Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

Maximal Sidon sets and matroids

A subset X of an abelian group Γ, written additively, is a Sidon set of orderh if whenever {(a_i,m_i):i∈I} and {(b_j,n_j):j∈J} are multisets of size h with elements in X and ∑_i∈Im_ia_i=∑_j∈Jn_jb_j, then {(a_i,m_i):i∈I}={(b_j,n_j):j∈J}. The set X is a generalized Sidon set of order(h,k) if whenever two such multisets have the same sum, then their multiset intersection has size at least k. It is proved that if X is a generalized Sidon set of order (2h−1,h−1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

Frame cellular automata: Configurations, generating sets and related matroids

We introduce a frame cellular automaton as a broad generalization of an earlier study on quasigroup-defined cellular automata. It consists of a triple (F,R,E_F) where, for a given finite set X of cells, the frame F is a family of subsets of X (called elementary frames, denoted S_i, i=1,…,n), which is a cover of X. A matching configuration is a map which attributes to each cell a state in a finite set G under restriction of a set of local rules R={R_i∣i=1,…n}, where R_i holds in the elementary frame S_i and is determined by an (|S_i|-1)-ary quasigroup over G. The frame associated map E_F models how a matching configuration can be grown iteratively from a certain initial cell-set. General properties of frames and related matroids are investigated. A generating set S⊂X is a set of cells such that there is a bijection between the collection of matching configurations and G^S. It is shown that for certain frames, the algebraically defined generating sets are bases of a related geometric-combinatorially defined matroid.     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

On the packing chromatic number of some lattices

For a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V₁,V₂,…,V_m where V_i is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z³ do not have finite packing chromatic numbers.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

Simultaneous singular value decomposition

We consider the following problem: Given a set of m×n real (or complex) matrices A₁,…,A_N, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P^*A₁Q,…,P^*A_NQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota-Kanno-Kojima-Kojima and Maehara-Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.     

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, αi∉(−1/2,1/2), the appropriately defined Riesz transforms , j=1,2,…,d, are Calderón-Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α∈(−1/2,1/2), by means of a local Calderón-Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy-Riemann type equations and to recover the Riesz-Laguerre transforms on the boundary. The two specific values of α, (−1/2,…,−1/2) and (1/2,…,1/2), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.     

Using Lagrangians of hypergraphs to find non-jumping numbers(II)

Let r?2 be an integer. A real number α∈[0,1) is a jump for r if for any ε>0 and any integer m?r, any r-uniform graph with n>n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c=c(α)>0 does not depend on ε and m. A result of Erd?s, Stone and Simonovits implies that every α∈[0,1) is a jump for r=2. Erd?s asked whether the same is true for r?3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumping numbers for every r?3. However, there are a lot of unknowns on determining whether or not a number is a jump for r?3. In this paper, we find two infinite sequences of non-jumping numbers for r=4, and extend one of the results to every r?4. Our approach is still based on the approach developed by Frankl and Rödl.     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0.5−?) for any ?>0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0.5−?) for any ?>0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

Faulhaber’s theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflective functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials.     

A unique decomposition result for HT factors with torsion free core

We prove that II₁ factors M have a unique (up to unitary conjugacy) cross-product type decomposition around “core subfactors” N⊂M satisfying the property HT of [S. Popa, On a class of type II₁ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006) 809-899] and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lαi(Z²)?Fni, i=1,2, for Fni⊂SL(2,Z) free groups of rank ni and αj=e2πitj, tj∉Q, implies n₁=n₂.     

Optimal interleaving schemes for correcting two-dimensional cluster errors

We present an elementary theory of optimal interleaving schemes for correcting cluster errors in two-dimensional digital data. It is assumed that each data page contains a fixed number of, say n, codewords with each codeword consisting of m code symbols and capable of correcting a single random error (or erasure). The goal is to interleave the codewords in the m×n array such that different symbols from each codeword are separated as much as possible, and consequently, an arbitrary error burst with size up to t can be corrected for the largest possible value of t. We show that, for any given m, n, the maximum possible interleaving distance, or equivalently, the largest size of correctable error bursts in an m×n array, is given by if n?⌈m²/2⌉, and t=m+⌊(n-⌈m²/2⌉)/m⌋ if n?⌈m²/2⌉. Furthermore, we develop a simple cyclic shifting algorithm that can provide a systematic construction of an m×n optimal interleaving array for arbitrary m and n. This extends important earlier work on the complementary problem of constructing interleaving arrays that, given the burst size t, minimize the interleaving degree, that is, the number of different codewords in a 2-D (or 3-D) array such that any error burst with given size t can be corrected. Our interleaving scheme thus provides the maximum burst error correcting power without requiring prior knowledge of the size or shape of an error burst.     

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.