displacement U=(u_{1},u_{2})
that satisfies (cf., e.g., Gurtin [4])
(1) U
+ ( + )
div U = 0 in R.
(2) [ (U
+ U^{T}) +
(div U) I] n = 0 on ðR.
Here R R^{2} is a regular region,
n the outward unit normal to the boundary, ðR, and ,
and are the (constant) Lamé
moduli.
It is well-known that (1) and (2) have a unique solution, modulo an infinitesimal
rigid deformation, provided that ≠ 0,
+
≠ 0, and 2 +
≠ 0.
The purpose of this note is to demonstrate that the above mentioned uniqueness
result fails when = - .
In fact we show that (1) and (2) have an infinite number of linearly independent
solutions (in spite of the ellipticity of the equations). The reason for this
unusual (for an elliptic system) behavior is that the boundary conditions fail
to satisfy the complementing (or Lopatinsky-Shapiro) conditions.
57 Complete integrability in statistical mechanics
and the Yang-Baxter equations
Craig Tracy
In this paper we give a differential formulation of the Yang-Baxter equations. This formulation leads to the introduction of the Yang-Baxter ideal I_{YB}, the basic geometric object in this formulation. These ideas are illustrated in the context of the Baxter model and the general eight-vertex model.
58 Boundedness of solutions of Duffing's equation
Tongren Ding
J. Littlewood, L. Markus, and J. Moser proposed independently the boundedness problem for solutions of Duffing's equation: x+g(x)=p(t), where p(t) is continuous and periodic and g(x) is superlinear at infinity. The purpose of this paper is to prove that all solutions of the above-mentioned Duffing's equation are bounded for tR when p(t) is even (or when p(t) is odd and g(x) is odd).
59 Workshop on price adjustment, quantity adjustment,
and business cycles
Abstract
The workshop dealt with economic models in which time plays an essential role, and both the description of adjustment to a static equilibrium and the description of equilibrium paths were considered. From a mathematical point of view, discrete dynamical systems and the dynamics of ordinary and partial differential equations played a major role.
60 The Coase theorem - An informational perspective
Rafael Rob
It is common knowledge these days that environmental policy plays a crucial role in a society characterized by a rapidly developing technology. While a world of highly mechanized productions methods offers its inhabitants a larger supply of goods and services, it raises at the same time serious questions about the ecological price that has to be paid for this increased abundance. The public concern expressed through the media, and governmental responses to this concern via budgetary provisions testify to the importance of the issue.
61 Approximate Newton methods and homotopy for stationary
operator equations
Joseph W. Jerome
A quadratically convergent algorithm, based upon a Newton-type iteration, is defined to approximate roots of operator equations in Banach spaces. Fréchet derivative operator invertibility is not required; approximate right inverses are used in a neighborhood of the root. This result, which requires an initially small residual, is sufficiently robust to yield existence; it may be viewed as a generalized version of the Kantorovich theorem. A second algorithm, based on continuation via single, Euler-predictor/Newton-corrector iterates, is also presented. It has the merit of controlling the residual until the homotopy terminates, at which point the first algorithm applies. This method is capable of yielding existence of a solution curve as well. An application is given for operators described by compact perturbations of the identity.
62 A note on competitive bidding with asymmetric
information
Rafael Robz
An interesting case of competitive bidding with an asymmetrical knowledge about the true value of the auctioned object is examined by R. Wilson [4]. The primary motivation for his study is the insight it provides about the value of information, or, more specifically, about the relative gains of the informed bidder vs. the uninformed bidder. As a by-product one can learn something about the ability of the seller to appropriate or realize the value of the item he offers for sale and about the identiy of the buyer. His analysis tells us, in short, about allocations and imputation under conditions of uncertainty and symmetric market positions - a fundamental question in economic theory. The purpose of this paper is expositional. By means of two alternative approaches, I will derive the equilibrium strategies and outcomes of the bidding game formulated by Wilson. A few flaws in his analysis will be corrected thererby. Additional examples illustrating the results will be offered. To be self-sufficient, let me start out by presenting the real-life situtation we wish to investigate and the model corresponding to it.
63 Equilibrium price distributions
Rafael Rob
Equilibrium price distributions (for a homogeneous product) consistent with individual incentives are investigated. They arise in informationally imperfect markets in which the only primitive datum is the distribution of search costs. It is shown that single, multi- and continuous price distributions are all viable long-run phenomena depending on the nature of search costs. A method for computing equilibrium price distributions is also provided.
64 The linearizing projection, global theories
William H. Ruckle
A linearizing operator or projection is a device which converts nonlinear
information into linear form. A well-known example of a linearizing projection
is the Shapley value, both in the descrete case (Shapley, 1953) and the continuous
case (Aumann and Shapley, 1974). A linearizing projection usually satisfies
certain axioms of rationality which insure that it is the "unique, fair" allocation
or distribution. Thus it is an admiriable bookkeeping device because bookkeeping
must be linear.
In Ruckle (1982) a first attempt was made to treat the Aumann-Shapley theory
of values in the setting of functionals defined in an arbitrary Banach space
E.
This paper continues the effort begun in Ruckle (1982) by constructing three
global theories of linearizing projections (or x_{0}-value). These theories
are called "global" because they refer to spaces of functionals which are defined
on the entire Banach space E. In Section 6 we shall describe what we mean by
a "local" theory and explain why such theories are needed.
65 Ergodic properties of linear dynamical systems
Russell Johnson, Kenneth Palmer, and George R. Sell
The Multiplicative Ergodic Theorem give information about the dynamical structure
of a cocycle , or a linear skew product
flow , over a suitable base space M. In
typical applications the base space M is either an attractor; a compact invariant
set; or the space of coefficients for a diffeomorphism, a differential equation,
or a vector field. This theorem asserts that for every invariant probability
measure on M there is a measurable decomposition
of the vector bundle over M into invariant measurable subbundles, and that every
solution with initial conditions in any of these subbundles has strong Lyapunov
exponets. These exponents, or growth rates, depend on the measure ,
and when is ergodic, they are constant
(almost everywhere) on M and form a finite set meas(), the measurable (Millionscikov-Oseledec) spectrum.
The main objective in this paper is to study the connection between the measurable
spectrum meas() and the dynamical spectrum dyn
introduced by Sacker and Sell (1975, 1978, 1980). (Also see Daletskii and Krein
(1974), as well as Selgrade (1975). The dynamical spectrum dyn
consists of those values
R for which the shifted flow _{ }
fails to have an exponential dichotomy over M. It follows from the Spectral
Theorem, Sacker and Sell (1978), that the dynamical spectrum is the finite union
of disjoint compact intervals when M is compact and dynamically connected.
66 How a network of processors can schedule its
work
Stanley Reiter
The problem addressed in this paper is to design a method by which a network of processors confronted with a flow of tasks may distribute the computing to be done among the processors so as to make effective use of them to perform the required computations. The method, and its variants, presented in this paper gives weight to the objectives of carrying out the prescribed tasks in short time, and to the relative urgencies associated with those tasks. This problem is reminiscent of the problem of scheduling the flow of jobs through a machine shop. The methods presented here are adapted from a method developed for that problem which were described in [1].
67 Linear subdivision is strictly a polynomial phenomenon
R.N. Goldman and D.C. Heath
In this paper we give an elementary proof that polynomial curves are the only differentiable curves which permit subdivision by standard linear techniques. Subdivision methods for rational polynomial curves are also discussed.
69 Realization and Nash implementation: Two aspects
of mechanism design
Steven R. Williams
In this paper we will show how a message process which "realizes" (or computes)
a given social choice rule F can be used to construct a game which implements
F in Nash equilibrium. Any efficient encoding of information that occurs in
the message process causes a corresponding reduction in the size of the strategy
space of the game which we will construct to implement F.
Necessary and (stronger) sufficient conditions on the message process will be
given for this construction.
70
Sufficient Conditions for Nash Implementation
Steven R. Williams
71
Equilibria in Banach lattices without ordered preferences
Nicholas C. Yannelis and William R. Zame
72
The reciprocals of solutions of linear ordinary differential equations
William A. Harris, Jr. and Yasutaka Sibuya
73
A dynamical meaning of fractal dimension
Steve Pelikan
74
Continuous-time portfolio management: Minimizing the expected time to reach
a goal
David C. Heath and William D. Sudderth
75
Information flows intrinsic to the stability of economic equilibrium
J.S. Jordan
84
Subjective
probability and expected utility without additivity
David Schmeidler
86
State
categories, closed categories, and the existence semi-continuous entropy functions
F. William Lawvere
87
Functional
Remarks on the General Concept of Chaos
F. William Lawvere
101
The
derivative of a tensor-valued function of a tensor
Donald E. Carlson and Anne Hoger
113
On
the derivatives of the principal invariants of a second-order tensor
D. Carlson and A. Hoger
171
On
Hadamard stability in finite elasticity
H.C. Simpson and S.J. Spector
239
Interaction of Shallow-Water Waves and Bottom Topography
B. Boczar-Karaki, J.L. Bona, and D.L. Cohen
286
Finite difference methods for the transient behavior of a semiconductor device
Jim Douglas, Jr. and Yuan Yirang
287
The extrapolation for boundary finite elements
Li Kaitai and Yan Ningning
288
Stochastic growth models
R. Durrett and R. Schonmann
289
Remarks about equilibrium configurations of crystals
David Kinderlehrer
290
Eventual C^{ }-regularity
and concavity for flows in one-dimensional porous media
D.G. Aronson and J.L. Vazquez
291
Distributed data structures for scientific computation
L.R. Scott, J.M. Boyle, and B. Bagheri
292
Simulation of flow in naturally fractured petroleum reservoirs
Jim Douglas, Jr., Paulo J. Paes Leme, Todd Arbogast,
and Tânia Schmitt
293
Optimal regularity for one-dimensional porous medium flow
D.G. Aronson and L.A. Caffarelli
294
Liquid crystals and energy estimates for -valued M maps
Haim Brezis
295
Analysis of the simulation of single phase flow through a naturally fractured
reservoir
Todd Arbogast
296
The coupling method of finite elements and boundary elements for radiation
problems
He Yinnian and Li Kaitai
297
Nonlinear effects in wave equation with a cubic restoring force
T. Cazenave, A. Haraux, L. Vazquez, and F.B. Weissler
298
Some blow-up results for a nonlinear parabolic equation with a gradient term
M. Chipot and F.B. Weissler
299
Perturbation solutions of simple and double bifurcation problems for Navier-Stokes
equations
Li Kaitai
300
The convergence on the multigrid algorithm for Navier-Stokes equations
Chen Zhangxin and Li Kaitai
301
Martingale approach for modeling DNA synthesis
A. Gerardi and G. Nappo
302
Regular inversion of the divergence operator with Dirichlet boundary conditions
on a polygon
Douglas N. Arnold, L. Ridgway Scott, and Michael Vogelius
303
Error analysis in ,
for mixed definite element methods for linear and quasi-linear ^M elliptic problems
Ricardo G. Duran
304
An efficient linear scheme to approximate parabolic free boundary problems:
Error estimates and implementation
Ricardo Nochetto and Claudio Verdi
305
Nonuniqueness for a hyperbolic system: Cavitation in nonlinear elastodynamics
K.A. Pericak-Spector and Scott J. Spector
306
q-series and orthogonal polynomials associated with Barnes' first Lemma
E.G. Kalnins and Willard Miller, Jr.
307
A uniformly accurate finite element method for Mindlin-Reissner plate
Douglas N. Arnold and Richard S. Falk
308
TVD properties of a class of modified ENO schemes for scalar conservation
laws
Chi-Wang Shu
309
A random boundary value problem modeling spatial variability in porous media
flow
Edmund Dikow and Ulrich Hornung
310
Compact attractors and singular perturbations
Jack K. Hale
311
The
TVD-projection method for solving implicit numeric schemes for scalar conservation
laws: A numerical study of a simple case
A. Bourgeat and B. Cockburn
312
Navier-Stokes computation of transonic vortices over a round leading edge
delta wing
Bernhard Muller and Arthur Rizzi
313
On the accuracy of vortex methods at large times
J. Thomas Beale
314
Decomposition methods for adherence problems in finite elasticity
P. Le Tallec and A. Lotfi
315
Approximation of waves in composite media
Jim Douglas, Jr. and Juan E. Santos
316
The double porosity model for single phase flow in naturally fractured reservoirs
Todd Arbogast
317
Two-phase immiscible flow in naturally fractured Reservoirs
Todd Arbogast, Jim Douglas, Jr., and Juan E. Santos
318
Numerical simulation of immiscible flow in porous media based on combining
the method of characteristics with mixed finite element procedures
Jim Douglas, Jr. and Y. Yirang
319
Sharp maximum norm error estimates for finite element approximations Of the
Stokes problem in 2-D
R. Durán, R.H. Nochetto, and J. Wan
320
A phase transition for a system of branching random walks in a random environment
Andreas Greven
321
Brownian models of open queueing networks with homogeneous customer populations
J.M. Harrison and R.J. Williams
322
Solutions et mesures invariantes pour des equations d'evolution Stochastiques
du type Navier-Stokes
Ana Bela Cruzeiro
323
Solutions et mesures invariantes pour des equations d'evolution Stochastiques
du type Navier-Stokes
Ana Bela Cruzeiro
324
Typical cluster size for 2-dim percolation processes (revised)
Bao Gia Nguyen
325
Stable defects of minimizers of constrained variational principles
Robert Hardt, David Kinderlehrer, and Fang-Hua Lin
326
Equilibrium configurations of crystals
Michel Chipot and David Kinderlehrer
327
Malliavin's C^{ } functionals
of a centered Gaussian system
Kiyosi Itô
328
Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau
model
Tadahisa Funaki
329
Schauder expansion by some quadratic base function
Masaya Yamaguti
330
Stabilized mixed methods for the Stokes problem
Franco Brezzi and Jim Douglas, Jr.
331
Inertial manifolds for reaction diffusion equations in higher space dimensions
J. Mallet-Paret and G.R. Sell
332
Relaxation methods for liquid crystal problems
San-Yih Lin and Mitchell Luskin
388
The
Runge-Kutta Local Projection P^{1}-Discontinuous-Galerkin Finite Element
Method for Scalar Conservation Laws
Bernardo Cockburn and Chi-Wang Shu
460
A quantitative model for lifespan curves
Matthew Witten
531
On
the Semilinear Elliptic Equation
in
Jong-Shenq Guo
633
Numerical approximation of the solutions of delay differential equations
on an infinite interval using piecewise constant arguments
K.L. Cooke and I. Györi
772
Computing centre conditions for certain cubic systems
N.G. Lloyd and J.M. Pearson
860
Quantum holography and neurocomputer architectures
Walter Schempp
987
Numerical methods for the regularization of descriptor systems by output
feedback
Angelika Bunse-Gerstner, Volker Mehrmann and Nancy
K. Nichols
1043
Protocol verification using discrete-event systems
Karen Rudie and W. Murray Wonham
1044
Nucleation, kinetics and admissibility criteria for propagating phase boundaries
Rohan Abeyaratne and James K. Knowles
1048
On the computation of suboptimal controllers for unstable infinite dimensional
systems
Onur Toker and Hitay Özbay
1051
A free boundary problem arising in the modeling of interanl oxidation of
binary alloys
Bei Hu and Jianhua Zhang
1055
Multiphase averaging for generalized flows on manifolds
H.S. Dumas, F. Golse, and P. Lochak
1056
Global solutions and quenching to a class of quasilinear parabolic equations
Bei Hu and Hong-Ming Yin
1063
Maximum principle for state-constrained optimal control problems governed
by quasilinear elliptic equations
Eduardo Casas and Jiongmin Yong
1064
Optimal control for degenerate parabolic equations with logistic growth
Suzanne M. Lenhart and Jiongmin Yong
1084
Fluids of differential type: Critical review and thermodynamic analysis
J.E. Dunn and K.R. Rajagopal
1231
Entropy maximization
K.B. Athreya
1316
Soliton's rebuilding in one-dimensional Schrödinger model with polynominal
nonlinearity
Valery E. Grikurov
1317
On self-similar solutions of the Navier-Stokes equations
J. Necas, M. Ruzicka, and V. Sverák
1318
Remarks on W^{2,p}-solutions of bilateral obstacle problems
Srdjan Stojanovic
1319
Pseudospectral vs. finite difference methods of initial value-problems with discontinuous
coeffcients
Erding Luo and Heinz-Otto Kreiss
1320 Soliton's
rebuilding in one-dimensional Schrödinger model with polynominal nonlinearity
Valery E. Grikurov
1321 A multiclass
closed queueing network with unconventional heavy traffic behavior
J.M. Harrison and R.J. Williams
1322 Microlocal
analysis on Morrey spaces
Michael E. Taylor
1323 Homogenization
of biharmonic equations in domains perforated with tiny holes
Chaocheng Huang
1324
An inverse obstacle problem: A uniqueness
theorem for spheres
Changmei Liu
1325 Approximation
of a laminated microstructure for a rotationally invariant, double well energy
density
Mitchell Luskin
1326
Haplotyping algorithms
Eric Sobel, Kenneth Lange, Jeffrey R. O'Connell, and
Daniel E. Weeks
1327
Haplotyping algorithms
Eric Sobel, Kenneth Lange, Jeffrey R. O'Connell, and
Daniel E. Weeks
1328 Haplotyping
algorithms
Eric Sobel, Kenneth Lange, Jeffrey R. O'Connell, and
Daniel E. Weeks
1329 Estimating
the number of asymptotic degrees of freedom for nonlinear dissipative systems
Bernardo Cockburn, Don A. Jones, and Edriss S. Titi
1330
Inverse SchrÃ¶dinger scattering on the line with partial knowledge of the
potential
Tuncay Aktosun
1331 Partition
of the potential of the one-dimensional Schrödinger equation
Tuncay Aktosun and Cornelis van der
1332
Convergence of the multigrid method with a wavelet coarse grid operator
Bjorn Engquist and Erding Luo
1333
Ergodic properties of the spin-boson system
V. Jaksic and C.-A. Pillet
1334 Recursive
solution for diffuse tomographic systems of arbitrary size
S.K. Patch
1335
Ergodic properties of the spin-boson system
V. Jaksic and C.-A. Pillet
1336
Bitangential structured interpolation
theory
Juan C. Cockburn
1337
The blow-up problem for exponential nonlinearities
Satyanad Kichenassamy
1338 How many
parameters can one solve for in diffuse tomography?
