Convolution of the Fourier coefficients of the Eisenstein-Maass series

The convolution is defined as the sum where for n≠0 and W₀,W₁ are arbitrary smooth functions. Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms. The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function.     

Stability of the eigenvalues of certain types of singular integral equations

In the paper one finds necessary and sufficient conditions for the existence of nontrivial solutions of the equation (A+C)f=f for all compact (or small in norm) perturbations C; for the unperturbed operators A one considers the operators of dual and conjugate to dual integral equations with kernels depending on the difference of arguments, Wiener-Hopf operators, and singular integral operators. The answer is given in terms of the symbol S (A) of the operator A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 127–136, 1984.     

Coordinare asymptotic behavior of Green's function of the scattering problem for the Helmholtz equation

A construction is described for the asymptotic behavior of Green's function G(x, x', ) of the scattering problem for the Schrödinger equation as x, which is uniform with respect to x'.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 236–238, 1974.The author is grateful to V. S. Buslaev for valuable discussions during the course of the work.     

A remark on convergence in the limit theorem for the distribution of random functionals

Estimates are established which characterize the rate of convergence in a global limit theorem for random functionals.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 197–206, 1979.The author is grateful to L. N. Dovbysh for assistance and to I. A. Ibragimov for valuable consultations.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

An Estimate for the Problem of Illumination of the Boundary of a Convex Body in E3

Let A be a convex body in Euclidean space E3. We denote by H(A) the smallest number of homothetic 'reduced copies of A by which it is possible to cover the whole of A. The conjecture of Hadwiger is H(A) 8. We prove that H(A) 16.     

Stability of the extreme point set of a polyhedron

This paper is focused on the stability properties of the extreme point set of a polyhedron. We consider a polyhedral setX(A,b) which is defined by a linear system of equality and inequality constraintsAxb, where the matrixA and the right-hand sideb are subject to perturbations. The extreme point setE(X(A,b)) of the polyhedronX(A,b) defines a multivalued map :(A,b)E(X(A,b)). In the paper, characterization of continuity and Lipschitz continuity of the map is obtained. Boundedness of the setX(A,b) is not assumed It is shown that lower Lipschitz continuity is equivalent to the lower semicontinuity of the map and to the Robinson and Mangasarian-Fromovitz constraint qualifications. Upper Lipschitz continuity is proved to be equivalent to the upper semicontinuity of the map . It appears that the upper semicontinuity of the map implies the lower semicontinuity of this map. Some examples of using the conditions obtained are provided.The author wishes to thank Dr. N. M. Novikova, Dr. S. K. Zavriev, and anonymous referees for their helpful comments and advice. The research described in this publication was made possible in part by Grant NJCU100 from the International Science Foundation and Russian Government, and by the Euler Grant, Deutsche Mathematiker Vereinigung.     

Formula for the differentiation of operator-valued functions depending on a parameter

A study of the convergence of the differentiation formula (f(A))=f (A)A+f ^m(A)/21[AA]+f ^m(A)/3l[[AA]A]'+... where [XY]=XY–YX, and A=A(t) is a function of the real variable t with values in a Banach algebra.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 207–218, August, 1971.The author wishes to thank A. G. Aslanyan for his comments concerning this work.     

Linear discrete operators on the disk algebra

Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that

and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.

     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

Nonlinear and quasilinear evolution equations: Existence,uniqueness, and comparison of solutions; the rate of convergence of the difference method

In the Banach space X one investigates the Cauchy problem where [u](t)=u|_{[o, t]}, f L₁ (0, T; X); for fixed t, w, the nonlinear operator A(t, w)=A is a pseudogenerating operator of the semigroup e^SA (s 0), and for u, v, w(r) Z_r(Z_n is a ball in ZX),; the conditions on the dependence of A(t, w) on w admit the occurrence of w in the leading terms. One proves local and global theorems of existence and uniqueness of the limit-difference solution of the Cauchy problem, one investigates its differentiability and its dependence on u_o and f. Similar results of Crandall-Pazy, Benilan, Crandall-Evans, Evans, Oharu, Pavel, etc. for the equations du(t)/dt=A(t)u(t)+f (t) with- dissipative operators are special cases of ours. In the quasilinear case, our results complement and generalize T. Kato's well-known theorem. In addition, one obtains estimates for the convergence rates of the difference method and estimates for the norm of the difference of the solutions of Cauchy problems with different operators A(t, w); these results are new also for the equations with dissipative operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 181–200, 1983.In conclusion, the author would like to express his sincere gratitude to O. A. Ladyzhenskaya for her interest in this paper.     

