Ricci curvature and betti numbers

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.     

Curvature,diameter and betti numbers

We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most manifolds admit no metrics of non-negative sectional curvature.     

Computation of betti numbers of monomial ideals associated with stacked polytopes

LetP(v, d) be a stackedd-polytope withv vertices, δ(P(v, d)) the boundary complex ofP(v, d), andk[Δ(P(v, d))] =A/I _Δ(P(v,d)) the Stanley-Reisner ring of Δ(P(v,d)) over a fieldk. We compute the Betti numbers which appear in a minimal free resolution ofk[Δ(P(v,d))] overA, and show that every Betti number depends only onv andd and is independent of the base fieldk. We also show that the Betti number sequences above are unimodal.     

Unimodal betti numbers for a class of nilpotent lie algebras

In this paper we prove that all finite dimens~onal complex nilpotent Lie algebras containing an abelian ideal of codimension 1 have unirnodal Betti numbers.     

Computation of betti numbers of monomial ideals associated with cyclic polytopes

We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ringk[Δ(P)]=A/I _Δ(P) of the boundary complex Δ(P) of an odd-dimensional cyclic polytopePover a fieldk. A corollary to the formula is that the Betti number sequence ofk[Δ(P)] is unimodal and does not depend on the base fieldk.     

The type numbers of closed geodesics

This is a short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics.     

Free quotients and the first betti number of some hyperbolic manifolds

In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections. In all these cases we actually show that Γ=π₁(M) has a finite index subgroup which is mapped onto a nonabelian free group.     

Approximation of smooth vectors of a closed operator by entire vectors of exponential type

Some classes of infinitely differentiable vectors of a normal operator in a Hilbert space are described in terms of the rate of convergence to zero of the best approximation of these vectors by entire vectors of exponential type. As a special case, we obtain many well-known results in the theory of approximations of continuous functions by algebraic or trigonometric polynomials or by entire functions of exponential type. The proofs of these results are essentially based on the theory of spaces with positive and negative norm developed by our teacher, Prof. Yu. M. Berezanskii, to whom we dedicate this paper as a sign of our deep gratitude.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 616–628, May, 1995.     

Asymptotic properties of type I elliptical random vectors

Let be an elliptical random vector with a non-singular square matrix and a spherical random vector in , and let be a sequence of vectors in such that . We assume in this paper that the associated random radius R _k =(S ₁ + S ₂ +...+S _k )^1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability for converging as to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of .      

Thue type problems for graphs, points, and numbers

A sequence S=s₁s₂…s_n is said to be nonrepetitive if no two adjacent blocks of S are the same. A celebrated 1906 theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set {0,1,2}. This result is the starting point of Combinatorics on Words—a wide area with many deep results, sophisticated methods, important applications and intriguing open problems.The main purpose of this survey is to present a range of new directions relating Thue sequences more closely to Graph Theory, Combinatorial Geometry, and Number Theory. For instance, one may consider graph colorings avoiding repetitions on paths, or colorings of points in the plane avoiding repetitions on straight lines. Besides presenting a variety of new challenges we also recall some older problems of this area.     

On a new type of distance Fibonacci numbers

In this paper, we define a new kind of Fibonacci numbers generalized in the distance sense. This generalization is related to distance Fibonacci numbers and distance Lucas numbers, introduced quite recently. We also study distinct properties of these numbers for negative integers. Their representations and interpretations in graphs are also studied.     

Whittaker vectors and conical vectors

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.