Direct and converse imbedding theorems for seminormed spaces

. n=2 . ., 103 498      

Exact bounds for norms of differentiable periodic functions in L metric

Exact values are obtained for the upper bounds on the norms,fL, on the classes W ^(r) H(r =0, 1, 2,...) of r-fold differentiable functions f(x), of periodicity2, for which (f^(r);t)(t),where (t) is a given convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 569–576, December, 1967.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

On the divergence of spherical sums of double Fourier-Haar series

. : [0, +) [0, +) - , u+ (u) (u)=o(u lnu). [0, 1]² f , ¦f¦ L([0, 1]²), - [0, 1]².     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

Subgroups of the general symplectic group containing the group of diagonal matrices

There are described the subgroups of the general symplectic group =GSp(2n, R) over a commutative semilocal ring R, containing the group of symplectic diagonal matrices. For each such subgroup P there is uniquely defined a symplectic D-net a such that ()pN(), where () is the net subgroup in corresponding to (cf. RZhMat, 1977, 5A288), and N() is its normalizer. The quotient group N × ()/() is calculated. There are also considered subgroups in Sp(2n, R). Analogous results for subgroups of the general linear group were obtained earlier in RZhMat, 1978, 9A237.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 31–47, 1980.     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Minimax detection of a signal in ℓp metrics

The problem of the minimax detection of a signal is considered in a Gaussian white noise of intensity 0. The set of the signals represents an ellipsoid in the _p-metric with lengths of semiaxes a_k ^–1 0 as k , from which a ball of radius 0 as 0 in the _p-metric has been removed. Asymptotically minimax tests have been constructed for the cases 1 < p and asymptotically sharp estimates of the minimax efficiency have been obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 152–168, 1990.     

Spectrum of an elliptic equation

It is shown that the spectrum for the first boundary-value problem for a second-order elliptic equation always lies in the half-plane ₀ Re z, where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from ₀, there are no other points of the spectrum on the straight line Re z=₀.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 495–502, April, 1970.The authors are grateful to V. S. Vladimirov for discussing the results of the present paper and for pointing out the proposition in [3], which made it possible to shorten the proof that the leading eigenvalue of the problem (1) is simple.     

Über Untergruppen Endlicher Algebraischer Gruppen

Let k be a commutative ring, GG finite affine algebraic k-groups, and HH the dual Hopfalgebras of the affine algebras of G resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H-linear, HH-colinear, unitary, augmented isomorphism HHH H, where HH is the coalgebra belonging to G/G. (II) If the k-submodule of the fixelements of (HH)^* is isomorphic to k (e.g. k principal or semilocal), then HH is a Frobeniusextension of the second kind.     

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 . . -      

On the notion of measurability

. , . , . , , - , . , , - .

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I wish to record my thanks to Dr. Kummer for his constant encouragement and many helpful suggestions in preparing this paper.     

On uniform approximations of almost periodic functions by entire functions of finite degree

We give a new proof of the well-known Bernshtein statement that, among entire functions of degree which realize the best uniform approximation (of degree ) of a periodic function on (–,), there is a trigonometric polynomial of degree . We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1274–1279, September, 1995.     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

d≤1 ∪d≥25 and constrained KP hierarchy from BRST invariance in thec≠3 topological algebra

The BRST invariance condition in a highest-weight representation of the topological ( twisted N=2) algebra captures the invariant content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the Kontsevich-Miwa-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into c=–26 reparametrization ghosts + matter + Liouville, while a similar splitting involving c=–2 ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the Liouville differ also by their ghost numbers, which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c3 provides a two-fold covering of the allowed regiond1 d25 of the matter central charge d via d=(c+1)(c+6)/(c–3). The Liouville field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 239–250, May, 1993.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

Generic continuity of restricted weak upper semi-continuous set-valued mappings

Set-valued mappings from a topological space into subsets of a Banach space which satisfy a restricted form of weak upper semi-continuity, have particularly noteworthy properties. We establish a selection theorem for certain set-valued mappings from a (-) unfavourable topological space into subsets of a Banach space and as a consequence derive the property that restricted weak upper semi-continuous set-valued mappings which satisfy a minimality condition, from a (-) unfavourable topological space into subsets of a Banach space are single-valued and norm upper semi-continuous at the points of a residual subset of their domain.     

A high-indices theorem for strong Cesàro summability of double series

[4] , [C,(, )]_;- (, 0 1) s, [C,(+,+)]- s , 0 0< , (i) , ; > > 1 , (ii) ,>0; >=1 ,>1-1/, [C,(,)- [C,(+,+)]- .

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This work was done while the author was a visiting researcher at the Steklov Mathematical Institute, Moscow, U.S.S.R.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.