Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

The sharp threshold for percolation on expander graphs

We consider a random subgraph G _n (p) of a finite graph family G _n = (V _n , E _n ) formed by retaining each edge of G _n independently with probability p. We show that if G _n is an expander graph with vertices of bounded degree, then for any c _n ∈ (0, 1) satisfying $c_n \gg {1 \mathord{\left/ {\vphantom {1 {\sqrt {\ln n} }}} \right. \kern-0em} {\sqrt {\ln n} }}$ and $\mathop {\lim \sup }\limits_{n \to \infty } c_n < 1$ , the property that the random subgraph contains a giant component of order c _n |V _n | has a sharp threshold.     

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT _t (n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α _t (G) ≤ f(n), where α _t (G) is the maximum number of vertices in a K _t -free induced subgraph of G. Erd?s, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT _t (n,K _s , o(n)), among them finding the minimum ? such that RT _t (n,K _t+? , o(n)) = Ω(n ²), where it is easy to see that RT _t (n,K _t+1, o(n)) = o(n ²). In this paper, we answer this question by proving that RT _t (n,K _t+2, o(n)) = Ω(n ²); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.     

Range of cube-indexed random walk

For given finite, connected, bipartite graphG=(V,E) with distinguishedν ₀ ∈V, set {fx189-1} Our main result says there is a fixedb so that whenG is a Hamming cube ({0, 1} ⁿ with the usual adjacency), andf is chosen uniformly fromF, the probability thatf takes more thanb values is at most e^?Ω(n). this settles in a very strong way a conjecture of I. Benjamini, O. Häggström and E. Mossel [2]. The proof is based on entropy considerations and a new log-concavity result.     

Hamiltonian cycles in Dirac graphs

We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$exp_2 [2h(G) - n\log e - o(n)],$

where h(G)=maxΣ _e x _e log(1/x _e ), the maximum over x: E → ?⁺ satisfying Σ _e?υ x _e = 1 for each υ ∈ V, and log =log₂. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) ⁿ . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) ⁿ , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.     

Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L ² (?π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (?∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L ² (?π, π) andE ² . When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities $$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$ hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ?^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(?∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence. Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].     

Constant-approximation for optimal data aggregation with physical interference

In this paper, we study the aggregation problem with power control under the physical interference. The maximum power is bounded. The goal is to assign power to nodes and schedule transmissions toward the sink without physical interferences such that the total number of time slots is minimized. Under the assumption that the unit disk graph G _{δ r} with transmission range δ r is connected for some constant 0 < δ ≤ 1/3^1/α , where r is the maximum transmission range determined by the maximum power, an approximation algorithm is presented with at most b ³(log₂ n + 6) + (R?1)(μ ₁ + μ ₂) time slots, where n is the number of nodes, R is the radius of graph G _{δ r} with respect to the sink, and b, μ ₁, μ ₂ are constants. Since both R and log₂ n are lower bounds for the optimal latency of aggregation in the unit disk graph G _{δ r} , our algorithm has a constant-approximation ratio for the aggregation problem in G _{δ r} .     

An example of asymmetry of convolution operators

Esistono un gruppo compatto non commutativoG ^～ ed un operatore di convoluzioneT tale che: perp∈[2,4] e perq∈[1,2),T∈L _p ^p (G ^～) eT?L _q ^q (G ^～).     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

Invariant random graphs with iid degrees in a general geography

Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287–298) and Deijfen and Jonasson (Electron Comm Probab 11:336–346), we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e., the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if ${\mathbb{E}[D\,R(D)] < \infty}$ . Here R(n) is the smallest possible radius of a combinatorial ball containing more than n vertices. With rapid growth we mean that the number of vertices in a ball of radius n is of at least order exp(n ^c ) for some c > 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for ${G\,=\,\mathbb {Z}}$ in [4] and proves, e.g., that with ${G\,=\,\mathbb {Z}^d}$ , there exists an invariant model with finite mean connections if and only if ${\mathbb {E}[D^{(d+1)/d}] < \infty}$ . When G has exponential growth, e.g., when G is a regular tree, the condition becomes ${\mathbb {E}[D\,\log\,D] < \infty}$ .     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

On bounds for solutions to nonlinear wave equations in Hilbert space with applications to nonlinear elastodynamics

For the nonlinear wave equationu _tt -Nu +G(t,u, u _t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ ₀ ^T ∥G(τ,u, u _τ)∥² dτ ≤ κ ∫ ₀ ^T ∥?(τ)∥² dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented.     

