This paper is a continuation of our previous work [Front. Math. China, 2016, 11(6): 1379–1418] where an efficient algorithm for computing the maximal eigenpair was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real (with some negative off-diagonal elements) or complex matrices.

Let Λ be a Koszul algebra, and let Mbe a graded Λ-bimodule. We prove that the trivial extension algebra of Λ by Mis also a Koszul algebra whenever Mis Koszul as a left Λ-module. Applications and examples are also provided.

Consider the general dispersive equation defined by (∗)$$\{\begin{array}{l}\mathrm{i}{{\displaystyle \partial}}_{t}u+\varphi (\sqrt{-\mathrm{\Delta}})u=\mathrm{0},(x,t)\in {\mathrm{?}}^{n}\times \mathrm{?},\\ u(x,\mathrm{0})=f(x),f\in \ell ({\mathrm{?}}^{n}),\end{array}$$where φ($\sqrt{-\mathrm{\Delta}}$) is a pseudo-differential operator with symbol φ(|ξ|). In this paper, for φ satisfying suitable growth conditions and the radial initial data f in Sobolev space, we give the local and global L^{q} estimate for the maximal operator ${{\displaystyle S}}_{\varphi}^{*}$ defined by ${{\displaystyle S}}_{\varphi}^{*}$f(x) = sup_{0<t<1}|S_{t,φ}f(x)|, where S_{t,φ}fis the solution of equation (∗). These estimates imply the a.e. convergence of the solution of equation (∗).

We study the counterparty risk for a credit default swap (CDS) in a regime-switching market driven by an underlying continuous-time Markov chain. We model the default dependence via some correlated Cox processes with regime-switching shot noise intensities containing common shock. Under the proposed model, the general bilateral counterparty risk pricing formula for CDS contracts with the possibility of joint defaults is presented. Based on some expressions for the conditional Laplace transform of the integrated intensity processes, semi-analytical solution for the bilateral credit valuation adjustment (CVA) is derived. When the model parameters satisfy some conditions, explicit formula for the bilateral CVA at time 0 is also given.

We consider the valuation of a correlation option, a two-factor analog of a European call option, under a Hull-White interest rate model with regime switching. More specifically, the model parameters are modulated by an observable, continuous-time, finite-state Markov chain. We obtain an integral pricing formula for the correlation option by adopting the techniques of measure changes and inverse Fourier transform. Numerical analysis, via the fast Fourier transform, is provided to illustrate the practical implementation of our model.

We prove that if ϕis a homogeneous harmonic map from a Riemann surface Minto a complex Grassmann manifold G(k, n),then the maps of the harmonic sequences generated by ϕare all homogeneous.

For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers (ADMM) has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function, ADMM or its direct extended version may not converge. In this paper, we consider the multi-block separable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

This paper is concerned with the contact process with random vertex weights on regular trees, and studies the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices xand yat rate λρ(x)ρ(y) for some λ>0,where {ρ(x), x∈T^{d}} are independent and identically distributed (i.i.d.) vertex weights. We show that when dis large enough, there is a phase transition at λ_{c}(d) ∈ (0,∞) such that for λ<λ_{c} (d),the contact process dies out, and for λ>λ_{c}(d),the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at λ_{e}(d) such that for λ<λ_{e}(d),the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as dincreases.

Let fbe a Maass cusp form for ${\Gamma}_{\mathrm{0}}(N)$ with Fourier coefficients ${\lambda}_{f}(n)$and Laplace eigenvalue $\frac{\mathrm{1}}{\mathrm{4}}}+{k}^{\mathrm{2}$.For real $\alpha \ne \mathrm{0}$ and β>0,consider the sum ${S}_{X}(f;\alpha ,\beta )={\displaystyle {\sum}_{n}{\lambda}_{f}(n)e(\alpha {n}^{\beta})\varphi (n/X)}$,where φis a smooth function of compact support. We prove bounds for the second spectral moment of ${S}_{X}(f;\alpha ,?\beta )$,with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X.This implies that if fhas its eigenvalue beyond ${X}^{\frac{\mathrm{1}}{\mathrm{2}}+\epsilon}$,the standard resonance main term for ${S}_{X}(f;\pm 2\sqrt{q},1/2),q\in {?}_{+}$,cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2)×GL(2).It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of ${K}^{\epsilon}\le L\le {K}^{1-\epsilon}$. The same bounds can be proved in a similar way for holomorphic cusp forms.

We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Z_{n}, extending the previous results for non-immigration cases in literature. As a by-product, the large deviations for Z_{n+}_{1}/Z_{n} are also studied. We can see that there is a phase transition in converging rates depending on the generating functions of both branching and immigration.

Diagnosability of a multiprocessor system is an important study topic. S. L. Peng, C. K. Lin, J. J. M. Tan, and L. H. Hsu [Appl. Math. Comput., 2012, 218(21): 10406–10412] proposed a new measure for fault diagnosis of the system, which is called the g-good-neighbor conditional diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a famous topological structure of interconnection networks, the n-dimensional star graph S_{n} has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of S_{n} under the PMC model and MM∗ model.

The boundedness of multilinear Calderón-Zygmund operators and their commutators with bounded mean oscillation (BMO) functions in variable exponent Morrey spaces are obtained.

We compute some generating series of integrals related to tautological bundles on Hilbert schemes of points on surfaces S[n], including the intersection numbers of two Chern classes of tautological bundles, and the Euler characteristics of Λ__{y}TS^{[}^{n}^{]}. We also propose some related conjectures, including an equivariant version of Lehn’s conjecture.

Let $\sigma =\{{{\displaystyle \sigma}}_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, and let G be a finite group. A set $\mathcal{H}$ of subgroups of G is said to be a complete Hall$\sigma $-set of G if every member $\ne \mathrm{1}$ of $H$ is a Hall σi-subgroup of G for somei ∈ I and $H$ contains exactly one Hall σi-subgroup of G for every i such that ${{\displaystyle \sigma}}_{i}\cap \pi (G)\ne \phi $. In this paper, we study the structure of G under the assuming that some subgroups of G permutes with all members of $\mathcal{H}$ .