Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

Quasigeodesics outside horoballs

Let be the path metric space obtained from a Hadamard manifold X by removing a collection of pairwise disjoint open horoballs from X. In case X is a rank one symmetric space, certain properties of quasigeodesics in have been used to derive results about lattices in X. For two such properties we give new proofs that simultaneously apply to the case where X is a Hadamard space with sectional curvature -a ²K-1,1a<2. In this case, as an application of one of these properties, we show that every lattice in X is biautomatic.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Linear Bilipschitz Extension Property

We give a sufficient geometric condition for a subset A of R ⁿ to enjoy the following property for a fixed C1: There is >0 such that for 0, each (1+)-bilipschitz map f: AR ⁿ extends to a (1+C)-bilipschitz map F: R ⁿ R ⁿ .     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

Rational chebyshev approximations of elementary functions

Explicit formulæ which should be useful in practical computations, are given for the functions e^–x, log x, (1 + x), arctg x/x, sin x/x, cos x and (2/x) arcsin (x/2) in the interval 0x1 except for log x, where the interval is 1/2x1 instead.     

On general problems with bounded state variables

This paper serves as the sequel to a recent article by the author concerned with a certain canonical control problem. The present paper extends the first-order necessary conditions obtained there to a general form of the control problem of Bolza with inequality state constraints of the type (t, x)0, =1,...,m.The preparation of this paper was sponsored in part by the U.S. Army Research Office under Grant No. DA-31-124-ARO(D)-355.     

Tensor products and Banach ideals of p-compact operators

A study is made of the norm w_p (1 p ) on the tensor product of two Banach spaces E and F. It is shown that w_p is a tensor norm, and a representation is deduced for the elements in the completion of E F equipped with w_p. Finally it is shown that the w_p-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly through^p (if 1 p ) or Co (if p=), with related equalities concerning the idea1 norms.     

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.     

The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, )=||^(x)(1< ₁(x)₂< +) is studied. Our main result: If(x) is Holder continuous, then the Lagrangianf(x, )=f(x, )=||^(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.Supported by the National Natural Science Foundation of China.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

Measure ratio of equidecomposable sets

Let n 2. There are Lebesgue measurable sets A and B in ³ such that (B)/(A)=r and A _n B if and only if 2/n r n/2.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Nearest q-Flat to m Points

Recently, Bradley and Mangasarian studied the problem of finding the nearest plane to m given points in ⁿ in the least square sense. They showed that the problem reduces to finding the least eigenvalue and associated eigenvector of a certain n×n symmetric positive-semidefinite matrix. We extend this result to the general problem of finding the nearest q-flat to m points, with 0qn–1.