Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

A compact formulation of a mixed-integer set

We consider mixed-integer sets of the form X = {(s, y) ∈ ?₊ × ? ⁿ : s + a _j y _j ≥ b _j , ?j ∈ N}. A polyhedral characterization for the case where X is defined by two inequalities is provided. From that characterization we give a compact formulation for the particular case where the coefficients of the integer variables can take only two possible integer values a ₁, a ₂ ∈ ?₊ : X _n,m = {(s, y) ∈ ?₊ × ? ^n+m : s + a ₁ y _j ≥ b _j , ?j ∈ N ₁, s + a ₂ y _j ≥ b _j , j ∈ N ₂} where N ₁ = {1, …, n}, N ₂ = {n + 1, …, n + m}. In addition, we discuss families of facet-defining inequalities for the convex hull of X _n,m .     

Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

Richardson's iteration for nonsymmetric matrices

To solve the linear N×N system (1) Ax=a for any nonsingular matrix A, Richardson's iteration (2) x_j+1=x_j-α_j(Ax_j-a), j=1,2,…,n, which is applied in a cyclic manner with cycle length n is investigated, where the α_j are free parameters. The objective is to minimize the error |x_n+1-x|, where x is the solution of (1). If the spectrum of A is known to lie in a compact set S, one is led to the Chebyshev-type approximation problem (3) min_p-1∈Vnmax_z∈S|p(z)|, where V_n is the linear span of z,z²,…,zⁿ. If p solves (3), then the reciprocals of the zeros of p are optimal iteration parameters α_j. It is shown that for a real problem (1) the iteration (2) can be carried out with real arithmetic alone, even when there are complex α_j. The stationary case n=1 is solved completely, i.e., for all compact sets S the problem (3) is solved explicitly. As a consequence, the converging stationary iteration processes can be characterized. For arbitrary closed disks S the problem (3) can be solved for all n∈$\text{N}$, and a simple proof is provided. The lemniscates associated with S are introduced. They appear as an important tool for studying the stability of the iteration (2).     

Disjoint skolem sequences and related disjoint structures

A Skolem sequence of order n is a sequence S = (s₁, s₂…, s²ⁿ) of 2n integers satisfying the following conditions: (1) for every k ∈ {1, 2,… n} there exist exactly two elements s_i,S_j such that S_i = S_j = k; (2) If s_i = s_j = k,i < j then j ? i = k. In this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. © 1993 John Wiley & Sons, Inc.     

Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

On convergence of LAD estimates in autoregression with infinite variance

The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a₁, …, a_q)′ for a stationary autoregressive model, are shown to have the property that N^σ(L(N) ? a) converge to zero in probability, for $\text{σ <}\text{1}\text{α}$, where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions.     

Estimates for the convergence rate in the central limit theorem for randomized separable statistics in a multinomial scheme

Assume that there are given random vectors ¯v₁, ..., ¯v_s, independent in their totality, where v_j=(v_1j, ... v_Nj) has the multinomial distribution M(nj, p1j, ..., P_Nj,. Assume further that _jm ^(N) (x₁, ..., x_s), j=1–k, k1 are random functions of s nonnegative integer arguments x₁, ..., x_s. One considers the multidimensional randomized separable statistic S_N=(S_N1, ..., S_Nk), where. One obtains estimates for the rate of convergence in the central limit theorem for S_N.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 315–337, 1986.     

Characters of representations of the quantum toroidal algebra : Plane partitions with “stands”

An expression for the generating function of plane partitions a _i,j subject to the constraints a _m,n = 0 and a _i,j ? k _j , 1 ? j ? n, which is the character of an irreducible representation of the quantum toroidal algebra , is obtained.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.     

Linear independence of values of Lerch functions

The number of linearly independent numbers among 1, Φ₁ (z, p/q), ...,Φ _a (z, p/q) is estimated depending on a natural number a, where Φ _s (z, p/q), s = 1, 2, ..., are Lerch functions.     

On the equivalence of the Euler-Pommier operators in spaces of analytic functions

In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L ₁=α _n z ⁿ Δ ⁿ + ... + α₁ zΔ+α₀ E and L ₂= z ⁿ a _n (z)Δ ⁿ + ... + za ₁(z)Δ+a ₀(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α _n ∈ ?, a _k (z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σ _j=s ^n?1 α _j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z ^s+1Δ+β(z)E, β(z) ∈ A _R , s ∈ ?, and z ^s+1 are equivalent in the spaces A _R, 0?R?-∞, if and only if β(z) = 0.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Canonical row-column-exchangeable arrays

Consider a standard row-column-exchangeable array X = (X_ij : i,j ≥ 1), i.e., X_ij = f(a, ξ_i, η_j, λ_ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′₁, ξ′₂,…, η′₁, η′₂,…, are conditionally independent given ∩_{n ≥ 1}σ(X′_ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.     

On the subpartitions of the ordinary partitions

Let a ₁≥a ₂≥???≥a _? be an ordinary partition. A subpartition with gap d of an ordinary partition is defined as the longest sequence satisfying a ₁>a ₂>???>a _s and a _s >a _s+1, where a _i ?a _j ≥d for all i<j≤s. This is a generalization of the Rogers–Ramanujan subpartition which was introduced by L. Kolitsch. In this note, we will study various properties of subpartitions, and as an application we will give a combinatorial proof of two entries which are in Ramanujan’s lost notebook.     

On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.     

Stability of solutions of functional equations connected with problems of characterizing probability distributions

The work contains some results pertaining to the solution ψ_j(x) of the functional equation $$\left| {\Sigma \Psi _j \left( {a_j^T t} \right)} \right| \leqslant \varepsilon ,$$ where a _j ^T =(a_1j, a_2j, ..., a_pj)∈ ?^p, all the coefficients a_ij are constant, t=(t₁, t₂, ..., t_p) ∈ ?^p, \(a_j^T t = \sum\limits_{i - 1}^p {a_{ij} t_i } ,p \geqslant 2\) and the relation (*) is satisfied for all Inequality (*) is connected with certain characterization theorems of probability theory and statistics. For simplicity, it is assumed that the ψ_j(x) are continuous functions, x∈?¹. The following basic ressult is obtained.     

Representations of lattices via neutral elements

For any neutral element a in a bounded latticeL, the mapping x→(x∧,x∨a) representsL as a subdirect product of [0, a]×[a, 1]. It is first shown that for certain neutral elements, the image ofL under this mapping is completely determined by a homomorphism of [0, a] into [a, 1]. Iterating this process, a representation ofL can be obtained as a subdirect product of the intervals [a_i, a_i+1] for any chain 0=a₀1... nn+1=1 where each a_i is such a neutral element relative to [0, a_i+1]. The image in this case is completely determined by a family of homomorphisms π_i,j:A_i →A_j(ii=[a_i, a_i+1].