Weight-monodromy conjecture for <Emphasis Type="Italic">p</Emphasis>-adically uniformized varieties

The aim of this paper is to prove the weight-monodromy conjecture (Delignes conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants. We also consider a p-adic analogue by using the weight spectral sequence of Mokrane.     

A topological approach to evasiveness

The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the graph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity Ω(v ²) wherev is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper the truth of Karp’s conjecture is shown to follow from another conjecture concerning group actions on topological spaces. A special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices. Supported in part by an NSF postdoctoral fellowship Supported in part by NSF under grant No. MCS-8102248     

Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension

We discuss the conjecture asserting that isolated equilibrium states of autonomous systems admitting invariant measures are unstable in spaces of odd dimension. This conjecture is proved for systems for which quasihomogeneous truncations with isolated singularities can be found. We consider a counterexample in the class of systems with infinitely differentiable right-hand sides and zero Maclaurin series at the equilibrium state. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 674–680, May, 1999.     

Self-adjoint differential vector-operators and matrix Hilbert spaces I

In the current work a generalization of the famous Weyl-Kodaira inversion formulas for the case of self-adjoint differential vector-operators is proved. A formula for spectral resolutions over an analytical defining set of solutions is discussed. The article is the first part of the planned two-part survey on the structural spectral theory of self-adjoint differential vector-operators in matrix Hilbert spaces. The work is dedicated to Professor Ravshan Ashurov on occasion of his 50-th anniversary.     

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.     

Hardy's inequalities in function spaces containing derivatives of noninteger order

A theorem on Hardy's inequality in function spaces containing derivatives of noninteger order is proved. Translated fromMatematichcskie Zametki, Vol. 63, No. 5, pp. 673–678, May, 1998. The author wishes to thank Professor V. A. Kondrat'ev for his attention to this work.     

Quadratic modules and the orthogonal group over polynomial rings

The following conjecture of Parimala is proved: Any quadratic space over a polynomial ring with coefficients from an algebraically closed field of characteristic different from 2 is extended from the coefficient field. In the case of an arbitrary field of characteristic different from 2, an analogous result is obtained for quadratic spaces whose Witt index is at least 2. Also proved are general cancellation theorems for quadratic modules and a stabilization theorem for the orthogonal group over arbitrary polynomial rings.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, im. V. A. Steklova AN SSSR, Vol. 71, pp. 216–250, 1977.     

Kählerity and Negativity of Weil–Petersson Metric on Square Integrable Teichmüller Space

The Weil–Petersson metric is a Hermitian metric originally defined on finite-dimensional Teichmüller spaces. Ahlfors proved that this metric is a Kähler metric and has some negative curvatures. Takhtajan and Teo showed that this result is also valid for the universal Teichmüller space equipped with a complex Hilbert manifold structure. In this paper, we stated that the Weil–Petersson metric can be also defined on a Hilbert manifold contained in the Teichmüller space of Fuchsian groups with Lehner’s condition, which we call the square integrable Teichmüller space, and proved that the results given by Ahlfors, Takhtajan, and Teo also hold in that case. Many parts of the proof were based on their ones. However, we needed more careful estimations in the infinite-dimensional case, which was achieved by two complex analytic characterizations of Lehner’s condition, by a certain integral equality for the partition of the upper half-plane by a Fuchisian group and by the invariant formula for the Bergman kernel.     

A new technique for obtaining diophantine representations via elimination of bounded universal quantifiers

M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 83–92. Original article Translated by Yu. V. Matiyasevich. The research described in this work was made possible in part by Grant No. 94-01-01030 from the Russsian Foundation for Fundamental Research and Grant No. R43000 from the International Science Foundation.     

Central limit theorem for positively associated stationary random fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis’s theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman’s conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.     

α-Admissibility of Observation Operators in Discrete and Continuous Time

In this paper a weighted form of the Weiss conjecture is studied. For certain weights, the conjecture is shown to hold for normal contraction operators related to discrete time linear systems. This is proved by an application of the Carleson measure theorem for weighted Dirichlet spaces. The result for discrete time systems is used to show that a weighted form of the Weiss conjecture holds for normal operators generating bounded C₀-semigroups. Previously, weighted admissibility has been characterised for generators of analytic semigroups. No such assumption of analyticity is made here. Additionally, results are presented regarding weighted Carleson measures, fractional powers of normal operators and weighted composition operators.     

