Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

The countability index of the endomorphism semigroup of a vector space

The countability index C(S) of a semigroup S is the least positive integer n, if such an integer exists, with the property that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists. C(S) is defined to be infinite. Let V be a vector space over a field F and denote by End V the endomorphism semigroup of V. In the two main results, it is determined precisely when C(End V)=2 and when C(End V)=x SpecificallyC(End V)=2 if and only if V is infinite dimensional or dim V=1 and F is finite and C(End V)=x if and only if F is infinite and dim V is an integer N≥1.     

Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T:X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ N:Tⁿ(x) ∈ V } has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅≠W(T) ∩ Tr(T) _≠^⊂W^*(T) ∩ Tr(T) _≠^⊂ QW(T) ∩ Tr(T) _≠^⊂ BR(T) ∩ Tr(T), in which W^*(T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W^*(T)\W(T) is residual in X. Moreover, we construct a point x ∈ BR\QW in symbol dynamical system, and demonstrate that the sets W(T), QW(T) and BR(T) of a dynamical system are all Borel sets.     

On mixtures of distributions of Markov chains

Let X be a chain with discrete state space I, and V be the matrix of entries V_i,n, where V_i,n denotes the position of the process immediately after the nth visit to i. We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V_i,n's are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.     

A note on convergence in Banach spaces of cotype p

Let B be a separable Banach space. The following is one of the results proved in this paper. The Banach space B is of cotype p if and only if

1. d_n, n 1, has no subsequence converging in probability, and

2. ∑_{n 1}|a_n|^p < ∞ whenever ∑_{n 1}a_nd_n converges almost surely are equivalent for every sequence d_n, n 1, of symmetric independent random elements taking values in B.

Chapter 3:inertia preservers

Let Vdenote either the space of n×n hermitian matrices or the space of n×nreal symmetric matrices, Given nonnegative integers r,s,t such that r+S+t=n, let G( r,s,r) denote the set of all matrices in V with inertia (r,s,t). We consider here linear operators on V which map G(r,s,t) into itself.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Polynomial identities from weighted lattice path counting

We give bijective proofs, using weighted lattice paths, of two multinomial identities concerning the generalized h-factorial polynomials of order n.

[x]ⁿ_h:=x(x + h)(x +2h)(x + (n − 1)h)

.

The first-one is the multinomial identity of order s verified by these polynomials. Using this identity (and its proof) as a lemma, we derive the main identity that generalizes previous results of Carlitz (1977), Egorychev (1974) and the classical identity of Banach.     

Chapter 4:algebraic sets, polynomials, and other functions

Let T be a linear operator on the vector space V ofn×n matrices over a field F. We discuss two types of problems in this chapter. First, what can we say about T if we assume that T maps a given algebraic set such as the special linear group into itself? Second, let p(x) be a polynomial function (such as det) on V into F. What can we say about T if Tpreserves p(x), i.e. p(T(X)) = p(X) for all X in V?     

On total covers of graphs

A total cover of a graph G is a subset of V(G)E(G) which covers all elements of V(G)E(G). The total covering number ₂(G) of a graph G is the minimum cardinality of a total cover in G. In [1], it is proven that ₂(G)[n/2] for a connected graph G of order n. Here we consider the extremal case and give some properties of connected graphs which have a total covering number [n/2]. We prove that such a graph with even order has a 1-factor and such a graph with odd order is factor-critical.     

Extending matchings in planar graphs V

A graph G on at least 2n + 2 vertices in n-extendable if every set of n independent edges extends to (i.e., is a subset of) a perfect matching in G. It is known that no planar graph is 3-extendable. In the present paper we continue to study 2-extendability in the plane. Suppose independent edges e₁ and e₂ are such that the removal of their endvertices leaves at least one odd component C_o. The subgraph G[V(C_o) V(e₁) V(e₂)] is called a generalized butterfly (or gbutterfly). Clearly, a 2-extendable graph can contain no gbutterfly. The converse, however, is false.

We improve upon a previous result by proving that if G is 4-connected, locally connected and planar with an even number of vertices and has no gbutterfly, it is 2-extendable. Sharpness with respect to the various hypotheses of this result is discussed.     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)∈F[x].LetxV → V be a linear operator. Let S_φbe the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.     

Bounds on a class of partial partitions of a vector space over gf(2):a graph theoretical approach

A collection Q of linearly independent w-suhicfs of the n-dimensional vector space V(n) over GF(2) is a w-quilt if whenever X and Y are distinct elements of Q, then X is disjoint from the linear span of Y. The main problem is to determine the maximum possibility cardinality of a w-quilt in V(n) for fixed w and n. Here a graph T(Q) is associated with each quilt Q. The connected components of T(Q) are shown to be complete graphs and the structure of the subquilts corresponding to these components is completely determined. By use of Ramsey type arguments these results are shown to lead to new upper bounds on the cardinality of a w-quilt in V(n) where n = w + 2, a case of particular interest.     

Extensions of n-Hom Lie algebras

n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Hom Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Hom Lie algebras. The concept of n_ρ-cocycle for an n-Hom Lie algebra (G, [,… , ], α) related to a G-module (V, ρ, β) is proposed, and a sufficient condition for the existence of the dual representation of an n-Hom Lie algebra is provided. From a G-module (V, ρ, β) and an n_ρ-cocycle θ, an n-Hom Lie algebra (T_θ(V ), [, … , ]θ, γ) is constructed on the vector space T_θ(V ) = G⊕V, which is called the T_θ-extension of an n-Hom Lie algebra (G, [, … , ], α) by the G-module (V, ρ, β).     

Irreducible components of L2 functions on hopf bundles

This paper provides an explicit decomposition of the L² function space on the unit sphereS^dn-1 for d = 1,2 and 4, into irreducible representations under the action of the Lie Groups K= SO(n) × SO(1)S(U(n) × U(1)), and Sp(n) × Sp(l), respectively. The decomposition is realized as the eigenspaces of the Laplacian acting on homogeneous polynomials over the reals, complex numbers and quaternions. For the quaternionic case, an additional differential operator that commutes with the Laplacian is used to find the decomposition.     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Circular permutation graph family with applications

In a circular permutation diagram, there are two sets of terminals on two concentric circles: C_in and C_out. Given a permutation Π = [π₁, π₂, …, π_n], terminal i on C_in and terminal π_i on C_out are connected by a wire. The intersection graph G_c of a circular permutation diagram D_c is called a circular permutation graph of a permutation Π corresponding to the diagram D_c. The set of all circular permutation graphs of a permutation Π is called the circular permutation graph family of permutation Π. In this paper, we propose the following: (1) an O(V + E) time algorithm to check if a labeled graph G = (V, E) is a labeled circular permutation graph. (2) An O(n log n + nt) time algorithm to find a maximum independent set of a family, where n = Π and t is the cardinality of the output. (Number t in the worst case is O(n). However, if Π is uniformly distributed (and independent from i), its expected value is O(√n).) (3) An O(min(δV_clog logV_c,V_clogV_c) + E_c) time algorithm for finding a maximum independent set of a circular permutation diagram D_c, where δ is the minimum degree of vertices in the intersection graph G_c = (V_c,E_c) of D_c. (4) An O(n log log n) time algorithm for finding a maximum clique and the chromatic number of a circular permutation diagram, where n is the number of wires in the diagram.