Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem

We develop a method of representation of distributive (V,0,1)-semilattices as semilattices of finitely generated ideals of locally matricial algebras. We use the method to reprove two representation results by G. M. Bergman and prove a new one that every distributive (0,1)-lattice is, as a semilattice, isomorphic to the semilattice of all finitely generated ideals of a locally matricial algebra. We apply this fact to solve the Γ-invariant problem. To the memory of Igor Slatkovsky Research supported by grant GAUK 12/97.     

The structure of uniformly Gâteaux smooth Banach spaces

It is shown that a Banach spaceX admits an equivalent uniformly Gateaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact. Supported by AV 101-97-02, AV 1019003 and GA ČR 201-98-1449. Supported by GA ČR 201-98-1449, AV 1019003 and GAUK 1/1998.     

Correction to: Killing spinor-valued forms and their integrability conditions

Annals of Global Analysis and Geometry - The authors gratefully acknowledge the support of the Grants GA19-06357S, GAUK 700217 and SVV-2017-260456.     

Algebraic theories, clones and their segments

Isomorphism and elementary equivalence of segments of clones of objects in concrete categories are investigated. A survey of results about the finitary case is presented and a new theorem about the infinitary case is proved.Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 and of the Grant Agency of the Charles University under the grant GAUK 349 is gratefully acknowledged.     

The Generalized Dolbeault Complex in Two Clifford Variables

The Penrose transform is used to construct a complex starting with the Dirac operator in two Clifford variables. The corresponding relative BGG complex and its direct image is computed for cohomology with values in line bundles induced by representations in singular infinitesimal character. The limit of the induced spectral sequence is computed in cases connected with the Dirac operator in two Clifford variables. The work presented here is a part of the research project MSM 0021620839 and was supported also by the grants GAUK 447/2004 and GA ČR 201/05/2117.     

PB∞-subgroups of Butler groups

It is well known [BF] that in the constructible universe (V=L) the class ofB ₁-groups is closed under prebalanced subgroups. A similar attempt at the increasing countable unions of prebalanced subgroups has been done in [B]. Here we give an affirmative answer to a similar question concerning the so-calledP B ^∞-subgroups. This research was initiated while the first author was on a short visit at the University of Colorado, Colorado Springs, in February 1994 and supported by Charles University grant GAUK 45/94.     

Application of MCMC to change point detection

A nonstandard approach to change point estimation is presented in this paper. Three models with random coefficients and Bayesian approach are used for modelling the year average temperatures measured in Prague Klementinum. The posterior distribution of the change point and other parameters are estimated from the random samples generated by the combination of the Metropolis-Hastings algorithm and the Gibbs sampler. (Supplement to the special issue of Appl. Math. 53 (2008), No. 3) The work is a part of the research project MSM 0021620839 financed by MŠMT. It was partially supported by grant GAČR 201/06/0186 (for the first author) and grant GAUK 135007-B (for the second author).     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

An honest stiff tree-like algebra

We construct in ZFC a compactificationb of integers such that the Boolean algebra Clopen(b) is tree-like and every one-to-one continuous mapping as well as every continuous onto mapping fromb tob is a homeomorphism, and every homeomorphism fromb onto itself is identity at all except finitely many points from .Presented by J. Mycielski.The research was supported by the grant GAUK 350.The author owes sincere thanks to the referee, whose comments bridged many gaps in the first version of the present paper.     

Dirac Operator in Two Variables from the Viewpoint of Parabolic Geometry

In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model G/P of a Cartan geometry. The first operator in this sequence is locally the Dirac operator in 2 Clifford variables, D = (D ₁, D ₂), where D _i = ∑ _j e _j . ∂ _ij . It follows from the construction that this operator is invariant with respect to the action of the group G. There are 2 other G-invariant differential operators following it so that the sequence of operators is exact. We compute the local expression of these operators and show that it coincides with the operators described in [2, 5, 6] by the tools of Clifford analysis. However, it follows from our approach that the operators are invariant. The work presented here was supported by the grants GAUK 447/2004 and GA ČR 201/05/H005.     

Lower semicontinuity of quasiconvex integrals

Summary New lower semicontinuity results for quasiconvex integrals. In particular, under certain structure conditions and growth 0≤F(ξ)≤C(∣≤∣ ^q ) the functional ∫ _Ω F(∇u)dx is proved to be lower semicontinuous onW ^1,q with respect to the weak convergence inW ^1,p, p≥q−1. Research supported by the grant No. 201/93/2171 of Czech Grant Agency (GAČR) and by the grant No. 364 of Charles University (GAUK). This article was processed by the author using the Springer-VerlagTex P Jourlg marco package 1991.     

