Return times, recurrence densities and entropy for actions of some discrete amenable groups

It is a theorem of Wyner and Ziv and Ornstein and Weiss that if one observes the initialk symbolsX ₀,…,X _k−1 of a typical realization of a finite valued ergodic process with entropyh, the waiting time until this sequence appears again in the same realization grows asymptotically like 2 ^hk [7, 12]. A similar result for random fields was obtained in [8]: in this case, one observes cubes in ℤ ^d instead of initial segments. In the present paper, we describe generalizations of this. We examine what happens when the set of possible return times is restricted. Fix an increasing sequence of sets of possible times {W _n } and defineR _k to be the firstn such thatX ₀,…,X _k−1 recurs at some time inW _n . It turns out that |W _{R k} | cannot drop below 2 ^hk asymptotically. We obtain conditions on the sequence {W _n } which ensure that |W _{R k} | is asymptotically equal to 2 ^hk . We consider also recurrence densities of initial blocks and derive a uniform Shannon-McMillan-Breiman theorem. Informally, ifU _k,n is the density of recurrences of the blockX ₀,…,X _k−1 inX _−n ,…,X _n , thenU _k,n grows at a rate of 2 ^hk , uniformly inn. We examine the conditions under which this is true when the recurrence times are again restricted to some sequence of sets {W _n }. The above questions are examined in the general context of finite-valued processes parametrized by discrete amenable groups. We show that many classes of groups have time-sequences {W _n } along which return times and recurrence densities behave as expected. An interesting feature here is that this can happen also when the time sequence lies in a small subgroup of the parameter group.     

Underlying graph and total length of optimal realizations of variable distance matrices

The general problem is to investigate, for acceptable values ofx, the optimal graph realization of a matrixD(x) obtained from a given tree-realizable distance matrixD as follows: we partition the index set ofD into two convex subsetsH andK, we subtractx from all entriesd _hk andd _kh whereh H andk K and we leave all other entries unchanged. We describe the optimal realization of the matrixD(x) and the behaviour of the total length of the optimal realization as a function ofx. This work was supported by the National Science Foundation Grant DMS-8401686 and by the PSC/CUNY Research Award Program.     

Variants of the Hajnal-Szemerédi Theorem

The Hajnal-Szemerédi Theorem [Hajnal & Szemerédi, Colloq Math Soc J Bolyai, 1970] states that a graph with hk vertices and minimum degree at least (h − 1)k contains k vertex disjoint copies of K_h. Its proof is not algorithmic. Here, we present an algorithm that, for a fixed h, finds in such a graph k − C(h) vertex disjoint copies of K_h in polynomial time in k, C(h) being a constant depending on h only. The proof suggests a variant of the theorem for h-partite graphs, which is conjectured here and proven in a slightly weaker form in some special cases. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 275–282, 1999     

Space $$ \mathcal{L} $$( X) of local dynamical systems and the Filippov space Aceu( X)

In the present paper, we establish a relationship between continuous local dynamical systems and spaces of the class A _ceu(X) of the Filippov theory. We suggest a construction method for a space of the class A _ceu(X) on the basis of a locally given dynamical system and conversely, a dynamical system is constructed locally in a specific way on the basis of a given space of the class A _ceu(X). The suggested construction method provides a homeomorphism between the space of all local dynamical systems on a locally compact metric space X and the space A _ceu(X). The obtained results generalize the Filippov theory to locally dynamical systems.     

Global dynamics of delay‐distributed HIV infection models with differential drug efficacy in cocirculating target cells

In this paper, we investigate the dynamical behaviors of three human immunodeficiency virus infection models with two types of cocirculating target cells and distributed intracellular delay. The models take into account both short‐lived infected cells and long‐lived chronically infected cells. In the two types of target cells, the drug efficacy is assumed to be different. The incidence rate of infection is given by bilinear and saturation functional responses in the first and second models, respectively, while it is given by a general function in the third model. Lyapunov functionals are constructed and LaSalle invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. We have derived the basic reproduction number R₀ for the three models. For the first two models, we have proven that the disease‐free equilibrium is globally asymptotically stable (GAS) when R₀≤1, and the endemic equilibrium is GAS when R₀>1. For the third model, we have established a set of sufficient conditions for global stability of both equilibria of the model. We have checked our theoretical results with numerical simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

D4型量子包络代数的Gelfand-Kirillov维数

本文研究了D₄ 型量子包络代数的Gelfand-Kirillov 维数的计算问题. 利用文献[1] 中给出的Gelfand-Kirillov 维数的计算方法和文献[2] 中给出的D₄ 型量子包络代数的Groebner-Shirshov 基计算了D₄型量子包络代数的Gelfand-Kirillov 维数, 得到的主要结果是D₄ 型量子包络代数的Gelfand-Kirillov 维数为28. 希望此结果为计算D_n型量子包络代数的Gelfand-Kirillov 维数提供一些思路.     

Finite Families of lp-Spaces and Multirectangular Characteristics

Under consideration are finite families of Banach spaces with unconditional basis. The problem of isomorphic classification of families is studied together with the related problem of uniqueness of an unconditional basis for a family. Complete solution of both problems for l ₁- and c ₀-families is given by using some geometric invariants combined with the corresponding results for l ₁ and c ₀.     

Is every cycle basis fundamental?

