bluuebunny/crossref_metadata_2025 huggingface dataset:
Brownian motion in a wedge with oblique reflection
Authors: Varadhan S. R. S. , Williams R. J.
Published: April 2007

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties:

The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion.

The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side.

The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero).

Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ1 and θ2 be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (‐½π < θ1, θ2 ½π), and set α = (θ1 + θ2)/ξ.

The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147‐225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)‐(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)‐(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2.

The general theory of multi‐dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks.
(i) The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).
Probabilistic approach to the neumann problem
Authors: Hsu Pei
Published: April 2007

The basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically:magnified image where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.
A factored implicit scheme for numerical weather prediction
Authors: Augenbaum J. M. , Cohn S. E. , Isaacson E. , Dee D. P. , Marchesin D.
Published: April 2007

The nonlinear partial differential equations of atmospheric dynamics govern motion on two time scales, a fast one and a slow one. Only the slow‐scale motions are relevant in predicting the evolution of large weather patterns. Implicit numerical methods are therefore attractive for weather prediction, since they permit a large time step chosen to resolve only the slow motions.

To develop an implicit method which is efficient for problems in more than one spatial dimension, one must approximate the problem by smaller, usually one‐dimensional problems. A popular way to do so is to approximately factor the multidimensional implicit operator into one‐dimensional operators. The factorization error incurred in such methods, however, is often unacceptably large for problems with multiple time scales.

We propose a new factorization method for numerical weather prediction which is based on factoring separately the fast and slow parts of the implicit operator. We show analytically that the new method has small factorization error, which is comparable to other discretization errors of the overall scheme. The analysis is based on properties of the shallow water equations, a simple two‐dimensional version of the fully three‐dimensional equations of atmospheric dynamics.
A simple one‐dimensional model for the three‐dimensional vorticity equation
Authors: Constantin P. , Lax P. D. , Majda A.
Published: April 2007

A simple qualitative one‐dimensional model for the 3‐D vorticity equation of incompressible fluid flow is developed. This simple model is solved exactly; despite its simplicity, this equation retains several of the most important structural features in the vorticity equations and its solutions exhibit some of the phenomena observed in numerical computations for breakdown for the 3‐D Euler equations.
Finite sections of toeplitz‐like matrices
Authors: Percus J. K.
Published: April 2007

Determinants of large finite sections of Toeplitz and Toeplitz‐like matrices are evaluated by an expansion in which the deviation from the identity is parametrically increased. Classical results are reproduced and the inverse matrix expanded as well. The expansion is asymptotically valid only for low‐order terms, and so a reordered expansion is introduced. It has the desired asymptotic character and suggests that sharp bounds are also available.
A random‐walk interpretation of the incompressible navier‐stokes equations
Authors: Peskin Charles S.
Published: April 2007

We introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid, and we consider the force (per unit volume) exerted by such a “molecule” on the rest of the fluid. Averaging this force over an ensemble of such molecules and taking an appropriate limit, we obtain the Navier‐Stokes equations. The origin of time‐irreversibility in these equations is discussed in terms of the model.
A uniqueness theorem for an inverse boundary value problem in electrical prospection
Authors: Sylvester John , Uhlmann Gunther
Published: April 2007

We show that a near constant conductivity of a two‐dimensional body can be uniquely determined by steady state direct current measurements at the boundary. Mathematically, we show that the coefficient γ in the operator ∇.γ∇ is uniquely determined by its Dirichlet integrals.
The toda flow on a generic orbit is integrable
Authors: Deift P. , Li L. C. , Nanda T. , Tomei C.
Published: April 2007

We construct action‐angle variables for the Toda flow on generic orbits of the coadjoint action of the lower triangular group on its dual Lie algebra.
The milne and kramers problems for the boltzmann equation of a hard sphere gas
Authors: Bardos Claude , Caflisch Russel E. , Nicolaenko Basil
Published: April 2007

Existence, uniqueness and properties of asymptotic behavior are proved for solutions of the Milne and Kramers problems for the linearized Boltzmann equation for a gas of hard spheres. The proof uses energy‐like estimates and follows the analysis of Bardos, Santos and Sentis for the equations of neutron transport.
Generalized voronoi diagrams for moving a ladder. I: Topological analysis
Authors: Ó'Dunlaing Colm , Sharir Micha , Yap Chee K.
Published: April 2007

Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line‐segment B (a [ladder]) of length d inside Ω. FP is a three‐dimensional manifold. A [Voronoi complex] in this manifold, a two‐dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one‐dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.
Monte carlo calculation of transport for three‐dimensional magnetohydrodynamic equilibria
Authors: Bauer F. , Betancourt O. , Garabedian P.
Published: April 2007

A Monte Carlo algorithm has been implemented which couples the results of a three‐dimensional magnetohydrodynamic equilibrium code with particle transport calculations similar to those of Boozer and Kuo‐Petravic. The equilibrium results are in a form which lends itself to easy and accurate computation of the coefficients required to follow particle orbits in flux variables at finite pressure. In particular, the parallel current and the Clebsch potential for the current are obtained from Fourier series solutions of first‐order partial differential equations with constant coefficients.

Confinement times are estimated from the exponential decay of expected values of appropriately chosen functionals of the particle distribution. The observation that these functionals satisfy boundary conditions helps us to compute confinement times over a wide range of collision frequencies, including cases where losses due to particle trapping are very high. Initial conditions are chosen to optimize the Monte Carlo calculation by using known information about the expected particle distribution.

Results for actual and proposed stellarator experiment are given. Electron and ion confinement times are compared, since this is relevant to the issue of the effect of ambipolar electric fields on stellarator confinement. A spectral method is given to determine the electric field from charge neutrality, and numerical evidence is presented to suggest that anomalous electron transport may be due to small resonant terms in the electric potential. Also, comparisons are made with a simplified theory of particle transport already incorporated in the equilibrium code.
Long time existence for a slightly perturbed vortex sheet
Authors: Caflisch Russel E. , Orellana Oscar F.
Published: April 2007

Consider a flat two‐dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ϵ and is analytic in a strip |𝒥m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ϵ is sufficiently small, with κ → 1 as ϵ → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time tc = |log ϵ| + O(log|log ϵ|. Our results prove existence for t < κ|log ϵ|, if ϵ is sufficiently small, with k κ → 1 as ϵ → 0. Thus our existence results are nearly optimal.
Geometric bases and topological equivalence
Authors: Szczepanski Susan
Published: April 2007

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê‐Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.
Dimensional reduction of integrals of orthogonal invariants
Authors: Percus J. K.
Published: April 2007

The integration of a function of scalar products ofp vectors in RD, where D > p, is reduced to that of a corresponding function of vectors in Rp.
Nonlinear waves in friedman‐robertson‐walker space‐times
Authors: Kovalyov Mikhail
Published: April 2007

In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman‐Robertson‐Walker space‐time, □ being the D'Alambertian in local coordinates of space‐time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.
Diffraction of a weak shock by a wedge
Authors: Harabetian Eduard
Published: April 2007

We match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock‐wedge diffraction problem.
Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation
Authors: Wolansky G.
Published: April 2007

We study a simplified model of barotropic flow. The governing equation is the two‐dimensional Euler equation for incompressible flow, subject to Ekman dissipation and arbitrary forcing, in a bounded domain. This equation is converted to an equivalent functional differential equation via a semi‐lagrangian formalism. We use Hölder Cα estimates and compactness arguments for the latter equation to deduce existence, uniqueness and asymptotic stability of stationary solutions under the assumption of small forcing with respect to the dissipation constant.

In case the forcing is of one sign only throughout the domain, we are able to deduce existence without any further assumptions. The same result holds in the limit of the dissipation constant approaching zero while the ratio of this constant to the forcing is fixed. Thus we get the existence of special types of stationary solutions to the inviscid Euler equation consistent with the given forcing.
Multiple impulse solutions to McKean's caricature of the nerve equation. I—existence
Authors: Wang Wei‐Ping
Published: April 2007

McKean's caricature of the nerve equation:

equation image

is considered. The H in (1) is the Heaviside function. We prove the existence of multiple impulse solutions consisting of any finite number of pulses. These solutions are referred to as n‐ple impulse solutions, where n is an arbitrary positive integer.
The modulator of the local time
Authors: Berman Simeon M.
Published: April 2007

Let x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, αm(x, 1) is integrable over B. The modulator is a function Mm(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the Lm(B)‐valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.
Dirichlet and neumann spectrum
Authors: Winkler Andrew M.
Published: April 2007

It is well known that the critical points of the Rayleigh quotient for the Dirichlet form relative to the square integral norm are eigenfunctions of the Laplace operator with Neumann boundary conditions and eigenvalue equal to the critical value.

