Quotient Complexes and Lexicographic Shellability

Let _n,k,k and _n,k,h , h < k, denote the intersection lattices of the k-equal subspace arrangement of type _n and the k,h-equal subspace arrangement of type _n respectively. Denote by the group of signed permutations. We show that ( _n,k,k )/ is collapsible. For ( _n,k,h )/ , h < k, we show the following. If n 0 (mod k), then it is homotopy equivalent to a sphere of dimension . If n h (mod k), then it is homotopy equivalent to a sphere of dimension . Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of on the homology groups of ( _n,k,k ) and ( _n,k,h ) are stated.The collapsibility of ( _n,k,k )/ is established using a discrete Morse function. The same method is used to show that ( _n,k,h )/ , h < k, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Björner and M. Wachs (Trans. Amer. Math. Soc. 4 (1996), 1299–1327) when the cell complex is an order complex of a poset.     

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.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Product Representations in ΠΣ-Fields

-fields are a very general class of difference fields that enable one to discover and prove multisum identities arising in combinatorics and special functions. In this article we focus on the problem how such multisums can be represented in terms of -fields. In particular we consider product representations and their simplifications in -fields.Received July 09, 2003     

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     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N –2),N 3 be the limiting Sobolev exponent and ^N a bounded smooth domain. We show that for H ^–1(),f satisfies some conditions then–u=c ₁ u ^p–1 +f(x,u) + admits at least two positive solutions.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Sufficient Conditions for the Existence of Disintegrations

Let be a probability space and a partition of . A necessary and sufficient condition is given for the existence of a -additive and measurable disintegration of P on . It is also shown that P admits a -additive (but not measurable) disintegration on whenever is a standard space and the set (₁, ₂):₁ and ₂ are in the same element of } is coanalytic in ×. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using -additive disintegrations as conditional probabilities.     

Grid structure of a ballistic function and a computing instrument with linear potentiometers for its generation

Zusammenfassung Die Arbeit befasst sich mit der ballistischen Beziehung zwischen Schrägdistanzs, Lagewinkel und Geschossflugzeitt zum Ziel im Vakuum.Der 1. Teil behandelt die Gewebestruktur dieser Beziehung. Es zeigt sich, dass sie annähernd ein Sechseckgewebe ist. Eine topologische TransformationS(s), (),T(t) wird so gefunden, dass eine Zerlegung in Funktionen einer Variablen in der FormT(t)=S(s)+() die Beziehungt(s,) mit befriedigender Genauigkeit darstellt (der maximale relative Fehler ist weniger als 1%).Im 2. Teil wird auf dieser Grundlage ein Rechengerät zur Darstellung der Funktiont(s, ) konstruiert, das lineare Potentiometer als Schaltelemente benutzt.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

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.     

On a problem of B. I. Golubov

. . . : s_n(x) — , _n-(, 1)- n L ². , . , : a _k l ² _n () [0,1] , (*) , (**) . a _k l ² u _n () [0,1] , (**), (*) .     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

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.