Treewidth computation and extremal combinatorics

For a given graph G and integers b,f ≥0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most $n\left( {_b^{b + f} } \right)$ such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G

1. compute the treewidth of G in time O(1.7549 ⁿ ) by making use of exponential space and in time O(2.6151 ⁿ ) and polynomial space
2. decide in time O(n ⁵· $_{k + 2}^{\left\lceil {(2n + k + 8)/3} \right\rceil } $ ) if the treewidth of G is at most k
3. list all minimal separators of G in time O(1.6181 ⁿ ) and all potential maximal cliques of G in time O(1.7549 ⁿ ).

This significantly improves previous algorithms for these problems.     

Two remarks on normality preserving Borel automorphisms of \boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}

Let T be a bijective map on ? ⁿ such that both T and T ^???1 are Borel measurable. For any θ?∈?? ⁿ and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? ⁿ with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ _j ???θ ₀, 1?≤?j?≤?n } is a basis for ? ⁿ . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε _j ?>???1 for every j and {u _j , 1?≤?j?≤?n } is a basis of unit vectors in ? ⁿ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u _j . Suppose N(0, I)T ^???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 ⁿ diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E _s } is a collection of 2 ⁿ Borel subsets of ? ⁿ such that { E _s } and {V s U (E _s )} are partitions of ? ⁿ modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E _s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).     

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Characterizations of Strength Extremal Graphs

With graphs considered as natural models for many network design problems, edge connectivity κ′(G) and maximum number of edge-disjoint spanning trees τ(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G (see Cunningham, in J ACM 32:549–561, 1985; Matula, in Proceedings of 28th Symposium Foundations of Computer Science, pp 249–251, 1987, among others). Mader (Math Ann 191:21–28, 1971) and Matula (J Appl Math 22:459–480, 1972) introduced the maximum subgraph edge connectivity \({\overline{\kappa'}(G) = {\rm max} \{\kappa'(H) : H {\rm is} \, {\rm a} \, {\rm subgraph} \, {\rm of} G \}}\) . Motivated by their applications in network design and by the established inequalities $$\overline{\kappa'}(G) \ge \kappa'(G) \ge \tau(G),$$ we present the following in this paper:

1. For each integer k > 0, a characterization for graphs G with the property that \({\overline{\kappa'}(G) \le k}\) but for any edge e not in G, \({\overline{\kappa'}(G + e) \ge k+1}\) .
2. For any integer n > 0, a characterization for graphs G with |V(G)| = n such that κ′(G) = τ(G) with |E(G)| minimized.

     

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

For a symmetric bounded measurable function W on [0, 1]² and a simple graph F, the homomorphism density $t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G _n ) of dense graphs is said to be convergent if the probability, t(F, G _n ), that a random map from V(F) into V(G _n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]². Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

On conjugate characteristic polynomial of a graph

Let $c=a+b\sqrt{m}$ and $\overline{c}=a-b\sqrt{m}$ , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A ^c =[c _ij ] is the conjugate adjacency matrix of a graph G if c _ij =c for any two adjacent vertices i and j, $c_{ij}=\overline{c}$ for any two nonadjacent vertices i and j, and c _ij =0 if i=j. Let P _G ^c (λ)=|λ I?A ^c | denote the conjugate characteristic polynomial of G. Further, let e=e(G) and Δ=Δ(G) be the number of edges and number of triangles of G, respectively. Let G and H be two graphs of order n and let e(G)=e(H). In this work we prove that c ₃(G)=c ₃(H) if and only if Δ(G)=Δ(H) and $\Delta(\overline{G})=\Delta(\overline{H})$ , where $\overline{G}$ denotes the complement of G and c _k is the coefficient which corresponds to λ ^n?k with respect to P _G ^c (λ). Besides, we here give the conjugate spectrum and conjugate characteristic polynomial of all connected graphs of order n=2,3,4,5, with respect to the constant $c=1+\sqrt{2}$ .