Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

Drachennetze mit geradlinigen Hauptdiagonalen

A C^s-net of curves N (s1) [3] in a regular C^s-2-surface Eⁿ (n2) is called a C^s-kite- net [4] if N and the net N₁ of its angular bisecting curves form a pair of diagonal nets [1] in such a way that each mesh of N-curves possessing two N₁-diagonals shows, with respect to one of these (calledmain diagonal), the same symmetry of angles and lengths as a rectilinear kite in E². Referring to the fact that the main diagonals of any C^s-kite-net N (s2) are geodesics in [5], we ask in this paper for all C^s-kite-nets and, more generally, C^s-D-nets [5] (s1) withstraight main diagonals. This leads, among other results, to a characterization of the skew ruled surfaces in Eⁿ (n3) with constant parameter of distribution and the constant striction /2.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag gewidmet     

On the kernel of intersecting families

Let be at-wises-intersecting family, i.e.,|F ₁ ... F _t | s holds for everyt members of. Then there exists a setY such that|F ₁ ... F _t Y| s still holds for everyF ₁,...,F _t . Here exponential lower and upper bounds are proven for the possible sizes ofY. This work was done while the authors visited Bell Communication Research, NJ 07960, and AT&T Bell Laboratories, Murray Hill, NJ 07974, USA, respectively.Research supported in part by Allon Fellowship and by Bat Sheva de Rothschild Foundation.     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Further results on the covering radii of the Reed-Muller codes

LetR(r, m) by therth order Reed-Muller code of length2 ^m , and let (r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. (r+1,m+2) 2(r, m)+2 if 0rm–2. This improves the successive use of the known inequalities (r+1,m+2)2(r+1,m+1) and (r+1,m+1) (r, m).2.(2, 7)44. Previously best known upper bound for (2, 7) was 46. 3. The covering radius ofR(1,m) inR(m–1,m) is the same as the covering radius ofR(1,m) inR(m–2,m) form4.     

Imbedding of pseudo-Riemannian manifolds in a pseudo-euclidean space

It is proved that every pseudo-Riemannian manifold M _{(p, q)} ⁿ with the C^k metric (3k) has an isometric C^k imbedding in the large in E _{(p, q)} ^{n(n+1)(3n+11)/2} , p(n+1)², q(n+1)².Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 193–198, February, 1971.     

There Exists a Simple Non Trivial t-design with an Arbitrarily Large Automorphism Group for Every t

In 1987, Teirlinckproved that if t and are two integers such that v t(mod(t + 1)!^(2t+1) and v t + 1 >0, then there exists a t - (v, t + 1, (t + 1)!^(2t+1)) design. We prove that if there exists a (t+1)-(v,k,)design and a t-(v-1,k-2, (k-t-1)/(v-k+1))design with t 2, then there exists a t-(v+1,k, (v-t+1)(v-t)/ (v-k+1)(k-t))design. Using this recursive construction, we prove that forany pair (t,n) of integers (t 2and n 0), there exists a simple non trivial t-(v,k,) design having an automorphism groupisomorphic to ⁿ ₂.     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Variations on Tilings in the Manhattan Metric

We investigate tilings of the integer lattice in the Euclidean n-dimensional space. The tiles considered here are the union of spheres defined by the Manhattan metric. We give a necessary condition for the existence of such a tiling for Z ⁿ when n 2. We prove that this condition is sufficient when n=2. Finally, we give some tilings of Z ⁿ when n 3.     

Abschnittspositive Matrizen und Positivitätsfaktoren in der Limitierungstheorie

Let A and B be normal matrices. In :={x=(x_k) ¦ x_k} we define the order relation _A by x_A0:<=> _k=0 ⁿ a_nkx_k0 (n ). Let T be a row-finite matrix. A is called T-section-positive, if _kt_mkx_ke^(k) _A0 (m ) for x_A0 (see [5]). We study the relation between T-sectional positivity and T-sectional boundedness. An (A,B)-summability factor sequence =(_k) is called positive, if (_kx_k)_B0 for each xc_A with x_A0. For B-section-positive matrices A we give a functional analytic characterization of positive (A,B)-summability factor sequences.

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Die Arbeit entstand während eines vom DAAD unterstützten Forschungsaufenthalts an der Fernuniversität-Gesamthochschule Hagen     

Codes from Affine Permutation Groups

It is shown that A(22,10) 50, A(23,10) 76, A(25,10) 166, A(26,10) 270, A(29,10) 1460, and A(28,12) 178, where A(n,d) denotes the maximum cardinality of a binary code of length n and minimum Hamming distance d. The constructed codes are invariant under permutations of some affine (or closely related) permutation group and have been found using computer search.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Toughness and the existence ofk-factors. II

In a paper with the same title [3], we proved Chvátal's conjecture thatk-tough graphs havek-factors if they satisfy trivial necessary conditions. In this paper, we prove the following stronger result: Suppose|V(G)| k + 1,k |V(G)| even, and|S| k w(G – S) – 7/8k ifw(G – S) 2, wherew(G – S) is the number of connected components ofG – S. ThenG has ak-factor.     

New Hadamard matrices and conference matrices obtained via Mathon's construction

We give a formulation, via (1, –1) matrices, of Mathon's construction for conference matrices and derive a new family of conference matrices of order 59^2t+1 + 1,t 0. This family produces a new conference matrix of order 3646 and a new Hadamard matrix of order 7292. In addition we construct new families of Hadamard matrices of orders 69^2t+1 + 2, 109^2t+1 + 2, 8499 ^t ,t 0;q ²(q + 3) + 2 whereq 3 (mod 4) is a prime power and 1/2(q + 5) is the order of a skew-Hadamard matrix); (q + 1)q ²9 ^t ,t 0 (whereq 7 (mod 8) is a prime power and 1/2(q + 1) is the order of an Hadamard matrix). We also give new constructions for Hadamard matrices of order 49 ^t 0 and (q + 1)q ² (whereq 3 (mod 4) is a prime power).This work was supported by grants from ARGS and ACRB.Dedicated to the memory of our esteemed friend Ernst Straus.     

Some best possible bounds concerning the traces of finite sets II

LetX be ann-element set and be a family of its subsets. Consider the family _x = {F – {x} : F } for a givenx X. We write(m, n) (m – k, n – 1), when for all with || m, there exists an elementx ofX such that| _x| m – k. We show that (m, n) (m – 10,n – 1) for allm 5n and (m, n) (m – 13,n – 1) for allm 29n/5.     

Editorial Policy

Consider the parametric linear complementarity problem w=Mz+q+p, w0, z0, w ^T z=0, where p0, 0q0, and 0. We show that a necessary condition for every complementary map z() to be isotone for every nonzero q0 and every p is that M be either a P-matrix or a -matrix. The Cottle necessary and sufficient conditions for strong and uniform isotonicity for P-matrices are restated, with slight modifications, for -matrices.     

Nonoscillation and asymptotic behaviour for third order nonlinear differential equations

In this paper we consider the equation y +q(t)y + p(t)h(y)=0, where p, q are real valued continuous functions on [0, ) such that q(t) 0, p(t) 0 and h(y) is continuous in (–, ) such that h(y)y > 0 for y 0. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.