Analytical and numerical investigation of the spectra of three-point difference operators

The properties of the spectra of discrete spatially one-dimensional problems of convection — diffusion type with constant coefficients and nonstandard boundary conditions are examined in the framework of stability of explicit algorithms for time-dependent problems of mathematical physics. An analytical method is proposed for finding isolated limit points of the operator spectrum. Limit points are determined for the difference transport equation with different versions of nonreflecting boundary conditions and for an approximation of the heat conduction equation on a grid with condensation near the boundary. Stability and other properties of the spectrum are also established numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 25–45, 2007.     

Variational and Asymptotic Methods for Finding Eigenvalues below the Continuous Spectrum Threshold

Considering the example of a mixed boundary value problem for the Helmholtz operator we discuss two methods for finding eigenvalues below the continuous spectrum threshold: one variational and the other—asymptotic. We construct asymptotics for the eigenvalue arising near the threshold as a small obstacle appears in the cylindrical waveguide. The resulting asymptotic formula, its derivation and justification differ substantially from the case of a bounded domain.     

The canonical-boundary representation for automorphism groups of locally compact countable state Markov shifts

We introduce the canonical-boundary representation and study its range. This conjugacy invariant homomorphism captures information about the symmetry of the Markov shift near its (canonical) boundary and exhibits which actions on the boundary can be realized by automorphisms. The path-structure at infinity — a relation on the set of orbits of the canonical boundary — is a new conjugacy invariant, which is stronger than the canonical boundary and the periodic data at infinity. Moreover we determine its influence on the range of the canonical-boundary representation and the extendability of automorphisms from subsystems (ascending sequences of shifts os finite type (SFTs) and infinite subsets of periodic points) to the entire Markov shift.     

Harmonic analysis on non-semisimple symmetric spaces

It is proven that theL ² spectrum for certain non-semisimple, non-nilpotent symmetric spaces is multiplicity-free. The spectrum and spectral measure are computed precisely for symmetric spaces corresponding to non-compact motion groups. Somewhat less complete results on theL ² spectrum — in both the Mackey Machine and Orbit Method modes — are given for general semidirect product symmetric spaces. The author was supported by the NSF through DMS84-00900-A01 and by a Senior Fulbright Fellowship.     

On a Stochastic Plate Equation

In this paper we discuss an initial—boundary value problem for an elastic plate driven by a space-time white noise. The existence and uniqueness of a weak solution is established. We use a specialized PDE method based upon the results for the deterministic equation. Accepted 2 February 2001. Online publication 4 May 2001.     

Atiyah—Patodi—Singer type index theorems for manifolds with splitting of $ \eta $-invariants

We construct self-adjoint extensions of Dirac operators on manifolds with corners of codimension 2, which generalize the Atiyah—Patodi—Singer boundary conditions. The boundary conditions are related to geometric constructions, which convert problems on manifolds with corners into problems on manifolds with boundary and wedge singularities. In the case, where the Dirac bundle is a super-bundle, we prove two general index theorems, which differ by the splitting formula for -invariants. Further we work out the de Rham, signature and twisted spin complex in closer detail. Finally we give a new proof of the splitting formula for the -invariant. Submitted: October 1999, Revised version: March 2001.     

On the Riemann Hilbert type problems in Clifford analysis

In this paper boundary value problems combining Jump — Riemann and Hilbert problems for monogenic functions in Ahlfors-David regular surfaces and in the upper half space respectively are investigated. The explicit formula of the solution is obtained.     

On the dynamics of predator-prey models with the Beddington–De Angelis functional response, under Robin boundary conditions

The paper concerns the Beddington–De Angelis predator-prey model, under Robin boundary conditions. General properties—such as boundedness, uniqueness and existence of invariant regions—are obtained. Linear stability (instability) threshold of the equilibrium state S (biologically meaningful) and diffusion-driven instability (Turing effect) are studied. In the framework of nonlinear L ²-energy stability, conditions guaranteeing stability and local attractiveness are obtained.      

Energy asymptotics for Type II superconductors

We study the Ginzburg–Landau functional in the parameter regime describing ‘Type II superconductors’. In the exact regime considered minimizers are localized to the boundary — i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behavior and give leading order energy asymptotics. This generalizes previous results by Lu–Pan, Helffer–Pan, and Pan.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Contour lines of the two-dimensional discrete Gaussian free field

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/λ - 1, b/λ - 1), a variant of SLE(4).     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in \mathbbC²{\mathbb{C}^{2}} near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range—roughly speaking—from being “mildly infinite-type” to very flat at the infinite-type points.     

On the falling of objects in non-Newtonian fluids

The falling of a lamina in between two parallel plates containing a fluid of second grade is studied. The velocity of the lamina and the fluid are determined by solving the mixed initial—boundary value problem using Laplace transform. Explicit exact solutions are obtained for the velocity of the lamina and the fluid. Next, the falling of a cylinder in a tube containing a fluid of second grade is analyzed using Laplace transform, and once again exact solutions are found.

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Sunto Si studia la caduta di una lamina fra due piastre parallele contenenti un fluido di secondo grado. La velocità della lamina e del fluido sono determinate risolvendo un problema misto al contorno—a valori iniziali per mezzo della trasformata di Laplace—.Si studia poi la caduta di un cilindro in un tubo contenente un fluido di secondo grado utilizzando ancora la trasformata di Laplace e anche in questo caso si determina la soluzione esatta.

     

Comparison and robustification of Bayes and Black-Litterman models

For determining an optimal portfolio allocation, parameters representing the underlying market—characterized by expected asset returns and the covariance matrix—are needed. Traditionally, these point estimates for the parameters are obtained from historical data samples, but as experts often have strong opinions about (some of) these values, approaches to combine sample information and experts’ views are sought for. The focus of this paper is on the two most popular of these frameworks—the Black-Litterman model and the Bayes approach. We will prove that—from the point of traditional portfolio optimization—the Black-Litterman is just a special case of the Bayes approach. In contrast to this, we will show that the extensions of both models to the robust portfolio framework yield two rather different robustified optimization problems.     

Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.     

Local characterization of invariant sets of an autonomous differential inclusion

Summary An application in robotics motivates us to characterize the evolution of a subset in state space due to a compact neighborhood of an arbitrary dynamical system—an instance of a differential inclusion. Earlier results of Blagodat·skikh and Filippov (1986) and Butkovskii (1982) characterize the boundary of theattainable set and theforward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection ofa compact regular subset of the state space. Let the collection of states such that the differential inclusion contains an equilibrium point be called asingular invariant set. We show that the fields at the boundary of the forward projection of a singular invariant set are degenerate under some regularity assumptions when the state-wise boundary of the differential inclusion is smooth. Consider instead those differential inclusions such that the state-wise boundary of the problem is a regular convex polytope—a piecewise smooth boundary rather than smooth. Our second result gives conditions for theuniqueness andexistence of the boundary of the forward projection of a singular invariant set. They characterize the bundle of unstable and stable manifolds of such a differential inclusion.     

The Sturm-Liouville problem with a nonlocal boundary condition

In this paper, we consider the Sturm-Liouville problem with one classical and another nonlocal boundary condition. We investigate general properties of the characteristic function and spectrum for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues, analyze the dependence of the spectrum on parameters of the boundary condition, and describe the qualitative behavior of all eigenvalues subject to of the nonlocal boundary condition. Dedicated to N. S. Bakhvalov (1934–2005) Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 410–428, July–September, 2007.     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.