Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

On the convergence of orthogonal series to +∞

For any sequence of numbers _n0, _n=1 a _n ² =, a uniformly bounded orthonormal system of continuous functions _n(x) which is complete in L₂ (0, 1), and a sequence of numbers b_n(0< b_na_n) are constructed such that _n=1 ^Emphasis> b_nn(x)= everywhere on (0, 1).Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 499–508, May, 1972.     

Shape-preserving Kolmogorov widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ₀ and let ₊ ^s be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t ₀,..., t _s I, the corresponding divided differences [x; t ₀,...,t _s ] of order s are nonnegative. Let ₊ ^s B _{p +} ^s B _p, 1 p where B _p is a unit ball in the space L _p, and let ₊ ^s L _{q +} ^s L _q, 1 q . For every s 3 and 1 q p , we determine the exact orders of the shape-preserving Kolmogorov widths {x - y} \right\ L_q , $$]]>, where M ⁿ is the collection of all affine linear manifolds M ⁿ in L _q such that dim M ⁿ n and M ⁿ ₊ ^s L _q .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.     

On convergence of function series

, a _n f _n (x) . .     

Lower central series and derived series of net subgroups of the general linear group

Over a commutative ring R with invertible element 2 and with radical , nets (i.e., tables =(_ij) of ideals _ij such that _{irrj ij}) such that _ii are considered. Such nets are called pseudoradical. The groups of the lower central series and the derived series are explicitly constructed for the corresponding net subgroups G () (of the general linear group GL (n,R)) in terms of .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 180–186, 1982.     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.     

A Local Approach to 1-Homogeneous Graphs

Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsC _i(x,y) = i–1(x) (y), A _i (x,y) = _i (x) (y) and B _i (x,y) = _i+1(x) (y).Then we say has the CAB_j property,if the partition CAB _i (x,y) = {C _i (x,y),A _i (x,y),B _i (x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CAB_j property then the parameters ofthe equitable partitions CAB i(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CAB _d property. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a ₁ 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c ₂ 2.     

Verschärfung einer Ungleichung von Ky Fan

Summary In this paper we prove the following:IfA _n ,G _n andH _n (resp.A _n ,G _n andH _n ) denote the arithmetic, geometric and harmonic means ofa ₁,, a _n (resp. 1 –a ₁,, 1 –a _n ) and ifa _i (0, 1/2],i = 1,,n, then(G _n /G _n ) ⁿ (A _n /A _n ) ^n-1 H _n /H _n , (^*) with equality holding forn = 1,2. Forn 3 equality holds if and only ifa ₁ = =a _n . The inequality (^*) sharpens the well-known inequality of Ky Fan:G _n /G _n A _n /A _n .

     

Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

Sample behavior of some random series

The scaled-sample property ({X ^j / _n } _j=1 ⁿ K a.s. asn, where _n <0 andX( ^j ) are i.i.d. with law ) is investigated for distributions which are the laws of random series of form _i=1 X _i v _i . Herev _i are fixed vectors in a Banach space, andX _i are i.i.d. real-valued with common law ₁. The main result presented here establishes this convergence for a large collection of possible distributions ₁, including some discrete examples. The main tools used are probability inequalities on the tails of the series, and concentration of measure arguments.Supported by NSF Grant No. DMS-9024961. This work will be included in the author's Ph.D. thesis.     

On the accuracy of empirical likelihood confidence regions for linear regression model

The coverage errors of the empirical likelihood confidence regions for in a linear regression model,Y _i=x _i+ _i , 1in, are of ordern ^–1. Bartlett corrections may be employed to reduce the order of magnitude of the coverage errors ton ^–2. For practical implementation of Bartlett correction, an empirical Bartlett correction is given.     

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 . . -      

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

On the method for numerical integration of Clenshaw and Curtis

Letf(x) be a function, defined and well behaved on the finite intervalaxb Clenshaw andCurtis [1] have given a method for the numerical integration off(x) froma tob, based on the approximation off(x) with a finite series of Chebyshev polynomials. We show that this method is asymptotically equivalent to using the trapezoïdal rule for integratingg(y)=f(cosy).     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

The central limit theorem for Chébli-Trimèche hypergroups

Let (S _nn>-1) be a random walk on a hypergroup ( ₊ , *), i.e., a Markov chain with transition kernelN(x, A) = _x * (A), where is a fixed probability measure on ₊ such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( ₊ , *) is of exponential growth then it is shown thatS _n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( ₊ , *) satisfies lim[A()/A()]=}, the normalized variablesS _n/[n Var()/(+1)]^1/2 converge to a Rayleigh distribution with parameter .     

Quasi-symmetric designs and the Smith Normal Form

We obtain necessary conditions for the existence of a 2 – (, k, ) design, for which the block intersection sizes s ₁, s ₂, ..., s _n satisfy s ₁ s ₂ ... s _n s (mod p ^e ),where p is a prime and the exponent e is odd. These conditions are obtained from restriction on the Smith Normal Form of the incidence matrix of the design. We also obtain restrictions on the action of the automorphism group of a 2 – (, k, ) design on points and on blocks.