Direct and converse imbedding theorems for seminormed spaces

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Zur Kennzeichnung Fanoscher Affin-Metrischer Geometrien

Let V be a vector space over the commutative field K such that char K 2 2 dim V , and let Q:V K be a quadratic form of rank 2. The pair (A(V,K),_Q), consisting of the affine space A(V,K) and the congruence relation _Q, defined by (a,b)_Q (c,d) Q(a–b) = Q(c–d) (a,b),(c,d) V×V, is called an affine-metric fano-space of rank 2. In this paper, such spaces are characterized by three simple geometrical properties.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

On uniform approximations of almost periodic functions by entire functions of finite degree

We give a new proof of the well-known Bernshtein statement that, among entire functions of degree which realize the best uniform approximation (of degree ) of a periodic function on (–,), there is a trigonometric polynomial of degree . We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1274–1279, September, 1995.     

Local Jacquet-Langlands correspondence and parametric degrees

Let F be a non-Archimedean local field with finite residue field. Let n be a positive integer, let G = GL_n(F), and let D be a central F-division algebra of dimension n². The Jacquet-Langlands correspondence gives a canonical bijection ^D from the set of equivalence classes of irreducible, smooth, essentially square-integrable representations of G to the set of equivalence classes of irreducible smooth representations of D^![![times;. We give a necessary and sufficient condition, in terms of dim, for an irreducible smooth representation of D^× to be of the form ^D, for an irreducible supercuspidal representation of G, thereby solving an old problem. This relies on the explicit classification of the irreducible smooth representations of G and the parallel classification of the irreducible representations of D^×.This paper was written while the first-named author was visiting, and partly supported by, Université de Paris-Sud. At that time, the second-named author was enjoying the hospitality of the IHES, during a stay at the CNRS granted by Université de Paris-Sud; he would like to thank all those institutions. The work was also partially supported by EU network Arithmetical Algebraic Geometry.Mathematics Subject Classification (2000): 22E50     

Subgroups of the general symplectic group containing the group of diagonal matrices

There are described the subgroups of the general symplectic group =GSp(2n, R) over a commutative semilocal ring R, containing the group of symplectic diagonal matrices. For each such subgroup P there is uniquely defined a symplectic D-net a such that ()pN(), where () is the net subgroup in corresponding to (cf. RZhMat, 1977, 5A288), and N() is its normalizer. The quotient group N × ()/() is calculated. There are also considered subgroups in Sp(2n, R). Analogous results for subgroups of the general linear group were obtained earlier in RZhMat, 1978, 9A237.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 31–47, 1980.     

Martingale transforms and Hardy spaces

Summary Burkholder's martingale transforms are especially useful in studying predictable martingale Hardy spaces. Characterizations of such spaces via martingale transforms are provided. In particular, it is shown that for 0<p<, a martingale inh ^p , defined by the conditioned square function, is the martingale transform of a bmo₂ martingale with a multiplier sequence whose maximal function is inL ^p .     

Estimation of density functionals

Given a sequence of independent random variables with densityf we estimate quantities of the form = (f(x))dx, a known function, by inserting histograms and kernel density estimators for the unknownf. We obtain conditions for consistency and asymptotic normality and discuss the choice of cell size and bandwidth.     

Minimaxtheoreme und das Integraldarstellungsproblem

In the present paper conditions for the strict determinateness of two-person zero-sum games are considered. In order to get such minimax theorems we first study games with concave-convex pay-off function. If a game does not have this convexity property one usually passes to a mixed extension where both players are allowed to use probability measures (-additive randomizations) or, more generally, probability contents (finitely additive randomizations) as mixed strategies. By means of a very general minimax theorem for such finitely additive randomizations it can be shown that the problem of strict determinateness of -additive randomizations is equivalent to an integral representation problem. The latter is investigated in the last paragraph.

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Diese Arbeit enthält einen Teil der Ergebnisse der Habilitationsschrift des Verfassers.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Conformal spaces

A conformal space is a non-singular metric vector space to which has been adjoined a null-cone of points at infinity. We define a conformal space in terms of a higher dimensional coordinate space, and then state and prove a fundamental theorem of conformal geometry.     

Clear visibility,starshaped sets,and finitely starlike sets

Let S be an arbitrary nonempty set in R^d. The following results are true for every k, 0kd: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional -neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional -neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R², S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S –R² and int cl SS=, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible.     

η%-Superconvergence of Finite Element Solutions and Error Estimators

This paper is a summary of our study on the superconvergence of the finite element solutions and error estimators. We will persent the analysis of %-superconvergence for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary. The %-superconvergence via local averaging will also be presented, and the error estimators are compared in the sense of %-superconvergence.     

Central Limit Theorems for dependent variables. I

Summary This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak mixing condition called -mixing. Roughly speaking, an array of real r.v.s is said to be -mixing if linear combinations of its past and future are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, -mixing and -mixing) are special cases of -mixing. Linear processes are shown to be -mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are nearly second order stationary.     

Imperfect and Nonideal Clutters: A Common Approach

We prove three theorems. First, Lovászs theorem about minimal imperfect clutters, including also Padbergs corollaries. Second, Lehmans result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a common denominator for Lovászs and Lehmans theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.* Visiting of the French Ministry of Research and Technology, laboratoire LEIBNIZ, Grenoble, November 1995—April 1996.     

Croissance lineaire et géométrie bornée

We show that on a noncompact manifold which has finite topology at infinity, there exists a Riemannian metric with bounded geometry and linear growth-type.     

Optimal transformations for prediction in continuous-time stochastic processes: finite past and future

In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the future of a stochastic process, and seeks its best predictor (in the L²-sense). In this paper we treat a variant of the prediction problem, whereby we seek the most predictable non-trivial functional of the future and its best predictor; we refer to such a pair (if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite past and future intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous-time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.This research was partially supported by ARO (MURI grant) DAAH04-96-1-0445, NSF grant DMS-0074276, and CNPq grant 301179/00-0.     

Exceptional sets and wavelet packets orthonormal bases

We give a partial positive answer to a problem posed by Coifman et al. in [1]. Indeed, starting from the transfer function m₀ arising from the Meyer wavelet and assuming m₀=1 only on [–/3, /3], we provide an example of pairwise disjoint dyadic intervals of the form I(n, q)=[2^qn, 2^q(n+1)), (n, q)EN×Z, which cover [0, +) except for a set A of Hausdorff dimension equal to 1/2, and such that the corresponding wavelet packets 2^q/2w_n (2^qx–k), kZ, (n, q)EN×Z form an orthonormal basis of L²(R).     

Augmented non-quadratic penalty algorithms

Auslender, Cominetti and Haddou have studied, in the convex case, a new family of penalty/barrier functions. In this paper, we analyze the asymptotic behavior of augmented penalty algorithms using those penalty functions under the usual second order sufficient optimality conditions, and present order of convergence results (superlinear convergence with order of convergence 4/3). Those results are related to the analysis of pure penalty algorithms, as well as augmented penalty using a quadratic penalty function. Limited numerical examples are presented to appreciate the practical impact of this local asymptotic analysis.This research was partially supported by NSERC grant OGP0005491     

Some New Transversality Properties

The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a selftransversality theorem in which the suitable line is determined by two other vertices. Here we present additional transversality properties in which the suitable line is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few sporadic cases.