Denote by D_m the dihedral group of order 2m. Let ℛ(D_m) 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.

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 ([inline-graphic not available: see fulltext], d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on [inline-graphic not available: see fulltext] × ⋯ × [inline-graphic not available: see fulltext] for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.

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.

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.

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