Statistical Convergence in Probability for a Sequence of Random Functions

In this paper, we introduce a new type of convergence for a sequence of random functions, namely, statistical convergence in probability, which is a natural generalization of convergence in probability. In this approach, we allow such a sequence to go far away from the limit point infinitely many times by presenting random deviations, provided that these deviations are negligible in some sense of measure. In this context, the set of values of a random function is considered as a probabilistic metric (PM) space of random variables, and some basic results are obtained using the tools of PM spaces.     

Moments of the Mean of Dubins–Freedman Random Probability Distributions

Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure . Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of . The support of the mean is also considered. It is shown that the support of M is connected whenever is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.     

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.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

Estimates for the Entropy Numbers of the Classes B p , θ Ω $$ {B}_{p,\theta}^{\varOmega } $$ of Periodic Multivariable Functions in the Uniform Metric

Ukrainian Mathematical Journal - We establish order estimates for the entropy numbers of the classes $$ {B}_{p,\theta}^{\varOmega } $$ of periodic multivariable functions in the uniform metric. For...     

Two-dimensional Meixner Random Vectors of Class ${mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

Asymptotics of Wave Functions of the Stationary Schrödinger Equation in the Weyl Chamber

We study stationary solutions of the Schrödinger equation with a monotonic potential U in a polyhedral angle (Weyl chamber) with the Dirichlet boundary condition. The potential has the form \(U\left( x \right) = \sum _{j = 1}^nV\left( {{x_j}} \right),x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}\), with a monotonically increasing function V (y). We construct semiclassical asymptotic formulas for eigenvalues and eigenfunctions in the form of the Slater determinant composed of Airy functions with arguments depending nonlinearly on x_j. We propose a method for implementing the Maslov canonical operator in the form of the Airy function based on canonical transformations.     

Asymptotics for Certain Harmonic Functions and the Martin Compactification on the Quaternionic Heisenberg Group

In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.     

Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions

We consider the Schrödinger operator on ${\mathbb{R}^2}$ with a locally square-integrable periodic potential V and give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm of V on a fundamental domain, provided that V is small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006).     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in $$mathbb {R}^3$$ R 3 Using Regularization and Blowup

Journal of Nonlinear Science - Two-fold singularities in a piecewise smooth (PWS) dynamical system in $$mathbb {R}^3$$ have long been the subject of intensive investigation. The interest stems...     

The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), where K is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of H(K) for all sufficiently small values of the zero-range attractive potentials is established.The asymptotics is found for the number of eigenvalues N(0,z) lying below . Moreover, for all sufficiently small nonzero values of the three-particle quasimomentum K, the finiteness of the number of eigenvalues below the essential spectrum of H(K) is established and the asymptotics of the number N(K,0) of eigenvalues of H(K) below zero is given.     

Asymptotics of the Fredholm Determinant Associated with the Correlation Functions of the Quantum Nonlinear Schrödinger Equation

The correlation functions of the quantum nonlinear Schrödinger equation can be presented in terms of the Fredholm determinant. An explicit expression for this determinant is found for large time and long distance. Bibliography: 6 titles.     

Recurrence Relationships for the Mean Number of Faces and Vertices for Random Convex Hulls

This paper studies the convex hull of n random points in R^d\mathsf{R}^{d} . A recently proved topological identity of the author is used in combination with identities of Efron and Buchta to find the expected number of vertices of the convex hull—yielding a new recurrence formula for all dimensions d. A recurrence for the expected number of facets and (d−2)-faces is also found, this analysis building on a technique of Rényi and Sulanke. Other relationships for the expected count of i-faces (1≤i<d) are found when d≤5, by applying the Dehn–Sommerville identities. A general recurrence identity (see (3) below) for this expected count is conjectured.     

Approximation of the Classes C β ψ $$ {C}_{beta}^{psi } $$ H�� By Biharmonic Poisson Integrals

Ukrainian Mathematical Journal - We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class...     

Equivalent Conditions of Asymptotics for the Density of the Supremum of a Random Walk in the Intermediate Case

This paper obtains some equivalent conditions about the asymptotics for the density of the supremum of a random walk with light-tailed increments in the intermediate case. To do this, the paper first corrects the proofs of some existing results about densities of random sums. On the basis of the above results, the paper obtains some equivalent conditions about the asymptotics for densities of ruin distributions in the intermediate case and densities of infinitely divisible distributions. In the above studies, some differences and relations between the results on a distribution and its corresponding density can be discovered.      

Boundaries of Holomorphic 1-Chains Within Holomorphic Line Bundles over $mathbb{CP}^{1}$

We show that boundaries of holomorphic 1-chains within holomorphic line bundles of mathbbCP¹mathbb{CP}^{1} can be characterized using a single generating function of Wermer moments. In the case of negative line bundles, a rationality condition on the generating function plus the vanishing moment condition together form an equivalent condition for bounding. We provide some examples which reveal that the vanishing moment condition is not sufficient by itself. These examples also can be used to demonstrate one point of caution about the use of birational maps in this topic.