Über Untergruppen Endlicher Algebraischer Gruppen

Let k be a commutative ring, GG finite affine algebraic k-groups, and HH the dual Hopfalgebras of the affine algebras of G resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H-linear, HH-colinear, unitary, augmented isomorphism HHH H, where HH is the coalgebra belonging to G/G. (II) If the k-submodule of the fixelements of (HH)^* is isomorphic to k (e.g. k principal or semilocal), then HH is a Frobeniusextension of the second kind.     

Spectral orders and convolution

. (R) _{– –} fg(y)h(x–y) dx dy _{– –} f ^{^} (x)g ^{^} (y)h ^{^} (x–y)dx dy (f,g0) —:f×gf ^{^} ×g ^{^}(f,g 0) f^{^} g^{^} f g -, X — . , - f ₁f ₂ , f ₁ ^{^} ×gf ₂×g 0g. .     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Some imbedding theorems concerning the moduli of Ditzian and Totik

, , 3. . . f ^*, f, , f . , .

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Research supported by Hungarian National Foundation for Scientific Research Grant No. 1801.     

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.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

Linearity properties of Shimura varieties, II

Let A=A_{g, 1, n} denote the moduli scheme over Z[1/N] of p.p. g-dimensional abelian varieties with a level n structure; its generic fibre can be described as a Shimura variety. We study its Shimura subvarieties. If x A is an ordinary moduli point in characteristic p, then we formulate a local linearity property in terms of the Serre–Tate group structure on the formal deformation space (= formal completion of A at x). We prove that an irreducible algebraic subvariety of A is a Shimura subvariety if, locally at an ordinary point x, it is formally linear. We show that there is a close connection to a differential-geometrical linearity property in characteristic 0.We apply our results to the study of Oort's conjecture on subvarieties Z A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.     

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.     

Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling

We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between strong and weak precedence for chains and trees in scheduling. We distinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different interpretations for chain and tree order relations in scheduling might have far reaching computational implications.     

Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

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.     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

Euler'sϕ-function in the context of IΔ 0

It is demonstrated that we can represent Euler's -function by means of a ₀-formula in such a way that the theory I ₀ proves the recursion equations that are characteristic for this function.     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

О сравнительной теории простЫх чисел

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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.