Minimal decompositions of graphs into mutually isomorphic subgraphs

SupposeG _n={G ₁, ...,G _k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG _n we mean a set of partitions of the edge setsE(G _t ) of theG _i , sayE(G _t )== \(\sum\limits_{j = 1}^r {E_{ij} } \) E _ij , such that for eachj, all theE _ij , 1≦i≦k, are isomorphic as graphs. Define the functionU(G _n) to be the least possible value ofr aU-decomposition ofG _n can have. Finally, letU _k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges. In an earlier paper, the authors showed that $$U_2 (n) = \frac{2}{3}n + o(n).$$ In this paper, the value ofU _k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU _k (n) does not depend onk. In particular we show $$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$      

Finite union theorem with restrictions

The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties:

1. ifS ₁,S ₂, ...,S _{k + 1} are distinct pariwise disjoint elements ofL then there exists nonemptyI ? {1, 2, ...,k + 1} with ∪ _i∈I S _i ?L
2. ifL =L ₁ ?...?L _r is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS ₁,S ₂, ...,S _k ∈L _j , such thatL _i∈I S _i ∈L _j for every nonemptyI ? {1, 2, ...,k}.

     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

Some remarks on the existence of optimal controls for quasilinear systems

Let a quasilinear control system having the state space \(\bar X \subseteq R^n \) be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x ₀ andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m.LetG:U ?R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ^?1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.     

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.     

On the uniqueness of solutions to linear complementarity problems

This paper characterizes the classU of all realn×n matricesM for which the linear complementarity problem (q, M) has a unique solution for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. It is shown that restricting the uniqueness property to the interior ofK(M) is necessary because whenU, the problem (q, M) has infinitely many solutions ifq belongs to the boundary of intK(M). It is shown thatM must have nonnegative principal minors whenU andK(M) is convex. Finally, it is shown that whenM has nonnegative principal minors, only one of which is 0, andK(M)≠R ⁿ , thenU andK(M) is a closed half-space.     

Quasi-factors in ergodic theory

Motivated by the notion of quasi-factor in topological dynamics, we introduce an analogous notion in the context of ergodic theory. For two processes,X andY , we haveX?Y if and only ifY has a factor which is isomorphic to a quasi-factor ofX. On the other hand, weakly mixing processes can have nontrivial quasifactors which are not w.m. We characterize those ergodic processes which admit only trivial continuous ergodic quasi-factors, and use this characterization to conclude that a process with minimal selfjoinings is of this type. From this we derive the fact that for every suchX and any ergodicY eitherX ⊥Y orY extends some symmetric product ofX.     

When is the radical associated with a module a torsion?

For an arbitrary R-module M we consider the radical (in the sense of Maranda)G _M, namely, the largest radical among all radicalsG, such thatG(M). We determine necessary and sufficient on M in order for the radicalG(M) to be a torsion. In particular,G(M) is a torsion if and only if M is a pseudo-injective module.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.     

Two results for monocompact measures

For every finite measure space (Ω,A, P) whereA is K₁-generated we prove the equivalence of compactness and monocompactness for P . Moreover, we prove the existence of a perfect, not monocompaot probability, thus answering an open question in [6]. Let P be a charge on the algebraA andK ?A be a monocompact class. We show that P is o-additive ifK ^S P-approximatesK ^S, the family of finite unions inK , needs not to be monocompact.     

Spaces of compact operators and their dual spaces

LetE andF be reflexive Banach spaces andC the space of all compact linear operators fromE toF. A representation of the dual space ofC is given and it is proved thatC is either reflexive or nonconjugate. Applications of these results are also given.     

Asymptotic behaviour of the minima to a class of optimization problems for differential inclusions

Denoting byS _k (ξ _k ) the set of solutions of the Cauchy problem $\dot x \in F_k (t,x),x(0) = \xi _k $ , fork∈N∪{∞}, we prove that, under appropriate assumptions, the sequence {S _k (ξ _k )} _{k ∈ N} converges toS _∞(∈_∞) in the Kuratowski sense as well as in the Mosco sense. This result together with some facts from Γ-convergence theory are used to prove a result concerning the existence and the asymptotic behavior of the minima to the optimization problem $$\min \int_0^T {[g_k (t,x(t)) + h_k (t,\dot x(t))]} dt + \psi _k (\xi ),x \in S_k (\xi ),\xi \in K$$ withK a compact subset ofR ⁿ .     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

A note on residually small varieties of semigroups

It is proved for any varietyG of groups that if the subdirectly irreducible groups inG form a set, and if the subdirectly irreducible representation algebras of groups inG form a set, then every finite group inG is Abelian. The result is essential for the characterization of residually finite varieties of semigroups.     

Solutions ofLu=u α dominated byL-harmonic functions

LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=u ^α and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). Putu∈U ⁰ ifu∈U and ifu≤h for someh∈H. To everyu∈U ⁰ there corresponds the minimalL-harmonic functionh _u which dominatesu andu→h _u is a 1–1 mapping fromU ⁰ onto a subsetH ⁰ ofH. The inverse mapping associates with everyh∈H ⁰ the maximal element ofU dominated byh. Supposeg(x, dy) is Green's kernel,k(x, y) is the Martin kernel and ?M is the Martin boundary associated withL. A subset Γ of ?M is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if $\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } } $ is equal to 0 or ∞ for everyc∈M and every measure ρ. Everyh∈H has a unique representation $h(x) = \int\limits_{\partial M} {k\left( {x,y} \right)v\left( {dy} \right)} $ where ρ is a measure concentrated on the minimal partM ^* of ?M. We show that the condition:

1. ρ(Γ)=0 for allR sets Γ is necessary and the condition:
2. ρ(Γ)=0 for allM-polar sets Γ is sufficient forh to belong toH ⁰. IfM is a bounded domain of classC ^{2, λ} in ? ^d , then conditions (a) and (b) are equivalent and therefore each of them characterizesH ⁰. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall forL=Δ, α=2 and domains of classC ⁵.