Clearly then, if we fix a critical point and consider the corresponding value and restrict the Rayleigh quotient to the space of eigenfunctions of the Laplace operator with that given eigenvalue, then the critical point remains a critical point of the restriction.

In this paper we answer the converse question. We show that if a given number occurs as a critical value on the subspace of eigenfunctions with that same number as eigenvalue, then either it is a critical value for the quotient on the entire space or it is a critical value on the subspace of all functions satisfying Dirichlet boundary conditions. Moreover, a precise relation holds between the corresponding critical points.

A similar relation holds for any elliptic operator in divergence form. The relation stems from a lattice identity induced by a certain indefinite quadratic form, involving certain subspaces.

This theory has been successfully applied to calculating the cuspidal spectrum in a certain range for the Laplace‐Beltrami operator for certain finite volume hyperbolic surfaces, addressing a conjecture of Roelcke and Selberg, and recent conjectures of Sarnak; see [1].

The numerical method used can be applied quite generally, typically to equations related to underlying symmetries where decompositions of the group give rise to lots of formal solutions.
A relative morse index for the symplectic action
Authors: Floer Andreas
Published: April 2007

The notion of a Morse index of a function on a finite‐dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.
A classification of well‐posed kinetic layer problems
Authors: Coron François , Golse François , Sulem Catherine
Published: April 2007

In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani [4], the problem is well‐posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively −c < u < 0 and u < −c).

In the second part, we show that the well‐posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.
On partial regularity results for the navier‐stokes equations
Authors: Struwe Michael
Published: April 2007

Looking at the regularity results of Scheffer, respectively, Caffarelli, Kohn and Nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the Navier‐Stokes equations.
Homogenization for semilinear hyperbolic systems with oscillatory data
Authors: Hou Thomas Y.
Published: April 2007

The behavior of multi‐dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple‐scale initial data is also established.
Linearized instability for nonlinear schrödinger and klein‐gordon equations
Authors: Grillakis Manoussos
Published: April 2007

In this paper I am introducing a new technique for proving instability of bound states for Hamiltonian systems. There are already two disparate types of instability results in the literature. The approach developed by Strauss‐Shatah [20] gave an instability criterion coming from the variational structure of the problem; on the other hand, Jones' approach [11] produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques. It turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results. Finally in order to demonstrate how this method works I apply the instability criterion in some specific examples.
The unregularized gradient flow of the symplectic action
Authors: Floer Andreas
Published: April 2007

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy‐Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.
Multiple impulse solutions to McKean's caricature of the nerve equation. II. Stability
Authors: Wang Wei‐Ping
Published: April 2007

We study McKean's caricature of a nerve conduction equation

where H is the Heaviside function. It is proved that an n‐ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n ‐ 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n‐ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n ‐ 1 unstable modes and all of them are of spacing type.
Morse index of some min‐max critical points. I. Application to multiplicity results
Authors: Bahri A. , Lions P. L.
Published: April 2007

We give here a general result on lower bounds for Morse indices of critical points obtained by some min‐max principles. Combining this information with a semi‐classical inequality yields sharp estimates on the growth of some critical values, from which we deduce new multiplicity results for solutions of semi‐linear second‐order elliptic equations.
On connectionist models
Authors: Hong Jiawei
Published: April 2007

We show that an arbitrary “connectionist” model of n neutrons, defined by an n X n real matrix, can be simulated by a system of O(n3log n) Boolean gates with an O(log n) time slow‐down factor. This establishes that, even though n2 real numbers possibly of high precision are required to define it, connectionist models do not possess any basic properties different from those of other (nonuniform) highly parallel hardware models.
On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's
Authors: Ishii Hitoshi
Published: April 2007

We prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second‐order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.‐L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions.
Fiber architecture of the left ventricular wall: An asymptotic analysis
Authors: Peskin Charles S.
Published: April 2007

