A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ^*. Define λ_c=1/(k?1)×(k/(k?1)) ^k . Then: (i) for λ≤λ_c, the Gibbs measure is unique (and coincides with the above measure μ^*), (ii) for λ>λ_c, in addition to μ^*, there exist two distinct translation-periodic measures, μ₊and μ_?, taken to each other by the unit space shift. Measures μ₊and μ_?are extreme ?λ>λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ^*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_eand λ_o, for even and odd sites. We discuss open problems and state several related conjectures.     

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.     

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.     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

On the maximum number of permutations with given maximal or minimal distance

Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number $\text{M}$ such that there exist $\text{M}$ different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(p^m, ? 2) = (p^m ? 2)!, R(p^m + 1, ? 3) = (p^m ? 2)!, $\text{R(k, ? k ? 3) =}\text{k!}\text{2}$, R(k, ? 0) = k, R(p^m, ? 1) = p^m(p^m ? 1), R(p^m + 1, ? 2) = (p^m + 1)p^m(p^m ? 1). The exact value of R(k, ? λ) is determined whenever k ? k₀(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k₀(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for $\text{λ = (}\text{k}\text{2}\text{) + O(}\text{k}\text{log}\text{k}\text{)}$.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Minimal cogenerators need not be unique

A_n = A_n(?) denotes the unique left distributive binary system on {0, 1,…,2ⁿ?1) that satisfies a ? 1 = a + 1 mod 2ⁿfor all a ? A_n, and o_n(a) = k indicates the period 2^kof a ? A_n (if b,c ? A_n, then a ? b = a ? c iff b ‵ c mod 2^kand if 0 < b < c < 2^kthen a < a ? b < a ? c). Amongs others, we prove that o_t2(2^t?2) ≤ t holds for every integer t ≥ 2, the equality taking place iff t is of the form 2^2? for an integer s ≥ 0.     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

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.     

The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For k ≥ 2, the k-generalized Fibonacci sequence (F _n ^(k) ) _n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F _m ^(k) = F _n ^(?) , with ? > k > 1, n > ? + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

Forbidding just one intersection

Following a conjecture of P. Erdös, we show that if $\text{F}$ is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ? 2l + 2 and n sufficiently large |$\text{F}$| ? (_{k ? l ? 1}^{n ? l ? 1}) with equality holding if and only if $\text{F}$ consists of all the k-sets containing a fixed (l + 1)-set. In general we show |$\text{F}$| ? d_kn^{max;{;l,k ? l ? 1};}, where d_k is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3).     

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.     

Conditional limit theorems for exponential families and finite versions of de Finetti's theorem

Consider an exponential familyP _λ which is maximal, smooth, and has uniformly bounded standardized fourth moments. Consider a sequenceX ₁,X ₂,... of i.i.d. random variables with parameter λ. LetQ _nsk be the law ofX ₁,...,X _k given thatS _n=X ₁+...+X _n=s. Choose λ so thatE _λ(X ₁)=s/n. Ifk andn→∞ butk/n→0, then $$\parallel Q_{nsk} - P_\lambda ^k \parallel = \gamma \frac{k}{n} + o\left( {\frac{k}{n}} \right)$$ where γ=1/2E{|1?Z ²|} andZ isN(0,1). The error term is uniform ins, the value ofS _n. Similar results are given fork/n→θ and for mixtures of theP _λ ^k . Versions of de Finetti's theorem follow.     

On λ-designs

A λ-design is a system of subsets S₁, S₂,…, S_n from an n-set S, n > 3, where |S_i ∩ S_j| = λ for i ≠ j, |S_j| = k_j > λ > 0, and not all k_j, are equal. Ryser [9] and Woodall [101 have shown that each element of S occurs either r₁, or r₂ times (r₁ ≠ r₂) among the sets S₁,…, S_n and r₁ +r₂ = n + 1. Here we: (i) mention most of what is currently known about λ-designs; (ii) provide simpler proofs of some known results; (iii) present several new general theorems; and (iv) apply our theorems and techniques to the calculation of all λ-designs for λ ? 5. In fact, this calculation has been done for all λ ?/ 9 and is available from the author.     

О разложениях в ряд ы о бобщенно-абсолютно-м онотонных функций

In an earlier paper [1] the notion of the so-called 〈?, GL_J>-absolutely monotonie functions was introduced, where ?≧1, {λ_{k k=0} ^∞ is an arbitrary non-increasing sequence of positive numbers. It was found that the condition \(\sum\limits_{\lambda _{k > 0} } {\lambda _k^{ - 1} } = + \infty \) is necessary in order to have the series expansion for any function f(x)∈〈?, λ_j). HereL ^k/? f/(x) are special integro-differential operators of fractional order, is a system of functions associated with the Mittag-Leffler type functions \(E_\varrho (z;\mu ) = \sum\limits_{k = 0}^\infty {z^n /\Gamma (\mu + \kappa /\varrho )} \) and with the sequence {λ_k}. In the present paper it is proved (in particular, see Theorem 3.2) that the expansion (*) is valid almost everywhere on (0,l) if ∑ λ _k ^?1 =+∞. This result contains, as a special case (when ?=1 and λ_k=0,k≧0) the known theorem of S. N. Bernstein on absolutely monotonic functions.     

Matrices and set intersections

The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AA^T=(k?λ)I+λJ, where A^T denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved.