Let $\mathfrak{N}$:=${{\displaystyle H}}_{n}\times {\u2102}^{n}$ be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where ${{\displaystyle H}}_{n}$ denotes the set of all $n\times n$ Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on $\mathfrak{N}$ and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on $\mathfrak{N}$ is a unitary operator from Sobolev space W^{n}^{;2} into L^{2}($\mathfrak{N}$):

We prove L^{p} estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the ${L}^{\mathrm{1}}({S}^{n-\mathrm{1}})$ integrability condition. The obtained L^{p} estimates resolve a problem left open in previous work. Our argument is based on duality technique and direct estimation of operators. As a consequence of our result, we deduce the L^{p} boundedness of a class of fractional Marcinkiewicz integral operators.

We frst give the definition of a vertex superalgebroid. Then we construct a family of vertex superalgebras associated to vertex superalgebroids. As the main result, we find a sufficient and necessary condition that these vertex superalgebras are semi-conformal. In addition, we give a concrete example of a semi-conformal vertex superalgebra and apply our results to this superalgebra.

Let $\tilde{tsns}$ denote the twisted N = 1 Schrodinger-Neveu-Schwarz algebra over the complex field $\u2102$. In this paper, we determine the superskewsymmetric super-biderivations of $\tilde{tsns}$. Furthermore, we prove that every super-skewsymmetric super-biderivation of $\tilde{tsns}$ is inner.

We study properties of graded maximal Cohen-Macaulay modules over an $\mathbb{N}$-graded locally finite, Auslander Gorenstein, and Cohen-Macaulay algebra of dimension two. As a consequence, we extend a part of the McKay correspondence in dimension two to a more general setting.

We consider the symmetric scan Gibbs sampler, and give some explicit estimates of convergence rates on the Wasserstein distance for this Markov chain Monte Carlo under the Dobrushin uniqueness condition.

Given a sequence ${\{{b}_{i}\}}_{i=\mathrm{1}}^{n}$ and a ratio $\lambda \in (\mathrm{0},\mathrm{1})$, let $E={\cup}_{i=\mathrm{1}}^{n}(\lambda E+{b}_{i})$ be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in E: Our main idea is from the multiple β-expansions.

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix, the equality cases of the bounds are completely characterized by graph theory methods. Applying these bounds to a nonnegative irreducible matrix or a connected graph (digraph), we can improve the results of L. H. You, Y. J. Shu, and P. Z. Yuan [Linear Multilinear Algebra, 2017, 65(1): 113–128], and obtain some new or known results. Applying these bounds to a uniform hypergraph, we obtain some new results and improve some known results of X. Y. Yuan, M. Zhang, and M. Lu [Linear Algebra Appl., 2015, 484: 540–549]. Finally, we give a characterization of a strongly connected k-uniform directed hypergraph, and obtain some new results by applying these bounds to a uniform directed hypergraph.

We prove that, with at most $O\left({N}^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon}\right)$ exceptions, all even positive integers up to Nare expressible in the form ${p}_{\mathrm{1}}^{\mathrm{2}}+{p}_{\mathrm{2}}^{\mathrm{2}}+{p}_{\mathrm{3}}^{\mathrm{3}}+{p}_{\mathrm{4}}^{\mathrm{3}}+{p}_{\mathrm{5}}^{\mathrm{4}}+{p}_{\mathrm{6}}^{\mathrm{4}}$,where ${p}_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}}$ are prime numbers. This gives large improvement of a recent result $O\left({N}^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon}\right)$ due to M. Zhang and J. J. Li.

We present an explicit and recursive representation for high order moments of the first hitting times of single death processes. Based on that, some necessary or sufficient conditions of exponential ergodicity as well as a criterion on$\mathit{\ell}$-ergodicity are obtained for single death processes, respectively.

A generalized Volterra lattice with a nonzero boundary condition is considered by virtue of the inverse scattering transform. The two-sheeted Riemann surface associated with the boundary problem is transformed into the Riemann sphere by introducing a suitable variable transformation. The associated spectral properties of the lattice in single-valued variable was discussed. The constraint condition about the nonzero boundary condition and the scattering data is found.