Cliques in Steiner systems

A partial Steiner (k,l)-system is a k-uniform hypergraph with the property that every l-element subset of V is contained in at most one edge of . In this paper we show that for given k,l and t there exists a partial Steiner (k,l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of this note, we establish asymptotic lower and upper bounds on the size of such cliques with respect to the order of Steiner systems. Research of the second author partially supported by NSERC grant OGP0025112.     

Transitive tournaments and self‐complementary graphs

A simple proof is given for a result of Sali and Simonyi on self‐complementary graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 111–112, 2001     

Anisotropic Operator Symbols Arising From Multivariate Jump Processes

It is shown that infinitesimal generators of certain multivariate pure jump Lévy copula processes give rise to a class of anisotropic symbols that extends the well-known classes of pseudo differential operators of H?rmander-type. In addition, we provide minimal regularity convergence analysis for a sparse tensor product finite element approximation to solutions of the corresponding stationary Kolmogorov equations . The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimension.      

The beta-algorithm

A set-monotonic non-gradient algorithm is proposed for finding global minima of general non-convex mathematical programming problems. The algorithm is based on the Cubic Algorithm (E. A. Galperin, J. Math. Anal. Appl.M, 112 1985, 635–640) equipped with a semi-certain distinction operator and the marginal comparison constant generator.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

Speeding up an algorithm to construct the hall collection formula

A program that contains a recursive procedure inadmissibly increasing the running time of an algorithm was applied earlier for computer construction of the collection formula for Burnside groups. A program that remedies this flaw is suggested. Bibliography:6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 106–112.     

Bahadur efficiency and local asymptotic optimality of certain nonparametric tests for independence

One computes the Bahadur efficiency of nonparametric tests for the verification of the independence hypothesis, based on a Kolmogorov type statistic.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 112–128, 1988.     

On quantization of general covariant theories. The dynamic quantization

A new method for quantizing general covariant theories, which includes regularization and a calculation scheme, is proposed. The method is based on the Dirac theory of quantizing constrained systems. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 132–141.     

Ranges of certain systems of functionals in a class of functions,convex in a certain direction

One finds the range of systems of functionals on classes of univalent functions which are convex in a certain direction.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 51–58, 1981.     

Local times for functions and Gaussian processes

Some necessary and sufficient conditions that a Gaussian process with continuous time have a local time are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 104–112, 1979.The author is grateful to the participants of the seminar on local times and to Yu. A. Davydov in particular.     

Nontrivial Solutions of Superquadratic Hamiltonian systems with Lagrangian Boundary Conditions and the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z（t） = J△↓H（t, z（t）） with Lagrangian boundary conditions, where ^H（t,z）=1/2（^B（t）z, z） ＋ ^H（t, z）,^B（t） is a semipositive symmetric continuous matrix and ^H（t, z） = satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

The base space of a strongly parabolic Riemannian manifold

A class of metrics that can be the base of only a trivial (cylindrical, cylinder-conical), strongly parabolic metric is isolated. The result has a local character. The main tool used in the investigation is the conullity operator.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 112–121, 1991.     

Asymptotic properties of nonparametric estimators of the characteristic function

One investigates the asymptotic properties of nonparametric estimators of the characteristic function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 97–112, 1984.     

A Higher Atiyah–Patodi–Singer Index Theorem for the Signature Operator on Galois Coverings

Let (N, g) be a closed Riemannianmanifold of dimension 2m – 1 and let Ñ N be a Galois covering of N. We assumethat is of polynomial growth with respect to a word metric and that _Ñ is L ²-invertible in degree m. By employing spectral sections with asymmetry property with respect to the -Hodge operator, we define the higher eta invariant associatedwith the signature operator on Ñ, thus extending previous work of Lott. If ₁(M) M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (M = N)then, under the above invertibility assumption on , andalways employing symmetric spectral sections, we define acanonical Atiyah–Patodi–Singer index class, in K ₀(C ^* _r ()), for the signature operator of . Using the higherAPS index theory developed in [6], we express the Chern character ofthis index class in terms of a local integral and of the higher etainvariant defined above, thus establishing a higher APS index theoremfor the signature operator on Galois coverings. We expect the notion ofa symmetric spectral section for the signature operator to have widerimplications in higher index theory for signatures operators.     

Asymptotic distribution of n-th-order integral matrices belonging to a given residue class

One obtains an asymptotic formula for the number of n-th-order integral matrices which belongs to a given domain of the discriminant surface and to a given residue class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 85–111, 1981.     

L ∞ stability of finite element approximations to elliptic gradient equations

Summary We examine theL stability of piecewise linear finite element approximationsU to the solutionu to elliptic gradient equations of the form –·[a(x)u]+f(x, u)=g(x) wheref is monotonically increasing inu. We identify a prioriL bounds for the finite element solutionU, which we call reduced bounds, and which are marginally weaker than those for the original differential equations. For the general,N-dimensionai, case we identify new conditions on the mesh, such that under the assumption thatf is Lipschitz continuous on a finite interval,U satisfies the reducedL bounds mentioned above. The new,N-dimensional regularity conditions preclude quasi-rectangular meshes.Moreover, we show thatU is stable inL in two dimensions for a discretization mesh on which –·[a(x)u] gives rise to anM-matrix, whileU is stable for any mesh in one dimension. The condition that the discretization of –·[a(x)u] has to be anM-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion.The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions forU are obtained by use of (nonexpansive) pointwise projection operators.The first author is supported by the National Science Foundation under grant EET-8719100Research of the second author supported by National Science Foundation grant DMS.8420192     

Correction to the paper “semisimplicity of the algebra of indecomposable representations of a finite commutative semigroup”

The result indicated in the title of this paper (Ref. Zh. Mat. 1976, 7A225) is false. The error is corrected.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 112–113, 1979.     

Quasimodes and exponential splitting of a hammock

The one-dimensional Schrödinger equation with a potential having two nondegenerate minima is considered. We formulate and prove a theorem on the behavior of the eigenvalue curves in the neighborhood of points where the curves approach each other exponentially.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 195, pp. 103–112, 1991.     

Tactical Decompositions of Steiner Systems and Orbits of Projective Groups

Block's lemma states that the numbers m of point-classes and n of block-classes in a tactical decomposition of a 2-(v, k, ) design with b blocks satisfy m n m + b – v. We present a strengthening of the upper bound for the case of Steiner systems (2-designs with = 1), together with results concerning the structure of the block-classes in both extreme cases. Applying the results to the Steiner systems of points and lines of projective space PG(N, q), we obtain a complete classification of the groups inducing decompositions satisfying the upper bound; answering the analog of a question raised by Cameron and Liebler (P.J. Cameron and R.A. Liebler, Lin. Alg. Appl. 46 (1982), 91–102) (and still open).     

Correlation functions in the quantum inverse scattering method

A method for calculating correlation functions for completely integrable models in the framework of the quantum inverse scattering method is proposed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 92–112, 1984.The authors thank L. D. Faddeev for discussions.