Asymptotic formulas for Toeplitz and Wiener-Hopf operators

For certain analytic functions f, the expression trace {f(T_n[]) – f(T_n[]T_n[])} is computed asymptotically. Here T_n[] is the finite Toeplitz matrix generated by the function . The analogous expression for Wiener-Hopf operators is also computed asymptotically. These results in turn yield information concerning the asymptotic behavior of determinants of finite Toeplitz and Wiener-Hopf operators with discontinuous generating function.     

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     

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.     

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.     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

Extremal partition and estimates for the coefficients in the class σ(r)

Let be the known class of functions f(z)=z+₀+₁z^–1+..., that are meromorphic and univalent in the region U^*={z ¦z¦ < 1}, and let (r) be the class of functions of for which f(U^*) does not contain a singly connected domain D_f, 0 D_f, with conformal radius r with respect to the coordinate origin, 0 < r < 1. Sharp inequalities are obtained for certain functionals, and sharp bounds are obtained for ¦ ₁ ¦ and ¦ ₂ ¦ in the class (r). The proof illustrates the possibility of using results with a symmetrization character in problems on extremal partition of a Riemannian sphere to obtain sharp bounds for coefficients in the classes and (r).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 101–116, 1991.     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), ¦z¦=r}, 1 < r < , be a level line of the function f(z) . Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class () for functions f(z)=z+₀+₁z^–1+... for which there exists a domain _f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(_f,0) , 0 < < 1. Also, a set of values is found for the coefficient ₁ in the class ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 138–153, 1991.     

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     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Sur une Classe de Fluides Non-Newtoniens

A constitutive law for a class of non newtonian fluids is considered. The stress-tension is defined as an element in the subgradient of a convex, l.s.c. function: (D);D is the tension of the rate of deformation. We give existence and uniqueness theorems. Some examples (Bingham fluid, pseudo-plastic and dilatant fluids) are also given.     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Non linear eigenvalues problems with a small parameter

In this work we study non linear eigenvalues problems like: [–² d²/dt²+(t²–)²+1]u=0 where. More precisely we study the spectrum of the operator: Q(;)=–²d²/dt²+(t²–)²+1 when 0, >0. Our method of proof consists in replacing our problem by a linear eigenvalue problem about a non self adjoint system.     

Flag-Transitive 2-Designs Arising from Line-Spreads in PG(2n-1,2)

A Singer cycle in GL(n,q) is an element of order q permuting cyclically all the nonzero vectors. Let be a Singer cycle in GL(2n,2). In this note we shall count the number of lines in PG (2n-1,2) whose orbit under the subgroup of index 3 in the Singer group is a spread. The lines constituting such a spread are permuted cyclically by the group ³, hence gives rise to a flag-transitive 2-(2²ⁿ ,4,1) design.     

Aperiodic transformations

Summary The concepts of periodicity and aperiodicity are extended so as to apply to any measurable non-singular transformation T in a -finite measure space (X, , ). Aperiodicity of T may then be characterized by the existence of sweep-out sets with certain properties and turns out to be closely related to the existence of a strong generator for the -algebra . This extends recent results of Parry and Rokhlin obtained for Lebesgue spaces.     

A characterization of subgroups of the orthogonal group

LetG be a subgroup of the general linear group GL_n(K), where charK 2. Put Kⁿ =V. AssumeG is generated by the setS of all elements inG for which dimV( – 1) = 1, and suppose ²=1_V for each inS. If {V(–1)¦S} contains a simplex, if – 1_V G, and if inG is a product of dim v(–1) elements inS wheneverV(–1) is not contained in the kernel of–1, thenG is a subgroup of an orthogonal group.This research was supported in part by NSERC Canada grant A7251.To Helmut Mäurer on his 60th birthday     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

Shift-coupling in continuous time

Summary The result linking shift-coupling to time-average total variation convergence and to the invariant -field is extended to continuous time and an analogous result established linking -couplings to smooth total variation convergence and to a smooth tail -field. Shift- and -coupling inequalities are presented.     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

Entire functions of exponential type and model subspaces in HP

Let W ^p be the space of entire fonctions in of exponential type , whose restrictions to the real line belong to L^p(). For an inner function in the upper half- plane ₊ we denote by K ^p (p 1) the invariant subspace of the inverse shift operator (the model subspace) in H^p, generated by the function K ^p =H^p , where H^p=H^p(₊) is the Hardy class. In the paper it is shown that many known properties of the spaces W ^p (some imbedding and uniqueness theorems, the Logvinenko-Sereda theorem on equivalent norms, S. N. Bernshtein's differential inequality) remain valid for the spaces K ^p if and only if the derivative is bounded. The classical results for entire functions are obtained for (x)= e^ix.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 81–100, 1991.