Multidimensional covariant random fields on commutative locally compact groups

Homogeneous in the wide sense, covariant random fields on commutative local compact groups with values in finite-dimensional complex Hilbert spaces are considered. The general formula for the correlation operator of such a field is proved, as well as the spectral representation of the field itself in the form of a series of stochastic integrals with respect to orthogonal random measures.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1505–1510, November, 1992.     

Disjointly strictly singular inclusions of symmetric spaces

In this paper, the disjoint strict singularity of inclusions of symmetric spaces of functions on an interval is considered. A condition for the presence of a “gap” between spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. The condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular (and similar conditions for Marcinkiewicz spaces) are obtained and certain other assertions are proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 3–14, January, 1999.     

Remarks on the convergence of Rosen's gradient projection method

The convergence of Rosen's gradient projection method is a long-standing problem in nonlinear programming. Recently, Zhang^[27] proved that it is convergent in the 3-dimensional space; Du and Zhang^[5] proved its convergence inn-dimensional space under a restriction on a paramater in Rosen's method. In this paper, we propose a linearly algebraic conjecture which can yield the convergence of Rosen's method without the restriction. By verifying this conjecture for some special cases, we prove that Rosen's method is convergent in 4-dimensional space.The work was done while the author was working as a postdoctoral member in the Mathematical Sciences Research at Berkely, USA, and supported in part by NSF Grant No. 8120790.     

Quantum Mechanics and Representation Theory: The New Synthesis

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C ^*-algebras. In particular, the study of the K-theory of the reduced C ^*-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.     

Estimates for solutions of retarded equations with variable coefficients

In this work, we study retarded-type differential-difference equations with variable coefficients. Using the adjoint equation, we obtain an integral representation of the solution. A number of results on the asymptotic behavior of the solutions is proved on the basis of this representation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 83–93, 2006.     

A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

A well‐known combinatorial theorem says that a set of n non‐collinear points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work, we prove a slightly stronger version of Chen and Chvátal conjecture for a family of graphs containing chordal graphs and distance‐hereditary graphs.     

Subspaces of sequential spaces

It is proved that the space of continuous functions on the ordinary closed interval with the topology of pointwise convergence is not subsequential. In sequential spaces satisfying certain conditions, subspaces dense-in-themselves without convergent sequences are found; such subspaces are constructed in certain sequential compact spaces and semitopological groups. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 407–413, September, 1998. The author thanks the participants of Arkhangel'skii's seminar held at Moscow State University for useful discussions of this work.     

Cohomology of Real Algebraic Varieties

A class of effective spaces with involution introduced by the author is studied. The cohomology ring of the fixed-point set of an effective space is completely determined by the spectral sequence of involution. Real algebraic varieties admitting a “cellular decomposition” are effective M-spaces. Under certain restrictions, one calculates the spectral sequence of involution and the total ℤ₂ Betti number of the real part for real subvarieties of real algebraic varieties that are effective GM-spaces. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2002, pp. 112–151.     

Icosahedral galois extensions and elliptic curves

Summary This paper is devoted to the last unsolved case of the Artin Conjecture in two dimensions. Given an irreducible 2-dimensional complex representation of the absolute Galois group of a number fieldF, the Artin Conjecture states that the associatedL-series is entire. The conjecture has been proved for all cases except the icosahedral one. In this paper we construct icosahedral representations of the absolute Galois group of ℚ(√5) by means of 5-torsion points of an elliptic curve defined over ℚ. We compute the L-series explicitely as an Euler product, giving algorithms for determining the factors at the difficult primes. We also prove a formula for the conductor of the elliptic representation. A feasible way of proving the Artin Conjecture in a given case is to construct a modular form whose L-series matches the one obtained from the representation. In this paper we obtain the following result: letρ be an elliptic Galois representation over ℚ(√5) of the type above, and letL(s, ρ) be the corresponding L-series. If there exists a Hilbert modular formf of weight one such thatL(s, f) ≡L(s, ρ) modulo a certain ideal above (√5), then the Artin conjecture is true forρ. This article was processed by the author using the LATEX style filecljour1m from Springer-Verlag.