Denote by D_m the dihedral group of order 2m. Let ?(Dm) be its complex representation ring, and let Δ(D_m) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δⁿ(D_m)/Δⁿ⁺¹(D_m) for each positive integer n.

Let (Mⁿ, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mⁿ, g) is a space form if it has sufficiently small L^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mⁿ, g) with positive scalar curvature.

In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the anticommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.

Let (X, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on L^p1(X) × · · · × L^pm(X) for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ?(t, s) = (t - s)^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938-950], the error analysis for this approach is carried out for 0<μ<1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

Kennaugh’s pseudo-eigenvalue equation is a basic equation that plays an extremely important role in radar polarimetry. In this paper, by means of real representation, we first present a necessary and sufficient condition for the general Kennaugh’s pseudo-eigenvalue equation having a solution, characterize the explicit form of the solution, and then study the solution of Kennaugh’s pseudo-eigenvalue equation. At last, we propose a new technique for finding the coneigenvalues and coneigenvectors of a complex matrix under appropriate conditions in radar polarimetry.

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

Correlations of active and passive random intersection graphs are studied in this paper. We present the joint probability generating function for degrees of G^active(n, m, p) and G^passive(n, m, p), which are generated by a random bipartite graph G^?(n, m, p) on n + m vertices.

Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any σ, π ∈ H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible imprimitive reflection group G(m, p, n) is a coset of the stabilizer of a point in {1, . . . , n} provided n is sufficiently large.

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A “turning point” technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian.

In this paper, we shall study the uniqueness problems on meromorphic functions sharing nonzero finite value or fixed point. We have answered some questions posed by Dyavanal. Our results improve or generalize a few of known results.

In this paper, for a given d×d expansive matrix M with |det M| = 2, we investigate the compactly supported M-wavelets for L2(?d). Starting with a pair of compactly supported refinable functions ? and ? ? satisfying a mild condition, we obtain an explicit construction of a compactly supported wavelet ψ such that {2j/2ψ(Mj·-k):j∈?, k∈?d} forms a Riesz basis for L2(?d). The (anti-)symmetry of such ψ is studied, and some examples are also provided.