Approximation of the conductivity coefficient in the heat equation

In this paper, we study the conductivity coefficient determination in the heat equation from observation of the lateral Dirichlet-to-Neumann map. We define a bilinear form function Q_γ associated to the boundary condition and the Dirichlet-to-Neumann map, and prove that the linearized problem d?Q_γ is injective. Based on the idea of complex geometrical optics solutions, we give two approximations to the conductivity coefficient by using the Fourier truncation method and the mollification method. Under the a priori assumption of the conductivity, we estimate the errors between the conductivity coefficient and its approximations by setting a suitable bound of the frequency.     

Chebyshev orthogonal polynomials and the Neumann problem

In a Banach space E, we consider the Neumann problem for a second-order homogeneous equation with a constant operator coefficient densely defined in E. Under a weakened strong positivity condition on the operator coefficient (we use the approach due to V.I. Gorbachuk and A.I. Knyazyuk to the study of a-positive operators), we present a criterion for the well-posed solvability of the problem. We obtain a representation of the solution via Chebyshev operator polynomials of the first and second kind.     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

Composition of an efficient portfolio in the Bielecki and Pliska market model

We study the continuous time portfolio optimization model due to Bielecki and Pliska where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors. We introduce the functional Q _γ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient γ to the variance when we keep the value of the factor levels fixed. The coefficient γ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for Q _γ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to γ.     

Inference on the variance and smoothing of the paths of diffusions

We give a hypothesis testing method to fit the diffusion coefficient σ of a d-dimensional stochastic differential equation on the basis of the observation of certain functionals of regularizations of the solution.     

Counting Lattice Points by Means of the Residue Theorem

We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.     

Dobrushin Coefficients of Ergodicity and Asymptotically Stable L 1-Contractions

We extend an inequality (which involves the Dobrushin coefficient of ergodicity; see Cohen et al.⁽⁴⁾) to any linear bounded operator with domain and codomain L ¹-spaces. We use the extended Dobrushin coefficient of ergodicity, that appears in the inequality, in order to obtain sufficient conditions for the uniform asymptotic stability of a positive contraction of an L ¹-space. We conclude the paper by studying a class of strongly asymptotically stable positive contractions.     

An Example of the Effect of Rescaling of the Reflection Coefficient on the Scattering Potential for the One-Dimensional Schrodinger Equation

Let us consider the one-dimensional Schrödinger equation with the scattering potential V(x)=2δ(x) and corresponding reflection coefficient b(k)=?i/(k + i). The potential satisfies a theorem of Deift and Trubowitz which states that non-negative measurable potentials V(x) satisfying a certain range condition have reflection coefficients b(k) such that b(0)=?1. We rescale the reflection coefficient for V(x)=2δ(x) by writing b(k)=?iv/(k + 1) where 0<v<1. It is shown how V(x) changes with v, through the use of the Gelfand-Levitan equation. This example illustrates how sensitive the potential is to rescaling of the reflection coefficient. In particular, the rescaling leads to a negative portion of V(x), as is expected from the Deift-Trubowitz theorem. The example of this paper will be used in a later paper to illustrate the nature of bounds on potentials obtained through the use of variational principles.     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

Linearization formula for the product of associated q-ultraspherical polynomials

We obtain an explicit formula for the linearization coefficient of the product of two associated q-ultraspherical polynomials in terms of a multiple of a balanced, terminating very-well-poised ₁₀φ₉ series. We also discuss the nonnegativity properties of the coefficients as well as some special cases. 2000 Mathematics Subject Classification Primary—33D45; Secondary—33D8 This work was supported in part by an NSERC grant A6197.     

Inverse problem for the strongly degenerate heat equation in a domain with free boundary

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as t → +0 as a power function t ^β , where β ≥ 1. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 28–43, January, 2009.     

Determination of the coefficient of heat output in the inverse problem for the heat equation

To determine the variable coefficient of heat output in a boundary condition of third kind we propose applying a representation of the solution of the one-sided heat conduction problem obtained by the symbolic operator method. We consider the cases of the one- and two-dimensional inverse problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 9–13.     

Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial _L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of _L of an m-component link L all of whose Milnor numbers μ_i1…ip vanish for pn. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.     

A matricial extension of the Helson-Sarason theorem and a characterization of some multivariate linearly completely regular processes

We generalize the theorems of Helson-Szegö and Helson-Sarason for matricial measures. We study two-weighted inequalities for the Hilbert transform in [0, 2π] and in R and give a characterization for the positivity of the angle between past and future of multivariate weakly stationary stochastic processes, in the discrete and the continuous case. We also characterize the multivariate weakly stationary stochastic processes that are linearly completely regular and study the rate of convergence of the maximal correlation coefficient.     

Bifurcation analysis of amathematical model for the growth of solid tumors in the presence of external inhibitors

We study bifurcations from radial solution of a free boundary problem modeling the dormant state of nonnecrotic solid tumors in the presence of external inhibitors. This problem consists in three linear elliptic equations with two Dirichlet and one Neumann boundary conditions and a fourth boundary condition coupling surface tension effects on free boundary. In this paper, surface tension coefficient γ plays the role of bifurcation parameter. We prove that in certain situations there exists a positive null point sequence for γ where bifurcation occurs from radial solution, while in the other situations, either bifurcation occurs at only finite many points of γ or even it does not occur for any γ > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Remarks on the Cauchy problem for first order hyperbolic systems with constant coefficient principal part

Abstract In this article we shall introduce the results obtained in [16], i.e., we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C^∞ well-posed under the maximal rank condition (see the condition (R) below). We shall also give a simple sufficient condition without any assumptions on the rank. Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part     

Remark on the dissipation of the magnetic helicity integral

We prove that the current helicity integral for a thin magnetic tube is directly proportional to the twist Tw of field lines, which appears in the Călugărianu formula as a summand. Using this approach, we analyze the law of the variation of magnetic helicity in a conducting medium with a nonzero magnetic diffusion coefficient and also the equation of the phase transition of the magnetic field under early universe conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 150–160, January, 2009.     

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls. Received: 22 September 1997 / Revised version: 15 January 1998     

Completeness of analytic functions and extremality of the coefficients of a Laurent series

We generalize Vitushkin's theorem on the fact that the completeness of the set of functions analytic on a compactum in the complex plane depends upon the extremality of the first coefficient of the Laurent series of the classes of functions connected with this compactum. We show that completeness is characterized by the extremality of the Laurent series coefficient with any fixed numbern, ns=1 The n-th analytic capacity considered, generalizing the concept of analytic capacity (n=1), also flexibly measures the set.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 277–283, August, 1977.