Free abelian covers, short loops, stable length, and systolic inequalities

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, , classified by f. We show that the finite-dimensionality of the rational homology of is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles. A. I. Suciu was supported by the National Science Foundation (grant DMS-0105342).     

Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case

Let S _n = X ₁ + ⋯ + X _n be a sum of independent random variables such that 0 ⩽ X _k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where X _k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d. case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related Hoeffding inequalities if 0 ⩽ X _k ⩽ 1. Our conditions are X _k ⩾ 0 and {ie237-05}. In particular, X _k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of X _k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities in Hoeffding [6] and Bentkus The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08.     

Approximation of classes of periodic multivariable functions by linear positive operators

In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m-and ((N − m))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 12–19, January, 2006.     

An Example of a Fiber in Fibrations Whose Serre Spectral Sequences Collapse

We give an example of a space X with the property that every orientable fibration with the fiber X is rationally totally non-cohomologous to zero, while there exists a nontrivial derivation of the rational cohomology of X of negative degree.     

Algebra of formal vector fields on the line and Buchstaber’s conjecture

We consider the Lie algebra L ₁ of formal vector fields on the line which vanish at the origin together with their first derivatives. V. M. Buchstaber and A. V. Shokurov showed that the universal enveloping algebra U(L ₁) is isomorphic to the Landweber-Novikov algebra S tensored with the reals. The cohomology H*(L ₁) = H*(U(L ₁)) was originally calculated by L. V. Goncharova. It follows from her computations that the multiplication in the cohomology H*(L ₁) is trivial. Buchstaber conjectured that the cohomology H*(L ₁) is generated with respect to nontrivial Massey products by one-dimensional cocycles. B. L. Feigin, D. B. Fuchs, and V. S. Retakh found a representation for additive generators of H*(L ₁) in the desired form, but the Massey products indicated by them later proved to contain the zero element. In the present paper, we prove that H*(L ₁) is recurrently generated with respect to nontrivial Massey products by two one-dimensional cocycles in H ¹(L ₁).     

Bergman kernels and local holomorphic Morse inequalities

Let (X,) be a hermitian manifold and let L^k be a high power of a hermitian holomorphic line bundle over X. Local versions of Demaillys holomorphic Morse inequalities (that give bounds on the dimension of the Dolbeault cohomology groups associated to L^k), are presented - after integration they give the usual holomorphic Morse inequalities. The local weak inequalities hold on any hermitian manifold (X,), regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in Mathematics Subject Classification (2000): 32A25, 32L10, 32L20in final form: 19 June 2003     

Lusternik‐Schnirelmann category and systolic category of low‐dimensional manifolds

We show that the geometry of a Riemannian manifold (M, ??) is sensitive to the apparently purely homotopy‐theoretic invariant of M known as the Lusternik‐Schnirelmann category, denoted cat_LS(M). Here we introduce a Riemannian analogue of cat_LS(M), called the systolic category of M. It is denoted cat_sys(M) and defined in terms of the existence of systolic inequalities satisfied by every metric ??, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_sysM ≤ cat_LSM is satisfied, which typically turns out to be an equality, e.g., in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to Massey products. The comparison with the value of cat_LS(M) leads us to prove or conjecture new systolic inequalities on M. © 2006 Wiley Periodicals, Inc.     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

On manifolds satisfying stable systolic inequalities

We show that for closed orientable manifolds the k-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree k that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg–Mac Lane space. Consequently, the stable k-systolic constant is completely determined by the multilinear intersection form on k-dimensional cohomology.     

Capacitary Criteria for Poincaré-Type Inequalities

The Poincaré-type inequality is a unification of various inequalities including the F-Sobolev inequalities, Sobolev-type inequalities, logarithmic Sobolev inequalities, and so on. The aim of this paper is to deduce some unified upper and lower bounds of the optimal constants in Poincaré-type inequalities for a large class of normed linear (Banach, Orlicz) spaces in terms of capacity. The lower and upper bounds differ only by a multiplicative constant, and so the capacitary criteria for the inequalities are also established. Both the transient and the ergodic cases are treated. Besides, the explicit lower and upper estimates in dimension one are computed. Mathematics Subject Classifications (2000) 60J55, 31C25, 60J35, 47D07.Research supported in part NSFC (No. 10121101) and 973 Project.     

Bounds for the Multiplicities of the Irreducible Factors of a Multivariate Polynomial

We provide explicit upper bounds for the multiplicities of the irreducible factors for some classes of polynomials in two variables X, Y over a field K, regarded as polynomials in Y with coefficients in K[X] whose degrees satisfy certain inequalities. We then obtain similar results for polynomials in an arbitrary number of variables over K.     

Inside S-inner product sets and Euclidean designs

A finite set X in the Euclidean space is called an s-inner product set if the set of the usual inner products of any two distinct points in X has size s. First, we give a special upper bound for the cardinality of an s-inner product set on concentric spheres. The upper bound coincides with the known lower bound for the size of a Euclidean 2s-design. Secondly, we prove the non-existence of 2- or 3-inner product sets on two concentric spheres attaining the upper bound for any d>1. The efficient property needed to prove the upper bound for an s-inner product set gives the new concept, inside s-inner product sets. We characterize the most known tight Euclidean designs as inside s-inner product sets attaining the upper bound.     

On Nordlander’s conjecture in the three-dimensional case

In the present paper we prove, that in the real normed space X, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space X is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that X is an inner product space.     

Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

Décompte dans une conjecture de Lang

Faltings has proven the following conjecture of Lang: if A is an abelian variety over a number field and X any subvariety then all rational points of X lie on a finite number N of translates, contained in X, of abelian subvarieties of A. We provide an upper bound for N whose main feature is uniformity in X since it does not depend on the height of X. Moreover, the bound is completely explicit. Together with the result of a previous paper, the heart of the proof is a suitable generalization of Mumford’s theorem for curves.

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Décompte dans une conjecture de Lang

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Oblatum 20-III-2000 & 3-V-2000?Published online: 16 August 2000     

Nonlinearities of S-boxes

We introduce an indicator of the non-balancedness of functions defined over Abelian groups, and deduce a new indicator, denoted by NB, of the nonlinearity of such functions. We prove an inequality relating NB and the classical indicator NL, introduced by Nyberg and studied by Chabaud and Vaudenay, of the nonlinearity of S-boxes. This inequality results in an upper bound on NL which unifies Sidelnikov–Chabaud–Vaudenay's bound and the covering radius bound. We also deduce from bounds on linear codes three new bounds on NL that improve upon Sidelnikov–Chabaud–Vaudenay's bound and the covering radius bound in many cases.     

A modular absolute bound condition for primitive association schemes

The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field. Partially supported by RFBR grants 07-01-00485, 08-01-00379 and 08-01-00640. The paper was done during the stay of the author at the Faculty of Science of Shinshu University.     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996