Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1–p _m . LetN(,T), 1/21,T3. denote the number of zeros =+i of the Riemann zeta function which fulfill and ||T. Letc2 andh0 be constants such thatN(,T)T ^c(1–) (logT) ^h holds true uniformly in 1/21. Let >0 be given. Then there is some constantK>0 such that      

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

On the asymptotic behavior of an unstable solution of an inhomogeneous stochastic diffusion equation

Sufficient conditions are obtained for the normalized trajectories of an unstable solution of the one-dimensional Itô stochastic differential equation with coefficientsa(t, x) and(t, x) to coincide with the normalized trajectories of the solution of the equation with coefficientsa(x) and(x) ast assuming that the coefficientsa(t, x) and(t, x) have a certain average closeness to the coefficientsa(x) and(x) over time ast.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 15, pp. 3–10, 1987.     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

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.     

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     

The connection between the zeros of the ζ-function and sequences(g(p)), p prime,mod 1

In this paper we give the connection between the zeros of the -function and sequences(g(p)), p prime, mod 1 ifg(x)=x for 0, >0 or ifg(X) is a polynomial in .     

Self-intersections of 1-dimensional random walks

Summary Consider a random walk S _n on the integers, where the steps _i have mean 0 and variance ². Let T be the time of first self-intersection of the random walk. It is shown that, as , T grows at rate ^2/3. More precisely, T^2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.Research supported by National Science Foundation Grant MCS80-02698.     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

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.     

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.     

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.

     

On the Convergence of Stewart's QLP Algorithm for Approximating the SVD

The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix A₀ is first factored as A₀=QR, and then the matrix R ^T₁ is factored as R ^T₁=PL ^T, resulting in A=Q₁ LP ^T₀ ^T, with Q and P orthogonal, L lower-triangular, and ₀ and ₁ permutation matrices. Stewart noted that the diagonal elements of L approximate the singular values of A with surprising accuracy. In this paper, we provide mathematical justification for this phenomenon. If there is a gap between _k and _k+1, partition the matrix L into diagonal blocks L ₁₁ and L ₂₂ and off-diagonal block L ₂₁, where L ₁₁ is k-by-k. We show that the convergence of ( _j (L ₁₁)^–1– _j ^–1)/ _j ^–1 for j=1,. . .,k, and of ( _j (L ₂₂)– _k+j )/ _k+j , for j=1,. . .,n–k, are all quadratic in the gap ratio _k+1/ _k . The worst case is therefore at the gap, where the absolute errors L ₁₁ ^–1– _k ^–1 and L ₂₂– _k+1 are thus cubic in _k ^–1 and _k+1, respectively. One order of convergence is due to the rank-revealing pivoting in the first step; then, because of the pivoting in the first step, two more orders are achieved in the second step. Our analysis assumes that ₁=I, that is, that pivoting is done only on the first step. Although our results explain some of the properties of the pivoted QLP decomposition, they hypothesize a gap in the singular values. However, a simple example shows that the decomposition can perform well even in the absence of a gap. Thus there is more to explain, and we hope that our paper encourages others to tackle the problem. The QLP algorithm can be continued beyond the first two steps, and we make some observations concerning the asymptotic convergence. For example, we point out that repeated singular values can accelerate convergence of individual elements. This, in addition to the relative convergence to all of the singular values being quadratic in the gap ratio, further indicates that the QLP decomposition can be powerful even when the ratios between neighboring singular values are close to one.     

Interpolation a plusieurs variables sur la sphere

Summary Given a functionf defined on the sphere , a continuation is considered which transformsf into a periodic function with period 2, accordingto each of the two usual variables , .Starting from this continuation, an explicit interpolation formula forf on in a set of trigonometric functions is obtained. A simple and numerically stable quadrature formula is given, which is accurate for a vast class of functions.Error-bounds for approximation and quadrature are given.     

Büschel linearer Abbildungen

If , , are linear mappings out of a projective space (P,G) into a projective space (P', G') and , then is said to belong to the pencil <,<> of linear mappings spanned by and if in the main (x), (x), (x) are collinear for all x P. We give some sufficient conditions for x P and , , such that (x) is uniquely determined by giving, and (z), z P.

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Herrn Prof. Dr.Helmut Karzel zum 60. Geburtstag gewidmet     

PAT – a Reliable Path-Following Algorithm

This paper presents a new technique for the reliable computation of the -pseudospectrum defined by (A)={zC : _min(A–zI)} where _min is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve (A)={zC : _min(A–zI)=} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to . The method is guaranteed to terminate, even in the presence of round-off errors.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.