Long paths and large cycles in finite graphs

Ore-type sufficient conditions ensuring the existence of a large cycle passing through any given path of length s for (s + 2)-connected graphs are given, and the extremal cases are characterized.     

On unavoidable digraphs in orientations of graphs

Let G be a graph, and let D be a directed graph. Write G → D to mean that, no matter how the edges of G are given orientations, a copy of D must appear as a subgraph of the resulting oriented graph. It is proved that among all G for which G → D, the minimum chromatic number is equal to the minimum k for which K_k → hom(D), where hom(D) is the set of homomorphs of D. Next, necessary and sufficient conditions are given for a directed graph to have a homomorphism into a given transitive tournament, directed path, or directed cycle. These results are then applied to various cases of the above theorem. In particular, the minimum chromatic number is evaluated whenever D is an oriented forest, and all D are characterized for which the minimum chromatic number is no more than three.     

Some new bibds with k = 6 and λ = 1

(v, 6,1) BIBDs are given for several new values of v. This reduces to 55 the number of v values for which existence of a (v,6,1) BIBD is in doubt. A new resolvable (565,5,1) BIBD is also given. © 1995 John Wiley & Sons, Inc.     

Function cones and interpolation

We consider interpolation of operators acting on functions that belong to a given cone Q with the so‐called decomposition property. The set of all positive functions whose level sets are the level sets of a given function is the main example, and the cone of all decreasing functions is a particular case. As applications, we obtain conditions for the identity (E₀ ∩ Q,E₁ ∩ Q)_θ,p = (E₀,E₁)_θ,p ∩ Q and interpolation results for operators which are bounded when restricted to a given family of characteristic funcions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global comparison of perturbed Green functions

Given a Green function G (e.g. with respect to (−Δ)^α/2, 0<α≤2) on a region X where G has a generalized triangle property and given a (G-bounded) signed measure μ on X, necessary and sufficient conditions are given for the existence of a perturbed Green function which is comparable with G. This is done in the general setting of measurable spaces. Applications to C^1,1-regions in R^d and, for the classical case α=2, to finitely connected regions in ℝ² are given.     

On the Cayley graphs of completely simple semigroups

In this paper, the Cayley graphs of completely simple semigroups are investigated. The basic structure and properties of this kind of Cayley graph are given, and a condition is given for a Cayley graph of a completely simple semigroup to be a disjoint union of complete graphs. We also describe all pairs (S,A) such that S is a completely simple semigroup, A⊆S, and Cay (S,A) is a strongly connected bipartite Cayley graph.     

Triangular Dissections, Aperiodic Tilings, and Jones Algebras

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type A_n determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of π/(n + 1). There are usually several possible infinite dissections compatible with a given n but a given one makes use of n/2 triangle types if n is even. Jones algebra with index [4 cos²(π/(n + 1))]⁻¹ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case n = 4, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The "tilings" associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints), using n/2 digits (if- n is even) and generalizing the Fibonacci numbers.     

Optimization of multiple covering of a bounded set with circles

Numerical algorithms for the optimization of multiple covering of a bounded set G in the plane P with equal circles are proposed. The variants in which G is a connected bounded set in P or a finite set in P are considered. The circles may be centered at arbitrary points of G or at points belonging to a given set. Minimization of the radius of the given number of circles and minimization of the number of circles of a given radius are considered. Models and solution algorithms are described, and estimates of the solutions provided by most variants are given. Numerical results are presented.     

Two operators on the lattice of completely regular semigroup varieties

In this paper, varieties of completely regular semigroups are studied. This paper is divided into six sections. Section 1 contains an introduction to varieties of completely regular semigroups and preliminaries. Most of the notation needed in this paper is given. In Section 2, the operators \La ( ) and \Ra ( ) on the lattice of subvarieties of varieties of completely regular semigroups are investigated. In Section 3, some further properties of the operators \La ( ) and \Ra ( ) are given. In Section 4, the semigroups generated by various subset of some operators are considered. In Section 5, the operators \La ( ) and \Ra ( ) are used in finding the join of two given varieties. The word problem for free objects in the variety OLBG is considered in Section 6 using the operator \La ( ) . June 1, 1999     

Gaussian Trivial Ring Extensions and FQP-rings

Let A be a commutative ring and E a nonzero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1, or ∞.     

Some new HSOLSSOMs of types hn and 1m u1

In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1^m u¹ and hⁿ are improved; for type 1^m u¹ there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hⁿ, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001     

Intervals of 1-Lipschitz aggregation operators,quasi-copulas,and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections. A significant part of this work was done during a visit of the second author at the Johannes Kepler University, Linz (Austria). The second author was supported by the grant VEGA 1/3012/06 and by the Science and Technology Assistance Agency (Contract No. APVT-20-003204). Both authors would like to thank the anonymous referee whose comments (including the two copulas C ₁ and C ₂ given in the Conclusion) not only solved the originally stated open problem in the negative, but also allowed them to formulate two more interesting open problems.     

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ? 2I and κ k = δ if D ? 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.     

Von neumaivn regularity of sf-rings

A ring R is called left (right) SF-ring if all simple left (right) R-modules are flat. It is proved that R is Von Neumann regular if R is a right SF-ring whoe maximal essential right ideals are ideals. This gives the positive answer to a qestion proposed by R. Yue Chi MIng in 1985, and a counterexample is given to settle the follwoing question in the negative: If R is an ERT ring which is one-sided V-ring, is R a left and right V-ring? Some other conditions are given for a SF-ring to be regular.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

Orthogonal latin square graphs

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.     

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.     

Einige unstetige stochastische Prozesse

Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG ^*(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx ₀ and ifF(x ₀)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.     

Covers and strong covers in directed bipartite graphs

Let G be a finite graph with directed bipartition (V⁺, V^?). Necessary and sufficient conditions are given for the existence of covers and strong covers that: (i) satisfy matching with respect to V⁺, and (ii) include a given set of edges that satisfies matching with respect to V⁺.