Convolutions of cantor measures without resonance

Denote by µ _a the distribution of the random sum \((1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} \), where P(ω _j = 0) = P(ω _j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ _a is supported on C _a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 ? 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ _a * (µ _b ° S _λ ^?1 ), where S _λ (x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ ≠ 0, then

$D({\mu _a} * ({\mu _b} \circ S_\lambda ^{ - 1})) = \min ({\dim _H}({C_a}) + {\dim _H}({C_b}),1)$

, where D denotes any of correlation, Hausdorff or packing dimension of a measure.

We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ_1/4* (µ_1/3 ° S _λ ^?1 ) is a singular measure, although dim _H (C _1/4) + dim _H (C _1/3) > 1 and log(1/3)/ log(1/4) is irrational.     

On likelihood ratio ordering of parallel systems with exponential components

Let T(λ₁,...,λ _n ) be the lifetime of a parallel system consisting of exponential components with hazard rates λ₁,...,λ _n , respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ₁, λ₂) majorizes (γ₁, γ₂), then, T(λ₁, λ₂) is larger than T(γ₁, γ₂) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ₁,...,λ _n ) is d-larger than (γ₁,...,γ _n ), then T(λ₁,...,λ _n ) is larger than T(γ₁,...,γ _n ) in likelihood ratio order.     

Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ?γ? = 1. Namely, all such unitary perturbations are operators U_γ:= U + (γ ? 1)( ·, b₁)_Hb, where b ∈ H, ∥b∥ = 1, b₁ = U^?1b, ?γ? = 1. For ?γ? < 1, the operators U_γ are contractions with one-dimensional defects.     

Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential

We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V _λ (x) = λx ^?2 as a representation of \(\widetilde{SL(2,\mathbb{R})}\). The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of \(\widetilde{SL(2,\mathbb{R})}\) ? H ₃, where H ₃ is the three-dimensional Heisenberg group.     

Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Consider real symmetric, complex Hermitian Toeplitz, and real symmetric Hankel band matrix models where the bandwidth b _N →∞ but b _N /N→b∈[0,1] as N→∞. We prove that the distributions of eigenvalues converge weakly to universal symmetric distributions γ _T (b) and γ _H (b). In the case b>0 or b=0 but with the addition of \(b_{N}\geq CN^{\frac{1}{2}+\epsilon_{0}}\) for some positive constants ε ₀ and C, we prove the almost sure convergence. The even moments of these distributions are the sums of some integrals related to certain pair partitions. In particular, when the bandwidth grows slowly, i.e., b=0, γ _T (0) is the standard Gaussian distribution, and γ _H (0) is the distribution |x|exp?(?x ²). In addition, from the fourth moments, we know that γ _T (b) are different for different b, γ _H (b) different for different \(b\in[0,\frac{1}{2}]\), and γ _H (b) different for different \(b\in [\frac{1}{2},1]\).     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

Polar functions of multiparameter bifractional Brownian sheets

Let B^H,K ： （B^H,K（t）, t ∈R＋^N} be an （N,d）-bifractional Brownian sheet with Hurst indices H = （H1,..., HN） ∈ （0, 1）^N and K = （K1,..., KN）∈ （0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov＇s entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b ₁ = k?λ?1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k ? 4b ₁ + 3 (and µ(u, w) + µ(u, z) = 2k ? 4b ₁ + 4). If k = 3b ₁ + γ with γ ≥ ?2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b ₁ = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

On Stability of Thomson’s Vortex <Emphasis Type="Italic">N</Emphasis>-gon in the Geostrophic Model of the Point Bessel Vortices

A stability analysis of the stationary rotation of a system of N identical point Bessel vortices lying uniformly on a circle of radius R is presented. The vortices have identical intensity Γ and length scale γ^?1 > 0. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for N = 2,..., 6 are studied sequentially. The case of odd N = 2?+1 ≥ 7 vortices and the case of even N = 2_n ≥ 8 vortices are considered separately. It is shown that the (2? + 1)-gon is exponentially unstable for 0 < γR<R_*(N). However, this (2? + 1)-gon is stable for γR ≥ R_*(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N = 2n ≥ 8 vortex 2n-gon is exponentially unstable for R > 0.     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.     

Traces of Higher Negative Orders for a String with a Singular Weight

We study the linear operator pencil A(λ) = L?λV, λ ∈ ?, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ₂ ^?1 [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λ_n, n ∈ ?, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L^?1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.     

1-Cocycles on the group of contactomorphisms on the supercircle S1|3 generalizing the Schwarzian derivative

The relative cohomology H_diff¹(K(1|3), osp(2, 3);D_γ,µ(S^1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators D_γ,µ(S^1|3) acting on tensor densities on S^1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(X_f) = D₁D₂D₃(_f)α₃^1/2, X^f ∈ K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3).     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

Oscillatory hyper Hilbert transforms along general curves

We consider the oscillatory hyper Hilbert transform H _γ,α,β f(x) = ∫ ₀ ^∞ f(x - Γ(t))e^it-β t^-(1+α)dt; where Γ(t) = (t, γ(t)) in ?² is a general curve. When γ is convex, we give a simple condition on γ such that H _γ,α,β is bounded on L ² when β > 3α, β > 0: As a corollary, under this condition, we obtain the L ^p -boundedness of H _γ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H _γ,α,β is bounded on L ²: As an application, we construct a class of strictly convex curves along which H _γ,α,β is bounded on L ² only if β > 2α > 0.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).