F.A. Grünbaum and S.K. Patch
1339
Reciprocal relations, bounds and size effects for composites with highly conducting
interface
Robert Lipton
1340 A global
nonexistence theorem for quasilinear evolution equations with dissipation
Howard A. Levine and James Serrin
1341 The conjugate
operator method: Application to DIRAC operators and to stratified media
Anne Boutet de Monvel and Radu Purice
1342 Stability
of matter through an electrostatic inequality
Gian Michele Graf
1343 Sharp
regularity estimates for solutions of the wave equation and their traces with
prescribed Neumann data
George Avalos
1344 The exponential
stability of a coupled hyperbolic/parabolic system arising in structural acoustics
George Avalos
1345
A differential Riccati equation for the active control of a problem in structural
acoustics
George Avalos and Irena Lasiecka
1346 Well-posedness
for a coupled hyperbolic/parabolic system seen in structural acoustics
George Avalos
1347 The strong
stability of a semigroup arising from a coupled hyperbolic/parabolic system
George Avalos and Irena Lasiecka
1348 Certain
optimal control problems for Navier-Stokes system with distributed control function
A.V. Fursikov
1349 One-dimensional
scattering theory for quantum systems with nontrivial spatial asymptotics
F. Gesztesy, R. Nowell, and W. Pötz
1350 On trace
formulas for SchrÃ¶dinger-type operators
F. Gesztesy and H. Holden
1351 Global
asymptotic limit of solutions of the Cahn-Hilliard equation
Xinfu Chen
1352 Lorenz
equations. Part I: Existence and nonexistence of homoclinic orbits
Xinfu Chen
1353 Lorenz
equations. Part II: "Randomly" rotated homoclinic orbits and chaotic trajectories
Xinfu Chen
1354 Lorenz
equations. Part III: Existence of hyperbolic sets
Xinfu Chen
1355 Kinetics
of materials with wiggly energies: Theory and application to the evolution of
twinning microstructures in a Cu-Al-Ni shape memory alloy
R. Abeyaratne, C. Chu, and R.D. James
1356
The Helmholtz equation on
Lipschitz domains
Changmei Liu
1357 Exponential
stability of a thermoelastic system without mechanical dissipation
George Avalos and Irena Lasiecka
1358
Heat conduction in fine scale mixtures with interfacial contact resistance
Robert Lipton
1359
Solvability of a nonlinear problem of Kirchhoff shell
Vladimir Odisharia and Jemal Peradze
1360
Affine invariant edge maps and active contours
Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum
1361 Hysteresis
in phase transformations
R.D. James
1362
A note on consistency and adjointness for numerical schemes
Alain Sei and William Symes
1363 Head-media
interaction in magnetic recording
Avner Friedman and Bei Hu
1364 Time-dependent
coating flows in a strip, Part I: The linearized problem
Avner Friedman and Juan J.L. Velázquez
1365 Young
measures in a nonlocal phase transition problem
Xiaofeng Ren and Matthias Winter
1366 Elastic
energy minimization and the recoverable strains of polycrystalline shape-memory
materials
Kaushik Bhattacharya and Robert V. Kohn
1367
Operator pencil and homogenization in the problem of vibration of fluid
in a vessel with a fine net on the surface
G.A. Chechkin
1368 On Poiseuille
flow of liquid crystals
M. Carme Calderer and Bagisa Mukherjee
1369
Pointwise Fourier inversion: A wave equation approach
Mark A. Pinsky and Michael E. Taylor
1370 Order
parameter models of elastic bars and precursor oscillations
Deborah Brandon and Robert C. Rogers
1371
A system of reaction diffusion equations arising in the theory of reinforced
random walks
Howard A. Levine and Brian D. Sleeman
1372
A priori error estimates for numerical
methods for scalar conservation laws. Part II: Flux-splitting monotone schemes
on irregular Cartesian grids
Bernardo Cockburn and Pierre-Alain Gremaud
1373 Finite
element analysis of microstructure for the cubic to tetragonal transformation
Bo Li and Mitchell Luskin
1374 On the
computation of crystalline microstructure
Mitchell Luskin
1375 On gradient
young measures supported on a point and a well
João P. Matos
1376 Scaling
properties of vortex ring formation at a circular tube opening
Monika Nitsche
1377 Decay
and analyticity of solitary waves
Jerry L. Bona and Yi A. Li
1378 On uniqueness
in a lateral cauchy problem with multiple characteristics
Victor Isakov
1379
Averaging for fundamental solutions of parabolic equations
Michael A. Kouritzin
1380
Integral equation methods for the inverse problem with discontinuous wavespeed
Tuncay Aktosun, Martin Klaus, and Cornelis van der
Mee
1381
Convergent spectral approximations for the thermomechanical processes in shape
memory alloys
Pedro Morin and Ruben D. Spies
1382
Interior estimates for a low order finite element method for the Reissner-Mindlin
plate model
Douglas N. Arnold and Xiaobo Liu
1383
Analysis of a linear-linear finite element for the Reissner-Mindlin plate
model
Douglas N. Arnold and Richard S. Falk
1384
Preconditioning in H(div) and applications
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther
1385
Nonlinear parabolic problems possessing solutions with unbounded gradients
M. Lavrentiev, Jr.
1386
Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure
Oscar P. Bruno and Peter Laurence
1387
The overall elastic energy of polycrystalline martensitic solids
Oscar P. Bruno, Fernando Reitich, and Perry H. Leo
1388
On critical exponents for a semilinear parabolic system coupled in an equation
and a boundary condition
Marek Fila and Howard A. Levine
1389
Optimal open-loop ram velocity profiles for isothermal forging: A variational
approach
Jordan M. Berg, W. Garth Frazier, Anil Chaudhary, and
Siva S. Banda
1390
Unfolding the zero structure of a linear control system
Jordan M. Berg and Harry G. Kwatny
1391
High order finite-difference approximations of the wave equation with absorbing
boundary conditions: A stability analysis
Alain Sei
1392
Small amplitude oscillatory forcing on two-layer plane channel flow
Adrian V. Coward and Yuriko Y. Renardy
1393
Approximation dynamics and the stability of invariant sets
Victor A. Pliss and George R. Sell
1394
A new computational model for heterojunction resonant tunneling diode
J. Gene Cao and Patrick Roblin
1395
Inverse obstacle problem: Local uniqueness for rougher obstacles and the identification
of a ball
Changmei Liu
1396
Dynamic cavitation with shocks in nonlinear elasticity
K.A. Pericak-Spector and Scott J. Spector
1397
Exponential stability of a thermoelastic system without mechanical dissipation
II: The case of simply supported boundary conditions
George Avalos and Irena Lasiecka
1398
Approximation of infima in the calculus
of variations
Bernard Brighi and Michel Chipot
1399
Concerning the well-posedness of a nonlinearly coupled semilinear wave and
beam-like equation
George Avalos
1400
Variational methods, bounds and size effects for composites with highly
conducting interface
Robert Lipton
1401
Non-classical shock waves in scalar conservation laws
Brian T. Hayes and Philippe G. LeFloch
1402
Boundary layers in weak solutions to hyperbolic conservation laws
K.T. Joseph and P.G. LeFloch
1403
Energies of knots
Y. Diao, C. Ernst, and E.J.J. Van Rensburg
1404
Multi-layer local minimum solutions of the bistable equation in an infinite
tube
Xiaofeng Ren
1405
Krylov sequences and orthogonal polynomials
Vlastimil Pták
1406
Factorization of scattering matrices due to partitioning of potentials in
one-dimensional SchrÃ¶dinger-type equations
Tuncay Aktosun, Martin Klaus, and Cornelis van der
Mee
1407
On the separation of stress-induced and texture-induced birefringence in
acoustoelasticity
Chi-Sing Man & Roberto Paroni
1408
Preconditioning discrete approximations of the Reissner-Mindlin plate model
Douglas N. Arnold, Richard S. Falk and Ragnar Winther
1409
On exact filters for continuous signals with discrete observations
Michael A. Kouritzin
1731 Runge-Kutta discontinuous Galerkin methods for
convection-dominated problems
Bernardo Cockburn and Chi-Wang Shu
In this paper, we review the development of the Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.
1732 Reaction-sheet jump conditions in
premixed flames
J.W. Dold, R.W. Thatcher, and A.A. Shah
The fundamental differences between the leading-order jump conditions, often assumed at a flame sheet in combustion theory, and the actual effect of a chemical reaction that satisfies Arrhenius kinetics with a finite activation temperature, need to be understood. These differences are "higher order" in terms of a large activation temperature analysis. However, they do provide a quantitative estimate of the errors that are inherent in adopting only the leading order version and they can indicate qualitative changes that may occur at finite activation temperatures in some cases. This paper derives two orders of asymptotic correction to the jump conditions normally used in describing premixed laminar combustion. An example involving steady, non-adiabatic flame-balls shows that the accepted asymptotic picture is limited to unusually large Zel'dovich numbers.
1733 Waveform relaxation
methods for stochastic differential equations
K.R. Schneider and H. Schurz
1734 Symbolic Representations of Iterated Maps
Xin-Chu Fu, Weiping Lu, Peter Ashwin, and Jinqiao Duan
This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general subshift over usually an uncountable alphabet. It is shown that at most the second order representation is enough for a continuous map. In particular, it is shown that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study of subshift structure of a general symbolic dynamics system. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps, piecewise continuous maps are discussed, and as examples, a representation of the Gauss map via a full shift over a countable alphabet and representations of interval exchange transformations as subshifts of infinite type are given.
1735 A multilevel discontinuous Galerkin
method
J. Gopalakrishnan and G. Kanschat
A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion problem, for which an error estimate is proved. A multigrid algorithm for this method is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.
1736 On the controllability in a mathematical model
of growth of tumors
J.I. Díaz and J.I. Tello
We study a model of growth of tumors with a free boundary, delaying the tumor region. We take into account the presence of inhhibitors and its interaction with the nutrients. We study the approximate controllability of the internal distribution of density of cells, that is propotional to concentration of nutrients, injecting inhibitor in a small inner region _{0} .
1737 On splitting up
method and stochastic partial differential equations
István Gyöngy and Nicolai Krylov
1738 Some qualitative properties for the total variational flow
F. Andreu, V. Caselles, J.I. Diaz, and J.M. Mazón
We prove the existence of a finite extinction time for the solutions of the Dirichiet problem for the total variational flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely different to the case of the problem associated to the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations.
1739 Completeness of superintegrability in two dimensional constant curvature spaces
We classify the Hamiltonians H=p_{x}^{2}+ p_{y}^{2} +V(x,y) of all classical superintegrable systems in two dimensional complex Euclidean space with second-order constants of the motion. We similarly classify the superintegrable Hamiltonians H=J_{1}^{2}+J_{2}^{2}+ J_{3}^{2}+V(x,y,z) on the complex 2-sphere where x^{2}+y^{2}+z^{2}=1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.
1740 Invariant Euler-Lagrange equations and the invariant variational
bicomplex
Irina A. Kogan and Peter J. Olver
In this paper, we derive an explicit group-invariant formula for the Euler-Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex that is based on a general moving frame construction and is of independent interest.
1741 Moment attractivity, stability and contractivity
exponents of stochastic dynamical systems
Henri Schurz (Communicated
by Shouchuan Hu)
Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference equations are in the center of this presentation in view of the asymptotic behavior of their moments. We study the exponential p-th mean growth behavior of their solutions as integration time tends to infinity. For this purpose, the concepts of attractivity, stability and contractivity exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behavior along certain Lyapunov-type functionals.
1742 A theory of finitely durable goods
monopoly with used-goods market and transaction costs
S. Huang, Y. Yang, and K. Anderson
We construct a dynamic game to model a monopoly of finitely durable goods. The solution concept is Markov perfect equilibria with general equilibria embedded in every time period. Our model is flexible enough to simultaneously explain or accommodate many commonly observed phenomena or stylized facts, such as concurrent leasing and selling, active secondary markets for used goods, heterogeneous consumers, endogenous consumption patterns, depreciation, an infinite time horizon, and non-trivial transaction costs. Within our model, consumers have incentives to segment themselves into various consumption classes according to their willingness to pay; and non-trivial transaction costs to sell used goods put strong constraints on consumers consumption sequences in time. As a direct consequence of the finite durability the market power of the monopolist remains intact. Leasing manifests itself as a facilitator of price discrimination, by de-bundling the durable good into new and used portions that are naturally bundled together under outright sales. The concurrent leasing and selling reflects the degree of the comparative advantage the monopolist has over consumers in disposing used goods. This comparative advantage, which is partially exploited by the monopolist and partially shared by the consumers, provides a sufficient mechanism to gain Pareto improvement on the market.
1743 Non-texture inpainting by curvature-driven
diffusions (CDD)
Tony F. Chan and Jianhong Shen
Inpainting refers to the practice of artists of restoring ancient paintings. Simply speaking, inpainting is to complete a painting by filling in the missing informa tion on prescribed domains. On such domains, the original painting has been damaged due to aging, scratching, or some other factors.
Inpainting and disocclusion in vision analysis are closely connected but also clearly different. Both try to recover the missing visual information from a given 2-D image, and mathematically, can be classified into the same category of inverse problems. The difference lies in both their goals and approaches.
The main goal of disocclusion is to model how human vision works to complete occluded objects in a given 2-D scene, and understand their physical or ders in the direction perpendicular to the imaging plane, and thus reconstruct approximately a meaningful 3-dimensional world (Nitzberg, Mumford, and Shiota [14]). The outputs from disocclusion are complete objects, and their relative orders or depth. Inpainting, on the other hand, is to complete a 2-D image which have certain regions missing. The output is still a 2-D image. (In applications, a missing region can indeed be the 2-D projection of a real object, such as the female statue in Figure 9.) Therefore, from the vision point of view, inpainting is a lower level process compared to disocclusion.
This fundamental difference naturally influences the approaches. The main approach for disocclusion is to segment the regions in a 2-D image, and then logically connect those which belong to the projection of a same physical object, and finally generate the order or depth for all the completed objects. Edge completion is one crucial step during the whole process. Disocclusion also often uses some high level information about objects (such as the near symmetry of human faces). For inpainting, an ideal scheme should be able to reconstruct an incomplete 2-D image in every detail so that it looks "complete" and "natural." More specifically, to inpaint, is not only to complete the broken edges, but also to connect each broken isophote (or level-line), so that the 2-D objects completed in such a way show their natural variation in intensity (or color for color images) [3, 6,11].
This comparison helps us understand better the real nature of the inpainting problem in a broader context.
The terminology of digital inpainting was first introduced by Bertalmio, Sapiro, Caselles, and Ballester [3]. Inspired by the real inpainting process of artists, the authors invented a successful digital inpainting scheme (referred to below as the BSCB inpainting scheme for convenience) based on the PDE method. The authors also deepened the interest in digital inpainting by demon strating its broad applications in text removal, restoring old photos, and creating special effects such as object disappearance from a scene.
Though a qualitative understanding based on the transportation mechanism can be well established, rigorous mathematical analysis on the BSCB scheme appears to be much more difficult. This has encouraged Chan and Shen [6] to develop a new inpainting model which is founded on the variational principle. Since the energy function is based on the total variational (TV) norm [6], the model is called TV inpainting. The TV inpainting scheme is surprisingly a close variation of the well known restoration model of Rudin, Osher and Fatemi (16, 17].
1744 Symmetric
real-valued orthonormal scaling functions with compact support in L_{2}(R
^{s}
)
Bin Han
1745 Fast and accurate algorithms for projective multi-image
structure from motion
John Oliensis and Yacup Genc
We describe algorithms for computing projective structure and motion from a multi-image sequence of tracked points. The algorithms are essentially linear, work for any motion of moderate size, and give accuracies similar to those of a maximum-likelihood estimate. They give better results than the Sturm/Triggs factorization approach and are equally fast, and they are much faster than bundle adjustment. Our experiments show that the (iterated) Sturm/Triggs approach often fails for linear camera motions. In adition, we study experimentally the common situation where the calibration is fixed and approximately known, comparing the projective versions of our algorithms to mixed projective/Euclidean strategies. We clarify the nature of dominant-plane compensation, showing that it can be considered a small-translation approximation rather than an approximation that the scene is planar. We show that projective algorithms accurately recover the (projected inverse depths and homographies despite the possibility of transforming the structure and motion by a projective transformation.
1746 Three algorithms for 2-image and 2-image structure
from motion
John Oliensis and Yacup Genc
We describe three approaches to 2-image and 2-image structure from motion. First, we present a new approximation to the least-squares image-reprojection error for 2 images. It depends only on the motion unknowns and is much more accurate than previous approximations such as the (weighted) coplanarity, especially for forward camera motions. We use this error to compute tight, rigorous upper and lower bounds on the true error and to study its properties experimentally. We demonstrate that the true error has many local minima for forward motions even when the motion is large. We propose and experimentally test a second approach, which is potentially more robust than bundle adjustment. Last, we describe algorithms for 2 images that reconstruct from the measured 2D affine deformations of image patches.
1747 Unified analysis of discontinuous Galerkin methods
for elliptic problems
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn,
and L. Donatella Marini
We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems by diverse communities over three decades.
1748 Growing Fitted Textures
Gabriele Gorla, Victoria Interrante, and Guillermo
Sapiro
In this paper, we address the problem of how to seamlessly and without repetition artifacts or visible projective distortion cover the surface of a polygonally-defined model with a texture pattern derived from an acquired 2D image such that the dominant orientation of the pattern will everywhere follow the surface shape in an aesthetically pleasing way. Specifically, we propose an efficient, automatic method for synthesizing, from a small sample swatch, patches of perceptually similar texture in which the pattern orientation may locally follow a specified vector field, such as the principal directions of curvature, at a per-pixel level, and in which the continuity of large and small scale features of the pattern is generally preserved across adjacent patches. We demonstrate the results of our method with a variety of texture swatches applied to standard graphics datasets.
1749 Inverse Scattering with Partial Information
on the Potential
Tuncay Aktosun and Ricardo Weder
The one-dimensional Schröddinger equation is considered when the potential is real valued and integrable and has a finite first moment. The recovery of such a potential is analyzed in terms of the scattering data consisting of a reflection coefficient, all the bound-state energies, knowledge of the potential on a finite interval, and all of the bound-state norming constants except one. It is shown that a missing norming constant in the data can cause at most a double nonuniqueness in the recovery. In the particular case when the missing norming constant in the data corresponds to the lowest-energy bound state, the necessary and sufficient conditions are obtained for the nonuniqueness, and the two norming constants and the corresponding potentials are determined. Some explicit examples are provided to illustrate the nonuniqueness.
1750
Chord uniqueness and controllability: The view from the boundary, I
Robert Gulliver and Walter Littman
1751 Uniqueness for the determination
of sound-soft defects in an inhomogeneous planar medium by acoustic boundary
measurements
Luca Rondi rondi@math.umn.edu
We consider the inverse problem of determining shape and location of sound-soft defects inside a known planar inhomogeneous and anisotropic medium through acoustic imaging at low frequency. We consider the case of acoustic boundary measurements, with different types of boundary conditions to be prescribed, and we prove that at most two, suitably chosen, measurements allow us to uniquely determine multiple defects under minimal regularity assumptions on the defects and the medium containing them. Finally we treat applications of these results to the case of inverse scattering.
1752 Mathematical Modeling in Industry - IMA Summer
Program for Graduate Students, July 19-28, 2000
Rachel Kuske and Fernando Reitich, Organizers
(There are six papers in this preprint)
1753 Visualization of high dynamic range images
Alvaro Pardo and Guillermo Sapiro
A novel paradigm for the visualization of high dynamic range images is presented in this paper. These images, real or synthetic, have luminance with typical ranges many orders of magnitude higher than that of standard output devices, thereby requiring some processing for visualization. In contrast with existent approaches, that compute a single image with reduced range, close in a given sense to the original data, we propose to look for a representative set of images. The goal is then to produce a minimal set of images capturing the information all over the high dynamic range data, while at the same time preserving a natural appearance for each one of the images in the set. A specific algorithm that achieves this goal is presented and tested on natural and synthetic data.
1754 Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces
An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and computationally optimal numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton-Jacobi based, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hyper-surfaces a computationally optimal technique. The approach can be extended to solve a more general class of Hamilton-Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows to perform the computations on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton-Jacobi equations are intrinsic to the implicit hyper-surface. For other surface representations like triangulated or unorganized points ones, the algorithm here introduced can be used after simple pre-processing of the data.
1755 Symmetry property and construction
of wavelets with a general dilation matrix
Bin Han
In this note, we are interested in the symmetry property of a refinable function with a general dilation matrix. We investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has certain kind of symmetry. Given two dilation matrices which produce the same lattice, we demonstrate that if a mask has certain kind of symmetry, then its associated refinable functions with respect to the two dilation matrices are the same; therefore, the two corresponding derived wavelet systems are essential the same. Finally, we illustrate that for any dilation matrix, orthogonal masks, as well as interpolatory masks having nonnegative symbols, can be easily constructed with any preassigned order of sum rules by employing a linear transform. Without solving any equation, the method in this note on constructing masks with certain desirable properties is simple, painless and general. Examples of quincunx wavelets are presented to illustrate the general theory.
1756 Projectable multivariate wavelets
Bin Han
We demonstrate that many multivariate wavelets are projectable wavelets; that is, they essentially carry the tensor product (separable) structure though themselves may be non-tensor product (nonseparable) wavelets. We show that a projectable wavelet can be replaced by a tensor product wavelet without loss of desirable properties such as spatial localization, smoothness and vanishing moments.
1757 Euler's elastica and curvature based
inpaintings
Tony F. Chan, Sang-Ha Kang, and Jianhong Shen
Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced by Mumford [21] to computer vision as a prior curve model. By functionalizaing the elastica energy, Masnou and Morel [19] proposed an elastica based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation, and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [2] and Chan and Shen [6,7]. A computational scheme based on numerical PDEs is presented, which allows the handling of topologically complex inpainting domains.
1758 Non-topological multivortex solutions
to the self-dual Maxwell-Chern-Simons-Higgs systems
Dongho Chae and Oleg Yu. Imanuvilov
In this paper we construct non-topological multivortex solutions to the non-relativistic self-dual Maxwell-Chern-Simons-Higgs system in $\Bbb R^2$ which make the energy functional finite. Moreover, our proof of the existence of solutions reveals precise asymptotic behavior of solutions near spatial infinity. Using exactly the same method, we also establish the existence of non-topological multivortex solutions to the relativistic self-dualMaxwell-Chern-Simons-Higgs system.
1759 Ergodicity of stochastically forced
large scale geophysical flows
Jinqiao Duan and Beniamin Goldys
We investigate the ergodicity of 2D large scale quasigeostrophic flows under random wind forcing. We show that the quasigeostrophic flows are ergodic under suitable conditions on the random forcing and on the fluid domain, and under no restrictions on viscosity, Ekman constant or Coriolis parameter. When these conditions are satisfied, then for any observable of the quasigeostrophic flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.
1760 Probabilistic dynamics of two-layer
geophysical flows
Igor Chueshov, Jinqiao Duan, and Bjö}rn Schmalfuss
The two-layer quasigeostrophic flow model is an intermidiate system between the single-layer 2D barotropic flow model and the continuously stratified, 3D baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. The wind forcing acts only on the top layer. We consider the two-layer quasigeostrophic model under stochastic wind forcing. We first transformed this system into a coupled system of random partial differential equations and then show that the asymptotic probabilistic dynamics of this system depends only on the top fluid layer. Namely, in the probability sense and asymptotically, the dynamics of the two-layer quasigeostrophic fluid system is determinied by the top fluid layer, or, the bottom fluid layer is slaved by the top fluid layer. This conclusion is true provided that the Wiener process and the fluid parameters satisfy a certain condition. In particular, this latter condition is satisfied when the trace of the covariance operator of the Wiener process is controled by a certain upper bound, and the Ekman constant r is sufficiently large. Note that the generalized time derivative of the Wiener process models the fluctuating part of the wind stress forcing on the top fluid layer, and the Ekman constant r measures the rate for vorticity decay due to the friction in the bottom Ekman layer.
1761 Report on the retrodigitization project
"archiv der mathematik"
Gerhard O. Michler
No postscript or pdf file.
