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.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Turán’s theorem inverted

In this note we complete an investigation started by Erd?s in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán’s theorem in extremal graph theory.Let be the complete r-partite graph with parts of sizes s₁≥2,s₂,…,s_r with an edge added to the first part. Letting t_r(n) be the number of edges of the r-partite Turán graph of order n, we prove that:For all r≥2 and all sufficiently small c>0, every graph of sufficiently large order n with t_r(n)+1 edges contains a .We also give a corresponding stability theorem and two supporting results of wider scope.     

A generalization of a conjecture due to Erd?s, Jacobson and Lehel

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is r-graphic if it is realizable by an r-graph on n vertices. Let be the smallest even integer such that each n-term r-graphic sequence with term sum of at least is realizable by an r-graph containing as a subgraph. In this paper, we determine the value of for sufficiently large n, which generalizes a conjecture due to Erd?s, Jacobson and Lehel.     

On the weak-approximate fixed point property

Let X be a Banach space and C a bounded, closed, convex subset of X. C is said to have the weak-approximate fixed point property if for any norm-continuous mapping , there exists a sequence {xn} in C such that (xn−fn(xn)) converges to 0 weakly. It is known that every infinite-dimensional Banach space with the Schur property does not have the weak-approximate fixed point property. In this article, we show that every Asplund space has the weak-approximate fixed point property. Applications to the asymptotic fixed point theory are given.     

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 .     

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.     

Some properties of rapidly varying sequences

We investigate the class R∞,s of rapidly varying sequences from different points of view. It is shown: (1) a sequence (cn)_n∈N of positive real numbers is rapidly varying if and only if the corresponding function x→c[x], x?1, is rapidly varying; (2) the class R∞,s satisfies some selection properties, as well as game-theoretical and Ramsey-theoretical conditions.     

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.     

Graphic sequences with a realization containing a generalized friendship graph

Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451-460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂+?+d_n≥σ(H,n) has a realization G containing H as a subgraph. Let F_t,r,k denote the generalized friendship graph on kt−kr+r vertices, that is, the graph of k copies of K_t meeting in a common r set, where K_t is the complete graph on t vertices and 0≤r≤t. In this paper, we determine σ(F_t,r,k,n) for k≥2, t≥3, 1≤r≤t−2 and n sufficiently large.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Degree-bounded factorizations of bipartite multigraphs and of pseudographs

For d≥1, s≥0 a (d,d+s)-graph is a graph whose degrees all lie in the interval {d,d+1,…,d+s}. For r≥1, a≥0 an (r,r+a)-factor of a graph G is a spanning (r,r+a)-subgraph of G. An (r,r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r,r+a)-factors. A graph is (r,r+a)-factorable if it has an (r,r+a)-factorization.We prove a number of results about (r,r+a)-factorizations of (d,d+s)-bipartite multigraphs and of (d,d+s)-pseudographs (multigraphs with loops permitted). For example, for t≥1 let β(r,s,a,t) be the least integer such that, if d≥β(r,s,a,t) then every (d,d+s)-bipartite multigraph G is (r,r+a)-factorable with x(r,r+a)-factors for at least t different values of x. Then we show that

     

Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s₁+s₂,s_i∈S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g?3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Kolountzakis’ idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo, Ruzsa and Trujillo, Jia, and Habsieger and Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.     

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.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H_D,n, motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H_D,n’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D) for a non-equilibrium qBp, by means of the Wigner function W, in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W_eq,D, which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W_eq,D is chosen to be the weight function which orthogonalizes the H_D,n’s. New results on W_eq,D and on the H_D,n’s are reported. We justify the key role of the H_D,n’s as follows. Using the H_D,n’s, moments W_eq,D,n and W_n are introduced for W_eq,D and W, respectively. At equilibrium, all moments W_eq,D,n except the lowest one (W_eq,D,0) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W_n (except the lowest moment W₀), will approach zero, while (ii) the lowest non-equilibrium moment W₀ will tend to W_eq,D,0(≠0). To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W₀ for the different models (D) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.     

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.     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.