Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

Hankel-Schur multipliers and multipliers of the space H1

In this paper there is given a sufficient condition for a Hankel matrix _F to belong to the space of Schur multipliers of all bounded operators in ² (or, what is the same, to the tensor algebra V²). It is shown that ifw is a nonnegative function on T, such that is a sequence of integers, {F_i}_j1 is a sequence of polynomials,) and, then _FV². It follows from this that under these conditions F is a multiplier of the space H¹, i.e.,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 113–119, 1984     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.     

Estimate of the norm ∥f(A1, A2)-f(B1, B2)∥ for pairs of self-adjoint commuting operators

In this note we announce the estimate for the difference mentioned in the title, where are the lengths of intervals containing the spectra of the operators A₁, B₂ and A₂, B₂; the function f is from the class Lip 1 (with constant 1). The outline of the proof of this estimate is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 175–177, 1984.     

Venkov's reduction theory of positive-quadratic forms

We prove the absolute finiteness of the number of faces (independent of the parameter) of Venkov's reduction domain (Izv. Akad. Nauk SSSR, Ser. Mat.4, 37–52 (1940)) ofn-ary positive quadratic forms. The casen=3 is given special consideration. We study the change of the reduction domain when changes along a line segment in the space of coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 108–116, 1983.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.     

Spherically symmetric solutions of Euclidean Yang-Mills equations

We consider the Euclidean Yang-Mills equations with structure groupSu(2). The action functional and the topological charge are invariant under the transformations, whereg runs over the set of unit length quaternions, andgx denotes the product ofg by the quaternionx=x ₄+ ix₁+jx_g+kx₃. ThisSu(2)-symmetry permits us to apply Coleman's principle. For the potentials A_µ, we obtain the following spherically symmetric Ansatz: while the Yang-Mills equations and the duality equations reduce to ordinary differential equations for the function. We show that every solution of the form (1) of the Yang-Mills equations with finite action and positive (negative) charge satisfies the duality equationsF=* F (respectively,F=–* F) and has charge 1 (respectively, –1). Moreover, we describe explicitly all solutions of form (1) of the duality equations; among them there are the 1-instanton solutions of Belavin, Polyakov, et al.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 126–132, 1984.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

Enumeration ofQ-acyclic simplicial complexes

Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( _k ⁿ⁻¹ )k-faces, (3)H _k (C)=0. We prove that for ,H _k−1(C) is a finite group, and our main result is: . This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.     

Isoperimetric inequalities for infinite hyperplane systems

Let be an infinite discrete system ofk-dimensional flats inn-dimensional Euclidean space. If the totalk-dimensional volume of the flats in intersected with the ball of center 0 and radiusr, divided by the volume of that ball, tends to a limit forr→∞, then this limit is called thedensity of . We consider isoperimetric problems of the following kind. If is a hyperplane system of positive density, find sharp upper bounds for the density of the system ofk-flats (k∈{0, ...,n−2}) that are generated as intersections of hyperplanes in . Ideas from the theory of uniform distribution of sequences are employed to define a large class of hyperplane systems, calleduniform, for which all necessary densities exist, isperimetric inequalities can be proved, and systems with extremal intersection densities can be characterized.     

Gauge group and gauge transformation in the continual theory of defects

A representation of the algebra (3)=t(3) S0(3, ) by differential Schaefer's operators is proposed, and an external algebra of (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 173–184, 1991.I wish to thank V. N. Popov for his interest.     

Spectral properties of one-dimensional disperse crystals

The connection between the spectral characteristics of the one-dimensional Schrödinger operator with periodic potential, asa, and the spectral characteristic of the Schrödinger operatorl(y)=–y+q(x)y with a decreasing potentialq(x) is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 197–211, 1984.     

Sul contatto di terz’ordine di due superficie algebriche lungo curve

Sunto Si dànno delle condizioni necessarie affinchè due superficie algebriche F, G d’una V ₃ algebrica non singolare possano presentare contatto d’ordine assegnato q −1 lungo una curva priva di punti multipli. Si approfondisce sopratuito il caso q=3, con applicazione al problema di determinare le superficie algebriche G_n di S ₃ che presentano contatto del 3^o ordine lungo una ⁿ con una superficie quartica F ₄ .     

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingx _i +x _j ≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope with integral vertices. We triangulate with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupB _d . This labeling gives rise to a shelling of the triangulation of , where theh-vector of (and the Ehrharth ^*-vector of can be computed as a descent statistic on a subset ofB _d defined in terms ofG. A recursive way of computing theh-vector of is also given, and a recursive formula for the volume of . This work was partially supported by grants from the Icelandic Council of Science and the Royal Swedish Academy of Sciences, respectively.     

Elementary equivalence and relatively free products of lattices

It is shown that if is a non-trivial variety of lattices, then there existA, B, C ∈ such thatB≡C but notA*B≡A*C. Except in the case when is the variety of all distributive lattices,A can be taken to consist of just one element. For the varietyD of all distributive lattices, it is shown that for anyB, C and any finiteA, B≡C if and only ifA*B≡A*C. The work of the first author was supported in part by NSF Grant GP-29129, and the work of the second author by a grant from the N.R.C. of Canada. Presented by C. C. Chang.     

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon with vertices inP of minimum area that satisfies one of the following properties: (1) is a convexk-gon, (2) is an empty convexk-gon, or (3) is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn ³). The space complexity isO(n) fork=4 andO(kn ²) fork5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.This paper includes work done while David Eppstein was at Columbia University, Department of Computer Science, and while Günter Rote and Gerhard Woeginger were at the Freie Universität Berlin, Fachbereich Mathematik, Institut für Informatik. Research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).     

Structures of polyhedra determined by submodular functions on crossing families

The present paper shows that for any submodular functionf on a crossing family with , if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique submodular function with such thatB(f) coincides with the base polyhedron associated with the submodular system . Here, iff is integer-valued, thenf ₁ is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.     

State-constrained control problems with neither coercivity norL 1 bounds on the controls

Summary A state-constrained, nonlinear, minimum problem is considered with dynamics depending sublinearly on a control which is not bounded in theL ¹ norm. Because of the lack of coercivity, the value map fails to be continuous, even in the unconstrained case. However, we prove that under suitable assumptions—which guarantee the continuity of the value maps of the problems withL ¹-bounded controls—the value map is upper semicontinuous and solves a Bellman equation with a continuous Hamiltonian. Moreover, the map obtained by by replacing its values at the horizon t=T with the values of the cost function turns out to be the maximal subsolution of the corresponding value problem. Entrata in Redazione il 31 dicembre 1997.     

Construction of representations of the Poincaré Lie algebra by extension of direct integrals of representations ofSL(2,C)

As a first application of a general method of construction, a class of representations of the Lie algebra of the Poincaré group , which includes Wigner’s representations with spin zero and positive mass, is obtained by forming suitable direct integrals of representations of class I of the principal series of SL(2, C) and extending them to by means of four additional momentum operators which are determined in explicit form. Lavoro eseguito nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le Applicazioni della Matematica alla Fisica ed all’Ingegneria, C.N.R. Entrata in Redazione il 9 marzo 1971.