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This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions \begin{eqnarray*} \begin{array}{rcl} \square \phi &=& F(\partial \phi)\,,\\[5pt] \phi(0,x) &=& f(x),\quad \partial_t \phi(0,x) = g(x)\,, \end{array} \end{eqnarray*} where $\square = \partial^2_t - \Delta$ is the wave operator, F is quadratic in ∂ ϕ with ∂ = (∂t, ∂x1,…, ∂xn). The minimal value of s is determined such that the above Cauchy problem is locally well-posed in Hs. It turns out that for the general equation s must satisfy \[ s > \max \left(\frac{n}{2}, \frac{n + 5}{4}\right). This is due to Ponce and Sideris (when n = 3) and Tataru (when n ≥ 5). The purpose of this paper is to supplement with a proof in the case n = 2,4.

Let G be a discrete group and (G, G+) an ordered group. Let (G, GF) be the minimal quasi-ordered group containing (G, G+). Let and be the corresponding Toeplitz algebras, and γGF,G+ the natural C*-algebra morphism from to . This paper studies the connection between Ker γGF, G+ and the minimal closed ideal of . It is proved that if G is amenable and GF≠ G+, then Ker γGF, G+ is exactly the minimal closed non-trivial ideal of . As an application, in the last part of this paper, a character of K-groups of Toeplitz algebras on ordered groups is clarified.

Let G be a discrete group, E1 and E2 be two subsets of G with E1⊆E2, and e∈E2. Denote by and the associated Toeplitz algebras. In this paper, it is proved that the natural morphism γE2,E1 from to exists as a C*-morphism if and only if E2 is finitely covariant-lifted by E1. Based on this necessary and sufficient condition, some applications are made.

In this paper, a complete classification of arc-transitive cubic graphs of order 4p is given.

Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general. This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g(t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.

In this paper the classical theorem of Zareckiĭ about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckiĭ theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one.

This paper characterizes ideal structure of the uniform Roe algebra B*(X) over simple cores X. A necessary and sufficient condition for a principal ideal of B*(X) to be spatial is given and an example of non-spatial ideal of B*(X) is constructed. By establishing an one-one correspondence between the ideals of B*(X) and the ω-filters on X, the maximal ideals of B*(X) are completely described by the corona of the Stone-Cech compactification of X.

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.

This paper is motivated by S. Park [10] in which the injective cover of left R[x]-module M[x-1] of inverse polynomials over a left R-module M was discussed. The author considers the Ω-covers of modules and shows that if η: P → M is an Ω-cover of M, then [ηS,≤] : [PS,≤] → [MS,≤] is an [ΩS,≤]-cover of left [[RS,≤]]-module [MS,≤], where Ω is a class of left R-modules and [MS,≤] is the left [[RS,≤]]-module of generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS,≤]]-module [MS,≤] are discussed.

This paper characterizes all connected graphs with exactly two Laplacian eigenvalues greater than two and all connected graphs with exactly one Laplacian eigenvalue greater than three.

This paper proves that: Let f be an entire function of finite order λ on Cn. Then (1) , where k(λ) is a nonnegative constant depending only on λ; (2) If , then λ is a positive integer and equals the lower order of f.

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.

Let G be a p-group (p odd prime) and let X=Cay(G,S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Aut(X) of X is isomorphic to the semidirect product GRAut(G, S), where GR is the right regular representation of G and Aut(G, S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a couterexample.

Supersaturated designs are useful in screening experiments. This paper discusses the topic or multi-level supersaturated design. Two quantities, E(d2) and Df, are proposed to evaluate the optimality of supersaturated designs. A lower bound of E(d2) is obtained with a necessary condition for achieving it. Some E(d2)-optimal supersaturated designs of 3, 4, and 5 levels are given.

Let L be a simple Lie algebra with irreducible root system Φ having roots of different length, F be a field of characteristic different from 2, G=L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φ. Set Gl= . It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.

Let D = (V,E) be a primitive digraph. The exponent of D, denoted by γ(D), is the least integer k such that for any u,v ∈ V there is a directed walk of length k from u to v. The local exponent of D at a vertex u ∈ V, denoted by expD(u), is the least integer k such that there is a directed walk of length k from u to v for each v ∈ V. Let V = {1, 2, …, n}. Following [1], the vertices of V are ordered so that expD(1) ≤ expD(2)≤ … ≤ expD(n) = γ (D). Let En(i): = {expD(i) | D ∈ PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n) = {γ(D) | D ∈ P Dn} has been completely settled by [7]. In 1998, En(1) was characterized by [5]. In this paper, the authors describe En(2) for all n ≥ 2.

In this paper, a conjecture about the existence of limit cycles in a multiply-connected region proposed by Ye Yanqian is proved.

This paper proves the existence of 4 families of nontrivial homotopy elements in the stable homotopy of spheres which are represented by α2bn,α2kn, α2gn and α2hnhm in the -terms of the Adams spectral sequence respectively, where α2, bn, kn, gn and hnhm are the known generators in the -terms whose internal degree are 4(p - 1) + 1, 2pn+1(p - 1), (4pn+1 + 2pn) (p - 1),(2pn+1 + 4pn)(p - 1), (2pn + 2pm)(p - 1) respectively and P ≥ 5 is a prime, m ≥ n + 2 ≥ 4.

In this paper, the author first gives a conjugacy-class version of Camina hypotheses and then applys the Camina group theory to discussing two classes of finite groups.