On the construction of the linear Hull of a set in a convexity space using only the extension operation

We use the operation of extension to construct the linear hull A of a set A in a convexity space: if A is n-dimensional (n>1) and₁A=A/A,_jA=_(j–1)A/_(j–1)A, we show that A=_jA if j=[log₂(n–1)]+3, [x] meaning the greatest integer contained in x.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh

Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this semi-cardinal box-spline subspace satisfy certain boundary conditions extending the not-a-knot end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4. AMS subject classification 41A15, 41A05, 41A25, 41A63, 39A70, 47B35     

Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S2, R is generated by its invertible elements, and the ideal of R generated by the differences –1 for all invertible coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e()PG(), where G is the net subgroup of () and E() is the subgroup generated by the transvections in G(). It is shown that E() is a normal subgroup of G(). The factor group G(/E()) is studied. The case of the special linear group is also considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 66–79, 1977.In conclusion,the author would like to thank Z. I. Borevich for posing the problem and for his continued interest, and A. A. Suslin for useful discussions leading to simplifications of some of the proofs.     

Unique solvability of the cauchy problem for the equations of discrete multidimensional chiral fields,taking values on the unit sphere

One considers a discrete model of the classical field theory, defined by the action with the constraints. Here the _i s are basis vectors in the dimensional integral lattice and the functions take values in. One proves that the Cauchy problem for the equations of the motion of this model has at least one C -solution for arbitrary initial data which are consistent with the constraints. The uniqueness of the solution is established under the condition of the uniform boundedness of. In the case =2, 3, 4, the uniqueness theorem is proved without this restriction.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 217–224, 1980.In conclusion, the author wishes to express his deep gratitude to his scientific advisor, O. A. Ladyzhenskaya, for suggesting the problem and for her assistance.     

On the typicality of linear Hamiltonian systems with exponential separateness

Linear Hamiltonian systems of ordinary differential equations,are considered, where f ^t is a continuous action of the group R on a complete metric space , A is an element of the space S _H ,consisting of the continuous bounded mappings of into the set of all pseudosymmetric matrices and endowed with the uniform convergence metric. By ₁ (A, x) _2n (A, x) we denote the Lyapunov exponents of such systems. The typicality (in the Baire sense) in the space S _H × is proved for those pairs (A, x) for which one has the alternative: either _k (A, x) = _k+1 (A, x) or the linear subspace of the solutions of the corresponding system with exponents less than _k (A, x) is exponentially separated from any of its algebraic complements in the space of all the solutions of the system. From here, in particular, there follows the typicality of the formulated alternative for linear Hamiltonian systems with continuous quasiperiodic coefficients (with the same frequency module) and also for linear Hamiltonian systems with continuous almost periodic coefficients. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 137–181, 1992.     

Contractions Satisfying the Absolute Value Property

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators which satisfy the absolute value condition . It is proved that if is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in , and it is shown that if normal subspaces of . It is proved that if are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation with the commutant of A* is quasinilpotent.     

Application of the ray method to the computation of complex eigenvalues

The ray method is used to obtain asymptotic equations for the complex eigenvalues of the Helmholtz equation. The problem of the transparent disk and oscillations of transparent whispering gallery type are considered in detail.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 228–235, 1974.The author is grateful to V. M. Babich for a detailed discussion of the work and to A. L. Alimov, V. S. Buldyrev, and V. N. Krasil'nikov for useful discussions.     

On the Subalgebra Lattice of Unary Algebras

We characterize pairs L, A, where Lis a lattice and Ais a unary partial algebra, such that the strong subalgebra lattice S_s(A) is isomorphic to L. Moreover, we find necessary and sufficient conditions for arbitrary unary partial algebras to have isomorphic strong subalgebra lattices. Observe, that for a total algebra A, the lattice S_s(A) is the usual well-known subalgebra lattice. Thus in particular we solve these two problems for total unary algebras and their lattices of (also total) subalgebras.For this purpose we apply some non-obvious connections between unary partial algebras and graphs from [9]. More precisely, we first characterize the pairs L, G, where Lis a lattice and Ga directed graph, such that the strong subdigraph lattice of Gis isomorphic to L. Next, we find a characterization of arbitrary digraphs with isomorphic strong subalgebra lattices. From these results we easily get solutions of our algebraic problems.