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).     

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).     

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 the completeness of algebraic polynomials in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(ℝ, <Emphasis Type="Italic">d</Emphasis>μ)

We prove that the theorem on the incompleteness of polynomials in the space C ₀ ^w established by de Branges in 1959 is not true for the space L _p (ℝ, dμ) if the support of the measure μ is sufficiently dense.     

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.     

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.     

The spectrum of (<Emphasis Type="Italic">gv</Emphasis>, <Emphasis Type="Italic">g</Emphasis>, 3, λ)-DF in <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">gv</Emphasis></Subscript>

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)-difference family in Z _gv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type g ^v without short orbits is determined. This work was supported by National Natural Science Foundation of China (Grants Nos. 10771013, 10831002)     

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.     

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     

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.     

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.     

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.     

Finite-dimensional <Emphasis Type="Italic">ℓ</Emphasis>-simple lattice-ordered algebras with a <Emphasis Type="Italic">d</Emphasis>-basis

It is shown that a unital finite-dimensional ℓ-simple ℓ-algebra with a distributive basis is isomorphic to a lattice-ordered matrix algebra with the entrywise lattice order over a lattice-ordered twisted group algebra of a finite group with the coordinatewise lattice order. It is also shown that the isomorphism is unique.     

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).     

On common values of <Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) and <Emphasis Type="Italic">σ</Emphasis>(<Emphasis Type="Italic">m</Emphasis>). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.     

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.