A collection of (simple) cycles in a graph is called fundamental if they form a basis for the cycle space and if they can be ordered such that C_j(C₁ U … U C_j-1) ≠ Ø for all j. We characterize by excluded minors those graphs for which every cycle basis is fundamental. We also give a constructive characterization that leads to a (polynomial) algorithm for recognizing these graphs. In addition, this algorithm can be used to determine if a graph has a cycle basis that covers every edge two or more times. An equivalent dual characterization for the cutset space is also given.     

Universal bases and greedy algorithms for anisotropic function classes

We suggest a three-step strategy to find a good basis (dictionary) for non-linear m-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) D _u ∈ D for a given pair (F, D) of collections: F of function classes and D of bases (dictionaries). This means that D_u provides near optimal approximation for each class F from a collection F. The third step deals with constructing a theoretical algorithm that realizes near best m-term approximation with regard to D _u for function classes from F. In this paper we work this strategy out in the model case of anisotropic function classes and the set of orthogonal bases. The results are positive. We construct a natural tensor-product-wavelet-type basis and prove that it is universal. Moreover, we prove that a greedy algorithm realizes near best m-term approximation with regard to this basis for all anisotropic function classes.     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

Let A be a commutative ring and M be a projective module of rank k with n generators. Let h=n-k. Standard computations show that M becomes free after localizations in comaximal elements (see Theorem 5). When the base ring A contains a field with at least hk+1 non-zero distinct elements we construct a comaximal family G with at most (hk+1)(nk+1) elements such that for each g∈G, the module M_g is free over A[1/g].     

On Besselian Schauder bases inl p (E)

By complex interpolation and tensor products, Schauder bases are constructed of the Banach sequence spacesl _p (E). In a general result, we study the Besselian property of the basis, and ifE is assumed to be theL _p (Lebesgue) andS _p (v. Neumann-Schatten) space, we obtain inequalities for the coefficient functionals associated to the basis which generalise other results given by Hausdorff—Young and Gohberg—Marcus. Finally, we construct non-Besselian and conditional bases ofl _p (E).     

On 321-Avoiding Permutations in Affine Weyl Groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A _{n – 1} by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GL _n ()     

Asymptotics of the 1/2-dimension,Discrete Airy Equation and its Approximating Differential Equation

A discrete finite difference model is constructed for the Airy equation using a nonstandard scheme formulated by Mickens and Ramadhani. The method of dominant balance is then applied to obtain a first-order difference equation for the solution that increases sufficiently fast as k→∞. We then calculate the corresponding approximating differential equation and obtain its exact solution as well as its “exact” discrete finite difference representation. The application of various symmetry operations allows the determination of the related rapidly decreasing solution and the oscillatory solutions for negative values of x k>=hk, where h=?x.     

Obstructions to lifting automorphisms of free algebras

ABSTRACT

A monomial basis and a filtration of subalgebras for the universal enveloping algebra 𝔘(𝔤 _l ) of a complex simple Lie algebra 𝔤 _l of type A _l is given in this note. In particular, a new multiplicity formula for the Weyl module V(λ) of 𝔘(𝔤 _l ) is obtained in this note.     

q-Deformed Euclidean algebras and their representations

A new q-deformed Euclidean algebra U_q (iso _n ), based on a definition of the algebra U_q (so _n ) different from the Drinfeld-Jimbo definition, is given. Infinite-dimensional representations T_a of this algebra, characterized by one complex number, is described. Explicit formulas for operators of these representations in an orthonormal basis are derived. The spectrum of the operator T_a(I_n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis is given. Contrary to the case of the classical Euclidean algebraiso _n, this spectrum is discrete and the spectrum points have one point of accumulation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 467–475, June, 1995.     

Threefolds of order one in the six-quadric

Consider the smooth quadric Q ₆ in ℙ⁷. The middle homology group H ₆(Q ₆, ℤ) is isomorphic to ℤ ⊕ ℤ, with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1, p) inside Q ₆.     

CALCULUS ON CANTOR TRIADIC SET （I）——DERIVATIVE

For real-valued functions defined on Cantor triadic ,set. a derivative with corresponding formula of Newton-Leihniz‘s type is given In particular, for the self-simltar functions and alter-nately jumping functions defined in this paper, their derivative and exceptional sets are studied ac-curately by using ergodic theory on Е2 and Duffin-Scbaeffer‘s theorem coneerning metric diophan-tine approximation. In addition, Haar basis of L2(Е2) is constructed and Flaar expansion of stan-drd self-similar function is given.     

Theorems of Peano's Type for Bivariate Functions and Optimal Recovery of Linear Functionals

The best recovery of a linear functional Lf, f = f (x, y), on the basis of given linear functionals L _j f, j = 1, 2, ···, N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a bivariate function by given scattered data has been obtained in a simple analytical form as a special case.     

On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation

Suppose that K ^d is compact and that we are given a function fC(K) together with distinct points x_iK, 1in. Radial basis interpolation consists of choosing a fixed (basis) function g : ⁺→ and looking for a linear combination of the translates g(|x−x_j|) which interpolates f at the given points. Specifically, we look for coefficients c_j such that has the property that F(x_i)=f(x_i), 1in. The Fekete-type points of this process are those for which the associated interpolation matrix [g(|x_i−x_j|)]_1i,jn has determinant as large as possible (in absolute value). In this work, we show that, in the univariate case, for a broad class of functions g, among all point sequences which are (strongly) asymptotically distributed according to a weight function, the equally spaced points give the asymptotically largest determinant. This gives strong evidence that the Fekete points themselves are indeed asymptotically equally spaced.