Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

Largest random component of ak-cube

LetC ^k denote the graph with vertices (? ₁, ...,? _k ),? _i =0,1 and vertices adjacent if they differ in exactly one coordinate. We callC ^k thek-cube. LetG=G _{k, p} denote the random subgraph ofC ^k defined by letting $$Prob(\{ i,j\} \in G) = p$$ for alli, j ∈ C ^k and letting these probabilities be mutually independent. We show that forp=λ/k, λ>1,G _{k, p} almost surely contains a connected component of sizec2 ^k ,c=c(λ). It is also true that the second largest component is of sizeo(2 ^k ).     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

On the hardness of sampling independent sets beyond the tree threshold

We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ^|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λ _c (d), where λ _c (d) is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the d-regular infinite tree. We show that for d ≥ 3, λ just above λ _c (d) with high probability over d-regular bipartite graphs, any local Markov chain Monte–Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for “replica” method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. A major theoretical challenge in recent years is to provide rigorous proofs for the correctness of such predictions. Our results establish such rigorous proofs for the case of hard-core model on bipartite graphs. We conjecture that λ _c is in fact the exact threshold for this computational problem, i.e., that for λ > λ _c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

An Inequality Between Dirichlet and Neumann Eigenvalues of the Heisenberg Laplacian

Let λ _k and μ _k be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in ? ^d , d > 1. Filonov has proved in a simple way that the inequality μ _k+1 < λ _k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions.     

Nonuniqueness of a Gibbs measure for the Ising ball model

We study a new model, the so-called Ising ball model on a Cayley tree of order k ≥ 2. We show that there exists a critical activity \(\lambda _{cr} = \sqrt[4]{{0.064}}\) such that at least one translation-invariant Gibbs measure exists for λ ≥ λ _cr , at least three translation-invariant Gibbs measures exist for 0 < λ < λ _cr , and for some λ, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor \(\hat G\) of index 2 of the group representation on the Cayley tree, we study \(\hat G\) -periodic Gibbs measures. We prove that there exists an uncountable set of \(\hat G\) -periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

On a parabolic equation in MEMS with fringing field

We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ _δ ^* > 0 such that if 0 < λ < λ _δ ^* , the equation has a global solution for some initial data; while for λ > λ _δ ^* , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Handcuffed designs

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parametersv, k, λ consists of a system of orderedk-subsets of av-set, called handcuffed blocks. In a block {A ₁,A ₂,?, A _k } each element is assumed to be handcuffed to its neighbours and the block containsk ? 1 handcuffed pairs (A ₁,A ₂), (A ₂,A ₃), ? (A _k?1,A _k ). These pairs are considered unordered. The collection of handcuffed blocks constitute a hundcuffed design if the following are satisfied: (1) each element of thev-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of thev-set are handcuffed in exactly λ of the blocks. If the total number of blocks isb and each element appears inr blocks the following conditions are necessary for the handcuffed design to exist:

1. λv(v?1) = (k?1) b,
2. rv = kb.

We denote byH(v, k, λ) the class of all handcuffed designs with parametersv, k, λ and sayH (v, k, λ) exists if there is a design with parametersv, k, λ. In this paper we prove that the necessary conditions forH (v, k, λ) exist are also sufficient in the following cases: (a)λ = 1 or 2; (b)k = 3; (c)k is evenk = 2h, and (λ, 2h ? 1) = 1; (d)k is odd,k = 2h + 1, and (λ, 4h)=2 or (λ, 4h)=1.     

Maximum two-dimensional （u × v, 4, 1, 3）-OOCs

Let Ф（u ×v, k, Aa, Ac） be the largest possible number of codewords among all two- dimensional （u ×v, k, λa, λc） optical orthogonal codes. A 2-D （u× v, k, λa, λ）-OOC with Ф（u× v, k, λa, λc） codewords is said to be maximum. In this paper, the number of codewords of a maximum 2-D （u × v, 4, 1, 3）-OOC has been determined.     

Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

LetP _n,k ^c denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[?1, 1], let $B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$ , andM _n,k ^* ?max{n(k+1),nlogn},M _n,k ?n(k+1). It is shown that $M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$ for everyx∈[?1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[?1,1] denotes the supremum norm on [?1,1]. This result should be compared with the inequalities $c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$ , for everyx∈[?1,1], wherec ₃>0 andc ₄>0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.     

Nontrivial expansions of zero and absolutely representing systems

В статье дается описа ние общего вида абсол ютно сходящегося в локаль но выпуклом пространствеH разлож ения нуля по системе ?_Λ:={e(λ_k)} _k=1 ^∞ , где ?λ∈CMe(λ) = λe(λ)M-линейный непрерывны й оператор вH иλ _k ∈C. При дополнительных пред положениях выясняет ся связь между наличием вH такого разложения нуля по системе ?_Λ и тем, что ?_Λ я вляется абсолютно представл яющей системой вH.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

Incidence graphs and subdesigns of semisymmetric designs

A semisymmetric design is a connected incidence structure satisfying; two points (blocks) are on 0 or λ blocks (points). Every block (point) is incident with k points (blocks). Properties of the incidence graph of these structures are investigated, leading to bounds on its diameter (d?k if λ = 2, d?[2k/(λ + 1)]+ 1 if λ > 2), and the number of points of these structures (υ?2^k-1 if λ = 2, υ?k2^{[2k/(λ + 1)]} if λ > 2). Bounds are also found for semisymmetric designs containing a subdesign. We give characterizations of semisymmetric designs with λ = 2 (semibiplanes) which contain a subdesign and achieve the bounds. This leads to a construction for a semibiplane with parameters υ = 2^r-1 (q²?1), k = q+q₁+?+q_r, where q_r is aprime power, q_i = q²_i+1 and q=q²₁.     

On the uniqueness of embedding a residual design

We prove that if a residual $\text{2-(k(k+λ?1)}\text{λ,k,λ)}$ design R has more than one embedding into a symmetric design then k ? λ(λ?1)². If equality holds then R has exactly two embeddings and the corresponding derived design is in both cases λ ? 1 identical copies of the design of points and lines of PG(3, λ ? 1). Using the main proposition from which these results follow we also prove that if a symmetric2-(v,k, λ) design has an axial non-central or central non-axial automorphism then k?λ(λ² ? 2λ + 2).     

Nontrivial solutions of a higher-order rational difference equation

We prove that, for every k ∈ ?, the following generalization of the Putnam difference equation $$ x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }} {{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 , $$ has a positive solution with the following asymptotics $$ x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } ) $$ for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ ^k+2 ? λ ? 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.     

Weighted tight frames of exponentials on a finite interval

Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL ²(I)→L _μ ² :f→f has finite codimension inL _μ ² , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.     

Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.