Hamiltonism and Partially Square Graphs

 Given a graph G, we define its partially square graph G ^* as the graph obtained by adding edges uv whenever the vertices u and v have a common neighbor x satisfying the condition N _G[x]⊆N _G[u]∪N _G [v], where N _G[x]=N _G(x)∪{x}. In particular, this condition is satisfied if x does not center a claw (an induced K _1,3). Obviously G⊆G ^*⊆G ², where G ² is the square of G. We prove that a k-connected graph (k≥2) G is hamiltonian if the independence number α(G ^*) of G ^* does not exceed k. If we replace G ^* by G we get a well known result of Chvátal and Erdo?s. If G is claw-free and G ^* is replaced by G ² then we obtain a result of Ainouche, Broersma and Veldman. Relationships between connectivity of G and independence number of G ^* for other hamiltonian properties are also given in this paper. Received: June 17, 1996 Revised: October 30, 1998     

Cycles in Partially Square Graphs

. In this work we consider finite undirected simple graphs. If G=(V,E) is a graph we denote by α(G) the stability number of G. For any vertex x let N[x] be the union of x and the neighborhood N(x). For each pair of vertices ab of G we associate the set J(a,b) as follows. J(a,b)={u∈N[a]∩N[b]∣N(u)⊆N[a]∪N[b]}. Given a graph G, its partially squareG ^* is the graph obtained by adding an edge uv for each pair u,v of vertices of G at distance 2 whenever J(u,v) is not empty. In the case G is a claw-free graph, G ^* is equal to G ². If G is k-connected, we cover the vertices of G by at most ⌈α(G ^*)/k⌉ cycles, where α(G ^*) is the stability number of the partially square graph of G. On the other hand we consider in G ^* conditions on the sum of the degrees. Let G be any 2-connected graph and t be any integer (t≥2). If ∑ _{x ∈ S} deg _G (x)≥|G|, for every t-stable set S⊆V(G) of G ^* then the vertex set of G can be covered with t−1 cycles. Different corollaries on covering by paths are given. Received: January 22, 1997 Final version received: February 15, 2000     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

Crossed products on the flow of groupoid with quasi-invariant measures

LetG be a second countable groupoid with Haar system {λ ^u },A be an abelian group which left invariant acts onG. Then we have aC ^*-dynamic system (C ^* (G, A, β). In this paper we have studied the existence of quasi-invariant measure with certain properties; using these measures some important results about crossed products and groupoidC ^*-algebras have been obtained. This work is supported by National Natural Science Foundation of China     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.     

ON SYMMETRIC UNITS IN GROUP ALGEBRAS

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g ^?1 of G can be extended linearly to an anti-automorphism a → a ^* of KG. Let S _* (KG) = {x ∈ U(KG) | x ^* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S _* (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.     

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ^∞(G)^* the conjugate ofl ^∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ^∞(G)^* such that γμ=0 whenever γ is an annihilator ofC ₀(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ^∞(G)^*, and each of them generates a left ideal ofl ^∞(G)^* that contains no minimal left ideal.     

Power cocentralizing generalized derivations on prime rings

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u)) ⁿ = 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s ₄ and char (R) = 2; (3) R satisfies s ₄ and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b) ⁿ = 0.     

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

Bounded Direction–Length Frameworks

A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ ^d for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G ⁺ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G ⁺,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G ⁺,p). We use this to characterize when a mixed graph is generically bounded in ℝ ^d . As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.     

Aspects of the conjugacy class structure of simple algebraic groups

Let G be an adjoint simple algebraic group over an algebraically closed field of characteristic p; let Φ be the root system of G, and take t∈ℕ. Lawther has proven that the dimension of the set G _[t]={g∈G:g ^t =1} depends only on Φ and t. In particular the value is independent of the characteristic p; this was observed for t small and prime by Liebeck. Since G _[t] is clearly a disjoint union of conjugacy classes the question arises as to whether a similar result holds if we replace G _[t] by one of those classes. This paper provides a partial answer to that question. A special case of what we have proven is the following. Take p,q to be distinct primes and G _p and G _q to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic p and q respectively. If s∈G _p has order q then there exists an element u∈G _q such that o(u)=o(s) and dimu^G_q=dims^G_p\dim u^{G_{q}}=\dim s^{G_{p}} .     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m (Ω), with m>N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs (u_e^*,y_e^*)(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast }) of (P _ε ) converges as ε goes to 0 to a pair (u ^*,y ^*) which realizes the minimum in (P), and y ^* is the solution to the original state system.     

On the Oka principle in a Banach space,II

 Let X be one of the Banach spaces c ₀ , ℓ _p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G ^* for structure group. We show that a suitable Runge-type approximation hypothesis on X, G ^* (which we also prove for G ^* a solvable Lie group) implies the vanishing of the sheaf cohomology groups H ^q (Ω, 𝒪 ^E ), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪^Γ (𝒞^Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G ^* for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G ^* (which we prove again for G, G ^* solvable Lie groups) implies the injectivity (and for X=ℓ₁ also the surjectivity) of the Grauert–Oka map H ¹ (Ω, 𝒪^Γ)→H ¹ (Ω, 𝒞^Γ) of multiplicative cohomology sets. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L20, 32L05, 46G20 RID="*" ID="*" Kedves Laci Móhan kisfiamnak. RID="*" ID="*" To my dear little Son