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 \({\Pi _{{q^n}}}\).     

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).     

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.     

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.     

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 \(B_p(s)\).     

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.     

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.     

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.     

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.     

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     

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.     

<Emphasis Type="Italic">q</Emphasis>-Blossoming and Hermite-Padé approximants to the <Emphasis Type="Italic">q</Emphasis>-exponential function

We derive explicit expressions for the type II Hermite-Padé approximants to the q-exponential function using the q-blossom of its partial sums. In particular, this derivation leads to an interpretation of the Padé approximants as control points of a q-Bézier curve defined in terms of the partial sums of the q-exponential function.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Sparse recovery in probability via <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-minimization with Weibull random matrices for 0 &lt; <Emphasis Type="Italic">q</Emphasis> ≤ 1

Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the l_2,q robust null space property that can weaken the D-RIP, and show that Weibull random matrices satisfy the l_2,q robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the l _q quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the l _q -minimization with Weibull random matrices, one is on the stability estimate when the measurement noise e ∈ ? ⁿ needs a priori ||e||₂ ≤ ?, the other is on the robustness estimate without needing to estimate the bound of ||e||₂. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.     

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.     

Potentially <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> — <Emphasis Type="Italic">G</Emphasis>-graphical sequences: A survey

The set of all non-increasing nonnegative integer sequences π = (d(v ₁), d(v ₂), …, d(v _n )) is denoted by NS _n . A sequence π ∈ NS _n is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π. The set of all graphic sequences in NS _n is denoted by GS _n . A graphical sequence π is potentially H-graphical if there is a realization of π containing H as a subgraph, while π is forcibly H-graphical if every realization of π contains H as a subgraph. Let K _k denote a complete graph on k vertices. Let K _m −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of K _m ). This paper summarizes briefly some recent results on potentially K _m −G-graphic sequences and give a useful classification for determining σ (H, n).     

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.