Nowadays many mathematical journals are published electronically, in general in portable document format (PDF). For example, the servers of the American Mathematical Society (AMS) contain at present more than 4700 articles published electronically from 1996 to date in the journals of the AMS that are accessible and searchable online to subscribers via the internet address: http://www.ams.org/journals. The PDF-format offers many advantages over the traditional paper format of a printed article. Authorized users can also link from journal articles references to the reviews in the Mathematical Reviews of the AMS, and in due course from there to the original quoted articles, if available in PDF. The online access also allows to make printouts of wanted articles at the user¹s printer and to view abstracts of recently posted articles to be published shortly. In view of the vaste literature in mathematics the search functions provided by the software of a distributed digital library for specific articles and/or specific results or concepts explained in such an article are very useful for researchers and graduate students. They also would like to have such a comfortable access to the back issues of the mathematical journals published only in paper format. The proofs of many important old theorems have never been incorporated into standard textbooks because they are still too complicated to be derived from generally known theoretical results. Therefore modern research articles have to quote arguments, methods, partial and final results of old papers.
Thus it is desirable to retrodigitize the back issues of a mathematical journal and combine them with the electronically published recent issues in such a way that all the volumes of the resulting digital journal are searchable and linked to "Mathematical Reviews" or "Zentralblatt Mathematik." Through these review journals the retrodigitized volumes could then be linked to all other mathematical journals belonging to a distributed digital mathematical library. In particular, an authorized user of such a library would be able to read on screen an original article and a cited article or a review at the same time.
This retrodigitization task is very demanding and requires a deep understanding of mathematics, computer science and the support of the publisher. It is the purpose of this survey article to describe a solution found by the author¹s study group in the pilot project "Retrodigitization of the journal Archiv der Mathematik" financially supported by the Deutsche Forschungsgemeinschaft from 1 April, 1997 until 80 September, 2000.
Using the IBM digital library database [4] and the MILESS software [7] developed by the Computer Center of Essen University we have been able to construct a prototype of a searchable and retrievable digital library "Archiv der Matheinatik." It contains
1) the retrodigitized volumes 60 (1998) - 67 (1996) in MVD format,
2) the electronically published volumes 68 (1997)- 73 (1999) in PDF format.
The cited journal articles of the references of all 13 volumes are recognized au- tomatically as well as the layout of the first page of each article. Thus it is possible to produce the bibliographic data including the ISSN of all journal articles of the 13 volumes and their references in XML format. In particular, all these articles and the cited journal articles of their references can be linked automatically to MathSciNet or Zentralblatt MATH.
The methods developed in this pilot project can be adjusted for the retrodigitization of other mathematical journals and older back issues of "Archiv der Matheinatik."
1762 Procreation of inner product space
for generalized B-function
Anand Singh and H.S. Dhami
In an attempt to mucilage the bridge connecting special functions with generalized hypergcometric functions, here an attempt is being made to procreate inner product space for generalized B-function.
1763 Capillarity driven spreading of power-law
fluids
S.I. Betelu and M.A. Fontelos
We investigate the spreading of thin liquid films of power-law rheology. We construct an explicit travelling wave solution and source-type similarity solutions. We show that when the nonlinearity exponent $\lambda$ for the rheology is larger than one, the governing dimensionless equation $h_t+(h^{\lambda+2}|h_{xxx}|^{\lambda-1}h_{xxx})_x=0$ admits solutions with compact support and moving fronts. We also show that the solutions have bounded energy dissipation rate.
1764 Performance of discontinuous Galerkin
methods for elliptic PDE's
Paul Castillo
In this paper, we compare the performance of the main discontinuous Galerkin (DG) methods for elliptic partial differential equations on a model problem. Theoretical estimates of the condition number of the stiffness matrix are given for DG methods whose bilinear form is symmetric, which are shown to be sharp numerically. Then, the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes.
1765 Words standarization by non-parametric
statistical methods
H.S. Dhami
No postscript or pdf file.
The author has dealt with amalgamation of pattern primitives and its justification in language description, in earlier studies [4] & [5]. Here an attempt is being made to apply distribution free techniques, more commonly known as non-parametric statistical methods in finding the confidence coefficient and adjustment factor for obtaining the standard form of words taken from different sub-dialects.
1766 Structure and texture filling-in
of missing image blocks in wireless transmission and compression applications
Shantanu D. Rane, Guillermo Sapiro, and Marcelo Bertalmio
An approach for filling-in blocks of missing data in wireless image transmission is presented in this paper. When compression algorithms such as JPEG are used as part of the wireless transmission process, images are first tiled into blocks of 8 × 8 pixels. When such images are transmitted over fading channels, the effects of noise can kill entire blocks of the image. Instead of using common retransmission query protocols, we aim to reconstruct the lost data using correlation between the lost block and its neighbors. If the lost block contained structure, it is reconstructed using an image inpainting algorithm, while texture synthesis is used for the textured blocks. The switch between the two schemes is done in a fully automatic fashion based on the surrounding available blocks. The performance of this method is tested for various images and combinations of lost blocks. The viability of this method for image compression, in association with lossy JPEG, is also discussed.
1767 Stability of solutions of chemotaxis
equations in reinforced random walks
Avner Friedman and J. Ignacio Tello
In this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation. The system arises in chemotaxis, a process whereby living organisms respond to chemical substance, or by aggregating or dispersing. We prove that stationary solutions of the system are asymptotically stable.
1768 A method for denoising textured surfaces
S. Betelu, A. Tannenbaum, and G. Sapiro
In this note, we present a simple method to denoise triangulated and implicit surfaces in a manner which preserves the 3D shape texture. The technique is based upon the synthesis of partial differential equations (PDE's), implicit surfaces, and Wiener filtering. The basic idea is to apply a computationally efficient local Wiener filter to an implicit representation of the surface. Such a representation can be directly given as the algorithm input or explicitly obtained via partial differential equation based implicitation techniques applied to the triangulated data. The proposed method has a computational complexity O(N log N).
1769 A geometric-optics proof of a theorem on boundary
control given a convex function
Michael Galbraith
In the area of boundary control of hyperbolic equations, the tools of geometric optics have sometimes proven to be very powerful. In geometric optics, authors including Littman [8] and Bardos, Lebeau and Rauch [1] have established under various circumstances that, if every bicharacteristic curve of the hyperbolic equation must cross a point on the boundary where the controls can be applied, then the equation can be controlled--- and the time required is just the maximum time needed for a bicharacteristic to reach that part of the boundary.
Now that these results are in place, they allow for theorems on boundary control which do not require new integral inequalities for particular situations. Rather, assumptions are made on the geometry of the domain of the equation. For instance, Gulliver and Littman [3] show that every bicharacteristic will cross the boundary, and hence control will be attained, so long as chords between points of the boundary are unique and the boundary is locally convex. They go on to give several examples of regions where this holds.
The present paper uses geometric optics to prove one of the main theorems in the important paper "Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients" by Lasiecka, Triggiani and Yao [5]. In that paper, the authors use Carleman estimates to show that the equation is controlled if there is a positive function v on which is strictly convex with respect to the metric defined by the coefficients of the equation, if that convex function has non-positive outward normal derivative on the uncontrolled part of the boundary. The time needed for control is a function of the maximum value of v on and a lower bound on its convexity. Here we will show that control in the same time is established by a simpler geometric optics argument---in fact it comes down to a short calculus computation on the value of v along a bicharacteristic of the equation.
1770 Morphologically invariant PDE inpaintings
Tony F. Chan and Jianhong Shen
This paper studies the PDE method for image inpaintings. Image inpainting is essentially an image interpolation problem, with wide applications in film and photo restoration, text removal, special effects in movies, disocclusion, digital zoom-in, and edge-based image compression and coding. Bertalmio, Sapiro, Caselles, and Ballester (2000) [3] first innovatively introduced the PDE method for the inpainting problem. Ever since, the authors of the present paper have worked along this line and developed the PDE method, mostly inspired by the Bayesian and variational method (especially by good image {\rm prior} models). The current paper has two major goals. First, by surveying all the recent PDE inpainting techniques, we intend to develop a unified viewpoint based on two infinitesimal mechanisms: transportation and curvature driven diffusions (CDD). Furthermore, based this knowledge, we construct a new class of third order inpainting PDEs, which is derived from the set of axioms (or principles) refined from the existing works: morphological invariance, rotational invariance, stability principle, and linearity principle.
1771 A simple proof of
a result of A. Novikov
N.V. Krylov
1772 Navier-Stokes, fluid dynamics, and image and video
inpainting
M. Bertalmio, A. Bertozzi, and G. Sapiro
Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. In this paper, we introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to propagate isophote lines continuously from the exterior into the region to be inpainted. The main idea is to think of the image intensity as a `stream function' for a two-dimensional incompressible flow. The Laplacian of the image intensity plays the role of the vorticity of the fluid; it is transported into the region to be inpainted by a vector field defined by the stream function. The resulting algorithm is designed to continue isophotes while matching gradient vectors at the boundary of the inpainting region. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis.
1773 Determining functionals for random partial differential
equations
Igor Chueshov, Jinqiao Duan, and Björn Schmalfuß
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results. We apply our results to the 2D Navier - Stokes equations forced by a white noise.
1774 The hp-local discontinuous Galerkin method for
low-frequency time-harmonic Maxwell's equations
Ilaria Perugia and Dominik Schötzau
The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in low-frequency regime is introduced and analyzed. We consider topologically non-trivial domains and heterogeneous media, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operators, based on a mixed formulation of the problem and on the introduction of the so-called numerical fluxes. An hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.
1775 Exact Controllability of Structural Acoustic Interactions
George Avalos and Irena Lasiecka
In this paper, we work to discern exact controllability properties of two coupled wave equations, one of which holds on the interior of a bounded open domain , and the other on a segment _{0} of the boundary \partial . Moreover, the coupling is accomplished through terms on the boundary. Because of the particular physical application involved-the attenuation of acoustic waves within a chamber by means of active controllers on the chamber walls--control is to be implemented on the boundary only. We give here concise results of exact controllability for this system of interactions, with the control functions being applied through \partial . In particular, it is seen that for special geometries, control may be exerted on the boundary segment _{0} only. We make use here of microlocal estimates derived for the Neumann-control of wave equations, as well as a special vector field which is now known to exist under certain geometrical situations.
1776 Spline subdivision schemes for compact sets with
metric averages
Nira Dyn and Elza Farkhi
To define spline subdivision schemes for general compact sets, we use the representation of spline subdivision schemes in terms of repeated averages, and replace the usual average (convex combination) by a binary averaging operation between two compact sets, introduced in [1] and termed here the "metric average." These schemes are shown to converge in the Hausdorff metric, and to provide O(h) approximation.
1777 Nonnecrotic tumor growth and the effect of vascularization.
I. Linear analysis and self-similar evolution
Vittorio Cristini, John Lowengrub, and Qing Nie
In this paper, we revisit the linear analysis of the transient evolution of a perturbed tumor interface in two and three dimensions. In Part II, we will study the full nonlinear problem using boundary-integral simulations. The tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation is developed that demonstrates that tumor evolution is described by a reduced set of two parameters and is qualitatively unaffected by the number of spatial dimensions. One parameter is related to the rate of mitosis. The other describes the balance between vascularization and apoptosis (programmed cell-death).
Three regimes of growth are identified with increasing degrees of vascularization: low (diffusion dominated), moderate and high vascularization. We demonstrate that parameter ranges exist for which the tumor evolves self-similarly (i.e., shape invariant) in the first two regimes. In the diffusion-dominated regime, vascularization is weak or absent and self-similar evolution leads to a nontrivial dormant state. In the second regime vascularization becomes significant with respect to apoptosis; self-similar growth is unbounded and is associated with critical conditions of vascularization. Away from these critical conditions, perturbations may either grow with respect to the unperturbed shape, and thus lead to invasive fingering into the external tissues and metastasization, or decay to zero. In the high-vascularization regime, we find that during unbounded growth the tumor shape always tends to the unperturbed shape and neither self-similar nor fingering evolution occur. This last result is in agreement with recent experimental observations of in vivo tumor growth and angiogenesis, and suggests that the metastatic growth of highly-vascularized tumors is associated to vascular and elastic anisotropies, which are not included in our model.
1778 Three-dimensional crystal growth. I. Linear analysis
and self-similar evolution
V. Cristini and J. Lowengrub
In this paper, Part I of our study, we revisit the linear analysis of the quasi-steady diffusional evolution of growing crystals in 3-D. We focus on a perturbed spherical solid crystal growing in an undercooled liquid with isotropic surface tension and interface kinetics. We investigate the relation between the far-field flux of temperature and undercooling in the far-field. In 3-D, the flux scales with the undercooling and with the instantaneous size of the crystal; this behavior is qualitatively different from 2-D, where there is no dependence on the size. As a consequence of this peculiarity, we demonstrate using linear analysis that in 3-D there exist critical conditions of flux at which self-similar evolution occurs. This leads to nonspherical, shape-invariant growing crystals. The critical flux increases with increasing wave-number of the perturbation, and separates regimes of stable and unstable growth, where stable growth implies that the perturbation decays with respect to the underlying sphere. The interfacial kinetics have a strong stabilizing effect, which is explored in detail here. These results demonstrate that the classical Mullins-Sekerka instability, that arises in the presence of constant undercooling, can be suppressed by maintaining near-critical flux conditions. Correspondingly, there is little creation of unstable modes during growth and unstable growth is very constrained or completely eliminated. Near-critical flux conditions can be achieved by appropriately varying the undercooling in time; thus this work has important implications for shape control in processing applications. Experiments are currently being designed (by Stefano Guido and coworkers at the University of Naples) to test this possibility. Moreover, in Part II of our study, we will investigate the nonlinear evolution using adaptive boundary-integral simulations.
1779 Lamellar microstructure of emulsions
V. Cristini, M. Simeone, S. Guido, R. Hooper, and C.W.
Macosko
Transient three-dimensional drop deformation is studied in dilute emulsions for large capillary numbers, corresponding to strong-flow or low-interfacial-tension conditions. Steady, planar linear flows are considered, described by the dimensionless vorticity orthogonal to the plane of flow. For drop-to-matrix viscosity ratios less than 1, drops widen along the vorticity direction due to the compressional component of the imposed flow. The drops are strongly elongated by the flow, and thus assume flat lamellar configurations, leading to remarkable interfacial area generation.
We analyze the limit of capillary number infinite first, and present analytic results demonstrating that in this case drop deformation is described by a universal function of the viscosity ratio, independent of the vorticity. Rigid-body rotation merely affects the time-evolution, i.e., with increasing vorticity, drops are more rotated away from the extensional direction of the flow thereby delaying deformation. We provide an exact solution that describes drop deformation far from the initial conditions. A constant drop width is achieved along the direction of vorticity, leading to the development of a stable lamellar morphology.
To explore the effect of interfacial tension (finite capillary number), drop evolution is then calculated using adaptive boundary-integral simulations and measured using video-microscopy. The experiments and the simulations are always found to be in good agreement. The extent of drop widening and interfacial area generation is strongly affected by a small but finite interfacial tension. Widening occurs only above a minimum capillary number, which increases with the viscosity ratio and the vorticity. The persistence of lamellar configurations is examined. As drops lengthen and flatten, local capillarity associated with high surface curvatures eventually becomes effective, and widening disappears. The development of a lamellar microstructure is thus a transitory phenomenon at finite capillary numbers: flattened drops evolve into slender cylindrical threads of fluid that finally break up.
1780 Modeling multiphase flows using a novel 3D adaptive
remeshing algorithm
Russell Hooper, Vittorio Cristini, Sundeep Shakya,
John S. Lowengrub, Jeffrey J. Derby, and Christopher W. Macosko
A novel three-dimensional adaptive remeshing algorithm is presented and applied to finite-element simulations of multiphase fluid flows. A three-dimensional domain enclosing another phase is discretized by an unstructured mesh of tetrahedra constructed from a triangulated surface of the phase boundaries. Complete remeshing is performed after each time step. The boundary mesh is reconstructed using an existing algorithm employing element addition/subtraction, edge swapping based on Delaunay triangulation and spring-like dynamical relaxation. The volume mesh is then generated from the boundary using the commercial software Hypermesh. The resulting adaptive discretization maintains resolution of prescribed local length scales.
1781 An adaptive mesh algorithm for evolving surfaces:
Simulations of drop breakup and coalescence
Vittorio Cristini, Jerzy Blawzdziewicz, and Michael
Loewenberg
An algorithm is presented for the adaptive restructuring of meshes on evolving surfaces. The resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations.
1782 Critical behavior of drops in linear flows: I.
Phenomenological theory for drop dynamics near critical stationary states
Jerzy Blawzdziewicz, Vittorio Cristini, and Michael
Loewenberg
The dynamics of viscous drops in linear creeping flows are investigated near the critical flow strength at which stationary drop shapes cease to exist. It is shown that the near-critical drop behavior is dominated by a single slow mode that evolves on the time scale diverging at the critical point with the exponent 1/2.
1783 Effect of inertia on drop breakup under
shear
Yuriko Y. Renardy and Vittorio Cristini
A spherical drop, placed in a second liquid of the same density and viscosity, is subjected to shear between parallel walls. The subsequent flow is investigated numerically with a volume-of-fluid continuous-surface-force algorithm. Inertially driven breakup is examined. The critical Reynolds numbers are examined for capillary numbers in the range where the drop does not break up in Stokes flow.
1784 Scalings for fragments produced from drop breakup
in shear flow with inertia
Yuriko Y. Renardy and Vittorio Cristini
When a drop is sheared in a matrix liquid, the largest daughter drops are produced by elongative end-pinching. The daughter drop size is found to scale with the critical drop size that would occur under the same flow conditions and fluid properties.
1787 Modelling pinchoff and reconnection in a Hele-Shaw
cell. Part I: The models and their calibration
Hyeong-Gi Lee, J.S. Lowengrub and J. Goodman
This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [1]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper (Part II [2]), we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [3] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in Part II [2].
1788 Modelling pinchoff and reconnection in a Hele-Shaw
cell. Part II: Analysis and simulation in the nonlinear regime
Hyeong-Gi Lee, J.S. Lowengrub, and J. Goodman
This is the second paper in a two part series in which we analyze two diffuse interface models to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. Diffusion between the components is limited if the components are macroscopically immiscible. In one of the systems (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we focus on buoyancy driven flow and the Rayleigh-Taylor instability. In the fully nonlinear regime before pinchoff, results from the HSCH and BHSCH models are compared to highly accurate boundary-integral simulations of the classical sharp interface system. In this case, we find that the diffuse interface models yield nearly identical results and we demonstrate convergence to the boundary-integral solutions as the interface thickness vanishes. We find that the break-up of an unstably stratified fluid layer is smoothly captured by both models. The HSCH model seems to be more diffusive than the BHSCH model and predicts an earlier pinchoff time which causes subtle differences between the two in the pinchoff region. Further, in the limit of zero interface thickness, we find that the effect of compressibility does not vanish at pinchoff. This distinguishes the HSCH model from all others in which compressibility effects are neglected. It may turn out, for example, that characterizing the limiting effect of compressibility at pinchoff may suggest a physically-based selection mechanism for cutting and reconnecting sharp interfaces. Varying the gravitational force and viscosities of the fluids yields different pinchoff times and numbers of satellite drops. Moreover, using the analysis of the linear growth rates from our first paper (Part I [1]), we confirm that the modified HSCH/BHSCH parameters suggested in that work lead to improved agreement with sharp interface results at finite interface thicknesses. Lastly, we also consider a case in which the fluid components are miscible. We find competition between buoyancy, viscous, diffusional and, at very early times, surface tension-like forces.
1789 On an elastically induced splitting instability
P.H. Leo, J.S. Lowengrub, and Qing Nie
We show that a morphological instability driven by deviatoric applied stresses can generate elastically induced particle splitting during phase transformations. The splitting instability occurs when the elastic fields are above some critical value. For elastic fields below critical, one observes a small perturbation of the particle shape consistent with splitting, but this perturbation is stabilized by surface tension. Both the onset of the splitting instability and the nonlinear evolution of the particle towards splitting depend on the precise form of the applied stress, the elastic constants of the precipitate and matrix, and the initial shape of the precipitate. We also investigate whether non-dilatational mistif strains can generate splitting instabilities in the absence of applied stress; however the results are inconclusive.
1790 Measurement and numerical analysis of freezing
in solutions enclosed in a small container
Ramachandra V. Devireddy, Perry H. Leo, John S. Lowengrub,
and John C. Bischof
The latent heat of fusion, L, of the cryobiological media (a solute laden aqueous solution) is a crucial parameter in the cryopreservation process and has often been approximated to that of pure water (335 mJ/mg). This study experimentally determines the magnitude and dynamics of latent heat during freezing of fourteen different pre-nucleated solute laden aqueous systems using a Differential Scanning Calorimeter (DSC). These solutions include NaCl-H_20, Phosphate Buffered Saline (PBS), serum free cell culture media (RPMI), glycerol and Anti Freeze Protein (AFP) in 1x PBS solutions. The latent heat of the solutions studied is found to be significantly less than that of pure water and is dependent on both the amount and type of solutes (or solids) in solution. DSC experiments are also performed at 1, 5 and 20 C/min in five representative cryobiological media to determine the kinetics of ice crystallization. The total magnitude of the latent heat release L is found to be independent of the cooling rate. However, the experimental data shows that at a fixed temperature, the fraction of heat released at higher cooling rates (5 and 20 C/min) is lower than at 1 C/min for all the solutions studied. We present a model to predict the experimentally measured behavior based on the full set of heat and mass transport equations during the freezing process in a DSC sample pan. Analysis of the parameters relevant to the transport processes reveals that the heat transport occurs much more rapidly than mass transport. The model also reveals the important physical parameters controlling the mass transport at the freezing interface and further elucidates the measured temperature and time dependence of the latent heat release.