The goal of this paper is to derive the fiber architecture of the left ventricle from first principles. The principles used are (i) mechanical equilibrium, (ii) a stress tensor that is the sum of a hydrostatic pressure and a fiber stress, (iii) the assumption that fiber tubes have constant cross‐sectional area along their length, (iv) axial symmetry, and (v) the approximation that the thickness of the ventricular wall is small in comparison with the other wall dimensions. From these hypotheses, it is deduced that the cardiac muscle fibers are approximate geodesics on a nested family of toroidal surfaces centered on a degenerate torus in the equatorial plane of the heart. (Exact geodesics are ruled out by the theory.) Formulae are derived for the fiber surfaces and the fiber‐angle distribution through the wall, and a differential equation is derived for the shape of the middle surface of the left ventricle. The results are in substantial agreement with experimental observations.
Matrix factorizations and integrable systems
Authors: Deift P. , Li L. C. , Tomei C.
Published: April 2007

We show that the QR, LU and Cholesky algorithms to compute the eigenvalues of real matrices are the integer time evaluations of completely integrable Hamiltonian flows.
Metastable patterns in solutions of ut = ϵ2uxx − f(u)
Authors: Carr J. , Pego R. L.
Published: April 2007

We consider the equation in question on the interval 0 ≦ x ≦ 1 having Neumann boundary conditions, with f(u) = F(u), where F is a double well energy density with equal minima at u = ±1. The only stable states of the system are patternless constant solutions. But given two‐phase initial data, a pattern of interfacial layers typically forms far out of equilibrium. The ensuing nonlinear relaxation process is extremely slow: patterns persist for exponentially long times proportional to exp{A±l/ϵ, where A = F(±1) and l is the minimum distance between layers. Physically, a tiny potential jump across a layer drives its motion.

We prove the existence and persistence of these metastable patterns, and characterise accurately the equations governing their motion. The point of view is reminiscent of center manifold theory: a manifold parametrising slowly evolving states is introduced, a neighbourhood is shown to be normally attracting, and the parallel flow is characterised to high relative accuracy. Proofs involve a detailed study of the Dirichlet problem, spectral gap analysis, and energy estimates.
Global weak solutions of Vlasov‐Maxwell systems
Authors: Diperna R. J. , Lions P. L.
Published: April 2007

We study here the Vlasov‐Maxwell system in its classical and relativistic form. We prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data. The main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work.
Tubuloglomerular feedback in a dynamic nephron
Authors: Pitman E. Bruce , Layton H. E.
Published: April 2007

A dynamic model for a short‐looped mammalian nephron is developed to study tubuloglomerular feedback (TGF). Evolution equations for salt and urea concentrations and for fluid flux in the nephron are derived and coupled to a resistance network that serves as a schematic model of the glomerulus and associated structures. The evolution equations, which are semi‐linear hyperbolic partial differential equations, are solved by the method of flux‐corrected transport. The implementation and testing of this method is described and numerical results are presented.

This investigation suggests that: (i) the concentrating nephron exhibits high gain, i.e., a small increase in single nephron glomerular filtration rate produces a large increase in the salt concentration of tubular fluid in the cortical thick ascending limb at the macula densa; (ii) the nephron, as a concentrating system, acts as a low‐pass filter, i.e., high frequency pressure oscillations (1 Hz) of a prescribed amplitude at the proximal tubule produce relatively low amplitude oscillations in tubular concentrations, while low frequency oscillations (1/30 Hz) produce relatively high amplitude oscillations in tubular concentrations; and (iii) as a consequence of long time delay in TGF, some perturbations in afferent arteriolar blood pressure induce sustained periodic oscillations similar to those observed in recent experiments.
Nonlinear singular sturm‐liouville problems and an application to transonic flow through a nozzle
Authors: Hsu Sze‐Bi , Liu Tai‐Ping
Published: April 2007

We consider a class of singular Sturm‐Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction‐diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.
On characterizing the set of possible effective tensors of composites: The variational method and the translation method
Authors: Milton Graeme W.
Published: April 2007

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization‐problem simplifies the derivation of the Hashin‐Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non‐local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two‐component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two‐components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet‐type variational principles and bounds for the effective tensors of general non‐selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.
Some existence results for fully nonlinear elliptic equations of monge‐ampère type
Authors: Li Yanyan
Published: April 2007