Isomorphic embeddings and harmonic behaviour of smooth operators

LetY be a Banach space, 1<p<∞. We give a simple criterion for embedding ℓ _p ⊂Y, namely it suffices that the positive cone ℓ _p ⁺ ⊂Y. This result is applied to the study of highly smooth operators from ℓ _p intoY (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless ℓ _p/K ⊂Y for someK ∈ ℕ. Supported by grants GAUK 277/2001, GAČR 201-01-1198, A1019205.     

Approximation in Sobolev spaces of nonlinear expressions involving the gradient

We investigate a problem of approximation of a large class of nonlinear expressionsf(x, u, ∇u), including polyconvex functions. Hereu: Ω→R ^m , Ω⊂R ⁿ , is a mapping from the Sobolev spaceW ^1,p . In particular, whenp=n, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in additionn=m, satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation off(x, u, ∇u) are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions. The first author was supported by KBN grant no. 2-PO3A-055-14, and by a scholarship from the Swedish Institute. The second author was supported by Research Project CEZ J13/98113200007 and grants GAČR 201/97/1161 and GAUK 170/99. This research originated during the stay of both authors at the Max-Planck Institute for Mathematics in the Sciences in Leipzig, 1998, and completed during their stay at the Mittag-Leffler Institute, Djursholm, 1999. They thank the institutes for the support and the hospitality.     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

Liftings of distributive lattices by locally matricial algebras with respect to the Id c functor

We study representations of distributive -lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive -lattices and of a subcategory of L_d corresponding to a partially ordered class which cannot be lifted with respect to the Id_c functor. On the other hand, we prove that there is such a lifting of every diagram in L_d or of a subcategory L_d1 of L_d whose objects are all distributive -lattices and whose morphisms are -embeddings. This paper is dedicated to Walter Taylor. Received February 8, 2005; accepted in final form August 11, 2005. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by INTAS project 03-51-4110, the grant GAUK 448/2004/B-MAT, and the post-doctoral grant GAČR 201/03/P140.     

Sur les ensembles représentés par les partitions d'un entier n

Let n = n₁ + n₂ + … + n_j a partition Π of n. One will say that this partition represents the integer a if there exists a subsum n_il + n_i2 + … + n_il equal to a. The set (Π) is defined as the set of all integers a represented by Π. Let be a subset of the set of positive integers. We denote by p( ,n) the number of partitions of n with parts in , and by (( ,n) the number of distinct sets represented by these partitions. Various estimates for ( ,n) are given. Two cases are more specially studied, when is the set {1, 2, 4, 8, 16, …} of powers of 2, and when is the set of all positive integers. Two partitions of n are said to be equivalent if they represent the same integers. We give some estimations for the minimal number of parts of a partition equivalent to a given partition.     

Pseudo-Hamiltonian-connected graphs

Given a graph G and a positive integer k, denote by G[k] the graph obtained from G by replacing each vertex of G with an independent set of size k. A graph G is called pseudo-k Hamiltonian-connected if G[k] is Hamiltonian-connected, i.e., every two distinct vertices of G[k] are connected by a Hamiltonian path. A graph G is called pseudo Hamiltonian-connected if it is pseudo-k Hamiltonian-connected for some positive integer k. This paper proves that a graph G is pseudo-Hamiltonian-connected if and only if for every non-empty proper subset X of V(G), |N(X)|>|X|. The proof of the characterization also provides a polynomial-time algorithm that decides whether or not a given graph is pseudo-Hamiltonian-connected. The characterization of pseudo-Hamiltonian-connected graphs also answers a question of Richard Nowakowski, which motivated this paper.     

Input control in nonnegative matrix equations

In this paper we consider the nonnegative matrix system c_iA+u_iB=c_i+1 where the nonnegative matrix A is allowed to vary, within bounds. The cone control problem is to find a nonnegative matrix B such that if C_i is a nonnegative vector in a specified cone, then there is a nonnegative vector u_i such that c_i+1 is in that cone. We extend this problem to input control by finding a B such that the cone, generated by the rows of B, is as small as possible. Thus, the percent distribution of ∣u_iB∣ through the states of the sustem by u_iB is either constant or varies little.     

A generalized Desargues configuration and the pure braid group

In this paper, a configuration with n = (₂^d) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of (₅^d) such configurations if d> 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz-Zariski theorem implied by results of Salvetti that the fundamental group π of the complexification of its line arrangement is Artin's pure (or coloured) braid group on d strands.

In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of π based on this particular configuration.