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).     

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.     

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.     

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.     

On the heritability of the property <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">π</Emphasis></Subscript> by subgroups

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.     

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.     

An efficient calculation of the Clausen functions Cl<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">θ</Emphasis>)(<Emphasis Type="Italic">n</Emphasis>≥2)

The Clausen functions appear in many problems, such as in the computation of singular integrals, quantum field theory, and so on. In this paper, we consider the Clausen functions Cl _n (θ) with n≥2. An efficient algorithm for evaluating them is suggested and the corresponding convergence analysis is established. Finally, some numerical examples are presented to show the efficiency of our algorithm.     

Loop subgroups of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">r</Emphasis></Subscript> and the image of their stabilizer subgroups in GL<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>(ℤ)

For a representative class of subgroups of F _r , the image of their stabilizer subgroup under the action of Aut(F _r ) in GL _r (ℤ) is calculated.     

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.     

Bilocal automorphisms of <Emphasis Type="Italic">T</Emphasis><Subscript>∞</Subscript>(<Emphasis Type="Italic">F</Emphasis>)

We prove that if F is a field such that |F| > 2, then every bilocal automorphism of T _∞(F) - the algebra of ? × ? upper triangular matrices over F, is an automorphism.     

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.     

Convergence in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>([0, <Emphasis Type="Italic">T</Emphasis>]) of Wavelet Expansions of <Emphasis Type="Italic">φ</Emphasis>-Sub-Gaussian Random Processes

The article presents new results on convergence in L _p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.     

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.     

A theorem of the alternatives for the equation |<Emphasis Type="Italic">Ax</Emphasis>| − |<Emphasis Type="Italic">B</Emphasis>||<Emphasis Type="Italic">x</Emphasis>| = <Emphasis Type="Italic">b</Emphasis>

A theorem of the alternatives for the equation \({|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}\) is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2 ⁿ solutions for each positive right-hand side b.     

A characterization of the algebra <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>(Ω)

We study some properties of algebras of continuous functions on a locally compact space, these algebras being equipped with the topology defined by a family of multiplication operators (β-uniform algebras). We prove an analog of a theorem due to Sheinberg for β-uniform algebras [see Uspekhi Mat. Nauk, 32:5 (197) (1977), 203–204].     

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

A characteristic property of the algebra <Emphasis Type="Italic">C</Emphasis>(Ω)<Subscript><Emphasis Type="Italic">ß</Emphasis></Subscript>

We study some properties of the algebras of continuous functions on a locally compact space whose topology is defined by the family of all multiplication operators (β-uniform algebras). We introduce the notion of a β-amenable algebra and show that a β-uniform algebra is β-amenable if and only if it coincides with the algebra of bounded functions on a locally compact space (an analog of M. V. She?nberg’s theorem for uniform algebras).     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.