1791 Boundary integral methods for multicomponent fluids
and multiphase materials
T.Y. Hou, J.S. Lowengrub, and M.J. Shelley
In this paper, we present an overview of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science. We focus on the recent development and outcome of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface and problems in Hele-Shaw flows involving pattern formation through the Saffman-Taylor instability, pattern selection and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations and the effects of the competition between surface and elastic energies on the microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time stepping methods.
1792 Focusing of an elongated hole in porous medium
flow
S.B. Angenent, D.G. Aronson, S.I. Betelu, and J.S.
Lowengrub
In the focusing problem we study solutions to the porous medium equation $u_t=\Delta u^m$ whose initial distributions are positive in the exterior of a compact two-dimensional region and zero inside. We assume that the initial interface is elongated and possesses reflectional symmetry with respect to both the x- and y- axes. We implement a numerical scheme that adapts the numerical grid around the interface so as to maintain a high resolution as the interface shrinks to a point. We find that as t tends to the focusing time T, the interface becomes oval-like with the lengths of the major and minor axes $O(\sqrt{T-t})$ and $O(T-t)$ respectively. Thus, the aspect ratio is $O(1/\sqrt{T-t})$. By scaling and formal asymptotic arguments, we derive an approximate solution which is valid for all m. This approximation indicates that the numerically observed power behavior for the major and minor axes is universal for all m>1.
1793 An hp-analysis of the local discontinuous Galerkin
method for diffusion problems
Ilaria Perugia and Dominik Schoetzau
We present an $hp$-analysis of the local discontinuous Galerkin method for diffusion problems, considering unstructured meshes with hanging nodes and two- and three-dimensional domains. Our estimates are optimal in the meshsize $h$ and slightly suboptimal in the polynomial approximation order $p$.
1794 Recent progress in the use of geometric integration
methods in micromagnetics and rigid body dynamics
Debra Lewis and Nilima Nigam
In this paper, we report further progress on our work on the use of Lie methods for integrating ordinary differential equations which evolve on manifolds. These algorithms better capture the qualitative behaviour of the trajectories since the numerical updates stay on the correct manifold. We study the effectiveness of higher order Lie methods in the context of rigid body dynamics, and for a problem in micromagnetics. This is work in progress.
1795 Computing the smoothness exponent of a symmetric
multivariate refinable function
Bin Han
Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increases the size of the matrix and therefore make the computation very expensive. In this paper, we shall present a simple algorithm to efficiently numerically compute the smoothness exponent of a symmetric refinable function with a general dilation matrix. By taking into account of symmetry of a refinable function, our algorithm greatly reduces the size of the matrix and enables us to numerically compute the Sobolev smoothness exponents of a large class of symmetric refinable functions. Step by step numerically stable algorithms and details of the numerical implementation are given. To illustrate our results by performing some numerical experiments, we construct a family of dyadic interpolatory masks in any dimension and we compute the smoothness exponents of their refinable functions in dimension three. Several examples will also be presented for computing smoothness exponents of symmetric refinable functions on the quincunx lattice and on the hexagonal lattice.
1796 The initial functions in a subdivision scheme
Bin Han
In this paper we shall study the initial functions in a subdivision scheme in a Sobolev space. By investigating the mutual relations among the initial functions in a subdivision scheme, we are able to study in a relatively unified approach several questions related to a subdivision scheme in a Sobolev space such as convergence, error estimate and convergence rate of a subdivision scheme in a Sobolev space with a general dilation matrix. A generalized definition of convergence of subdivision schemes in Banach spaces is also introduced.
1797 An adaptive finite-difference method for traveltimes
and amplitudes
Jianliang Qian and William W. Symes
The point source traveltime field has an upwind singularity at the source point. Consequently, all formally high-order finite-difference eikonal solvers exhibit first-order convergence and relatively large errors. Adaptive upwind finite-difference methods based on high-order Weighted Essentially NonOscillatory (WENO) Runge-Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than does the fixed-grid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as takeoff angles and geometrical spreading factors.
1798 Numerical analysis of the Cahn-Hilliard equation
and approximation for the Hele-Shaw problem, Part I: Error analysis under minimum
regularities
Xiaobing Feng and Andreas Prohl
In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semi-discrete (in time) scheme and a fully discrete mixed finite element scheme for the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ arising from phase transition in materials science, where $\vepsi$ is a small parameter known as an ``interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on $\varepsilon$. Quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size $h$ and the local time step size $k_m$ of the stretched time grid, and minimum regularity assumptions on the initial function $u_0$ and domain $\Omega$. In particular, all our error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and to establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on the stretched time grid. It is this polynomial dependency of the error bounds that paves the way for us to establish convergence of the numerical solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem (as $\varepsilon \searrow 0$) in Part II \cite{XA3} of the series.
1799 Numerical analysis of the Cahn-Hilliard equation
and approximation for the Hele-Shaw problem, Part II: Error analysis and convergence
of the interface
Xiaobing Feng and Andreas Prohl
In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ as $\varepsilon \searrow 0$. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [21] to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the Hele-Shaw (Mullins-Sekerka) problem has a global (in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an $L^\infty(L^\infty)$-error estimate, and making full use of the convergence result of [2]. Like in [20, 21], the cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco [3] and Chen [12], and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.
1800 Wave focusing on the line
Tuncay Aktosun and James H. Rose
Focusing of waves in one dimension is analyzed for the plasma-wave equation and the wave equation with variable speed. The existence of focusing causal solutions to these equations is established, and such wave solutions are constructed explicitly by deriving an orthogonality relation for the time-independent Schrödinger equation. The connection between wave focusing and inverse scattering is studied. The potential at any point is recovered from the incident wave that leads to focusing to that point. It is shown that focusing waves satisfy certain temporal-antisymmetry and support properties. Discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the Schrödinger equation. The theory is illustrated with some explicit examples.
1801 Ghost Symmetries
Peter J. Olver, Jan A. Sanders, and Jing Ping
Wang
We introduce the notion of a ghost symmetry for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for nonlocal vector fields.
1802 Interferometric GPS ambiguity resolution
T. Craig Poling and Aleksandar Zatezalo
The Maximum a Posteriori Ambiguity Search (MAPAS) method for GPS ambiguity resolution is generalized to accommodate: (1) satellite switches caused by satellites rising or falling below the horizon or obstructing terrain, and (2) cycle slips due to temporary loss of lock on satellite signals. It is shown that MAPAS and generalized MAPAS are equivalent to Bayesian estimation. The generalized MAPAS method is successfully applied to real GPS satellite data with cycle slips and satellite switches due to satellite obstruction.
1803 On a nonlinear partial differential
equation arising in magnetic resonance electrical impedance tomography
Sungwhan Kim, Ohin Kwon, Jin Keun Seo, and Jeong-Rock
Yoon
This paper considers the fundamental questions, such as existence and uniqueness, of a mathematical model arising in MREIT system, which is electrical impedance tomography technique integrated with magnetic resonance imaging. The mathematical model for MREIT is the Neumann problem of a nonlinear elliptic partial differential equation $\div\left(\frac{a(x)}{|\na u(x)|}\na u(x)\right)=0$. We show that this Neumann problem belongs to one of two cases: either infinitely many solutions or no solution exist. This explains rigorously the reason why we have used the modified model in [7] which is a system of the Neumann problem associated with two different Neumann data. For this modified system, we prove a uniqueness result on the edge detection of a piecewise continuous conductivity distribution.
1804 Finite-difference quasi-P traveltimes
for anisotropic media
Jianliang Qian and William W. Symes
The first-arrival quasi-P wave traveltime field in an anisotropic elastic solid solves a first-order nonlinear partial differential equation, the qP eikonal equation. The difficulty in solving this eikonal equation by a finite-difference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty. Furthermore, the solution of the paraxial qP eikonal equation, an evolution equation in depth, gives the first-arrival traveltime along downward propagating rays. A second-order upwind finite-difference scheme solves this paraxial eikonal equation in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2-D and 3-D transversely isotropic models demonstrate the accuracy of the scheme.
1805 On the Hölder continuity of
solutions of a certain system related to Maxwell's equations
KyungKeun Kang and Seick Kim
In this paper, we prove the Hölder continuity of weak solutions of a certain system arising from the Maxwell's equations in a quasi-stationary electromagnetic field.
1806 Wavelet-Domain Reconstruction of
Lost Blocks in Wireless Image Transmission and Packet-Switched Networks
Shantanu D. Rane, Jeremiah Remus, and Guillermo Sapiro
A fast scheme for wavelet-domain interpolation of lost image blocks in wireless image transmission is presented in this paper. The algorithm is designed to be compatible with the wavelet-based JPEG2000 image compression standard. In the transmission of block-coded images, fading in wireless channels and congestion in packet-switched networks can cause entire blocks to be lost. Instead of using common retransmission query protocols, we reconstruct the lost block in the wavelet-domain using the correlation between the lost block and its neighbors. The algorithm first uses a simple method to determine the presence or absence of edges in the lost block. This is followed by an interpolation scheme, designed to minimize the blockiness effect, while preserving the edges or texture in the interior of the block. The interpolation scheme minimizes the square of the error between the border coefficients of the lost block and those of its neighbors, at each transform scale. The performance of the algorithm on standard test images, its low computational overhead at the decoder, and its performance vis-a-vis other reconstruction schemes, is discussed.
1807 Enstrophy dynamics of stochastically forced
large-scale geophysical flows
Dirk Blömker, Jinqiao Duan, and Thomas Wanner
Enstrophy is an averaged measure of fluid vorticity. This quantity is particularly important in {\em rotating} geophysical flows. We investigate the dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under random wind forcing. We obtain upper bounds on the enstrophy, as well as results establishing its Hölder continuity and describing the small-time asymptotics.
1808 Dynamics of the thermohaline circulation under
wind forcing
Hongjun Gao and Jinqiao Duan
The ocean thermohaline circulation, also called meridional overturning circulation, is caused by water density contrasts. This circulation has large capacity of carrying heat around the globe and it thus affects the energy budget and further affects the climate. We consider a thermohaline circulation model in the meridional plane under external wind forcing. We show that, when there is no wind forcing, the stream function and the density fluctuation (under appropriate metrics) tend to zero exponentially fast as time goes to infinity. With rapidly oscillating wind forcing, we obtain an averaging principle for the thermohaline circulation model. This averaging principle provides convergence results and comparison estimates between the original thermohaline circulation and the averaged thermohaline circulation, where the wind forcing is replaced by its time average. This establishes the validity for using the averaged thermohaline circulation model for numerical simulations at long time scales.
1809 A generalization of Helgason's support
theorem
Takashi Takiguchi
We discuss a generalization of Helgason's support theorem for the Radon transform. In this theorem, the assumption of rapid decay of functions is essential. We restrict this rapid decay condition to an open cone and give a generalization. We also mention that our generalization is not possible with no global decay condition, to prove which we construct a counterexample.
1810 Examinations on a three-dimensional differentiable vector
field that equals its own curl
Biao Ou
Consider the differential equation curl f = f for a 3-dimensional differentiable vector field f. We prove that f is analytic and then prove an existence and uniqueness theorem for the differential equation with a prescribed boundary data. We also outline with a few variations Professor J. Ericksen's work on a unit vector field that equals its own curl.
1811 Stabilized interior penalty
methods for the time-harmonic Maxwell equations
I. Perugia, D. Schötzau, and P. Monk
We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time--harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach.
1812 Digital inpainting based on the Mumford-Shah-Euler
image model
Selim Esedoglu and Jianhong Shen
Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang, and Shen's recent inpainting schemes based on the bounded variation and the elastica image models. In the present paper, we propose two novel inpainting models based on the Mumford-Shah image models and the its high order correction -- the Mumford-Shah-Euler image model. We also present their efficient numerical realization based on the Gamma-convergence approximations of Ambrosio and Tortorelli, and De Giorgi.
1813 A uniqueness theorem of the 3-dimensional
acoustic scattering problem in a shallow ocean with a fluid-like seabed
Robert P. Gilbert and Miao-jung Ou
This paper shows that under the assumption of the out-going radiation conditions at infinity, the time-harmonic acoustic scattered field off a sound-soft solid in a shallow ocean with a fluid-like seabed is unique in C^{2} (M_{1}) C^{2} < (M_{2}) C(R_{h}^{3} \ ). Here M_{1} is the water part, M_{2} the seabed, R_{h}^{3} the waveguide and the solid object. The associated modal problem is studied and a representation formula for the solution in terms of the Green's function is derived.
1814 Geometric integration algorithms
on homogeneous manifolds
Debra Lewis and Peter J. Olver
Given an ordinary differential equation on a homogeneous manifold, one can construct a "geometric integrator'' by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is "full,'' then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere.
1815 Euler, Jacobi, and missions to comets
and asteroid
Michael Efroimsky
Whenever a freely spinning body is found in a complex rotational state, this means that either the body experienced some interaction within its relaxation-time span, or that it was recently "prepared'' in a non-principal state. Both options are encountered in astronomy where a wobbling rotator is either a recent victim of an impact or a tidal interaction, or is a fragment of a disrupted progenitor. Another factor (relevant for comets) is outgassing. By now, the optical and radar observational programmes have disclosed that complex rotation is hardly a rare phenomenon among the small bodies. Due to impacts, tidal forces and outgassing, the asteroidal and cometary precession must be a generic phenomenon: while some rotators are in the state of visible tumbling, a much larger amount of objects must be performing narrow-cone precession not so easily observable from the Earth.
The internal dissipation in a freely precessing top leads to relaxation (gradual damping of the precession) and sometimes to spontaneous changes in the rotation axis. Recently developed theory of dissipative precession of a rigid body reveals that this is a highly nonlinear process: while the body is precessing at an angular rate , the precession-caused stresses and strains in the body contain components oscillating at other frequencies. Dependent upon the spin state, those frequencies may be higher or, most remarkably, lower than the precession rate. In many states dissipation at the harmonics is comparable to or even exceeds that at the principal frequency.
For this and other reasons, in many spin states the damping of asteroidal and cometary wobble happens faster, by several orders, than believed previously. This makes it possible to measure the precession-damping rate. The narrowing of the precession cone through the period of about a year can be registered by the currently available spacecraft-based observational means. We propose an appropriate observational scheme that could be accomplished by comet and asteroid-aimed missions. Improved understanding of damping of excited rotation will directly enhance understanding of the current distribution of small-body spin states. It also will constrain the structure and composition of excited rotators.
However, in the near-separatrix spin states a precessing rotator can considerably slow down its relaxation. This lingering effect is similar to the one discovered in 1968 by Russian spacecraft engineers who studied free wobble of a tank with viscous fuel.
1816 Dynamics of quasi-geostrophic fluid motions
with rapidly oscillating Coriolis force
Hongjun Gao and Jinqiao Duan
An averaging principle for quasi-geostrophic fluid motions with rapidly oscillating Coriolis force is proved. This result includes comparison estimate and convergence result between quasi-geostrophic fluid motions and its averaged fluid motions. This averaging principle provides an autonomous system as an approximation for the nonautonomous quasi-geostrophic flows with rapidly oscillating Coriolis force.
1817 Analysis of a fully discrete finite
element method for the phase field model and approximation of its sharp
interface limits
Xiaobing Feng and Andreas Prohl
We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter $\varepsilon$, known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size $h$ and the time step size k. In particular, it is shown that all error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen [Comm. PDE, 1371-1395, 1994] and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.
1818 Matrix generalizations
of multiple hypergeometric functions
Lalit Mohan Upadhyaya and H.S. Dhami
1819 On regularity of stationary Stokes and Navier-Stokes
equations near boundary
Kyungkeun Kang
We obtain local estimates of the steady-state Stokes system "without pressure'' near boundary. As an application of the local estimates, we prove the partial regularity up to the boundary for the stationary Navier-Stokes equations in a smooth domain in five dimension.
1820 Dynamics of a Coupled Atmosphere-Ocean
Model
Hongjun Gao and Jinqiao Duan
The coupled atmosphere-ocean system defines the environment we live. The research of this complex, nonlinear and multiscale system is not only scientifically challenging but also practically important.
We consider a coupled atmosphere-ocean model, which involves hydrodynamics, thermodynamics and nonautonomous interaction at the air-sea interface. First, we show that the coupled atmosphere-ocean system is stable under the external fluctuation in the atmospheric energy balance relation. Then, we estimate the atmospheric temperature feedback in terms of the freshwater flux, heat flux and the external fluctuation at the air-sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Finally, we prove that the coupled atmosphere-ocean system has time-periodic, quasiperiodic and almost periodic motions, whenever the external fluctuation in the atmospheric energy balance relation is time-periodic, quasiperiodic and almost periodic, respectively. 25,1
1821 On some multiple
hypergeometric functions of several matrix arguments
Lalit Mohan Upadhyaya and H.S. Dhami
1822 Invariant manifolds for stochastic
partial differential equations
Jinqiao Duan, Kening Lu, and
Björn Schmalfuss
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional {\em random} dynamical systems generated by stochastic partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.
1823 Stability properties of Perona-Malik
scheme
Selim Esedoglu
The Perona-Malik scheme is a numerical technique for de-noising digital images without blurring object boundaries (edges). In general, solutions generated by this scheme do not satisfy a comparison principle. We identify conditions under which two solutions initially ordered remain ordered, and state (restricted) comparison principles. These allow us to study stability properties of the scheme. We also explore what these results say in the limit as the discretization size goes to 0.
1824 Control of the wave equation by
time-dependent coefficient
Antonin Chambolle and Fadil Santosa
We study an initial boundary-value problem for a wave equation with time-dependent soundspeed. In the control problem, we wish to determine a soundspeed function which damps the vibration of the system. We consider the case where the soundspeed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead energy decay. We illustrate the rich behavior of this problem in numerical examples.
1825 From 2-D to 3-D: Algorithms to recreate
a real-world scene from flat photographs
Jessica Conway
The goal of this paper is to provide a simple and efficient algorithm for the recovery of a three-dimensional scene from two-dimensional images of the same object or scene. To this end, we present an outline of an approach to extracting depth information from two-dimensional images, and then a direct featureless method to recover the 15 parameters of the exact projective coordinate transformation between two images. When we say exact, we are operating under the assumptions of static scene and no parallax, although we suspect and hope in the future to show that our methods are robust under deviations from these assumptions. Future work includes numerical experiments, and comparisons with real data.
1826 Regularity of axially symmetric
flows in a half-space in three dimension
Kyungkeun Kang
We study axially symmetric solutions with no swirl of the three dimensional Navier-Stokes equations in a half-space. We prove that suitable weak solutions in this case are Hölder continuous up to the boundary at all points except for the origin. For interior points this implies smoothness in the spatial variables. Hölder continuity at the origin remains as an open problem.
1827 Solving variational problems and
partial differential equations mapping into general target manifolds
Facundo Mémoli, Guillermo Sapiro, and Stanley
Osher
A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [3]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system of this new embedding function. In the case of variational problem, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we implicitly represent all the equation characteristics. We then obtain a set of equations that while defined on the whole Euclidean space, they are intrinsic to the implicit target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional surfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is an hypersphere; probability maps, where the target manifold is an hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional surfaces.
1828 Mechanical alignment of suprathermal
paramagnetic cosmic-dust granules: the cross-section mechanism
Michael Efroimsky
Mechanical alignment of suprathermal paramagnetic cosmic-dust granules: the cross-section mechanism
We develop a comprehensive quantitative description of the cross-section mechanism discovered several years ago by Lazarian. This is one of the processes that determine grain orientation in clouds of suprathermal cosmic dust. The cross-section mechanism manifests itself when an ensemble of suprathermal paramagnetic granules is placed in a magnetic field and is subject to ultrasonic gas bombardment. The mechanism yields dust alignment whose efficiency depends upon two factors: the geometric shape of the granules, and the angle between the magnetic line and the gas flow. We calculate the quantitative measure of this alignment, and study its dependence upon the said factors. It turns out that, irrelevant of the grain shape, the action of a flux does not lead to alignment if = arccos (1/3).
1829 A good image model eases restoration
- on the contribution of Rudin-Osher-Fatmi's BV image model
Tony F. Chan and Jianhong Shen
What we believe images are determines how we take actions in image and low-level vision analysis. In the Bayesian framework, it is known as the importance of a good image prior model. This paper intends to give a concise overview on the vision foundation, mathematical theory, computational algorithms, and various classical as well as unexpected new applications of the BV (bounded variation) image model, first introduced into image processing by Rudin, Osher, and Fatemi in 1992 [Physica D, 60:259-268].
1830 Stochastic dynamics of a coupled
atmosphere-ocean model
Jinqiao Duan, Hongjung Gao, and Björn Schmalfuß
The investigation of the coupled atmosphere-ocean system is not only scientifically challenging but also practically important.
