Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

Sufficiency and duality for multiobjective control problems under generalized (<Emphasis Type="Italic">B</Emphasis>, <Emphasis Type="Italic">ρ</Emphasis>)-type I functions

In this paper, we introduce the classes of (B, ρ)-type I and generalized (B, ρ)-type I, and derive various sufficient optimality conditions and mixed type duality results for multiobjective control problems under (B, ρ)-type I and generalized (B, ρ)-type I assumptions.     

On Kazhdan-Lusztig cells in type <Emphasis Type="Italic">B</Emphasis>

We prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type B are unions of combinatorial cells (defined using the domino insertion algorithm).     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Equivalence classes in the Weyl groups of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We consider two families of equivalence classes in the Weyl groups of type B _n which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer r. It has been shown that the first family coincides with left cells corresponding to the equal parameter Iwahori-Hecke algebra when r=0; the equivalence classes in the second family agree with left cells corresponding to a special class of choices of unequal parameters when r is sufficiently large. Our main result shows that the two families of equivalence classes coincide, suggesting the structure of left cells for remaining choices of the Iwahori-Hecke algebra parameters.      

Automorphisms of Chevalley groups of types <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> over local rings

In the paper, we prove that every automorphism of any adjoint Chevalley group of type B ₂ or G ₂ is standard, i.e., it is a composition of an “inner” automorphism, a ring automorphism, and a central automorphism. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 3–29, 2007.     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

The Characteristic Intersection Property of Line-Free Choquet Simplices in <Emphasis Type="Italic">E</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

   Abstract. Generalizing the characteristic intersection property of Choquet simplices, it is proved that for line-free convex bodies B ₁ and B ₂ in E ^d , the following conditions are equivalent: (i) there is a line-free convex body B ⊂ E ^d such that every nonempty intersection B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , is a homothetic copy of B , (ii) both B ₁ and B ₂ are Choquet simplices and the nonempty intersections B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , are homothetic copies of a Choquet simplex B . All such triplets B ₁ ,B ₂ ,B are described.     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

2-(<Emphasis Type="Italic">v,k</Emphasis>,1) Designs and PSL (3,<Emphasis Type="Italic">q</Emphasis>) where <Emphasis Type="Italic">q</Emphasis> is ODD

Let G be a block-primitive automorphism group of a 2-(v,k,1) design. If G is isomorphic to PSL (3,q) where q is odd, then G is also point-primitive. Supported by the National Natural Science Foundation of China (10171089).     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

<Emphasis Type="Italic">Eω</Emphasis>-Clifford congruences and <Emphasis Type="Italic">Eω</Emphasis>-<Emphasis Type="Italic">E</Emphasis>-reflexive congruences on an inverse semigroup

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω-E-reflexive semigroups, are studied. ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are found to be the least Eω-Clifford and Eω-E-reflexive congruences on S, respectively. Characterizations of the congruences ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are developed. The lattices of all Eω-Clifford and Eω-E-reflexive congruences are also studied.     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

Generalization of the Kneser theorem on zeros of solutions of the equation <Emphasis Type="Italic">y</Emphasis>″ + <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)<Emphasis Type="Italic">y</Emphasis> = 0

We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 571–576, April, 2007.     

Lacunary Strong (<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript><Emphasis Type="Italic">, p</Emphasis>)-Convergence

The definition of lacunary strongly convergence is extended to the definition of lacunary strong (A _σ , p)-convergence with respect to invariant mean when A is an infinite matrix and p = (p _i ) is a strictly positive sequence. We study some properties and inclusion relations.     

Monomial Realization of Crystal Bases <Emphasis Type="Italic">B</Emphasis>(∞) for the Quantum Finite Algebras

In this paper, we give a new realization of the crystal basis B(∞) using modified Nakajima monomials for the quantum finite algebras. Moreover, as an application, we obtain the image of the Kashiwara embedding Ψ _ι from this realization of B(∞). Presented by Alain Verschoren.