On the Gosper’s <Emphasis Type="Italic">q</Emphasis>-constant Π<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

Gosper introduced the functions sin_qz and cos_qz as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by \({\Pi _{{q^n}}}\) (n ∈ N). Gosper noticed that all his formulas on these constants have more than two of the \({\Pi _{{q^n}}}\). So, it is natural to raise the question of establishing identities involving only two of the \({\Pi _{{q^n}}}\). In this paper, our main goal is to give examples of such formulas in only two \({\Pi _{{q^n}}}\).     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

A Characterization of <Emphasis Type="Italic">PSU</Emphasis><Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis> ≥ 5

Based on the prime graph of a finite simple group, its order is the product of its order components (see [4]). We prove that the simple groups PSU₃(q) are uniquely determined by their order components. Our result immediately implies that the Thompsons conjecture and the Wujie Shis conjecture [16] are valid for these groups.AMS Subject Classification: 20D05, 20D60     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

Congruences for Ramanujan’s <Emphasis Type="Italic">ω</Emphasis>(<Emphasis Type="Italic">q</Emphasis>)

Recently, Bruinier and Ono investigated the arithmetic of the coefficients of Ramanujan’s mock theta function ω(q). In Ramanujan J. (submitted) they obtained congruences with respect to the modulus 512. Here we show that ω(q) modulo 5 is dictated by an elliptic curve.     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

The critical group of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> × <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

In this paper, the structure of the critical group of the graph K _m × C _n is determined, where m, n ≥ 3.     

Bishop’s theorem and differentiability of a subspace of <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">b</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let K be a Hausdorff space and C _b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gateaux and Fréchet differentiability of subspaces of C _b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C _b (K) is a dense G _δ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

Extended Srivastava’s triple hypergeometric <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">A,p,q</Emphasis></Subscript> function and related bounding inequalities

In this paper, motivated by certain recent extensions of the Euler’s beta, Gauss’ hypergeometric and confluent hypergeometric functions (see [4]), we extend the Srivastava’s triple hypergeometric function H _A by making use of two additional parameters in the integrand. Systematic investigation of its properties including, among others, various integral representations of Euler and Laplace type, Mellin transforms, Laguerre polynomial representation, transformation formulas and a recurrence relation, is presented. Also, by virtue of Luke’s bounds for hypergeometric functions and various bounds upon the Bessel functions appearing in the kernels of the newly established integral representations, we deduce a set of bounding inequalities for the extended Srivastava’s triple hypergeometric function H _A,p,q .     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Systematic Construction of <Emphasis Type="Italic">q</Emphasis>-Analogs of <Emphasis Type="Italic">t</Emphasis>−(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,λ)-Designs

In the present paper we consider a q-analog of t–(v,k,)-designs. It is canonic since it arises by replacing sets by vector spaces over GF(q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom.Dedicata vol. 63, pp. 247–253 (1996)] they are called t –(v,k,;q)- designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package. &nbsp; In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t –(v,k,)-designs on sets, obtaining this way new results on the existence of t–(v,k,;q)-designs on spaces for further quintuples (t,v,k,;q) of parameters. We present several 2–(6,3,;2)-designs, 2–(7,3,;2)-designs and, as far as we know, the very first 3-designs over GF(q).classification 05B05     

Pointwise multipliers and decomposition theorems in <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">∞,q</Emphasis></Superscript><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript>

In this paper we give a characterization of the pointwise multipliers of the holomorphic Triebel-Lizorkin spaces F^{, q}_s, and we apply these results to prove a corona decomposition theorem for these spaces.Mathematics Subject Classification (1991):32A37Both authors partially supported by DGICYT Grant BMF2002-04072-C02-01 and CIRIT Grant 2001-SGR-00172     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

Ramanujan complexes of type<Emphasis Type="Italic">Ã</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of typeà _d?1 associated with PGL _d (F) whereF is a local field of positive characteristic.     

The crossing number of <Emphasis Type="Italic">G</Emphasis> ▭ <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> for the graph <Emphasis Type="Italic">G</Emphasis> on six vertices

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C _n for some graphs G on five and six vertices and the cycle C _n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C _n , where G is a specific graph on six vertices.     

Some properties of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">φ</Emphasis></Subscript>-splines

A general approach to the construction of asymptotics of coordinate (not necessarily polynomial) B _φ -splines of an arbitrary order is proposed. Asymptotic representations for Lagrange type third order B _φ -splines are obtained. Bibliography: 4 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 11–21.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.