We consider a coupled atmosphere-ocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the air-sea interface. We reformulate this model as a random dynamical system. First, we have shown that the asymptotic dynamics of the coupled atmosphere-ocean model is described by a random climatic attractor. Second, we have estimated the atmospheric temperature evolution under oceanic feedback, in terms of the freshwater flux, heat flux and the external fluctuation at the air-sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Third, we have demonstrated that this system has finite degree of freedom by presenting a finite set of determining functionals in probability. Finally, we have proved that the coupled atmosphere-ocean model is ergodic under suitable conditions for physical parameters and randomness, and thus for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.
1831 Multiscale resolution in the computation
of crystalline microstructure
Sören Bartels and Andreas Prohl
This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and simple austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optimal.
1832 On
Kampé De Fériet and Lauricella functions of matrix arguments - I
Lalit Mohan Upadhyaya and H.S. Dhami
1833 Frequency response of uncertain systems:
strong Kharitonov-like results
Long Wang
In this paper, we study the frequency response of uncertain systems using Kharitonov stability theory on first order complex polynomial set. For an interval transfer function, we show that the minimal real part of the frequency response at any fixed frequency is attained at some prescribed vertex transfer functions. By further geometric and algebraic analysis, we identify an index for strict positive realness of interval transfer functions. Some extensions and applications in positivity verification and robust absolute stability of feedback control systems are also presented.
1834 On
the number of positive solutions to a class of integral equations
Long Wang, Wensheng Yu, and Lin Zhang
1835 Performance evaluation of switched
discrete event systems
Long Wang
This paper discusses the asymptotic periodic behavior of a class of switched discrete event systems, and shows how to evaluate the asymptotic performance of such systems.
1836 On
Lauricella and related functions of matrix arguments-II
Lalit Mohan Upadhyaya and H.S. Dhami
1837 Unbounded normal derivative for the
Stokes system near boundary
Kyungkeun Kang
We study local boundary regularity for the Stokes system. We show that, unlike in the interior case, non-local effects can lead to a violation of local regularity in the spatial variables near the boundary.
1838 Hamiltonian methods for geophysical
fluid dynamics: an introduction
Peter Lynch
The value of general Hamiltonian methods in geophysical fluid dynamics has become clear over recent years. This paper provides an introduction to some of the key ideas necessary for fruitful application of these methods to problems in atmosphere and ocean dynamics. Hamiltonian dynamics is reviewed in the context of simple particle dynamics. The non-canonical formalism which is required for fluid dynamics is introduced first in the finite-dimensional case. The Lagrangian and Eulerian formulations of the fluid dynamical equations are then considered, and the method of reduction from Lagrangian to Eulerian form is described. Rotational effects are introduced in the context of the shallow water equations, and these equations are expressed in Hamiltonian form in both Lagrangian and Eulerian variables. Finally, simple balanced systems are derived, in which constraints are imposed on the fluid motion by applying least action principles to Lagrangians modified either by additional terms with Lagrange multipliers or by direct approximation.
1839 Stability of polytopic polynomial
matrices
Long Wang, Zhizhen Wang, and Wensheng Yu
This paper gives a necessary and sufficient condition for robust D-stability of Polytopic Polynomial Matrices.
1840 A recipe for construction of the
critical vertices for left-sector stability of interval polynomial
Long Wang
For the left-sector stability of interval polynomials, it suffices to check a subset of its vertex polynomials. This paper provides a recipe for construction of these critical vertices. Illustrative examples are presented.
1841 Geometric characterization of strictly
positive real regions and its applications
Long Wang and Wensheng Yu
Strict positive realness (SPR) is an important concept in absolute stability theory, adaptive control, system identification, etc. This paper characterizes the strictly positive real (SPR) regions in coefficient space and presents a robust design method for SPR transfer functions. We first introduce the concepts of SPR regions and weak SPR regions and show that the SPR region associated with a fixed polynomial is unbounded, whereas the weak SPR region is bounded. We then prove that the intersection of several weak SPR regions associated with different polynomials can not be a single point. Furthermore, we show how to construct a point in the SPR region from a point in the weak SPR region. Based on these theoretical development, we propose an algorithm for robust design of SPR transfer functions. This algorithm works well for both low order and high order polynomial families. Illustrative examples are provided to show the effectiveness of this algorithm.
1842 Robust performance of a class of
control systems
Long Wang
Some Kharitonov-like robust Hurwitz stability criteria are established for a class of complex polynomial families with nonlinearly correlated perturbations. These results are extended to the polynomial matrix case and non-interval D-stability case. Applications of these results in testing of robust strict positive realness of real and complex interval transfer function families are also presented.
1843 Robust SPR synthesis for low-order
polynomial segments and interval polynomials
Long Wang and Wensheng Yu
We prove that, for low-order (n 4) stable polynomial segments or interval polynomials, there always exists a fixed polynomial such that their ratio is SPR-invariant, thereby providing a rigorous proof of Anderson's claim on SPR synthesis for the fourth-order stable interval polynomials. Moreover, the relationship between SPR synthesis for low-order polynomial segments and SPR synthesis for low-order interval polynomials is also discussed.
1844 Equations for the Keplerian elements:
Hidden symmetry as an unexpected source of numerical error
Michael Efroimsky
We revisit the Lagrange's system of equations for the six osculating elements, in the context of long-term planetary-orbit integration. An accurate re-examination of the derivation of Lagrange's system shows that, in fact, the orbit is always located not in the 6-dimensional space of the osculating elements, but in a certain 3-dimensional submanifold. If an analytic solution to Lagrange's system were available, it would obey this demand. However, whatever numerical integrator will cause drift away from this submanifold. This will result in a new type of accumulating numerical error that will be especially significant at long time spans. We point out an adjustment to be instilled in the integrator, that would eliminate this error.
We point out that the choice of the said submanifold is mathematically equivalent to fixing a gauge in field theory. The existing freedom of subminifold choice (~=~freedom of gauge fixing) reveals a symmetry (and a fibre bundle structure) hiding behind Lagrange's system. Just as a choice of the convenient gauge simplifies calculations in electrodynamics, the freedom in choice of the submanifold may, potentially, lead to simpler schemes of orbit integration.
1845 On
superintegrable symmetry-breaking potentials in N-dimensional Euclidean space
E.G. Kalnins, G.C. Williams, W. Miller, Jr.,
and G.S. Pogosyan
1846 Complete
sets of invariants for dynamical systems that admit a separation of variables
E.G. Kalnins, J.M. Kress, G. Pogosyan, and W.
Miller, Jr.
1847 Micro- and macro-scopic models of
rock fracture
Donald L. Turcotte, William I. Newman, and Robert
Shcherbakov
The anelastic deformation of solids is often treated using continuum damage mechanics. An alternative approach to the brittle failure of a solid is provided by the discrete fiber-bundle model. Here we show that the continuum damage model can give exactly the same solution for material failure as the fiber-bundle model. We compare both models with laboratory experiments on the time dependent failure of chipboard and fiberglass. The power-law scaling obtained in both models and in the experiments is consistent with the power-law seismic activation observed prior to some earthquakes.
1848 Appel's
and Humbert's functions of matrix arguments - I
Lalit Mohan Upadhyaya and H.S. Dhami
1849 Error of the network approximation
for densely packed composites with irregular geometry
Leonid Berlyand and Alexei Novikov
We apply a discrete network approximation to the problem of the effective conductivity of the high contrast, highly packed composites. The inclusions are irregularly (randomly) distributed in the hosting medium, so that a significant fraction of them may not participate in the conducting spanning cluster. For this class of inclusion distributions we derive a discrete network approximation and obtain an a priori error estimate for this approximation in which all the constants are explicitly computed. Explicit dependence on the irregular geometry of the inclusions' array is obtained.
We use variational techniques to provide rigorous mathematical justification for the approximation and its error estimate.
1850 On
boundary regularity of the Navier-Stokes equations
Kyungkeun Kang
1851 Weber's law and weberized TV restoration
Jianhong Shen
Most conventional image processors consider little the influence of human vision psychology. Weber's Law in psychology and psychophysics claims that human's perception and response to the intensity fluctuation of visual signals are weighted by the background stimulus, instead of being plainly uniform. This paper attempts to integrate this well known perceptual law into the classical total variation (TV) image restoration model of Rudin, Osher, and Fatemi [Physica D, 60:259-268, 1992]. We study the issues of existence and uniqueness for the proposed Weberized nonlinear TV restoration model, making use of the direct method in the space of functions with bounded variations. We also propose an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equation.
1852 The
Navier-Stokes equations and backward uniqueness
G. Seregin and V. Sverák
1853 Appel's
and Humbert's Functions of Matrix Arguments - II
Lalit Mohan Upadhyaya and H.S. Dhami
1854 A
transmission problem for fluid-structure interaction in the exterior of a
thin domain
G.C. Hsiao and N. Nigam
1855 On some quantum and analytical properties
of fractional Fourier transforms
Jianhong Shen
Fractional Fourier transforms (FrFT) are a natural one-parameter family of unitary transforms that have the ordinary Fourier transform embedded as a special case. In this paper, following the efforts of several authors, we explore the theory and applications of FrFT, from the standpoints of both quantum mechanics and analysis. These include the phase plane interpretation of FrFT, FrFT's role in the order reduction of certain classes of differential equations, the integral representation of FrFT, and its Paley-Wiener theorem and Heisenberg uncertainty principle. Our two major tools are quantum operator algebra and asymptotic analysis such as the singular perturbation theory and the stationary phase technique.
1856
Humbert's functions of matrix arguments-I
Lalit Mohan Upadhyaya and H.S. Dhami
1857 Anisotropic inverse conductivity
and scattering problems
Kiwoon Kwon and Dongwoo Sheen
Uniqueness in inverse conductivity and scattering problems is considered. In case the medium consists of two discontinuous constant anisotropic conductive parts, the measurements of potential and induced currents on the boundary of surrounding body are enough to guarantee uniqueness to determine conductivity and region of embedded unknown material under a very weak condition. The analogous uniqueness result is also obtained for an inverse scattering problem in the case that the medium is composed of two anisotropic and homogeneous materials.
1858 On the significance of the
Titius-Bode Law for the distribution of the planets
Peter Lynch
The radii of the planetary and satellite orbits are in approximate agreement with geometric progressions. The question of whether the observed patterns have some physical basis or are due to chance may be addressed using a Monte Carlo approach. We find that the estimated probability of chance occurrence depends sensitively on the restrictions imposed on the population of orbits. We argue that it is not possible to conclude unequivocally that laws of Titius-Bode type are, or are not, significant. Therefore, the possibility of a physical explanation for the observed distributions remains open.
1859 Constructing stationary
Gaussian processes from deterministic processes with random initial conditions
P.F. Tupper
We consider a family of stationary Gaussian processes that includes the stationary Ornstein-Uhlenbeck process. We show that processes in this family can be attained as the limit of a sequence of deterministic processes with random initial conditions. Weak convergence in the supremum norm on finite time-intervals is shown. We also establish the convergence of a wide variety of long-term statistics. Our construction provides a rigorous example of how macroscopic stochastic dynamics can be derived from microscopic deterministic dynamics.
1860 Is image steganography natural?
Guillermo Sapiro
Steganography is the art of secret communication [1, 2]. Its purpose is to hide the presence of information, using for example images as covers. After embedding the secret message into the cover image, a stego-image is obtained. While steganography algorithms create stego-images that are perceptually natural, we questioned if they are statistically natural [3, 4]. We show that stego-images violate recent models of natural images, and discuss the implications of this both in the art of steganography and in the mathematical modeling of natural images.
1861 Morse description and geometric
encoding of digital elevation maps
Andrés Solé, Vicent Caselles, Guillermo
Sapiro, and Francisco Arándiga
Two complementary geometric structures for the topographic representation of an image are developed in this work. The first one computes a description of the Morse-topological structure of the image, while the second one computes a simplified version of its drainage structure. The topographic significance of the Morse and drainage structures of Digital Elevation Maps (DEM) suggests that they can been used as the basis of an efficient encoding scheme. As an application we combine this geometric representation with an interpolation algorithm and lossless data compression schemes to develop a compression scheme for DEM. This algorithm achieves high compression while controlling the maximum error in the decoded elevation map, a property that is necessary for the majority of applications dealing with DEM. We present the underlying theory and compression results for standard DEM data.
1862 On the regularity of solutions
to a parabolic system related to Maxwell's equations
Kyungkeun Kang, Seick Kim, and Aurelia Minut
The goal of this paper is to establish H\"older estimates for the solutions of a certain parabolic system related to Maxwell's equations. Such an estimate is employed to get the local H\"older continuity of the magnetic field arising from Maxwell's equations in a quasi-stationary electromagnetic field, provided the resistivity of the material is continuous in time.
1863
Optimal blowup rates for the minimal energy null control for the structurally
damped abstract wave equation
George Avalos and Irena Lasiecka
1864
Analysis of total variation flow and its finite element approximations
Xiaobing Feng and Andreas Prohl
1865
Humbert's Functions of Matrix Arguments-II
Lalit Mohan Upadhyaya and H.S. Dhami
1866-1 Modeling planarization
in chemical-mechanical polishing
Dilek Alagoz, Stephanie Hoogendoorn, Satyanarayana
Kakollu, Maria Reznikoff, Richard Schugart, and Michael Sostarecz (Leonard
Borucki, mentor)
A mathematical model for chemical-mechanical polishing is developed. The effects of pad bending, fluid flow, and friction are considered. Fluid flow and friction effects are determined to be insignificant in the current model. Numerical results for the model including pad bending are presented and compared to experimental data.
1866-2 Vehicle networks: achieving
regular formation
Madalena Chaves, Robert Day, Lucia Gomez Ramos,
Parthasarathi Nag, Anca Williams, and Wei Zhang (Sonja Glavaski, mentor)
In this paper we will consider a network of vehicles exchanging information among themselves with the intention of achieving a specified polygonal formation. The network achieves the formation through decentralized feedback control, which is constructed from the available information. Several information flow laws are considered in order to improve the performance of the vehicle network. A stochastic model for information flow is also considered, allowing for the randomly breaking of the communication links among the vehicles.
1866-3 Designing airplane struts
using minimal surfaces
Sara Del Valle, Todd Moeller, Siva Kumar Natarajan
, Gergina V. Pencheva, Jason C. Sherman, and Steven M. Wise (Thomas Grandine,
mentor)
A model for minimizing the effects of skin drag and pressure drag is constructed. We show that a simple scaling technique can be used to transform a dual, constrained minimization problem into a volume constrained surface area minimization. We discuss some successes and failures with implementing numerical methods for the problem.
1866-4 Mobility management in
cellular telephony
Benjamin P. Cooke, Darongsae Kwon, Dmitry Glotov,
Simon Schurr, Daniel Taylor, and Todd Wittman (David F. Shallcross, mentor)
In the world of cellular telephony there is a hierarchy of controlling devices. Cellular telephones communicate with Base Transceiver Satations (BTS) or transceivers, which is turn are assigned to Base Station Controllers (BSC) or controllers. All controllers are connected to a Mobile Switching Centers (MSC). As a user of a cellular telephone moves around, the call must be transfered from transceiver to transceiver, and sometimes from controller to controller. We are interested in the problem of minimizing the cost of these transfers from controller to controller. In the above hierarchy, we consider a subtree emanating from one Mobile Switching Center. The two sets corresponding to the subtree are I, the set of transceivers, and J, the set of controllers. The problem is to assign each transceiver from the set I to a controller from the set J optimally subject to certain constraints...
1866-5 Optimal design for a
varying environment
Serguei Lapin, Xuan Hien Nguyen, Jiyeon Oh, Daniel
Vasiliu, Pei Yin, and Ningyi Zhang (David Misemer, mentor)
Lasers are currently used in many processes in which materials
are manipulated, including abalation of polymers, cutting of both metals and
nonmetals, and annealing of semiconductors. In many applications, the processes
include not only changing the material properties but patterning them as well.
Computer-generated holograms are diffractive optical elements (DOE) that permit
very general changes in phase and amplitude of an incoming wave. By adjusting
the local phase function, one can create the desired target intensity in the
image plane. The DOE responsible for the phase shifts is called a phase mask.
Such optical elements can be used to shape a beam with Gaussian intensity
profile into a uniform top hat shape. Although it is now possible
to create continuous masks to adjust the phase, the discrete ones are more
common and inexpenisve...
1866-6 Modeling the economics
of differentiated durable-goods markets
Miyuki Breen, George Chikhladze, Jose Figueroa-Lopez,
Yaniv Gershon, Yanto Muliadi, and Ivy Prendergast (Suzhou Huang, mentor)
Leasing has traditionally been one of the tools that firm employs to increase market share. It is not intuitive that this strategy would actually be beneficial. It is certainly not true if we consider only perishable goods. However, there are goods in the market that does not perish the instant we comsume it or durable goods. Having leasing option in durable goods created another market for used goods as a residue when the lease term ends. In the past, we have seen companies implementing this policy exclusively even abandoning their selling option. Here we are only considering the case where firms never sells their new goods due to the high transaction cost incurred. Durable goods is interesting on its own accord since we are faced with a dynamic problem in which each period depends on the previous periods action since we might still have an option to reuse the product we purchased from the previous periods. Thus the decision process for the market participants are intrinsically dynamic...
1867 The motion of a tracer particle
in a one-dimensional system: Analysis and simulation
P.F. Tupper
Our goal is to obtain a test system for the evaluation of time-stepping methods in molecular dynamics. We consider a family of deterministic systems consisting of a finite number of particles interacting on a compact interval. The particles are given random initial conditions and interact through instantaneous energy- and momentum-conserving collisions. As the number of particles, the particle density, and the mean particle speed go to infinity, the trajectory of a tracer particle is shown to converge to a stationary Gaussian process. We simulate the system with two numerical methods, one symplectic, the other energy-conserving, and assess the methods' ability to recapture the system's limiting statistics.
1868 Variational image inpainting
Tony F. Chan and Jianhong Shen
Inpainting is an image interpolation problem, with broad applications in image and vision analysis. This paper presents our recent efforts in developing inpainting models based on the Bayesian and variational principles. We discuss several geometric image models, their role in the construction of variational inpainting models, and the associated Euler-Lagrange PDEs and their numerical computation.
1869 Simultaneous structure and
texture image inpainting
Marcelo Bertalmio, Luminita Vese, Guillermo Sapiro,
and Stanley Osher
An algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two sub-images. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with inpainting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.
1870 Variational PDE models in
image processing
Tony F. Chan, Jianhong Shen, and Luminita Vese
In this article, we intend to give a broad picture of mathematical image processing through one of the most recent and very successful approaches -- the variational PDE method. We first discuss two crucial ingredients for image processing: image modeling or representation, and processor modeling. We then focus on the variational PDE method. The backbone of this article consists of two major problems in image processing -- inpainting and segmentation.
1871 Harnack inequality for nondivergent
elliptic operators on Riemannian manifolds
Seick Kim
We consider second-order linear elliptic operators of nondivergence type which is intrinsically defined on Riemannian manifolds. Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré's result and, as a consequence, we give another proof to Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator.
1872
A note on boundary blow-up problem of u=u ^{p}
Seick Kim
1873 On the range of applicability
of the Reissner-Mindlin and Kirchhoff-Love plate bending models
Douglas N. Arnold, Alexandre L. Madureira, and
Sheng Zhang
We show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchhoff-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff-Love model fails.
1874 Nonconforming mixed elements
for elasticity
Douglas N. Arnold and Ragnar Winther
We construct first order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger-Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degrees of freedom.
1875 Differential complexes and
numerical stability
Douglas N. Arnold
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.
1876
Generalized Horn's functions of matrix arguments
Lalit Mohan Upadhyaya and H.S. Dhami
1877 Complex rotation with internal
dissipation. Applications to cosmic-dust alignment and to wobbling comets
and asteroids
Michael Efroimsky, Alex Lazarian, and Vladislav
Sidorenko
Neutron stars, asteroids, comets, cosmic-dust granules, spacecraft, as well as whatever other freely spinning body dissipate energy when they rotate about any axis different from principal. We discuss the internal-dissipation-caused relaxation of a freely precessing rotator towards its minimal-energy mode (mode that corresponds to the spin about the maximal-inertia axis). We show that this simple system contains in itself some quite unexpected physics. While the body nutates at some rate, the internal stresses and strains within the body oscillate at frequencies both higher and (what is especially surprising) lower than this rate. The internal dissipation takes place not so much at the frequency of nutation but rather at the second and higher harmonics. In other words, this mechanical system provides an example of an extreme non-linerity. Issues like chaos and separatrix also come into play. The earlier estimates, that ignored non-linearity, considerably underestimated the efficiency of the internal relaxation of wobbling asteroids and comets. At the same time, owing to the non-linearlity of inelastic relaxation, small-angle nutations can persist for very long time spans. The latter circumstance is important for the analysis and interpretation of NEAR's data on Eros' rotation state. Regarding the comets, estimates show that the currently available angular resolution of spacecraft-based instruments makes it possible to observe wobble damping within year- or maybe even month-long spans of time. Our review also covers pertinent topics from the cosmic-dust astrophysics; in particular, the role played by precession damping in the dust alignment. We show that this damping provides coupling of the grain's rotational and vibrational degrees of freedom; this entails occasional flipping of dust grains due to thermal fluctuations. During such a flip, grain preserves its angular momentum, but the direction of torques arising from H_{2} formation reverses. As a result, flipping grain will not rotate fast in spite of the action of uncompensated H_{2} formation torques. The grains get ``thermally trapped,'' and their alignment is marginal. Inelastic relaxation competes with the nuclear and Barnett relaxations, so we define the range of sizes for which the inelastic relaxation dominates.
