Matrix identities and the pigeonhole principle

We show that short bounded-depth Frege proofs of matrix identities, such as PQ=IQP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.     

G-analysis of high-dimensional observations

Consistent and asymptotically normal G-estimators are obtained for generalized variance and normalized spectral function of the covariance matrix when Kolmogorov's condition is satisfied. The proofs are based on the application of limit theorems for random determinants and resolvents of random matrices.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 115–121, 1986.     

Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property

Under some nondegeneracy condition, we show that sequences of entropy and approximate solutions of a semilinear ultra-parabolic equation are strongly precompact in the general case of a Caratheodory flux vector and a diffusion matrix. The proofs are based on localization principles for the parabolic H-measures corresponding to sequences of measure-valued functions. Bibliography: 21 titles. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 49–92.     

Girth and Euclidean distortion

Let G be a k-regular graph, , with girth g. We prove that every embedding has distortion . Two proofs are given, one based on Markov type [B] and the other on quadratic programming. In the core of both proofs are some Poincaré-type inequalities on graph metrics. Submitted: July 2001, Revised: September 2001.     

Absolute continuity and summability of transport densities: simpler proofs and new estimates

The paper presents some short proofs for transport density absolute continuity and L ^p estimates. Most of the previously existing results which were proven by geometric arguments are re-proved through a strategy based on displacement interpolation and on approximation by discrete measures; some of them are partially extended.     

Modeling diseases with latency and nonlinear incidence rates: global dynamics of a multi‐group model

In this paper, we perform global stability analysis of a multi‐group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease‐free equilibrium is globally asymptotically stable if R₀≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if R₀>1. The proofs of global stability of equilibria exploit a matrix‐theoretic method using Perron eigenvetor, a graph‐theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotics for Global Measures of Accuracy of Splines

We obtain central limit theorems for the stochastic parts of L_p-norms of smoothed cubic spline estimators. The proofs are based on the observation that the variance term of the cubic spline is approximately of a form corresponding to a kernel estimator.     

Simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost <Emphasis Type="Italic">f</Emphasis>-algebras

The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f-algebras^[5], based on a sequence approximation.      

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m ^d+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. Supported in part by NSF grant DMS-9205181     

0-Dual Closures for Several Classes of Graphs

We prove that for almost all sufficient conditions based on degree sums or neighborhood unions of 3-independent sets for a graph G to be hamiltonian imply that the 0-dual closure of G is complete. The proofs are very short.     

Structure of Chevalley groups: The proof from the book

Different geometric proofs of the main structure theorems for Chevalley groups over commutative rings are described and compared. Known geometric proofs, published by I. Z. Golubchik, N. A. Vavilov, A. V. Stepanov, and E. B. Plotkin, such as A₂ and A₃ proofs for classical groups, A₅ and D₅ proofs for E₆, A₇ and D₆ proofs for E₇, and a D₈ proof for E₈ are given in outline. After that, A₂ proofs for exceptional groups of types F₄, E₆, and E₇, based on the multiple commutation, are discussed in more detail. This new proof, the proof from the Book, provides better bounds than any previously known proof. Moreover, it does not use results for fields, the factorization with respect to the radical, or any specific information concerning the structure constants and the equations defining exceptional Chevalley groups. Bibliography: 71 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 36–76.     

Sufficient conditions for permutation equivalence to a WHS-matrix

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A ^?1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.     

On the existence of a complete set of solutions of the Lur’e equation

The algebraic Lur’e equations are considered in the general case of an arbitrary pair (A, B) and a nonsingular matrix Γ of quadratic form. The necessary and sufficient conditions for the existence of a complete set of solutions of such equations are obtained. These conditions are other than in the standard case of a definite matrix Γ. For the standard case, the constraints on the pair (A, B) are maximally relaxed. Then the results are extended to the case of a singular matrix Γ. A special representation of Hamiltonian matrices, which forms the basis for the proofs, is developed. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ &lt; A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A &lt; B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.     

The Number of Spanning Trees in <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-Complements of Quasi-Threshold Graphs

In this paper we examine the classes of graphs whose K_n-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K_n, the K_n-complement of H is the graph K_n–H which is obtained from K_n by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form K_n–H admitting formulas for the number of their spanning trees.Final version received: March 18, 2004     

On the <Emphasis Type="Italic">KO</Emphasis> Characteristic Cycle of a Spin<Superscript><Emphasis Type="Italic">c</Emphasis></Superscript> Manifold

We compute the KO-characteristic numbers of a characteristic submanifold of a Spin^c manifold in terms of its K-characteristic numbers. The proof is based on the geometry of the Thom class in K-theory and is simpler than the existing proofs of several previously known special cases.     

A geometric perspective on counting non-negative integer solutions and combinatorial identities

We consider the effect of constraints on the number of non-negative integer solutions of x+y+z = n, relating the number of solutions to linear combinations of triangular numbers. Our approach is geometric and may be viewed as an introduction to proofs without words. We use this geometrical perspective to prove identities by counting the number of solutions in two different ways, thereby combining combinatorial proofs and proofs without words.     

Some Characteristic Properties of the Fisher Information Matrix via Cacoullos-Type Inequalities

Some lower bounds for the variance of a function g of a random vector X are extended to a wider class of distributions. Using these bounds, some useful inequalities for the Fisher information are obtained for convolutions and linear combinations of random variables. Finally, using these inequalities, simple proofs are given of classical characterizations of the normal distribution, under certain restrictions, including the matrix analogue of the Darmois-Skitovich result.     

Some new symmetry results for elliptic problems on the sphere and in Euclidean space

We prove some new symmetry results for positive solutions of elliptic problems in ℝ ⁿ and on the sphere. The proofs are based on the moving plane method, rearrangement arguments and stereographic projection.     

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.