Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

Comparison of Normal Linear Experiments by Quadratic Forms

Let X and Y be observation vectors in normal linear experiments =N(A, V) and F = N(B, W). We write > Fif for any quadratic form YGY there exists a quadratic formXHX such that E(XHX) = E(Y'GY) and var(X'HX) var(Y'GY).The relation > is characterized by the matrices A, B, V and W. Moreoversome connections with known orderings of linear experiments are given.     

Metric σ-Frames versus Metric Lindelof Spaces

The adjoint relation between the category RegFrm, of regular -frames, Alex, of Alexandroff spaces, are studied in [9]. Here, we introduce the category MFrm, of metric -frames and give the adjoint relation between this category and the category MLSp, of metric Lindelof spaces, and show that MLSp is dually equivalent to the category of Alexandroff metric -frames.AMS Subject Classification: 06D99-54B30     

Messbare Mengen von Massen und Inhalten

Let X be a completely regular space. The customary -field is the coarsest -field on the space of Bairemeasures on X which makes (A) measurable for any Baire set A. We compare the customary -field with the Baire and Borel -field induced by the weak* topology which lies on the dual space C(X). In (2.3) it is shown that the customary -field is just the Baire -field. In part 3 necessary and sufficient conditions are given under which the set of -smooth measures is measurable with respect to the Borel -field which lies on the positive cone of the space of finitely additive, regular measures C(X). Finally, a decomposition theorem for generalized kernels is proved. The -smooth part of a generalized kernel is a kernel again if certain conditions are fulfilled.     

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.     

Approximation of functions by partial sums of Fourier series in polynomials orthogonal on an interval

For certain weight functions p(t) and q(t), upper bounds are obtained for the difference between partial sums of Fourier series of a function/ with respect to the systems _p and _q of polynomials orthogonal on [–1, 1] (a comparison theorem is incidentally proved for the systems _p and _q). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (forf W^r H) of the remainder in a Fourier-Chebyshev series, we establish corresponding results for Fourier series with respect to a system _p.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 431–441, October, 1970.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

On the Iterates of the Sum of Unitary Divisors

Let (n) be the sum of unitary divisors of n, and k(n) be its k-fold iterate. We prove that (n) ¦ 2(n) 1 as n on a set of density one.     

Projektive G-Faserräume

Let > 0 be an integer. A projective -fibre space is formed by a covering of a projective geometry with -1 isomorphic geometries. The double elliptic space (Sphärischer Raum) is an example of a 2-fibre space. This note deals with projective -fibre spaces which are structured by multy valued orderfunctions (This notion was introduced by W. JUNKERS [2] for projective geometries) the range of which is a group G. If such an ordered -fibre space has the property ¦G¦=, it is called projective G-fibre space. It is proved that the desarguesian projective G-fibre spaces V are exactly those, which are induced by a vector space S over a field K (commutative or not) having a normal subgroup P K^*(·) with K^* P such that GK^*/P and SV^*/P. This theorem is a generalization of the well-known case P=K^*.     

Net determinant over a Bezoutian local ring

Let be any D-net of ideals of order n over a commutative local Bezoutian ring R and denote by G() the corresponding net subgroup in the general linear group of degree n over R (RZhMat, 1977, 2A280). We give an explicit computation of the factor group G()/E(), where E() is the subgroup generated by all elementary transvections in G().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 5–13, 1982.     

Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum

This paper continues the work started by Basu and Ghosh (J. Mult. Anal. (1978), 8, 413–429), by Gilliland and Hannan (J. Amer. Stat. Assoc. (1980), 75, No. 371, 651–654), and then continued on by Mukherjea and Stephens (Prob. Theory and Rel. Fields (1990), 84, 289–296), and Elnaggar and Mukherjea (J. Stat. Planning and Inference (1990), 78, 23–37). Let (X₁, X₂,..., X_n) be a multivariate normal vector with zero means, a common correlation and variances ² ₁, ² ₂,..., ² _n such that the parameters , ² ₁, ² ₂,..., s² _n are unknown, but the distribution of the max{X_i: 1in} (or equivalently, the distribution of the min{X_i: 1in}) is known. The problem is whether the parameters are identifiable and then how to determine the (unknown) parameters in terms of the distribution of the maximum (or its density). Here, we solve this problem for general n. Earlier, this problem was considered only for n3. Identifiability problems in related contexts were considered earlier by numerous authors including: T. W. Anderson and S. G. Ghurye, A. A. Tsiatis, H. A. David, S. M. Berman, A. Nadas, and many others. We also consider here the case where the X_i's have a common covariance instead of a common correlation.     