We study the existence and uniqueness of solutions of Monge‐Ampère‐type equations. This type of equations has been studied extensively by Caffarelli, Nirenberg, Spruck and many others. (See [5] through [8] and the references therein.) We present some existence and uniqueness results for this type of equations on compact Riemannian manifolds with non‐negative sectional curvature. We have also generalized some results in [7].
Convergence of the point vortex method for the 2‐D euler equations
Authors: Goodman Jonathan , Hou Thomas Y. , Lowengrub John
Published: April 2007

We prove consistency, stability and convergence of the point vortex approximation to the 2‐D incompressible Euler equations with smooth solutions. We first show that the discretization error is second‐order accurate. Then we show that the method is stable in lp norm. Consequently the method converges in lp norm for all time. The convergence is also illustrated by a numerical experiment.
A boundary‐value problem for the stationary vlasov‐poisson equations: The plane diode
Authors: Greengard Claude , Raviart P.‐A.
Published: April 2007

The stationary Vlasov‐Poisson boundary value problem in a spatially one‐dimensional domain is studied. The equations describe the flow of electrons in a plane diode. Existence is proved when the boundary condition (the cathode emission distribution) is a bounded function which decays super‐linearly or a Dirac mass. Uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable. It is shown that uniqueness does not always hold for the Dirac mass boundary conditions.
Existence and stability of stationary profiles of the lw scheme
Authors: Smyrlis Yiorgos Sokratis
Published: April 2007

In this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws

magnified image

When a difference scheme introduces artificial numerical diffusion, for example the Lax‐Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax‐Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock.

In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first‐order accurate; they are known to be contractive in the sense of any lp norm. Jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes.

Here we study similar questions for the Lax‐Wendroff scheme without artificial viscosity; this is a nonmonotone, second‐order accurate scheme. We prove existence of a one‐parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm. The proof relies on studying the linearized Lax‐Wendroff scheme.
Analysis of the backward‐euler/langevin method for molecular dynamics
Authors: Peskin Charles S.
Published: April 2007

This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward‐Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ωc, which may be set equal to kT/h to simulate quantum‐mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward‐Euler/Langevin method is considered, an integral equation for the equilibrium phase‐space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second‐order partial differential equation for the equilibrium phase‐space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.
A lagrangian finite element method for the 2‐D euler equations
Authors: Chacon‐Rebollo Tomas , Hou Thomas Y.
Published: April 2007

A grid free Lagrangian finite element method is introduced for the 2‐D incompressible Euler equations. The method is derived based on the observation that the product of the Biot‐Savart kernel and a polynomial can be integrated analytically over any triangle. This enables us to obtain a numerically stable Lagrangian method without using numerical smoothing. Moreover, we show that the method converges uniformly with second‐order accuracy. Actually, we establish a l∞ stability result which applies to kernels that are more singular than the Biot‐Savart kernel, as long as the kernel is L integrable. Another useful result is that we prove convergence of our method when using local regridding, which allows the method to run for longer time even with a fixed mesh. The second‐order convergence is also illustrated by our numerical experiments.
The effective conductivity of an infinitely interchangeable mixture
Authors: Bruno Oscar P.
Published: April 2007

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability.

The usual way to approach cell materials is through n‐point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.
Homogenization and convergence of the vortex method for 2‐D euler equations with oscillatory vorticity fields
Authors: Weinan E , Hou Thomas Y.
Published: April 2007

The 2‐D incompressible Euler equations with oscillatory vorticity fields are studied. A homogenization result for 2‐D Euler equations in velocity‐vorticity formulation is obtained and weak continuity of the equations is proved. Convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles. Numerical results are given. Comparisons are made with the corresponding finite difference approximation.
A fast algorithm for the evaluation of heat potentials
Authors: Greengard Leslie , Strain John
Published: April 2007

Numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials. When M points are used in the discretization of the boundary and N time steps are computed, an amount of work of the order O(N2M2) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NM), and we observe speedups of five orders of magnitude for large‐scale problems. Thus, the method makes it possible to solve the heat equation by potential theory in practical situations.
Rotationally invariant hyperbolic waves
Authors: Brio M. , Hunter J. K.
Published: April 2007

We use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves. The equation can include weak dissipative, dispersive, or diffractive effects. We give applications to equations from magnetohydrodynamics, elasticity, and viscoelasticity.
Exact solitary water waves with capillary ripples at infinity
Authors: Beale J. Thomas
Published: April 2007

We prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.
On the numerical solution of two‐point boundary value problems
Authors: Greengard L. , Rokhlin V.
Published: April 2007

In this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O (N·p2) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.
Inverse problem for N × N hyperbolic systems on the plane and the N‐wave interactions
Authors: Sung L. Y. , Fokas A. S.
Published: April 2007

We study regiorously the solvability of the direct and inverse problems associated with Ψx–JΨy = QΨ,(x,y) ∈ ℝ2, where (i) Ψ is an N × N‐matrix‐valued function on ℝ2 (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J1 > J2 > …︁ > JN and (iii) Q is off‐diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N‐wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N‐wave interations equation has a unique global solution.
Homogenization of the Navier‐Stokes equations with a slip boundary condition
Authors: Allaire Grégoires
Published: April 2007

This paper deals with the homogenization of the Stokes or Navier‐Stokes equations in a domain containing periodically distributed obstacles, with a slip boundary condition (i.e., the normal component of the velocity is equal to zero, while the tangential velocity is proportional to the tangential component of the normal stress). We generalize our previous results (see [1]) established in the case of a Dirichlet boundary condition; in particular, for a so‐called critical size of the obstacles (equal to ε3 in the three‐dimensional case, ε being the inter‐hole distance), we prove the convergence of the homogenization process to a Brinkman‐type law.
Optimal use of the translation method and relaxations of variational problems
Authors: Firoozye Nikan B.
Published: April 2007

The translation method has been used with great success in bounding the effective moduli of composite materials. We consider here the analogous method for bounding the relaxations of variational problems. We optimize the bound over the set of all available translations. Our method is to cast this in the form of a minmax problem. Using techniques of nonsmooth analysis, we are able to identify the optimal translation bound, meanwhile proving the existence of at least one optimal combination rank‐one convex quadratic and null‐Lagrangian translation. The optimal translation bound proves to be a better general lower bound on relaxations of variational problems than is the polyconvexification in three dimensions. In two dimensions, we discuss the negative result that the optimal translation bound is exactly the polyconvexification. Several examples of optimal applications of translation bounds to non‐convex nonlinear variational problems are given.
The fluid‐dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks
Authors: Xin Zhouping
Published: April 2007

This paper studies the asymptotic equivalence of the Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path ε. It is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow, although there are thin shock layers in which the convergence does not hold. More precisely, by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, we can show that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order (ε) as the mean free path ε goes to zero. For the proof, we first construct a formal solution for the Broadwell equation by matching the truncated Hilbert expansion and shock layer expansion. Then the existence of Broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution. We use an energy method which makes full use of the inner structure of time dependent shock profiles for the Broadwell equations.
A class of homogenization problems in the calculus of variations
Authors: Weinan E.
Published: April 2007

We study a class of integral functionals for which the integrand fe(x, u, ∇u) is an oscillatory function of both x and u. Our method is based on the concept of Γ‐convergenee. Technical difficulties arise because fe(x, u, ∇u) is not convex or equi‐continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton‐Jacobi equations.
The distance to the stomach for feeding tube placement in children predicted from regression on height
Authors: Beckstrand Jan , Ellett Marsha , Welch Joyce , Dye Janice , Games Connie , Henrie Sue , Barlow Roberta S.
Published: April 2007

Nurses use several external measures referenced to the head and chest to gauge the insertion distance for orogastric and nasogastric (NG) tubes. Few of the measures have been tested. However, in previous studies height was the external measure most correlated with esophageal length both in children and adults. In this study, the ability of previously published regression equations on height to predict esophageal length for NG–tube insertion in 107 children was evaluated. The regression equations were examined for stability, predictive performance, and the likely positions of the tube. The data were heights and esophageal lengths obtained from esophageal manometry records and hospital charts. The predicted values for nasal insertions were biased and averaged 2.4 cm too long (R=.56, n=30). Prediction errors greater than 5 cm in absolute value occurred in 25% of the nasally‐referenced sample. The predicted values represented overestimates in 18 nasally referenced cases that were 11.5% longer on the average than the measured esophageal lengths. In contrast, the predicted values for the oral insertions were unbiased (R=.92, n=77), and gave accurate predictions in the majority of cases. Eighty percent of the oral predictions were within ±1.5 cm of the measured esophageal length and represented percentage errors between 0 and 7% (M=3%, n=50).