Ultrafilter Convergence in Ordered Topological Spaces

We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter D, the notions of D-compactness and of D-pseudocompactness are equivalent. Any product of initially λ-compact generalized ordered topological spaces is still initially λ-compact. On the other hand, preservation under products of certain compactness properties is independent from the usual axioms for set theory.     

Asymptotic quadratic convergence of the parallel block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices

The proof of the asymptotic quadratic convergence is provided for the parallel two-sided block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices. The discussion covers the case of well-separated eigenvalues as well as clusters of eigenvalues. Having p processors, each parallel iteration step consists of zeroing 2p off-diagonal blocks chosen by dynamic ordering with the aim to maximize the decrease of the off-diagonal Frobenius norm. Numerical experiments illustrate and confirm the developed theory.     

On eigenvalues of random complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model X^k(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices

Let X ⁽ⁿ⁾=(X _ij ) be a p×n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R ⁽ⁿ⁾=(ρ _ij ) be the p×p sample correlation coefficient matrix of X ⁽ⁿ⁾, and \(S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}\) be the sample covariance matrix of X ⁽ⁿ⁾, where \(\bar{X}\) is the mean vector of the n observations. Assuming that X _ij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R ⁽ⁿ⁾ converges almost surely to the limit \((1-\sqrt{c}\,)^{2}\) as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S ⁽ⁿ⁾ converges almost surely to \((1-\sqrt{c}\,)^{2}\).     

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern p is denoted A v _P (p). We extend a proof of Simion and Schmidt to show that A v _P (132)=A v _P (123) for any poset P, and we exactly classify the posets for which equality holds.     

The critical boundary RSOS <Emphasis Type="Italic">M</Emphasis>(3,5) model

We consider the critical nonunitary minimal model M(3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz (TBA) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A₄restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the (r, s) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of (m, n) systems.     

On expansions and extensions of powerful digraphs

We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph ?-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list p ₁(x), ..., p _n (x) of all nonprincipal 1-types in an LCC theory, if all types r(x ₁, ..., x _m ) containing \(p_{i_1 } \) (x ₁) ∪ ... ∪ \(p_{i_m } \)(x _m ) are dominated by some type q then q is a powerful type.     

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

On the Nörlund means of Vilenkin-Fourier series

We prove and discuss some new (H _p ,L _p )-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q _k : k ? 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results.     

Stable CLTs and Rates for Power Variation of <Emphasis Type="Italic">α</Emphasis>-Stable Lévy Processes

In a central limit type result it has been shown that the pth power variations of an α-stable Lévy process along sequences of equidistant partitions of a given time interval have \(\frac{\alpha}{p}\)-stable limits. In this paper we give precise orders of convergence for the distances of the approximate power variations computed for partitions with mesh of order \(\frac{1}{n}\) and the limiting law, measured in terms of the Kolmogorov-Smirnov metric. In case 2α?<?p the convergence rate is seen to be of order \(\frac{1}{n}\), in case α?<?p?<?2α the order is \(n^{1-\frac{p}{\alpha}}.\)     

Higher Order Riesz Transforms in the Ultraspherical Setting as Principal Value Integral Operators

In this paper we represent the kth Riesz transform in the ultraspherical setting as a principal value integral operator for every \({k \in \mathbb N}\). We also measure the speed of convergence of the limit by proving L ^p -boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

Detection and Analysis of Spikes in a Random Sequence

Motivated by the more frequent natural and anthropogenic hazards, we revisit the problem of assessing whether an apparent temporal clustering in a sequence of randomly occurring events is a genuine surprise and should call for an examination. We study the problem in both discrete and continuous time formulation. In the discrete formulation, the problem reduces to deriving the probability that p independent people all have birthdays within d days of each other. We provide an analytical expression for a warning limit such that if a subset of p people among n are observed to have birthdays within d days of each other and d is smaller than our warning limit, then it should be treated as a surprising cluster. In the continuous time framework, three different sets of results are given. First, we provide an asymptotic analysis of the problem by embedding it into an extreme value problem for high order spacings of iid samples from the U[0, 1] density. Second, a novel analytical nonasymptotic bound is derived by using certain tools of empirical process theory. Finally, the required probability is approximated by using various bounds and asymptotic results on the supremum of the scanning process of a one dimensional stationary Poisson process. We apply the theories to climate change related datasets, datasets on temperatures, and mass shooting records in the United States. These real data applications of our theoretical methods lead to supporting evidence for climate change and recent spikes in gun violence.     

Algorithms with high order convergence speed for blind source extraction

A rigorous convergence analysis for the fixed point ICA algorithm of Hyvärinen and Oja is provided and a generalization of it involving cumulants of an arbitrary order is presented. We consider a specific optimization problem OP(p), p>3, integer, arising from a Blind Source Extraction problem (BSE) and prove that every local maximum of OP(p) is a solution of (BSE) in sense that it extracts one source signal from a linear mixture of unknown statistically independent signals. An algorithm for solving OP(p) is constructed, which has a rate of convergence p?1.     

Some Poisson structures and Lax equations associated with the Toeplitz lattice and the Schur lattice

The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this paper, we unveil the origins of this Poisson structure and derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety H _n of GL _n (C), which we view as a real or complex Poisson–Lie group whose Poisson structure comes from a quadratic R-bracket on gl _n (C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on H _n , combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety H _n ^τ of U _n which is itself a Poisson–Dirac subvariety of GL _n ^R (C). We then construct a Hamiltonian for the Poisson structure induced on H _n ^τ , corresponding to another system which derives from the Toeplitz lattice the modified Schur lattice. Thanks to the properties of Poisson–Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.     

Inductive and projective limits of Banach spaces of measurable functions with order unities with respect to power parameter

We prove that a measurable function f is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order unities f^α and f^β with α > β > 0. We show that it is natural to understand the limit of ordered vector spaces with order unities f^α (α approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.     

Convergence of greedy algorithm in Walsh system in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

Convergence of the greedy algorithm in Walsh system in L _p , p > 1 is studied. It is proved that there exists a function in L _p , 1 < p < 2, with greedy algorithm not converging in measure to that function. A continuous function with divergent in L _p , p > 2, greedy algorithm is constructed and sufficient conditions for convergence of the greedy algorithm in L _p , p > 1 are given.