1878 Backward uniqueness for
the heat operator in half space
L. Escauriaza, G. Seregin, and V. Sverák
We prove a backward uniqueness result for the heat operator with variable lower order terms in a half space. The main point of the result is that the boundary conditions are not controlled by the assumptions.
1879 Invariant manifolds in a
dynamical model for gene transcription
Martin Caberlin, Michael Mackey, and Nilima Nigam
We present some recent results concerning stiffness in the Satillán-Mackey model of the tryptophan operon. In particular, we describe the existence of invariant manifolds in this system, and describe their biological significance.
1880 Special canonical models
for multidimensional data analysis with applications and implications
Vithanage Pemajayantha
Deterministic and stochastic forms of linear and non-linear ``prior'' models were used to develop a new multidimensional data analysis within the classical canonical analysis. Detection of outliers with the new model is discussed. While the new model opens up a variety of research problem, it has potential straightforward applications in data mining in science, economics, commerce and industry.
1881 An equality for the curvature
function of a simple, closed curve on the plane
Biao Ou
We prove an equality for the curvature function of a simple, closed curve on the plane. This equality leads to another proof of the four-vertex theorem in differential geometry.
1882
On Exton's generalized quadruple hypergeometric functions and Chandel's
function of matrix arguments
Lalit Mohan Upadhyaya and H.S. Dhami
1883 Competition model for two
exotic species and one native species
Daniel L. Kern
The spread of two exotic plant species and the corresponding replacement of a single native species is examined as a competition model with spatial considerations. The general model is a system of three Lotka-Volterra type nonlinear reaction-diffusion equations. The traveling wave solution is examined, giving conditions for minimum wave speed for the exotic species. The work is based on the case of Russian olive trees and tamarisks in the cottonwood woodlands of New Mexico.
1884 Conservative multigrid methods
for Cahn-Hilliard fluids
Junseok Kim, Kyungkeun Kang, and John Lowengrub
We develop a conservative, second order accurate fully implicit discretization in two dimensions of the Navier-Stokes NS and Cahn-Hilliard CH system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [44, 4]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We analyze and prove convergence of the scheme in the absence of flow. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an external shear flow, the evolution of the flow is nontrivial and the flow morphology repeats itself in time as multiple pinchoff and reconnection events occur. Eventually, the periodic motion ceases and the system relaxes to a global equilibrium. The equilibria we observe appears has a similar structure in all cases although the dynamics of the evolution is quite different. We view the work presented in this paper as preparatory for the detailed investigation of liquid/liquid interfaces with surface tension where the interfaces separate two immiscible fluids [37]. To this end, we include a simulation of the pinchoff of a liquid thread under the Rayleigh instability at finite Reynolds number.
1885 Morse and drainage description
and encoding of image
Vicent Caselles, Guillermo Sapiro, and Andres Solé
In this paper we develop and analyze basic geometric structures for the topographic representation of images. One component of the geometric description is based on the Morse structure of the image, while a second one is connected to its drainage structure. These fundamental descriptors could be used as building blocks for a geometric multiscale representation of images in general and Digital Elevation Models (DEM) in particular. The topographic significance of the Morse and drainage structures of DEMs suggests that they can be used as the basis of an efficient encoding scheme. Therefore, we combine this geometric representation with partial differential equations based interpolation algorithms and lossless data compression techniques to develop a compression scheme for DEM. This algorithm permits to obtain compression results while controlling the maximum error in the decoded elevation map, a property that is necessary for the majority of applications dealing with DEM. We present the underlying theory and compression results for standard DEM data.
1886 Acoustic wave propagation
in a composite of two different poroelastic materials with a very rough periodic
interface: a homogenization approach
Robert P. Gilbert and Miao-jung Ou
Homogenization is used to analyze the system of Biot-type partial differential equations in a domain of two different poroelastic materials with a very rough periodic interface. It is shown that by using homogenization, such a rough interface can be replaced by an equivalent flat layer within which a system of modified differential equations holds. The coefficients of this new system of equations are certain ``effective'' parameters. These coefficients are determined by solutions of the auxiliary problems which involve the detailed structure of the interface. In this paper, the auxiliary problems are derived and the homogenized system of equations is given.
1887
On some generalized multiple hypergeometric functions of matrix arguments
Lalit Mohan Upadhyaya and H.S. Dhami
1888 General solution to the
robust strictly positive real synthesis problem for polynomial segments
Yuwensheng Wanglong
This paper constructively solves a long standing open problem in modern control theory. Namely, for any two n-th order polynomials a(s) and b(s), the Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial c(s) such that c(s) / a(s) and c(s) / b(s) are both strictly positive real.
1889 Improved results on robust
stability of multivariable interval control systems
Zhizhen Wang, Long Wang, and Wensheng Yu
For interval polynomial matrices, we identify the minimal testing set, whose stability can guarantee that of the whole uncertain set. Our results improve the conclusions given by Kamal and Dahleh.
1890 H^{ }
performance of interval systems
Long Wang
In this paper, we study H^{ } performance of interval systems. We prove that, for an interval system, the maximal H^{ } norm of its sensitivity function is achieved at twelve (out of sixteen) Kharitonov vertices.
1891 Robust strictly positive
real synthesis for convex combination of sixth-order polynomials
Wensheng Yu and Long Wang
For the two sixth-order polynomials a(s) and b(s), Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial c(s) such that c(s) / a(s) and c(s) / b(s) are both strictly positive real. Our reasoning method is constructive, and is insightful and helpful in solving the general robust strictly positive real synthesis problem.
1892 Robust D-stability
of uncertain MIMO systems: LMI criteria
Long Wang, Zhizhen Wang, Baoyu Wu, and Wensheng
Yu
The focal point of this paper is to provide some simple and efficient criteria to judge the D-stability of two families of polynomials, i.e., an interval multilinear polynomial matrix family and a polytopic polynomial family. Taking advantage of the uncertain parameter information, we analyze these two classes of uncertain models and give some LMI conditions for the robust stability of the two families. Two examples illustrate the effectiveness of our results.
1893 Robust strictly positive
real synthesis for polynomial families of arbitrary order
Wensheng Yu and Long Wang
For any two n-th order polynomials a(s) and b(s), the Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial c(s) such that c(s) / a(s) and c(s) / b(s) are both strictly positive real.
1894 Edge theorem for multivariable
systems
Long Wang, Zhizhen Wang, Lin Zhang, and Wensheng
Yu
This paper studies robustness of multivariable systems with parametric uncertainties, and establishes a multivariable version of Edge Theorem. An illustrative example is presented.
1895 Non-Photorealistic rendering
from stereo
A. Bartesaghi and G. Sapiro
A new paradigm for automatic non-photorealistic rendering is introduced in this paper. Non-photorealistic rendering (NPR) provides an alternative way to render complex scenes by emphasizing high level or salient perceptual features. Particularly, the pen-and-ink rendering style produces sketchy-like drawings that can effectively communicate shape and geometry. This is achieved by combining drawing primitives that mimic ink patterns used by artists. Existing NPR approaches can be categorized in two groups depending on the type of input they use: image based and object based. Image based NPR techniques use 2D images to produce the renderings. Object based techniques work directly on given 3D models and make use of the full volumetric representation. In this paper we propose to enjoy the best of both worlds developing an hybrid model that simultaneously uses information from the image and object domains. These two sources of information are provided by a calibrated stereoscopic system. Given a pair of stereo images and the calibration data we solve the stereo problem in order to extract the normal and principal direction fields, which are fundamental to guide a texture synthesis algorithm that generates the NPR renderings. In particular, normals guide tonal variations, while principal directions determine the orientation of stroke-like texture patterns. We describe a particular, fully automatic, implementation of these ideas and present a number of examples.
1896 The method of variation
of constants and multiple time scales in orbital mechanics
William I. Newman and Michael Efroimsky
The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in orbital mechanics. This methodology assumes that certain ``constants'' associated with a homogeneous problem will vary in time in response to an external force. It also introduces one or more constraint equations motivated by the nature of the time-dependent driver. We show that these constraints can be generalized, in analogy to gauge theories in physics, and that different constraints can offer conceptual advances and methodological benefits to the solution of the underlying problem. Examples are given from linear ordinary differential equation theory and from orbital mechanics. However, a slow driving force in the presence of multiple time scales contained in the underlying (homogeneous) problem nevertheless requires special care, and this has strong implications to the analytic and numerical solutions of problems ranging from celestial mechanics to molecular dynamics.
1897 A characterization of hybridized
mixed methods for second order elliptic problems
Bernardo Cockburn and Jayadeep Gopalakrishnan
In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second-order, self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart-Thomas and the Brezzi-Douglas-Marini methods of similar order are identical.
1898 Numerical simulation of
deformable drops with soluble surfactant: Pair interactions and coalescence
in shear flow
H. Zhou, V. Cristini, and C.W. Macosko
We study numerically the dynamics of deformable drops in the presence of surfactant species both on the drop-matrix interfaces and in the bulk fluids using a novel 3D adaptive finite-element method. The method is based on unstructured adaptive triangulated and tetrahedral meshes that discretize the interfaces and the bulk respectively, and on an efficient parallelization of the numerical solvers. We use this method to investigate the effects of surfactants on drop-drop interactions in shear flow. The simulations account for surfactant effects through a nonlinear Langmuir equation of state and through adsorption/desorption laws describing the transport between bulk and interface. Van der Waals forces responsible for coalescence are included. For clean drops (no surfactant), our simulations confirm (for the first time to our knowledge) a well known theoretical result [1] for the dependence of the critical capillary number-below which coalescence occurs-on the drop radius with an exponent -4/9. Our results reveal a non-monotonic dependence of the critical capillary number Ca_{c} on the surface coverage of surfactant. Marangoni stresses prevent drop approach thus decreasing Ca_{c} with respect to the clean-drop case. However, at large coverages close to the maximum packing of surfactant molecules, surfactant redistribution is prohibited (the surfactant is nearly incompressible) and thus the effect of Marangoni stresses is weakened, leading to an increase of Ca_{c}. In some cases, Ca_{c} at high coverages is even larger than in the clean-drop case: surfactant near-incompressibility hinders drop deformation and thus coalescence can occur at higher capillary number. Finally, our results also reveal a non-monotonic dependence of Ca_{c} on surfactant solubility in the bulk. At moderate surfactant concentration, diffusion in the bulk decreases surfactant redistribution on the interface and thus weakens Marangoni stresses resulting in higher Ca_{c} than in the insoluble case. However, when the surfactant bulk concentration is large, high adsorption fluxes maintain a higher surface concentration in equilibrium than for the insoluble case, thus resulting in larger drop deformation and in lower Ca_{c}.
1899
Lauricella-Saran triple hypergeometric functions of matrix arguments-I
Lalit Mohan Upadhyaya and H.S. Dhami
1900 Bayesian video dejittering
by BV image model
Jianhong Shen
Line jittering, or random horizontal displacement in video images, occurs when the synchronization signals are corrupted in video storage media, or by electromagnetic interference in wireless video transmission. The goal of intrinsic video dejittering is to recover the ideal video directly from the observed jittered and often noisy frames. The existing approaches in the literature are mostly based on local or semi-local filtering techniques and autoregressive image models, and complemented by various image processing tools. In this paper, based on the statistical rationale of Bayesian inference, we propose the first variational dejittering model based on the bounded variation (BV) image model, which is global, clean and self-contained, and intrinsically combines dejittering with denoising. The mathematical properties of the model are studied based on the direct method in Calculus of Variations. We design one effective algorithm and present its computational implementation based on techniques from numerical partial differential equations (PDE) and nonlinear optimizations.
1901 On the foundations of vision
modeling. II. Mining of mirror symmetry of 2-D shapes
Jianhong Shen
Vision can be considered as a feature mining problem. Visually meaningful features are often geometrical, e.g., boundaries (or edges), corners, T-junctions, and symmetries. Mirror symmetry or near mirror symmetry is very common and useful in image and vision analysis. The current paper proposes several different approaches to extract the symmetry mirrors of 2-dimensional (2-D) mirror symmetric shapes. Proper mirror symmetry metrics are introduced based on Lebesgue measures, Hausdorff distance, and lower-dimensional feature sets. Theory and computation of these approaches and measures are studied.
1902 Distance functions and geodesics
on points clouds
Facundo Mémoli and Guillermo Sapiro
An algorithm for computing intrinsic distance functions and geodesics on sub-manifolds of R^{d} given by point clouds is introduced in this paper. The basic idea is that, as shown in this paper, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of R^{d} can be accurately approximated by the extrinsic Euclidean ones computed in a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We then use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on sub-manifolds of R^{d}, a computationally optimal approach. For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of point clouds representing noisy samples of a sub-manifold of Euclidean space is studied as well. All the underlying theoretical results are presented, together with experimental examples, and comparisons to graph-based distance algorithms.
1903 Signal and noise in tropical
Pacific sea level height analyses
A. Kaplan, M.A. Cane, D. Chen, D.L. Witter, and
R.E. Cheney
(Related to IMA workshop "Data
Assimilation in the Oceanic and Atmospheric Sciences," April 29th-May
3rd, 2002)
Monthly interannual anomalies of tropical Pacific sea level height from Topex/Poseidon altimetry are compared with simulation and assimilation products from a variety of models, ranging from a simple linear long wave approximation to ocean general circulation models. Major spatial similarities in the error patterns are identified. These include zonally elongated maxima in the northwest and southwest tropical Pacific Ocean, a narrow band of high values near 10^{°}N which is slightly inclined towards the equator from the Central American coast, and low values on the equator and in the southeastern tropical Pacific. These features are also present in the pattern of small-scale variability of sea level height. Spatial and temporal components of this small-scale variability are analyzed for predominant variability types. Monte Carlo experiments identify the areas where high small-scale sea level height variability is wind-driven, caused by a similar pattern of variability in the wind stress. Model products systematically underestimate signal variance in such areas. Variability in other areas is due to the instability of ocean currents. The major component of uncertainty in the gridded satellite altimeter analyses is due to sampling error, for which estimates are developed and verified.
1904 On L_{ 3,}-solutions to the Navier-Stokes
equations and backward uniqueness
L. Escauriaza, G. Seregin, and V. Sverák
This is an expository paper on the regularity of solutions of the incompressible, three-dimensional Navier-Stokes equations in the critical space L_{ 3,}
1905 Inpainting surface holes
Joan Verdera, Vincent Caselles, Marcelo Bertalmio,
and Guillermo Sapiro
An algorithm for filling-in surface holes is introduced in this paper. The basic idea is to represent the surface of interest in implicit form, and fill-in the holes with a system of geometric partial differential equations derived from image inpainting algorithms. The framework and examples with synthetic and real data are presented.
1906 Color histogram equalization
through mesh deformation
Eric Pichon, Marc Niethammer, and Guillermo Sapiro
In this paper we propose an extension of grayscale histogram equalization for color images. For aesthetic reasons, previously proposed color histogram equalization techniques do not generate uniform color histograms. Our method will always generate an almost uniform color histogram thus making an optimal use of the color space. This is particularly interesting for pseudo-color scientific visualization. The method is based on deforming a mesh in color space to fit the existing histogram and then map it to a uniform histogram. It is a natural extension of grayscale histogram equalization and it can be applied to spatial and color space of any dimension.
1907 Population set based global
optimization algorithms: Some modifications and numerical studies
M.M. Ali and A. Törn
This paper studies the efficiency and robustness of some recent and well known population set based direct search global optimization methods such as Controlled Random Search, Differential Evolution, and the Genetic Algorithm. Some modifications are made to Differential Evolution and to the Genetic Algorithm to improve their efficiency and robustness. All methods are tested on two sets of test problems, one composed of easy but commonly used problems and the other of a number of relatively difficult problems.
1908 Wave propagation in a 3-D
optical waveguide
Oleg Alexandrov and Giulio Ciraolo
In this work we study the problem of wave propagation in a
3-D optical fiber. (We will use the terms optical waveguide and optical fiber
interchangeably.) The goal is to obtain a solution for the time-harmonic field
caused by a source in a cylindrically symmetric waveguide. The geometry of
the problem, corresponding to an open waveguide, makes the problem challenging.
To solve it, we construct a transform theory which is a nontrivial generalization
of a method for solving a 2-D version of this problem given in [M-S].
The extension to 3-D is made complicated by the fact that the resulting eigenvalue
problem defining the transform kernel is singular both at the origin and at
infinity. The singularities require the investigation of the behavior of the
solutions of the eigenvalue problem. Moreover, the derivation of the transform
formulas needed to solve the wave propagation problem involve nontrivial calculations.
The paper provides a complete description on how to construct the solution
to the wave propagation problem in a 3-D optical waveguide with cylindrical
symmetry. An numerical example, that of computing the field caused by a point
source in a step index fiber, is provided.
1909 Time averaging and turbulence
terms in meteorology
Douglas P. Dokken and Mikhail M. Shvartsman
We discuss averaging in time for the planetary boundary layer in the Boussinesq approximation. We introduce a notion of instantaneous turbulent kinetic energy (ITKE) and then derive a balance equation for ITKE.
1910 Nominal stability of the
real-time iteration scheme for nonlinear model predictive control
Moritz Diehl, Rolf Findeisen, Frank Allgöwer,
Hans Georg Bock, and Johannes Schlöder
We present and investigate a Newton type method for online optimization in nonlinear model predictive control, the so called ``real-time iteration scheme''. In this scheme only one Newton type iteration is performed per sampling instant, and the control of the system and the solution of the optimal control problem are performed in parallel. In the resulting combined dynamics of system and optimizer, the actual feedback control in each step is based on the current solution estimate, and the solution estimates are at each sampling instant refined and transferred to the next optimization problem by a specially designed transition. This approach yields an efficient online optimization algorithm that has already been successfully tested in several applications. Due to the close dovetailing of system and optimizer dynamics, however, stability of the closed-loop system is not implied by standard nonlinear model predictive control results. In this paper, we give a proof of nominal stability of the scheme which builds on concepts from both, NMPC stability theory and convergence analysis of Newton type methods. The principal result is that -- under some reasonable assumptions -- the combined system-optimizer dynamics can be guaranteed to converge towards the origin from significantly disturbed system-optimizer states.
1911 Quantum constants of the
motion for two-dimensional systems
E.G. Kalnins, W. Miller, Jr., and G. Pogosyan
Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis Q, P where P_{1} = H, P_{2}, are the other 2nd-order constants of the motion associated with the separable coordinates, and [Q_{i}, Q_{j}] = [P_{i}, P_{j}] = 0, [Q_{i}, P_{j}] = _{ij}. The 3 operators Q_{2}, P_{1}, P_{2} form a basis for the invariants. In general these are infinite-order differential operators. We shed some light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. We go further and consider all cases where the Hamilton-Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.
1912 On the use of quadratic
models in unconstrained minimization without derivatives
M.J.D. Powell
Quadratic approximations to the objective function provide a way of estimating first and second derivatives in iterative algorithms for unconstrained minimization. Therefore we address the construction of suitable quadratic models Q by interpolating values of the objective function F. On a typical iteration, the objective function is calculated at the point that minimizes the current quadratic model subject to a trust region bound, and we find that these values of F provide good information for the updating of Q, except that a few extra values are needed occasionally to avoid degeneracy. The number of interpolation points and their positions can be controlled adequately by deleting one of the current points to make room for each new one. An algorithm is described that works in this way. It is applied to some optimization calculations that have between 10 and 160 variables. The numerical results suggest that, if m = 2n + 1, then the number of evaluations of F is only of magnitude n, where m and n are the number of interpolation conditions of each model and the number of variables, respectively. This success is due to the technique that updates Q. It minimizes the Frobenius norm of the change to ^{2} Q, subject to the interpolation conditions that have been mentioned.
1913 Eddy kinetic energy and
small-scale sea level height variability
A. Kaplan
A mathematical connection is established between the ocean near-surface geostrophic kinetic energy and the small-scale variance of its surface height. The latter is defined as the spatial variance of sea surface height inside a given grid box and represents a basic statistical characteristic of the field, necessary for estimating its vulnerability to sampling error. The former is also computed from sea surface height fields and, being an important dynamical attribute of the ocean, is often used to describe its mesoscale variability, or eddy energy. Under the condition of isotropic distribution of mesoscale energy, simple formulas connecting the two are obtained for the long- and short-wave (compared to the grid scale) portions of the ocean power spectrum. Without these simplifying assumptions, a factor depending on the actual location-dependent two-dimensional wavenumber power spectrum enters the equation. Approximations based on the Stammer (1997) one-dimensional power spectrum estimates are developed. They are verified by application to the Ducet et al. (2000) gridded satellite altimetry fields.
1914 An iterative global optimization
algorithm for potential energy minimization
N.P. Moloi and M.M. Ali
In this paper we propose an algorithm for the minimization of potential energy functions. The new algorithm is based on the differential evolution algorithm of Storn and Price [1]. The algorithm is tested on two different potential energy functions. The first function is the Lennard Jones energy function and the second function is the many-body potential energy function of Tersoff [2, 3]. The first problem is a pair potential and the second problem is a semi-empirical many-body potential energy function considered for silicon-silicon atomic interactions. The minimum binding energies of up to atoms are reported.
