The Picard iteration and its application

The convergence behavior of the Picard iteration X_k+1=AX_k+B and the weighted case Y_k=X_k/b_k is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {D^NT^N }.     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Generalized discrepancies on the sphere

Generalized discrepancies are a class of discrepancies introduced in the seminal paper [1] to measure uniformity of points over the unit sphere in ℝ³. However, convergence to 0 of this quantity has been shown only in the case of spherical t –designs. In the following, we completely characterize sequences for which convergence to 0 of D (𝒫_N ; A ) holds. The interest of this result is that, when evaluating uniformity on the sphere, generalized discrepancies are much simpler to compute than the well-known spherical cap discrepancy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large bipartite graphs with given degree and diameter

We give constructions of bipartite graphs with maximum Δ, diameter D on B vertices, such that for every D ≥ 2 the lim inf_Δ→∞B. Δ^1-D = b_D > 0. We also improve similar results on ordinary graphs, for example, we prove that lim_Δ→∞N · Δ^?D = 1 if D is 3 or 5. This is a partial answer to a problem of Bollobás.     

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.     

Distances in random graphs with finite variance degrees

In this paper we study a random graph with N nodes, where node j has degree D_j and {D_j} are i.i.d. with ?(D_j ≤ x) = F(x). We assume that 1 ? F(x) ≤ cx^?τ+1 for some τ > 3 and some constant c > 0. This graph model is a variant of the so‐called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log_ν N, when the base of the logarithm equals ν = ??[D_j(D_j ? 1)]/??[D_j] > 1. This confirms the heuristic argument of Newman, Strogatz, and Watts [Phys Rev E 64 (2002), 026118, 1–17]. In addition, the random fluctuations around this asymptotic mean log_ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Characterization and Transformation of Invariants

We consider difference equations of order k _n+k ≥ 2 of the form: y_n+k = f(yn,…,yn+k-1), n= 0,1,2,… where f: D ^k → D is a continuous function, and D?R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x ₁,…,x_k ) ∈C^∞[D^k,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.     

Approximate D-optimal designs of experiments on the convex hull of a finite set of information matrices

In the paper we solve the problem of D _ℋ-optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D _ℋ-optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D _ℋ-optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D _ℋ-optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.     

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, V _D ^m (r) the volume of a small geodesic ball for the connection D and C C _D ¹ the first non-trivial term of the Taylor expansion of V _D ^m (r). NK-manifolds are characterized in terms of C C _D ¹ and a family of Hermitian manifolds for which _M C C _D ¹ dvol is a spectral invariant is given and one proves that C C _D ¹ and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.     

Approximation of the lauricella hypergeometric functionsF D (N) by branched continued fractions

Applying recursion relations for the Lauricella hypergeometric functions F _D ^Nl , we construct an expansion of a ratio of these functions in branched continued fractions. We study the convergence of the resulting expansion in the case of real parameters. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 70–74.     

Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

We consider a second‐order differential operator A( x )=?∑?_ia_ij( x )?_j+ ∑?_j(b_j( x )·)+c( x ) on ?^d, on a bounded domain D with Dirichlet boundary conditions on ?D, under mild assumptions on the coefficients of the diffusion tensor a_ij. The object is to construct monotone numerical schemes to approximate the solution of the problem A( x )u( x )=µ( x ), x ∈D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness introducing function spaces needed to prove convergence results. Then, we define non‐standard stencils on grid‐knots that lead to extended discretization schemes by matrices possessing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W‐convergence in detail, and mention convergence in C and L₁ spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley & Sons, Ltd.     

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998     

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D _?1 → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, ?) and Hom(?, 𝒞_𝒲) exact. In this article, we show that C = … → C ^?1 → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries.     

Holomorphic Dependence of the Heins Map

Let D be a bounded convex domain and Hol _c (D,D) the set of holomorphic maps from D to C ⁿ with image relatively compact in D. Consider Hol _c (D,D) as a open set in the complex Banach space H ^∞ _n (D) of bounded holomorphic maps from D to C ⁿ . We show that the map τ: Hol _c (D,D) → D (called the Heins map for D equals to the unit disc of C) which associates to ? ∈ Hol _c (D,D) its unique fixed point τ? ∈ D is holomorphic and its differential is given by dτ_?(v) = (Id-df_τ(?))^?1 v(τ(?)) for v ∈ H ^∞ _n (D).     

Recognition by prime graph of $^2 D_{2^m + 1} (3)$

As shown in [1] the simple group ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(² D_{2^m + 1} (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ ² D_{2^m + 1} (3)G \cong ^2 D_{2^m + 1} (3).