A remark on covering graphs

Every graph G may be transformed into a covering graph either by deletion of edges or by subdivision. Let _E (G) and _V (G) denote corresponding minimal numbers. We prove _E (G) = _V (G) for every graph G.     

Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let be the fundamental group of a closed orientable surface of genus g 1, and let R(, G)/G be the space of conjugacy classes of representations of into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map ¯ on R(, G)/G and investigate whether the fibres of ¯ are isotropic with respect to the natural symplectic structure on R(, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of ¯ are also isotropic, but we give an example to show that in general they are not.     

The odd ordér analogue of Dempwolff's B-group problem

Let be a translation plane of odd order p^2R for p a prime. Let admit at least two p-groups, B_i, i=1, 2, B₁B₂ of orders > p^R/2 which fix Baer subplanes _i, i = 1,2, ₁₂ pointwise. It is shown that under these assumptions must be a Hall plane.     

Über lokalarchimedische Anordnungen projektiver Ebenen

The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that is Archimedean, i.e. that every ternary ring of is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes.     

Schwarz-Pick Inequalities for Hyperbolic Domains in the Extended Plane

Let and be two hyperbolic simply connected domains in the extended complex plane C¯ = C¯ {}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: at a point z ₀ . The bounds depend on the densities (z ₀) and (f(z ₀)) of the Poincaré metrics and on the hyperbolic distances of the points z ₀ and f(z ₀) to the point .     

Π 2 1 -Logic and uniformization in the analytical hierarchy

We give a simple proof of ₁ ¹ and ₂ ¹ uniformization results, which is based on the use of ₂ ¹ -Logic.     

Über die Tonneliertheit von lokalkonvexen Tensorprodukten

It is known that the inductive tensor product of two barrelled spaces is barrelled and that the projective tensor product of two barrelled metrizable spaces or barrelled (DF)-spaces is barrelled. In this note it will be shown by a counterexample that the projective tensor product EF of two barrelled spaces E and F in general is not barrelled, even if E is (DF)-Montel-space and F (F)-Montel-space. Furthermore we show that the -tensor product of two (B)-spaces in general is not barrelled. It follows from the fact that an (F)-space E is nuclear if and only if the -tensor product E l ⁴ is barrelled.     

The cohomology of invariant variational bicomplexes

Let: E M be a fiber bundle and let be an infinitesimal Lie transformation group acting onE. We announce various new results concerning the cohomology of the invariant variational bicomplex ( ^*,* (J(E)), d_H, d_V) and the associated invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups.     

The Oscillator Character Formula,for isometry groups of split forms in deep stable range

Let and be a reductive dual pair of the type mentioned in the title, with the smaller member. Let and be unitary representations of , which occur in Howe's correspondence. We express the distribution character of in terms of the character of via an explicit integral kernel operator.Oblatum 4-I-1995Research partially supported by the¹UMK Grant 514-M, and the²NSF Grant DMS 9204488.     

Hilbert Series of Group Representations and Gröbner Bases for Generic Modules

Each matrix representation :G GL_n() of a finite Group G over a field induces an action of G on the module ⁿ over the polynomial algebra The graded -submodule M() of _n generated by the orbit of is studied. A decomposition of M() into generic modules is given. Relations between the numerical invariants of and those of M(), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of can be read off from the Hilbert series of M(Pi;). It is proved that determinantal relations form Gröbner bases for the syzygies on generic matrices with respect to any lexicographic order. Gröbner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(Pi;) is obtained for an arbitrary representation.     

An intersection property of the simple random walks inZ d

Let {S _d (n)} _n0 be the simple random walk inZ ^d , and ^(d)(a,b)={S _d (n)Z ^d :anb}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE _n ^(d) ={^(d)(0,n)^(d)(n+f(n),)}. In this paper, a necessary and sufficient condition to ensureP(E _n ^d) ,i.o.)=0, or 1 is derived ford=3, 4. This problem was first studied by P. Erdös and S.J. Taylor.This work is partly supported by the National Natural Sciences Foundation of China.     

Standard completeness theorem for ΠMTL

MTL is a schematic extension of the monoidal t-norm based logic (MTL) by the characteristic axioms of product logic. In this paper we prove that MTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of MTL-algebras (bounded commutative cancellative residuated l-monoids) in the real unit interval [0,1] generates the variety of all MTL-algebras.The work was supported by the Grant Agency of the Czech Republic under projects GACR 201/02/1540, 401/03/H047, and by Net CEEPUS SK-042.Set offprint requests to: Rostislav Horík     

A Note on the Place Topology of Projective Planes

It is shown that the place topology induced by a proper epimorphism of a projective plane , which is known to make a Lenz-topological plane, makes even a topological projective plane, if the extended radical of some underlying ternary field is bounded.     

A note on theΠ 2 0 -induction rule

It is well-known (due to C. Parsons) that the extension of primitive recursive arithmeticPRA by first-order predicate logic and the rule of ₂ ⁰ -induction ₂ ⁰ -IR is ₂ ⁰ -conservative overPRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbersAC ^0,0-qf. More precisely we show that :=PRA ² + ₂ ⁰ -IR+AC ^0,0-qf proves the totality of the Ackermann function, wherePRA ² is the extension ofPRA by number and function quantifiers and ₂ ⁰ -IR may contain function parameters.This is true even forPRA ² + ₁ ⁰ -IR+ ₂ ⁰ -IR ^–+AC ^0,0-qf, where ₂ ⁰ -IR ^– is the restriction of ₂ ⁰ -IR without function parameters.I am grateful to an anonymous referee whose suggestions led to an improved discussion of our results     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

A Solution of a Problem of A. E. Brouwer

Let q=p^e 1(mod 4) be a prime power, and let ^(q) be the Paley graph over the finite field . Denote by ^(q) the subgraph of ^(q) induced on the set of non-zero squares of . In this paper the full automorphism group of ^(q) is determined affirming the conjecture of Brouwer [Des. Codes Cryptograph. 21, 69–76 (2000)]. The proof combines spectral and Schur ring techniques.     

On simple polytopes

Summary LetP be a simpled-polytope ind-dimensional euclidean space , and let (P) be the subalgebra of the polytope algebra generated by the classes of summands ofP. It is shown that the dimensions of the weight spaces _r(P) of (P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality between _r (P) and _d-r (P). Moreover, (P) admits a Lefschetz decomposition under multiplication by the element of ₁(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between (P) and the face ring of the dual simplicial polytopeP ^* enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP ^* is also established.Oblatum 1-X-1992     

Extensions of Laguerre operators in indefinite inner product spaces

The Laguerre-Sonin polynomialsL _n ⁽⁾ are orthogonal in linear spaces with indefinite inner product if<–1. We construct the completion () of this space and describe self-adjoint extensions of the Laguerre operatorl(y)=xy+(1+–x)y,<–1, in the space (). In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctions coincide with the Laguerre-Sonin polynomials and form an orthogonal basis in ().Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 509–521, April, 1998.This research was partially supported by the INTAS foundation under grant No. 93-02449.     

Convergence of the method of orthogonal polynomials for approximate solution of integral equations of the first kind with Π-kernels

The method of orthogonal polynomials for approxiamte solution of integral equations of the first kind with -kernels is substantiated. The solvability of the corresponding algebraic systems is proved, and estimates for the rate of the convergence of approxiamte solutions to exact ones are established. The applicability of the method of orthogonal polynomials to approximate solution of integral equations of the second kind with -kernels is shown, as well as its high effectiveness in solving integral equations of the first kind with -kernels. Concrete -kernels, most often occurring in applications, are presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 223–229, February, 1991.