1915 Word problem and genesis
of a free group for english alphabet
Raj Kishor Bisht and H.S. Dhami
With an aim to find applications of the elements of the fundamental groups in computerizing the group operations, an attempt has been made in the present paper to discuss the word problem in the form of finding the generators of the English alphabet. The generating set has been utilized in the genesis of free group.
1916 Logarithmic lower bounds
for Néel walls
Christof Melcher
Most mathematical models for interfaces and transition layers in materials science exhibit sharply localized and rapidly decaying transition profiles. We show that this behavior can largely change when non-local interactions dominate and internal length scales fail to be determined by dimensional analysis: we consider a reduced model for the micromagnetic N\'eel wall which is observed in thin films. The typical phenomenon associated with this wall type is the very long logarithmic tail of transition profiles. Logarithmic upper bounds were recently derived by the author. In the present article we prove that the latter result is indeed optimal. In particular, we show that N\'eel wall profiles are supported by explicitly known comparison profiles that minimize relaxed variational principles and exhibit logarithmic decay behavior. This lower bound is established by a comparison argument based on a global maximum principle for the non-local field operator and the qualitative decay behavior of comparison profiles.
1917 Effective reformulations
of the truss topology design problem
Michal Kocvara and Jirí V. Outrata
We present a new formulation of the truss topology problem that results in unique design and unique displacements of the optimal truss. This is reached by adding an upper level to the original optimization problem and formulating the new problem as an MPCC (Mathematical Program with Complementarity Constraints). We derive optimality conditions for this problem and present several techniques for its numerical solution. Finally, we compare two of these techniques on a series of numerical examples.
1918 Finite volume methods on
spheres and spherical centroidal Voronoi meshes
Qiang Du and Lili Ju
We study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular, the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the L^{2} error of the approximate solution is of quadratic order when the underlying Voronoi mesh is given by a spherical centroidal Voronoi mesh. We also demonstrates numerically the high accuracy and the superconvergence of the approximate solutions.
1919 Continuous dependence and
error estimation for viscosity methods
Bernardo Cockburn
In this paper, we review some ideas on continuous dependence results for the entropy solution of hyperbolic scalar conservation laws. They lead to a complete L^{} (L^{1})-approximation theory with which error estimates for numerical methods for this type of equations can be obtained. The approach we consider consists in obtaining continuous dependence results for the solutions of parabolic conservation laws and deducing from them the corresponding results for the entropy solution. This is a natural approach as the entropy solution is nothing but the limit of solutions of parabolic scalar conservation laws as the viscosity coefficient goes to zero.
1920
Analysis of gradient flow of a regularized Mumford-Shah functional for
image segmentation and image inpainting
Xiaobing Feng and Andreas Prohl
1921 Discontinuous Galerkin methods
B. Cockburn
This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.
1922 On the foundations of vision
modeling III. Pattern-theoretic analysis of Hopf and Turing's reaction-diffusion
patterns
Jianhong Shen and Yoon Mo Jung
After Turing's ingenious work on the chemical basis of morphogenesis fifty years ago, reaction-diffusion patterns have been extensively studied in terms of modelling and analysis of pattern formations (both in chemistry and biology), pattern growing in complex laboratory environments, and novel applications in computer graphics. But one of the most fundamental elements has still been missing in the literature. That is, what do we mean exactly by (reaction-diffusion) {\em patterns}? When presented to human vision and visual system, the patterns usually look deceptively simple and are often tagged by household names like {\em spots} or {\em stripes}. But are such split-second pattern identification and classification equally simple for a computer vision system? The answer does not seem to be confirmative, just as in the case of face recognition, one of the greatest challenges in contemporary A.I. and computer vision research. Inspired and fuelled by the recent advancement in mathematical image and vision analysis (Miva), as well as modern {\em pattern theory}, the current paper develops both statistical and geometrical tools and frameworks for identifying, classifying, and characterizing common reaction-diffusion patterns and pattern formations. In essence, it presents a data mining theory for the scientific simulations of reaction-diffusion patterns.
1923
Exton's quadruple hypergeometric functions of matrix arguments-I
Lalit Mohan Upadhyaya and H.S. Dhami
1924 Dynamic shapes average
Pierre Maurel and Guillermo Sapiro
A framework for computing shape statistics in general, and average in particular, for dynamic shapes is introduced in this paper. Given a metric d(·,·) on the set of static shapes, the empirical mean of N static shapes, C_{1},...,C_{N}, is defined by arg min_{C} 1/N _{i=1}^{N} d(C,C_{i})^{2}. The purpose of this paper is to extend this shape average work to the case of N dynamic shapes and to give an efficient algorithm to compute it. The key concept is to combine the static shape statistics approach with a time-alignment step. To align the time scale while performing the shape average we use dynamic time warping, adapted to deal with dynamic shapes. The proposed technique is independent of the particular choice of the shape metric d(·,·). We present the underlying concepts, a number of examples, and conclude with a variational formulation to address the dynamic shape average problem. We also demonstrate how to use these results for comparing different types of dynamics. Although only average is addressed in this paper, other shape statistics can be similarly obtained following the framework here proposed.
1925 Distance functions and geodesics
on point clouds
Facundo Mémoli and Guillermo Sapiro
A new paradigm for computing intrinsic distance functions and geodesics on sub-manifolds of R^{d} given by point clouds is introduced in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of R^{d} can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold...
1926 Construction of the half-line
potential from the Jost function
Tuncay Aktosun
For the one-dimensional Schrödinger equation, the analysis is provided to recover the portion of the potential lying to the right (left) of any chosen point. The scattering data used consists of the left (right) Jost solution or its spatial derivative evaluated at that point, or the amplitudes of such functions. Various uniqueness and nonuniqueness results are established, and the recovery is illustrated with some explicit examples.
1927 Inverse scattering transform,
KdV, and solitons
Tuncay Aktosun
In this review paper, the Korteweg-de Vries equation (KdV) is considered, and it is derived by using the Lax method and the AKNS method. An outline of the inverse scattering problem and of its solution is presented for the associated Schrödinger equation on the line. The inverse scattering transform is described to solve the initial-value problem for the KdV, and the time evolution of the corresponding scattering data is obtained. Soliton solutions to the KdV are derived in several ways.
1928 The structure of optimal
solutions to the submodular function minimization problem
Collette Coullard
In this paper, we study the structure of optimal solutions to the submodular function minimization problem. We introduce prime sets and pseudo-prime sets as basic building block of minimizer sets, and investigate composition, decomposition, recognition, and certification of prime sets. We show how Schrijver's submodular function minimization algorithm can be modified to construct in polynomial time a prime or pseudoprime decomposition of the ground set V. We also show that the final vector x obtained by this algorithm is an extreme point of the polyhedron P:= { x <= 0 : x(A) <= f(A), for all subsets A of V }.
1929 Superintegrable systems
in Darboux spaces
E.G. Kalnins, J.M. Kress, W. Miller, Jr., and P.
Winternitz
Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via ``coupling constant metamorphosis'' (or equivalently, via Stäckel multiplier transformations). We present tables of the results.
1930 On Parameters Repeated Estimation
Methods (PREM's Method) and its applications in data mining
Vithanage Pemajayantha
A new class of estimation of parameters is proposed for data
mining, analysis and modeling of massive datasets.
With the expansion of Information Technology, the present problem with many
scientists is the analysis and modeling with extremely large databases, sometime
refers to as data mining or knowledge discovery in databases. It was found
that many attempts used to solve this problem were based on classical approaches
such as regression, classification and multivariate techniques, and even summary
statistics such as mean and standard deviations are still having problem of
estimation with extremely large datasets. Because classical statistical approaches
were developed historically to cater the limited availability of data, they
do not intend to solve the problem with massive dataset.
In this study, certain properties of sub-totaling and repeated estimation
of population parameters were used to establish a new statistical method for
estimating summary characteristics of populations, and relationships between
variables with extremely large datasets. While the method has straightforward
applications in data mining and analysis of large databases, it poses the
significance of further statistical research.
1931 Optimization of a telecommunication
network with financial considerations
M.M. Ali
In this paper we have presented a methodology for a rural and semi-urban telecommunication network placement. In order to optimally place the network and to ensure that the network is realistic and viable, we address four key issues, namely the demographic and socio-economic issues, geographical estimation, optimization of the network placement and financial optimization. A digital representation of the map of the region where the network has to be placed is used. A continuous optimization algorithm is applied to optimally place the backbone rings, and a combinatorial optimization algorithm is applied to obtain the optimal rollout order for the network. Mathematical formulations for both the optimization problems are presented. Optimal financial indicators are obtained.
1932 Recovery of a potential
from the ratio of reflection and transmission coefficients
Tuncay Aktosun and Vassilis G. Papanicolaou
For the one-dimensional Schrödinger equation, the analysis is provided to recover the potential from the data consisting of the ratio of a reflection coefficient to the transmission coefficient. It is investigated whether such data uniquely constructs a reflection coefficient, the number of bound states, bound-state energies, bound-state norming constants, and a corresponding potential. In all the three cases when there is no knowledge of the support of the potential, the support of the potential is confined to a half line, and the support is confined to a finite interval, various uniqueness and nonuniqueness results are established, the precise criteria are provided for the uniqueness and the nonuniqueness and the degree of nonuniqueness, and the recovery is illustrated with some explicit examples.
1933
Lauricella-Saran triple hypergeometric functions of matrix arguments-II
Lalit Mohan Upadhyaya and H.S. Dhami
1934
Pattern search methods for linearly constrained minimization in the presence
of degeneracy
Olga A. Brezhneva and J.E. Dennis Jr.
1935 Connections for general
group actions
Debra Lewis, Nilima Nigam, and Peter J. Olver
Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps determining projections of the tangent bundle onto the partial connection; this approach eliminates many of the complications arising from the presence of isotropy. A connection form taking values in the dual of the Lie algebra is smooth even at singular points of the action, while analogs of the classical algebra-valued connection form are necessarily discontinuous at such points. The curvature of a partial connection form can be defined under mild technical hypotheses; the interpretation of curvature as a measure of the lack of involutivity of the (partial) connection carries over to this general setting.
1936 Domain wall motion in ferromagnetic
layers
Christof Melcher
We consider the dynamics of one-dimensional micromagnetic domain walls in layers of uniaxial anisotropy. In the regime of bulk materials, i.e. when the thickness is assumed to be infinite, and the magnetostatic interaction terms appear as local quantities, explicit traveling wave solutions for the corresponding Landau-Lifshitz equation, known as Walker exact solutions, can be constructed. A natural question is whether this construction can be perturbed to the non-local regime of layers of finite thickness. Our stability analysis gives an affirmative answer.
1937 A family of discontinuous
Galerkin finite elements for the Reissner-Mindlin plate
Douglas N. Arnold, Franco Brezzi, and L. Donatella
Marini
We develop a family of locking-free elements for the Reissner-Mindlin plate using Discontinuous Galerkin techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the element of Arnold and Falk to higher degree.
1938 Quadrilateral H(div) finite
elements
Douglas N. Arnold, Daniele Boffi, and Richard S.
Falk
We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficientcondition for approximation of order r+1 in L^{2} is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms,the situation is more complicated and we give a precise characterization of what is needed for optimal order L^{2}-approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element aproximations of the solution of Poisson's equation.
1939 New first-order formulation
for the Einstein equations
Alexander M. Alekseenko and Douglas N. Arnold
We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3+1 decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration.
1940 Asymptotic estimates of
hierarchical modeling
Douglas N. Arnold and Alexandre L. Madureira
In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).
1941 Procreation of distribution
for words
Raj Kishore Bisht and H.S. Dhami
In the present paper an attempt has been made to find the probability distributions based on the frequency of words in a small sample of text. The test of goodness of fit of the distributions has been also worked out on exemplary basis for some selected words.
1942 Numerical evaluation of
H-function by continued fraction
B.S. Rana and H.S. Dham
In the present paper an attempt has been made to evaluate for different values of parmetes m, n, p, q and the variable Z in the range 0.1 to 10.0 by the application of continued fractions.
1943
Matrix generalizations of multiple hypergeometric functions by using Mathai's
matrix transform techniques
Lalit Mohan Upadhyaya
1944
Theory of computation of multidimensional entropy with an application
to the monomer-dimer problem
Shmuel Friedland and Uri N. Peled
1945 Existence of partially regular
solutions for Landau-Lifshitz equations in R^{3}
Christof Melcher
We establish existence of partially regular weak solutions for the Landau-Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. We show that the singular set of such a solution has locally finite 3-dimensional parabolic Hausdorff measure. The construction relies on an approximation based on the Ginzburg-Landau energy.
1946
Periodic solutions to a hysteresis model in micromagnetics
Martin Kruzík
1947
Inverse scattering on the line with incomplete scattering data
Tuncay Aktosun
1948
A simultaneous reconstruction of missing data in DNA microarrays
Shmuel Friedland, Amir Niknejad, and Laura Chihara
1949 On the foundations of vision
modeling IV. Weberized Mumford-Shah model with Bose-Einstein photon noise:
Light adapted segmentation inspired by vision psychology, retinal physiology,
and quantum statistics
Jianhong Shen and Yoon-Mo Jung
Human vision works equally well in a large dynamic range of
light intensities, from only a few photons to typical midday sunlight. Contributing
to such remarkable flexibility is a famous law in perceptual (both visual
and aural) psychology and psychophysics known as Weber's Law. There
has been a great deal of efforts in mathematical biology as well to simulate
and interpret the law in the cellular and molecular level, and by using linear
and nonlinear system modelling tools. In terms of image and vision analysis,
it is the first author who has emphasized the significance of the law in faithfully
modelling both human and computer vision, and attempted to integrate it into
visual processors such as image denoising ( Physica D, 175,
pp. 241-251, 2003).
The current paper develops a new segmentation model based on the integration
of both Weber's Law and the celebrated Mumford-Shah segmentation model (
Comm. Pure Applied Math., 42, pp. 577-685, 1989). Explained in
details are issues concerning why the classical Mumford-Shah model lacks light
adaptivity, and why its ``weberized" version can more faithfully reflect human
vision's superior segmentation capability in a variety of illuminance conditions
from dawn to dusk. It is also argued that the popular Gaussian noise model
is physically inappropriate for the weberization procedure. As a result, the
intrinsic thermal noise of photon ensembles is introduced based on Bose and
Einstein's distribution in quantum statistics, which turns out to be compatible
with weberization both analytically and computationally.
The current paper then focuses on both the theory and computation of the weberized
Mumford-Shah model with Bose-Einstein noise. In particular, Ambrosio-Tortorelli's
Gamma-convergence approximation theory is adapted (Boll. Un. Mat. Ital.,
6-B, pp. 105-123,1992), and stable numerical algorithms are developed
for the associated pair of nonlinear Euler-Lagrange PDEs. Numerical results
confirm and highlight the light adaptivity feature of the new model.
1950
A new approach for 3D segmentation of cellular tomograms obtained using
three-dimensional electron microscopy
A. Bartesaghi, G. Sapiro, S. Lee, J. Lefman, and
S. Subramaniam
1951
Approximation theorems for random permanents and associated stochastic
processes
Grzegorz A. Rempala and Jacek Wesolowski
1952
Singular value decomposition in DNA microarrays
Shmuel Friedland
1953
A Poincaré inequality on R^{n} and its application to potential
fluid flows in space
Guozhen Lu and Biao Ou
1954
Fast numerical solution of parabolic integro-differential equations with
applications in finance
Ana-Maria Matache, Christoph Schwab, and Thomas
P. Wihler
1955
Inverse scattering on the line for a generalized nonlinear Schrödinger
equation
Tuncay Aktosun, Vassilis G. Papanicolaou, and Vassilis
Zisis
1956
Numerical evaluation of G-function
B.S. Rana and H.S. Dhami
1957
Lightfield completion
Liron Yatziv, Guillermo Sapiro, and Marc Levoy
1958
Automatic image decompostion
Kedar A. Patwardhan and Guillermo Sapiro
1959
Discrete network approximation for highly-packed composites with irregular
geometry in three dimensions
Leonid Berlyand, Yuliya Gorb, and Alexei Novikov
1960
Inverse spectral-scattering problem with two sets of discrete spectra
for the radial Schrödinger equation
Tuncay Aktosun and Ricardo Weder
1961
Global well-posedness and scattering for the energy-critical nonlinear
Schrödinger equation in R^{3}
J. Colliander, M. Keel, G. Staffilani, H. Takaoka,
and T. Tao
1962
Evaluation of G-function by multiplication and division techniques of
continued fractions
B.S. Rana and H.S. Dhami
1963
Asymptotic properties of a two sample randomized test for partially dependent
data
Grzegorz A. Rempala and Stephen W. Looney
1964
Statistical analysis of RNA backbone
Guillermo Sapiro, Eli Hershkovitz, Allen Tannenbaum,
and Loren Dean Williams
1965
Is image steganography natural?
Alvaro Martín, Guillermo Sapiro, and Gadiel
Seroussi
1966
Several related models for multilayer sandwich plates
Scott W. Hansen
1967
The general state vector linear model for sustainable ecodevelopment applied
on illustrative basis to a sample valley village of Almora district
H.S. Dhami, A.K. Singh, G.S. Negi, and Anubha Shah
1968
A note on the almost sure central limit theorem for the product of partial
sums
Khurelbaatar Gonchigdanzan and Grzegorz A. Rempala
1969
Preprocessing sparse semidefinite programs via matrix completion
Katsuki Fujisawa, Mituhiro Fukuda, and Kazuhide
Nakata
1970
On the foundations of vision modeling V. Noncommutative monoids of occlusive
preimages
Jianhong Shen
1971
Expansion of power of multiple product of trigonometrical functions in
terms of sum of multiple angles
A.S. Uniyal and Amarendra Behera
1972
Reduction formula for complicated functions in terms of known results
A.S. Uniyal and Amarendra Behera
1973
Convergence of products of matrices in projective spaces
Shmuel Friedland
1974
Prediction/estimation with simple linear models: Is it really that simple?
Yuhong Yang
1975
On multivariate interpolation
Peter J. Olver
1976
Estimation of bias and relative error from the aggregation model developed
for a sample valley village of Almora district
H.S. Dhami, A.K. Singh, G.S. Negi, and Bhupendra
Singh
1977
Meshless geometric subdivision
Carsten Moenning, Facundo Mémoli, Guillermo
Sapiro, Nira Dyn, and Neil A. Dodgson
1978
Comparing point clouds
Facundo Mémoli and Guillermo Sapiro
1979
Inpainting the colors
Guillermo Sapiro
1980
A theoretical and computational framework for isometry invariant recognition
of point cloud data
Facundo Mémoli and Guillermo Sapiro
1981
Statistical analysis of RNA backbone
Eli Hershkovitz, Guillermo Sapiro, Allen Tannenbaum,
and Loren Dean Williams
1982
Nonabelian algebraic topology
Ronald Brown
1983
A mixed finite element method for elasticity in three dimensions
Scot Adams and Bernardo Cockburn
1984
Boundary value problems and regularity on polyhedral domains
Constantin Bacuta, Victor Nistor, and Ludmil T.