On Upper and Lower Almost β-continuous Multifunctions

In this paper, the authors define a multifunction F : X Y to be upper (lower) almost -continuous if F+(V) (F- (V)) is -open in X for every regular open set V of Y. They obtain some characterizations and several properties concerning upper (lower) almost -continuous multifunctions.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Growth along a ray of a subharmonic function,having a mass distributed on the negative axis

Let U be a subharmonic function in C with a Riesz mass , distributed on the negative semiaxis without some neighborhood of zero, let and be its order and lower order, and let B(r, U) be the maximum of U(z) for ¦z¦=r. Estimates are obtained for the measure of sets of those values of r 0 for which certain inequalities hold. The following result is typical. LetE = {r:u(re ^l)–cosB<(r,U) > 0}. If < < 1, ¦¦=., then the lower logarithmic density of the set E is at least 1 – /. If < > 1,¦¦ ., then the upper logarithmic density of the set E is at least 1 – /.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 31–38, 1988.     

The ring of integers of an Abelian extension of an algebraic number field as a Galois module

The ringO of integers of a finite Abelian extension K of an algebraic number field k is studied as a module over the group ring =[G], where is the ring of integers of k and G is the Galois group of K/k. It is proved that the ring is a decomposable -module if and only if there exists in K/k an intermediate extension K/F. FK, whose degree divides the different.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta. im. V. A. Steklova AN SSSR, Vol. 71, pp. 80–84, 1977.     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Über gewisse diophantische Gleichungen mit unendlich vielen Lösungen

Let be a real number with 1<<12/11. Denote by [A] the integer part of the real number A. Using van der Corput-type estimates for trigonometrical sums, we prove that there exist infinitely many squares of the form [x] with natural x; actually, we give an asymptotic formula for the number of natural numbers x]=y² for some natural y. Furthermore, for every natural number d, the (Pell-like) equation [x]–dy²=1 has infinitely many solutions in natural numbers x and y · Also, every natural number n>n() can be written (in many ways) as n=[x]+y² with natural x and y. Instead of squares we study integer-valued polynomials of degree at least two.

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Unterstützt durch NSF-grant 9038     

Spektrale Invarianz bei verallgemeinerten Eigenfunktionsentwicklungen

Let T be a selfadjoint operator in a Gelfand triplet H. Examples show that there can occur generalized eigenvalues of T which are not in the Hilbert space spectrum (T) of T. Moreover, the fuction d, which assigns to each real number s the dimension of the generalized eigenspace corresponding to s, can be essentially greater then the von Neumann multiplicity function of T. We therefore construct a new triplet H, closely related to the given Gelfand triplet, according to which the set of generalized eigenvalues of T is contained in (T), and the function d essentially equals the von Neumann multiplicity function of T. Then, in particular, the closure of the set of generalized eigenvalues equals (T). The expansion theorems in H are transferred to H.

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Grundlage dieser Arbeit ist ein Teil meiner Dissertation. Ich danke Herrn Prof. Dr. H.G. Tillmann für die Anregung hierzu und für viele wertvolle Hinweise.     

Equicontinuity and convergence of measures

It is shown in this paper that each family of measures with values in an abelian topological group which is equicontinuous on a ring is equicontinuous on the generated -ring. A family of measures is equicontinuous iff the corresponding family of semivariations is equicontinuous. It is furthermore shown that a family of measures which is equicontinuous and Cauchy convergent on a ring is Cauchy convergent on the generated -ring. A family of measures which is Cauchy convergent for all countable sums of elements of a ring is Cauchy convergent on the generated -ring.