A limit theorem for matrix-solutions of Hamiltonian systems

The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A ^n?1) B(x)≠N,B(x)∈C _n(R), andB(x)(A ^T)^j-1 C(x)∈C _n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx ₀∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A ^TV, and whenever η_i(x)∈R ^N satisfy η_i(x ₀)=U(x ₀)d _i ∈ imageU(x ₀) η′_i-Aη_ni(x) ∈ imageB(x),B(x)(η′_i(x)-Aη_i(x)) ∈C _n-1 R fori=1,2.     

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.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

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.     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

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 .     

Multifractal analysis of the divergence of Fourier series: The extreme cases

We study the size, in terms of the Hausdorff dimension, of the subsets of T such that the Fourier series of a generic function in L ¹(T), L ^p (T), or C(T) may behave badly. Genericity is related to the Baire Category Theorem or the notion of prevalence. This paper is a continuation of [3].     

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 ^*.     

On cleavability overT i,ρ spaces

Si danno risposte, per le principali classiP di spazi topologici separati, al seguente problema: “SiaX uno spazio topologico spezzabile sulla classeP. È vero o no cheX∈P?”. In particolare si studia il problema per le classiP of spaziT _i,ρ (i=2,3,4,5), sotto particolari tipi di spezzabilità.     

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.     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

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.     

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.     

k-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operatorT∈L(⊕ ₁ ² H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT ₁ is hyponormal andT ₃ ^k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.