Weak convergence with random indices

Suppose {X_nn?-0} are random variables such that for normalizing constants a_n>0, b_n, n?0 we have Y_n(·)=(X[_{n, ·}]-b_n/a_n ? Y(·) in D(0.∞) . Then a_n and b_n must in specific ways and the process Y possesses a scaling property. If {N_n} are positive integer valued random variables we discuss when Y_Nn → Y and Y'_n=(X[_Nn]-b_n)/a_n ? Y'. Results given subsume random index limit theorems for convergence to Brownian motion, stable processes and extremal processes.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Optimum super-simple mixed covering arrays of type <Emphasis Type="Italic">a</Emphasis><Superscript>1</Superscript><Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>−1</Superscript>

A mixed covering array (MCA) of type (_v 1, v ₂,..., v _k ), denoted by MCA_λ (N; t, k, (v ₁, v ₂,..., v _k )), is an N × k array with entries in the i-th column from a set _{V i} of v _i symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ i ≤ k. An MCA _λ (N; t, k, (_v 1, v ₂,..., v _k )) is said to be super-simple, if each of its N × (t + 1) sub-arrays contains each (t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type (a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type (a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type (a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA₂(2b ²; 2, 4, (a, b, b, b))′s exist, is completely determined, where 2 ≤ a ≤ b.     

On the additive completion of sets of integers

Denote by k = k(N) the least integer for which there exists integers b₁, b₂, …, b_k satisfying 0 ≤ b₁ ≤ b₂ ≤ … ≤ b_k ≤ N such that every integer in |1, N| can be written in the form i² + b_j. It is shown that for all sufficiently large N, k ≥ (1.147)√N.     

When does the zero-one <Emphasis Type="Italic">k</Emphasis>-law fail?

The limit probabilities of the first-order properties of a random graph in the Erd?s–Rényi model G(n, n^?α), α ∈ (0, 1), are studied. A random graph G(n, n^?α) is said to obey the zero-one k-law if, given any property expressed by a formula of quantifier depth at most k, the probability of this property tends to either 0 or 1. As is known, for α = 1? 1/(2^k?1 + a/b), where a > 2^k?1, the zero-one k-law holds. Moreover, this law does not hold for b = 1 and a ≤ 2^k?1 ? 2. It is proved that the k-law also fails for b > 1 and a ≤ 2^k?1 ? (b + 1)².     

Bernstein operators for extended Chebyshev systems

Let U_n ⊂ Cⁿ[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f₀ ∈ U_n is strictly positive and f₁ ∈ U_n has the property that f₁/f₀ is strictly increasing. We search for conditions ensuring the existence of points t₀, …, t_n ∈ [a, b] and positive coefficients α₀, …, α_n such that for all f ∈ C[a, b], the operator B_n:C[a, b] → U_n defined by satisfies B_nf₀ = f₀ and B_nf₁ = f₁. Here it is assumed that p_n,k, k = 0, …, n, is a Bernstein basis, defined by the property that each p_n,k has a zero of order k at a and a zero of order n − k at b.     

The Product of Like-Indexed Terms in Binary Recurrences

In recent work by Hajdu and Szalay, Diophantine equations of the form (a^k−1)(b^k−1)=x² were completely solved for a few pairs (a, b). In this paper, a general finiteness theorem for equations of the form u_kv_k=xⁿ is proved, where u_k and v_k are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type (a^k−1)(b^k−1)=x² is described, and this approach is used to completely solve such equations for almost all (a,b) in the range 1<a<b?100. In the final section of this paper, it is shown that the abc conjecture implies much stronger results on these types of Diophantine problems.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

Extended Jacobson density theorem for rings with derivations and automorphisms

A deterministic version of the Itô calculus is presented. We consider a modelY _t=H(N _t ,t) with a deterministic Brownian N _t and an unknown functionH. We predictY _c from the observation {Y _t;t ∈ [a, b]}, wherea. We prove that there exists an estimatorY _t based on the observation such thatE[(? _t?Y _c)²]=O((c?b)²) asc ↓b.     

Completely strong path-connected tournaments

Let T = (V, A) be a tournament with p vertices. T is called completely strong path-connected if for each arc (a, b) ∈ A and k (k = 2, 3,…, p), there is a path from b to a of length k (denoted by P_k(a, b)) and a path from a to b of length k (denoted by P′_k(a, b)). In this paper, we prove that T is completely strong path-connected if and only if for each arc (a, b) ∈ A, there exist P₂(a, b), P′₂(a, b) in T, and T satisfies one of the following conditions: (a) T/T₀-type graph, (b) T is 2-connected, (c) for each arc (a, b) ∈ A, there exists a P′_p?1(a, b) in T.     

