A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k

We obtain a necessary condition on the coefficients of a formal power series, which is a formal solution of a nontrivial singular linear differential equation of order k, with analytic coefficients and prove a “uniqueness” theorem for the differential equation.     

Existence of solutions to derivative-dependent,nonlinear singular boundary value problems

This article examines two-point boundary value problems (BVPs) for second-order, singular ordinary differential equations where the right-hand-side of the differential equation may depend on the derivative of the solution. We introduce a method to obtain a priori bounds on all potential solutions, including their “derivatives”, to the singular BVP under consideration. The approach is based on the application of differential inequalities of singular type. The ideas are then applied to yield new existence results for solutions.     

Estimates of the Nash–Aronson type for degenerating parabolic equations

We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L ₂ with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”     

一类高阶超双曲型方程的中量定理及其逆定理

Asgeirsson中量定理表明超双曲型方程的Cauchy问题一般是不适定的,对Asgeirsson中量定理的推广就有重要意义。目前关于高阶方程解的中量只有初步探讨,尚未得到具体结果,本文直接利用Asgeirsson中量定理结果和积分、微分的性质与关系,得到了高阶方程解的中量满足广义双轴对称位势方程,同时还证明了其逆定理。利用关于广义双轴对称位势方程正则解的表达式及雅可比多项式的特殊性质,得到了高阶方程解的中量公式,从而使得关于解的拓展性和适定性的讨论将有可能。     

Spaces of Wald-Berestovskii curvature bounded below

We consider inner metric spaces of curvature bounded below in the sense of Wald, without assuming local compactness or existence of minimal curves. We first extend the Hopf-Rinow theorem by proving existence, uniqueness, and “almost extendability” of minimal curves from any point to a denseG _δ subset. An immediate consequence is that Alexandrov’s comparisons are meaningful in this setting. We then prove Toponogov’s theorem in this generality, and a rigidity theorem which characterizes spheres. Finally, we use our characterization to show the existence of spheres in the space of directions at points in a denseG _δ set. This allows us to define a notion of “local dimension” of the space using the dimension of such spheres. If the local dimension is finite, the space is an Alexandrov space.     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

Tensor Approximation Approach to Calculation of Singular Values and Vectors for SVD Problem

A new method of calculation of singular values and left and right singular vectors of arbitrary nonsquare matrices is proposed. The method allows one to avoid solutions of high rank systems of linear equations of singular value decomposition problems, which makes it not sensitive to ill-conditioness of the decomposed matrix. On the base of the Eckart–Young theorem, it was shown that each second order r-rank tensor can be represented as a sum of the first rank r-order “coordinate” tensors. A new system of equations for “coordinate” tensor generator vectors was obtained. An iterative method of solution of the system was elaborated. Results of the method were compared with classical methods of solutions of singular value decomposition problems.     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

Existence of weak positive solutions to a nonlinear PDE system around a triple phase boundary, coupling domain and boundary variables

We consider a 2D nonlinear system of PDEs representing a simplified model of processes near a triple-phase boundary (TPB) in cathode catalyst layer of hydrogen fuel cells. The particularity of this system is the coupling of a variable satisfying a PDE in the interior of the domain with another variable satisfying a differential equation (DE) defined only on the boundary, through an adsorption-desorption equilibrium mechanism. The system includes also an isolated singular boundary condition which models the flux continuity at the contact of the TPB with a subdomain. By freezing certain terms we transform the nonlinear PDE system to an equation, which has a variational formulation. We prove several L^∞ and W1,p a priori estimates and then by using Schauder fixed point theorem we prove the existence of a weak positive bounded solution.     

A “planar” representation for generalized transition kernels

In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of “Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is “sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ?: an ordinate set (in which vertical sections have Lebesgue measure), an “atomic” set contained inX × (??), and a “singular” set with null sections. The method incidentally produces and exploits a “reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing “uniformly σ-finite” measures (in the “standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2].     

The complete classification of fibres in pencils of curves of genus two

This article contains geometrical classification of all fibres in pencils of curves of genus two, which is essentially different from the numerical one given by Ogg ([11]) and Iitaka ([7]). Given a family π:X→D of curves of genus two which is smooth overD′=D?{0}, we define a multivalued holomorphic mapT _π fromD′ into the Siegel upper half plane of degree two, and three invariants called “monodromy”, “modulus point” and “degree”. We assert that the family π is completely determined byT _π, and its singular fibre by these three invariants. Hence all types of fibres are classified by these invariants and we list them up in a table, which is the main part of this article.     

On the nature of the de Branges Hamiltonian

We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developing the theory of Hilbert spaces of entire functions (we call them Krein-de Branges spaces), de Branges arrived at a certain class of canonical equations of phase dimension 2. He showed that, for any given Krein-de Branges space, there exists a canonical equation of the class indicated that restores a chain of Krein-de Branges spaces imbedded one into another. The Hamiltonians of such canonical equations are called de Branges Hamiltonians. The following question arises: Under what conditions will the Hamiltonian of a certain canonical equation be a de Branges Hamiltonian? The main theorem of the present work, together with Theorem 1 of the paper cited above, gives an answer to this question.     

Finite difference solution of a singular boundary value problem for the p-Laplace operator

We are concerned about a singular boundary value problem for a second order nonlinear ordinary differential equation. The differential operator of this equation is the radial part of the so-called N-dimensional p-Laplacian (where p?>?1), which reduces to the classical Laplacian when p?=?2. We introduce a finite difference method to obtain a numerical solution and, in order to improve the accuracy of this method, we use a smoothing variable substitution that takes into account the behavior of the solution in the neighborhood of the singular points.     

On the support of complete intersection zero-cycles

On an algebraic varietyY ? ? _N we will call complete intersection a 0-cycle when it is the intersection of Y with a codimension n complete intersection of ? _N . We consider the following problem: Let E?Y be given. Does E contain the support of a complete intersection 0-cycle? The two main theorems shown in this article give the answers in some cases: first, a negative answer for E some “big” subset of a singular irreducible algebraic variety; secondly, a positive answer for some “small” subset, on any algebraic variety.     

Chromatic number and skewness

In this note, we solve the problem of determing the chromatic number of planar graphs to which a certain number μ of “extra” edges are attached. We obtain a (best-possible) theorem when μ < ₂^k for every integer k ≥ 3. The statement of the theorem for k = 2 is the Four-Color Conjecture.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.     

A factorization theorem for a certain class of graphs

In this note we give a necessary and sufficient condition for factorization of graphs satisfying the “odd cycle property”. We show that a graph G with the odd cycle property contains a [k_i] factor if and only if the sequence [H]+[k_i] is graphical for all subgraphs H of the complement of G.A similar theorem is shown to be true for all digraphs.     

On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions

We prove a lower bound expansion on the probability that a random ±1 matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form e _i ±e _j , which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form e _i ±e _j ±e _k ±e _? , which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the “novel partitions” which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are “first place and runners up” in relation to the Erd?s-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.     

A model-data weak formulation for simultaneous estimation of state and model bias

We introduce a Petrov–Galerkin regularized saddle approximation which incorporates a “model” (partial differential equation) and “data” (M experimental observations) to yield estimates for both state and model bias. We provide an a priori theory that identifies two distinct contributions to the reduction in the error in state as a function of the number of observations, M: the stability constant increases with M; the model-bias best-fit error decreases with M. We present results for a synthetic Helmholtz problem and an actual acoustics system.     

The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ?_t|?u|², which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.