Zikatanov
1985
Energy norm a posteriori error estimation of hp-adaptive discontinuous
Galerkin methods for elliptic problems
Paul Houston, Dominik Schötzau, and Thomas
P. Wihler
1986
Interior numerical approximation of boundary value problems with a distributional
data
Ivo Babuska and Victor Nistor
1987
New results for H-function and G-function by the application of fractional
calculus
M.K. Gaira and H.S. Dhami
1988
Curvature function in the recognition of people's handwriting
Biao Ou
1989
On a restricted weak lower semicontinuity for smooth functional on Sobolev
spaces
Daniel Vasiliu and Baisheng Yan
1990
Linear complexity solution of parabolic integro-differential equations
Ana-Maria Matache, Christoph Schwab, and Thomas
P. Wihler
1991
Conformal mapping methods for interfacial dynamics
Martin Z. Bazant and Darren Crowdy
1992
Trivariate spline approximation of divergence-free vector fields
Gerard Awanou and Ming-Jun Lai
1993
An adaptive method with rigorous error control for the Hamilton-Jacobi
equations. Part I: The one-dimensional steady state case
Bernardo Cockburn and Bayram Yenikaya
1994
An adaptive method with rigorous error control for the Hamilton-Jacobi
equations. Part II: The two-dimensional steady state case
Bernardo Cockburn and Bayram Yenikaya
1995
A note on heat kernel estimates for second-order elliptic operators
Seick Kim
1996
Inverse scattering for vowel articulation with frequency-domain data
Tuncay Aktosun
1997
Spontaneous superconducting islands and Hall voltage in clean superconductors
Jorge Berger
1998
Recent developments in modeling, analysis and numerics of ferromagnetism
Martin Kruzík and Andreas Prohl
1999
Fractional integral formulae involving the product of a general class
of polynomials and the multivariate H-function
M.K. Gaira and H.S. Dhami
2000
Fractional derivative operator involving products of special functions
and general class of polynomials
M.K. Gaira and H.S. Dhami
2001
Meshless geometric subdivision
Carsten Moenning, Facundo Mémoli, Guillermo
Sapiro, Nira Dyn, and Neil A. Dodgson
2002
Optimal control of a semilinear PDE with nonlocal radiation interface
conditions
C. Meyer, P. Philip, and F. Tröltzsch
2003
Mesoscopic model of microstructure evolution in shape memory alloys with
applications to NiMnGa
Martin Kruzík and Tomás Roubícek
2004
Expressions for H-function in terms of product of elementry special functions
by the applications of fractional calculus
M.K. Gaira and H.S. Dhami
2005
Area density and regularity for soap film-like surfaces spanning graphs
Robert Gulliver and Sumio Yamada
2006
Design of an effective numerical method for a reaction-diffusion system
with internal and transient layers
Ana Maria Soane, Matthias K. Gobbert, and Thomas
I. Seidman
2007
On explicit exact solutions for the Liénard equation and its application
to the complex Ginzburg-Landau equation with higher-order terms
Emmanuel Yomba and Timoléon Crépin
Kofané
2008
Null controllability of the von Kármán thermoelastic plates
under the clamped or free mechanical boundary conditions
George Avalos
2009
An energy-based three dimensional segmentation approach for the quantitative
interpretation of electron tomograms
Alberto Bartesaghi, Guillermo Sapiro, and Sriram
Subramaniam
2010
Fast image and video colorization using chrominance blending
Liron Yatziv and Guillermo Sapiro
2011
Frameable non-stationary processes and volatility applications
Enrico Capobianco
2012
Elliptic problems on networks with constrictions
Jacob Rubinstein, Peter Sternberg, and Gershon
Wolansky
2013
Exploration and reduction of high dimensional spaces with independent
component analysis
Enrico Capobianco
2014
General projective Riccati equations method and exact solutions for a
class of nonlinear partial differential equations
Emmanuel Yomba
2015
Tracking of moving objects under severe and total occlusions
Alberto Bartesaghi and Guillermo Sapiro
2016
Video inpainting of occluding and occluded objects
Kedar A. Patwardhan, Guillermo Sapiro, and Marcelo
Bertalmio
2017
Exact reachability of finite energy states for an acoustic wave/plate
interaction under the influence of boundary and localized controls
George Avalos and Irena Lasiecka
2018
On the asymptotics of some large Hankel determinants generated by Fisher-Hartwig
symbols defined on the real line
T.M. Garoni
2019
Error control and analysis in coarse-graining of stochastic lattice dynamics
Markos A. Katsoulakis, Petr Plechác, and
Alexandros Sopasakis
2020
Randomized volatility estimation from semimartingales
Enrico Capobianco
2021
O(N) implementation of the fast marching algorithm
Liron Yatziv, Alberto Bartesaghi, and Guillermo
Sapiro
2022
L^{}-bounds for weak solutions
of an evolutionary equation with the p-Laplacian
Peter Takác
2023
Differential complexes and stability of finite element methods. I. The
de Rham complex
Douglas N. Arnold, Richard S. Falk, and Ragnar
Winther
2024
Differential complexes and stability of finite element methods. II. The
elasticity complex
Douglas N. Arnold, Richard S. Falk, and Ragnar
Winther
2025
Rectangular mixed finite elements for elasticity
Douglas N. Arnold and Gerard Awanou
2026
On the control of an evolutionary equilibrium in micromagnetics
Michal Kocvara, Martin Kruzík and Jirí
V. Outrata
2027
Divorcing pressure from viscosity in incompressible Navier-Stokes dynamics
Jian-Guo Liu, Jie Liu, and Robert L. Pego
2028
The extended F-expansion method and its application for solving the nonlinear
wave, CKGZ, GDS and GZ equations
Emmanuel Yomba
2029
On geometric variational models for inpainting surface holes
V. Caselles, G. Haro, G. Sapiro, and J. Verdera
2030
The gradient flow motion of boundary vortices
Matthias Kurzke
2031
Non-linear functional approximation of heterogeneous dynamics
Enrico Capobianco
2032
Inverse scattering to determine the shape of a vocal tract
Tuncay Aktosun
2033
Fourth order partial differential equations on general geometries
John B. Greer, Andrea L. Bertozzi, and Guillermo
Sapiro
2034
Explicit exact solutions for the generalized non-conservative ultrashort
pulse propagation system
Emmanuel Yomba and Timoléon Crépin
Kofané
2035
Partitions for spectral (finite) volume reconstruction in the tetrahedron
Qian-Yong Chen
2036
Remodelling of general state vector linear model for sustainable ecodevelopment
applicable to a sample valley village of Himalayan region
H.S. Dhami, A.K. Singh, G.S. Negi, and Bhupendra
Singh
2037
Non-overlapping domain decomposition method for a nodal finite element
method
A. Bendali and Y. Boubendir
2038
Stripe--domains in nematic elastomers: old and new
Antonio Desimone and Georg Dolzmann
2039
Convergence to equilibrium of a multiscale model for suspensions
E. Cancès and C. Le Bris
2040
Theoretical and numerical comparison of some sampling methods for molecular
dynamics
Eric Cancès, Frédéric Legoll,
and Gabriel Stoltz
2041
The new solitary wave solution for nonlinear wave, CKGZ, GDS, DS and GZ
equations
Emmanuel Yomba
2042
Strong stability of PDE semigroups via a resolvent criterion of Y. Tomilov
George Avalos
2043
Optimization of composite structures subject to local stress constraints
Robert Lipton and Michael Stuebner
2044
Alternative proofs on the radial symmetry and monotonicity of positive
regular solutions to a singular integral equation
Wenxiong Chen, Congming Li, and Biao Ou
2045
Surface and bulk growth unified
Antonio DiCarlo
2046
Numerical simulation of heat transfer in materials with
anisotropic thermal conductivity: A finite volume scheme to
handle complex geometries
Jürgen Geiser, Olaf Klein, and Peter Philip
2047
Isovector fields and similarity solutions for 1-D linear
poroelasticity
Miao-jung Ou
2048
Influence of anisotropic thermal conductivity in
the apparatus insulation for sublimation growth of SiC:
Numerical investigation of heat transfer
Jürgen Geiser, Olaf Klein, and Peter Philip
2049
Constrained regularization of digital terrain elevation data
Anish Mohan and Guillermo Sapiro
2050
Fission, fusion and annihilation in the interaction of
localized structures for the (2+1)-dimensional generalized
Broer-Kaup system
Emmanuel Yomba and Yan-ze Peng
2051
On the remodelling of anisotropic bone tissue
Antonio DiCarlo, Salah Naili, and Sara Quiligotti
2052
Fast image and video denoising via non-local
means of similar neighborhoods
Mona Mahmoudi and Guillermo Sapiro
2053
Nonaffine correlations in random elastic media
B.A. DiDonna and T.C. Lubensky
2054
Computing absolute and essential spectra using
continuation
Jens D.M. Rademacher, Björn Sandstede, and
Arnd Scheel
2055
Fission and fusion in the new localized structures to the
integrable (2+1)-dimensional higher order Broer-Kaup system
Emmanuel Yomba and Yan-ze Peng
2056
Multiscale asymptotics of partial hedging
Gerard Awanou
2057
Piecewise H^{-1} + H^{0} + H^{1}
images and the Mumford-Shah-Sobolev
model for segmented image decomposition
Jianhong (Jackie) Shen
2058
Fair polyline networks for constrained smoothing of digital terrain
elevation data
Michael Hofer, Guillermo Sapiro, and Johannes Wallner
2059
A mixed finite volume method for elliptic problems
Ilya D. Mishev and Qian-Yong Chen
2060
Texture mixing via universal simulation
Gustavo Brown, Guillermo Sapiro, and Gadiel Seroussi
2061
Note on: Exact solitary waves of the Fisher equation
Emmanuel Yomba
2062
A stochastic-variational model for soft Mumford-Shah segmentation
Jianhong (Jackie) Shen
2064
Algorithms for differential invariants of
symmetry groups of differential equations
Jeongoo Cheh, Peter J. Olver, and Juha Pohjanpelto
2065
Boundary conditions for the Einstein-Christoffel
formulation of Einstein's equations
Douglas N. Arnold and Nicolae Tarfulea
2066
The cardinality of sets of k independent
vectors over finite fields
S.B. Damelin, G. Michalski, and G.L. Mullen
2067
On point energies, separation radius and
mesh norm for s-extremal configurations on
compact sets in R^{n}
S.B. Damelin and V. Maymeskul
2068
On the support of the equilibrium measure for
arcs of the unit circle and for real intervals
D. Benko, P.D. Dragnev, and S.B. Damelin
2069
Oscillations and concentrations in
sequences of gradients
Agnieszka Kalamajska and Martin Kruzík
2070
The Borg-Marchenko theorem with a continuous
spectrum
Tuncay Aktosun and Ricardo Weder
2071
Non-overlapping domain decomposition method and
nodal finite element method
Y. Boubendir, A. Bendali, and M.B. Fares
2072
Constrained localization in sensor networks
Mona Mahmoudi and Guillermo Sapiro
2073
Minimal discrete energy problems and numerical
integration on compact sets in Euclidean spaces
S.B. Damelin and V. Maymeskul
2074
On existence and uniqueness of the forward and inverse problem in
refractive index based optical tomography
Taufiquar Khan, Alan Thomas, and Jeong-Rock Yoon
2075
Mixed finite element methods for linear elasticity with
weakly imposed symmetry
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther
2076
A variational approach to blending based on
warping for non-overlapped images
Chang-Ock Lee, Kiwan Jeon, Youngsoo Ha and
Jooyoung Hahn
2077
Asymptotic stability of a fluid-structure semigroup
George Avalos
2078
A geometric method for automatic extraction of
sulcal fundi
C.-Y. Kao, M. Hofer, G. Sapiro, J. Stern, and
D.A. Rottenberg
2079
Transient numerical study of temperature gradients during sublimation
growth of SiC: Dependence on apparatus design
Jürgen Geiser, Olaf Klein, and Peter Philip
2080
Pixel layering and layer propagation for video modelling and foreground
detection
Kedar A. Patwardhan, Guillermo Sapiro, and Vassilios Morellas
2081
A sharp natural characterization of
A^{1}
Winston Chih-Wei Ou
2082
Energy estimates and the Weyl criterion on compact homogeneous manifolds
S.B. Damelin, J. Levesley, and X. Sun
2083
The fundamental solution of the time-dependent system of crystal optics
Robert Burridge and Jianliang Qian
2084
Locking-free Reissner-Mindlin elements without reduced integration
Douglas N. Arnold, Franco Brezzi, Richard S. Falk, and
L. Donatella Marini
2085
Boundary value problems in spaces of distributions on smooth and
polygonal domains
Ivo Babuska and Victor Nistor
2086
Mapping properties of heat kernels, maximal regularity, and semi-linear
parabolic equations on noncompact manifolds
Anna Mazzucato and Victor Nistor
2087
Weighted Sobolev spaces and regularity for
polyhedral domains
Bernd Ammann and Victor Nistor
2088
Restoration and zoom of irregularly sampled, blurred and noisy images
by accurate total variation minimization with local constraints
Andrés Almansa, Vicent Caselles, Gloria Haro, and Bernard
Rougé
2089
On geometric variational models for inpainting surface holes
Vicent Caselles, Gloria Haro, Guillermo Sapiro, and Joan Verdera
2090
Region based segmentation using the tree of shapes
Juan Cardelino, Gregory Randall, Marcelo Bertalmío, and
Vicent Caselles
2091
Constrained localization in static and dynamic sensor networks
Mona Mahmoudi and Guillermo Sapiro
2092
Brain and surface warping via minimizing Lipschitz extensions
Facundo Mémoli, Guillermo Sapiro, and Paul Thompson
2093
A discrepancy theorem for harmonic functions on the $d$ dimensional
sphere with applications to scatterings of point clouds
S.B. Damelin
2094
Finite element exterior calculus, homological techniques,
and applications
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther
2095
Mesh-independence and fast local convergence of
a primal-dual active-set method for mixed control-state
constrained elliptic control problems
Michael Hintermüller
2096
Approximate Dirichlet boundary conditions in the
generalized finite element method
Ivo Babuska, Victor Nistor, and Nicolae Tarfulea
2097
A direct tracking method for a grounded conductor
inside a pipeline from capacitance measurements
Hyenkyun Woo, Sungwhan Kim, Jin Keun Seo, William Lionheart,
and Eung Je Woo
2098
Level set based bimodal segmentation with stationary
global minimum
Suk-Ho Lee and Jin Keun Seo
2099
Mathematical framework for current density imaging due
to discharge of electro-muscular disruption devices
Jeehyun Lee, Jin Keun Seo, and Eung Je Woo
2100
Locally analytic schemes: A link between diffusion filtering and wavelet
shrinkage
Martin Welk, Gabriele Steidl, and Joachim Weickert
2101
Least-square halftoning via human vision system and Markov gradient
descent (LS-MGD): Algorithm and analysis
Jianhong (Jackie) Shen
2102
Convergence of numerical approximations to a phase field bending
elasticity model of membrane deformations
Qiang Du and Xiaoqiang Wang
2103
Statistical characterization of protein ensembles
Diego Rother, Guillermo Sapiro, and Vijay Pande
Qiang Du and Xiaoqiang Wang
2104
Rotating line cameras: Model and calibration
Fay Huang, Shou Kang Wei, and Reinhard Klette
2105
Rotating line cameras: Epipolar geometry and
spatial sampling
Fay Huang, Shou Kang Wei, and Reinhard Klette
2106
Total variation semi-blind deconvolution using shock filters
James H. Money and Sung Ha Kang
2107
Geodesic flow and two (super) component analog of
the Camassa-Holm equation
Partha Guha, and Peter J. Olver
2108
Young measures and order-disorder transition in stationary
flow of liquid crystals
M. Carme Calderer and Alexander Panchenko
2109
Analysis of nonlocal electrostatic effects in chiral
smectic C liquid crystals
Jinhae Park and M. Carme Calderer
2110
Rotating line cameras: Image quality control and
camera design
Shou Kang Wei, Fay Huang, and Reinhard Klette
2111
Gamma limit of the non self-dual Chern-Simons-Higgs energy
Matthias Kurzke and Daniel Spirn
2112
On discrete moments of unbounded order
Reinhard Klette and Jovisa Zunic
2113
Calculation of the number of tunnels
Fajie Li and Reinhard Klette
2114
Improving the rate of convergence of `high order finite
elements' on polyhedra II: Mesh refinements and interpolation
Constantin Bacuta, Victor Nistor, and Ludmil T. Zikatanov
2115
A geometric method for automatic extraction of
sulcal fundi
Chiu-Yen Kao, Michael Hofer, Guillermo Sapiro, Josh Stern,
Kelly Rehm, and David A. Rottenberg
2116
Modelling and simulations of multi-component lipid membranes
and open membranes via diffusive interface approaches
Xiaoqiang Wang and Qiang Du
2117
Boundary and local null controllability of structurally damped
elastic systems
George Avalos and Paul Cokeley
2118
Activity recognition via classification constrained
diffusion maps
Y. Ma and S.B. Damelin
2119
A TV based restoration model with local constraints
A. Almansa, C. Ballester, V. Caselles, and G. Haro
2120
Solitary waves and their linear stability in weakly
coupled KdV equations
J. Douglas Wright and Arnd Scheel
2121
On the convergence of the harmonic B_{z} algorithm
in magnetic resonance electrical impedance tomography
J.J. Liu, J.K. Seo, M. Sini, and E.J. Woo
2122
Explicit solutions to the Korteweg-de Vries equation
on the half line
Tuncay Aktosun and Cornelis van der Mee
2123
Spatially-coherent non-linear dimensionality reduction
and segmentation of hyper-spectral images
Anish Mohan, Guillermo Sapiro, and Edward Bosch
2124
Multiphase image segmentation via Modica-Mortola phase transition
Yoon Mo Jung, Sung Ha Kang, and Jianhong Shen
2125
Highly accurate segmentation using geometric attraction-driven flow in
edge-regions
Jooyoung Hahn and Chang-Ock Lee (An eye for aphids)
2126
Comparative study of search strategies for the direct binary search image
halftoning algorithm
Sagar Bhatt, John Sabino, John Harlim, Joel Lepak, Robert Ronkese, and
Chai Wah Wu
2127
Reconstructing thin shapes from boundary electrical measurements with
level sets
D. Álvarez, O. Dorn, and M. Moscoso
2128
Two-dimensional offsets and medial axis transform
Hyeong In Choi, Sung Woo Choi, Chang Yong Han, Tae-wan Kim, Song-Hwa
Kwon, Hwan Pyo Moon, Kyeong Hah Roh, and Nam-Sook Wee
2129
Explicit nonflat time evolution for PDE-based image restoration
Seongjai Kim and Song-Hwa Kwon
2130
Scaling limits of the Chern-Simons-Higgs energy
Matthias Kurzke and Daniel Spirn
2131
Topological criterion for selection of quintic Pythagorean-hodograph
Hermite interpolants
Hyeong In Choi, Rida T. Farouki, Song-Hwa Kwon, and Hwan Pyo Moon
2132
Interactive image segmentation via adaptive
weighted distances
Alexis Protiere and Guillermo Sapiro
2133
Mathematical modeling in industry X - IMA Workshop for graduate students
(August 9-18, 2006)
Fadil Santosa, Richard J. Braun, and Fernando Reitich, organizers
(There are six papers in this preprint)
2134
Movie denoising by average of warped lines
Marcelo BertalmÃo, Vicent Caselles, and Ãlvaro Par
2135
Stratification learning: Detecting mixed density and dimensionality in high
dimensional point clouds
Gloria Haro, Gregory Randall, and Guillermo Sapiro
2136
The finite element method for a class of degenerate elliptic equations
Hengguang Li
2137
Binomial D-modules
Alicia Dickenstein, Laura Felicia Matusevich, and
Ezra Miller
2138
Variational models for image colorization via chromaticity
and brightness decomposition
Sung Ha Kang and Riccardo March
2139
Sparse representation for color image restoration
Julien Mairal, Michael Elad, and Guillermo Sapiro
2140
Flux-gradient and source term balancing for
certain high resolution shock-capturing schemes
Vicent Caselles, Rosa Donat, and Gloria Haro
2141
Exact and approximate algorithms for the
calculation of shortest paths
Fajie Li and Reinhard Klette
2142
Well posedness and regularity for the elasticity
equation with mixed boundary conditions on polyhedral
domains and domains with cracks
Anna Mazzucato and Victor Nistor
2143
A basic inequality for the Stokes operator related to
the Navier boundary condition
Luan Thach Hoang
2144
Regularization for inverting the Radon transform with wedge
consideration
I. Aganj, A. Bartesaghi, M. Borgnia, H.Y. Liao,
G. Sapiro, and S. Subramaniam
2145
From gigabytes to bytes: Automated denoising and
feature identification in electron tomograms of intact
bacterial cells
Rajesh Narasimha, Iman Aganj, Mario Borgnia,
Guillermo Sapiro, Steven McLaughlin, Jacqueline Milne,
and Sriram Subramaniam
2146
An analysis of the minimal dissipation local discontinuous
Galerkin method for convection-diffusion problems
Bernardo Cockburn and Bo Dong
2147
Optimal convergence of the original DG method for the
transport-reaction equation on special meshes
Bernardo Cockburn, Bo Dong, and Johnny Guzmán
2148
Primary decomposition of zero-dimensional ideals
over finite fields
Shuhong Gao, Daqing Wan, and Mingsheng Wang
2149
On global existence of solutions to
a cross-diffusion system
Phan Van Tuôc
2150
Exact and quasi-exact solvability of second order
superintegrable quantum systems. II. Relation to
separation of variables
E.G. Kalnins, W. Miller, Jr., and G.S. Pogosyan
2151
Nondegenerate 2D complex Euclidean superintegrable
systems and algebraic varieties
E.G. Kalnins, J.M. Kress, and W. Miller, Jr.
2152
Multiscale sparse image representation with
learned dictionaries
Julien Mairal, Guillermo Sapiro, and Michael Elad
2153
Distancecut: Interactive real-time segmentation and
matting of images and videos
Xue Bai and Guillermo Sapiro
2154
A graph-based foreground representation and its
application in example based people matching in video
Kedar A. Patwardhan, Guillermo Sapiro, and
Vassilios Morellas
2155
On the active response of soft living tissues
P. Nardinocchi and L. Teresi
2156
Multivariate moment matrices with zeros in their inverse
J.W. Helton, J.B. Lasserre, and M. Putinar
2157
Exact solutions to the focusing nonlinear Schrödinger equation
Tuncay Aktosun, Francesco Demontis, and Cornelis van der Mee
2158
A globally convergent flow for computing the best low rank
approximation of a matrix
Kenneth R. Driessel
2159
Differential invariants of equi-affine surfaces
Peter J. Olver
2160
Generating differential invariants
Peter J. Olver
2161
Numerical invariantization for morphological PDE schemes
Martin Welk, Pilwon Kim, and Peter J. Olver
2162
Positive polynomials on projective limits of
real algebraic varieties
Salma Kuhlmann and Mihai Putinar
2163
An image processing approach to computing distances
between RNA secondary structures Dot Plots
Tor Ivry, Shahar Michal, Danny Barash, and Guillermo Sapiro
2164
On the energy of superconductors in large and small domains
Matthias Kurzke and Daniel Spirn
2165
Undecidability in a free *-algebra
Mihai Putinar
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