Revisiting queueing output processes: a point process viewpoint

After some historical notes concerning queueing output processes N _dep??, the paper discusses methods for establishing asymptotic linear relations for var??N _dep??(0,t], whether in the crude form B ₁ t or the more detailed form B ₁ t+B ₀+o(1) for t→∞. The crude form holds whenever the process N _adm of customers admitted to service has a linear asymptote, and then (var??N _dep??(0,t])/t and (var??N _adm(0,t])/t share a common limit (that may be infinite) in stationary G/G/k/K systems. A standard integral formula for the variance of a stationary orderly point process shows that, if N _dep?? is a renewal process whose generic lifetime X has finite second moment, then B ₁=(var??X)/([E(X)]²), and the more detailed linear asymptote holds when E(X ³) is finite. Geometric ergodicity of the queue size process Q(?) in stationary M/M/k/K systems establishes that the more detailed linear asymptote is true for them. It is conjectured that var??N(0,t]～B ₁ t for any stationary point process N possessing an embedded regenerative structure.     

The non-gap sequence of a subcode of a generalized Reed–Solomon code

This paper addresses the question how often the square code of an arbitrary l-dimensional subcode of the code GRS _k (a, b) is exactly the code GRS_2k-1(a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first used and stated by Wieschebrink where he applied the Sidelnikov–Shestakov attack to break the Berger–Loidreau cryptosystem.     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

The fundamental constituents of iteration digraphs of finite commutative rings

For a finite commutative ring R and a positive integer k ? 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a ^k . Let R = R ₁ ⊕ … ⊕ R _s , where s > 1 and R _i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R ₁, …, R _s } (it is possible that N is the empty set \(\not 0\) ). We define the fundamental constituents G _N ^* (R, k) of G(R, k) induced by the vertices which are of the form {(a ₁, …, a _s ) ∈ R: a _i ∈ D(R _i ) if R _i ∈ N, otherwise a _i ∈ U(R _i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* _N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.     

Resolvable balanced bipartite designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets, B¹ and B², where ∣Bⁱ∣ = k_i, i = 1 or 2. Two elements are said to be linked in B if and only if they belong to different subsets of B. A balanced bipartite design, BBD(v, k₁, k₂, λ), is an arrangement of v elements into b blocks, each containing k elements such that each element occurs in exactly r blocks and any two distinct elements are linked in exactly λ blocks. A resolvable balanced bipartite design, RBBD(v, k₁, k₂, λ), is a BBD(v, k₁, k₂, λ), the b blocks of which can be divided into r sets which are called complete replications, such that each complete replication contains all the v elements of the design.Necessary conditions for the existence of RBBD(v, 1, k₂, λ) and RBBD(v, n, n, λ) are obtained and it is shown that some of the conditions are also sufficient. In particular, necessary and sufficient conditions for the existence of RBBD(v, 1, k₂, λ), where k₂ is odd or equal to two, and of RBBD(v, n, n, λ), where n is even and 2n ? 1 is a prime power, are given.     

The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let B_k denote the subset of B determined by the designs with block size k. We present some information about B and the sets B_k. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k^′ such that the equation b^′=b+h has infinitely many solutions b,b^′ satisfying bB_k and b^′B_k^′. The study is restricted to the case λ=1.     

Balanced bipartite weighing designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets B¹ and B², where |Bⁱ| = k_i, i = 1 or 2. Two elements of B are said to be linked or n-linked in B if and only if they belong to different subsets or the same subset of B respectively. A balanced bipartite weighing design, (briefly BBWD (υ, k₁, k₂, λ₁)) is an arrangement of υ elements into b blocks, each containing k elements, such that each element occurs in exactly r blocks, any two distinct elements are linked in exactly λ₁ blocks and n-linked in exactly λ₂ blocks.Given fixed k₁ and k₂, there is always a minimal value of λ₁ such that the necessary conditions for the existence of a BBWD are satisfied for same υ. It is proved that in many cases, the necessary conditions are also sufficient. Some general methods for constructing BBWD's as well as a table of all designs with υ ? 13 are obtained.