Semigroups satisfying (xy)m = xmym

In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)^m = x^my^m as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)^m = x^my^m by a nil semigroup satisfying (xy)^m = x^my^m. Regular semigroups satisfying (xy)² = x²y² are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)² = x²y² if and only if it is a retract extension of a regular semigroup satisfying (xy)² = x²y² by a nil semigroup satisfying (xy)² = x²y²     

The pinched Veronese is Koszul

In this paper we prove that the coordinate ring of the pinched Veronese, k[X ³,X ² Y,XY ²,Y ³,X ² Z,Y ² Z,XZ ²,YZ ²,Z ³], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

Partition relations for uncountable ordinals

Partition relations of the form α→(α,m)², where α is an ordinal andm is a positive integer, are considered. Let κ be a cardinal. The following are proved: If κ is singular and 2^K=K ⁺ then (K⁺)²?((K⁺)²,3)². If κ is a strong limit cardinal, then², iff ((cfκ)²→((cfκ)²,m)². If κ is regular and K²→(K²,3)², then the κ-Souslin hypothesis holds. If K^ω<α+ and cfα=cfκ>ω, then α?(α,3)².     

Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B^?,A^?) such that B^?A^?=(AB)⁺_MN, and a class of pairs (B^?,A^? of this property is shown. A necessary and sufficient condition for (AB)^? to have the expression B^?A^? is also given.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Patroids

A matroid M over a set E of elements is semiseparated by a partition {S¹, S²} of E iff rank E = rank S¹ + rank S² + 1. Such a semiseparation defines in each Sⁱ a pair of matroids or patroid Pⁱ = (Mⁱ, mⁱ); the two patroids P¹, P² weld to form M. The operations of removing and contracting a non-degenerate element of a matroid produce a patroid. The properties of patroids, their bases, and circuits are discussed.     

On the homogenization of some linear problems in domains weakly connected by a system of traps

The aim of the paper is to study the asymptotic behaviour of solutions of second‐order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain Ω^ε weakly connected by a system of traps ??^ε, where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain Ω^ε ?Ω a bounded open set of ?³ such that Ω^ε =Ω₁^ε ∪Ω₂^ε ∪??^ε ∪ W^ε, where Ω₁^ε, Ω₂^ε are non‐intersecting subdomains strongly connected with respect to Ω, ??^ε is a system of traps and meas W^ε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain Ω^ε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology. Copyright © 2007 John Wiley & Sons, Ltd.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

广义逆多项式模的Attached素理想

设R是有单位元1的结合环,(S,≤)是严格全序Artin幺半群,M_R是右R-模,Att(M_R)与Att([M~(S,≤)]_([[R~(S,≤)]]))分别表示模M_R与广义逆多项式模[M~(S,≤)]_([[R~(S,≤)]])的所有Attached素理想组成的集合.该文主要讨论了广义幂级数环[[R~(S;≤)]]广义逆多项式模[[R~(S;≤)]]的Attached素理想的相关性质,证明了在一定条件下,有Att([M~(S,≤)]_([[R~(S,≤)]])={[[PR~(S;≤)]]P∈Att(M_R)}.这一结论表明广义逆多项式模([M~(S,≤)]_([[R~(S,≤)]])的Attached素理想在一定条件下可以用模M_R的Attached素理想来刻画,推广了Annin S在文献[1]中关于斜多项式环上逆多项式模的Attached素理想的相关结论.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Uniqueness of mild solutions of the Navier-Stokes equation and maximal Lp-regularity

In this Note, we give a new proof of the uniqueness of mild solutions of the Navier-Stokes equation in C([0,T) ; (L³(ℝ³))³). The main tool of the proof is the maximal L^p-regularity of the Laplacian in (L³ (ℝ³))³.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

Quasirecognition by prime graph of 2 D n (3α) where n = 4m + 1 ≥ 21 and α is odd

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D _n (3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D _n (3^α). We also show that if G is a finite group satisfying |G| = |² D _n (3^α)|, and Γ(G) = Γ(² D _n (3^α)), then G ? ² D _n (3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D _n (3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D _n (3^α) is quasirecognizable by the spectrum.