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Frontiers of Mathematics in China


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, Volume 12 Issue 5 Previous Issue   
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Global algorithms for maximal eigenpair
Front. Math. China. 2017, 12 (5): 1023-1043.   DOI: 10.1007/s11464-017-0658-8
Abstract   PDF (219KB)

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.

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Trivial extension of Koszul algebras
Front. Math. China. 2017, 12 (5): 1045-1056.   DOI: 10.1007/s11464-017-0655-y
Abstract   PDF (160KB)

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.

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Maximal estimate for solutions to a class of dispersive equation with radial initial value
Yong DING, Yaoming NIU
Front. Math. China. 2017, 12 (5): 1057-1084.   DOI: 10.1007/s11464-017-0654-z
Abstract   PDF (252KB)

Consider the general dispersive equation defined by {itu+ϕ(Δ)u=0,(x,t)?n×?,u(x,0)=f(x),f(?n),where φ(Δ) 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 Lq estimate for the maximal operator Sϕ* defined by Sϕ*f(x) = sup0<t<1|St,φf(x)|, where St,φfis the solution of equation (∗). These estimates imply the a.e. convergence of the solution of equation (∗).

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Valuation of CDS counterparty risk under a reduced-form model with regime-switching shot noise default intensities
Yinghui DONG, Kam Chuen YUEN, Guojing WANG
Front. Math. China. 2017, 12 (5): 1085-1112.   DOI: 10.1007/s11464-017-0656-x
Abstract   PDF (445KB)

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.

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Valuation of correlation options under a stochastic interest rate model with regime switching
Kun FAN, Rongming WANG
Front. Math. China. 2017, 12 (5): 1113-1130.   DOI: 10.1007/s11464-017-0608-5
Abstract   PDF (757KB)

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.

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Homogeneity-preserving property of harmonic sequences from surfaces into complex Grassmann manifolds
Jie FEI, Wenjuan ZHANG
Front. Math. China. 2017, 12 (5): 1131-1137.   DOI: 10.1007/s11464-017-0639-y
Abstract   PDF (118KB)

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.

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Convergence of ADMM for multi-block nonconvex separable optimization models
Ke GUO, Deren HAN, David Z. W. WANG, Tingting WU
Front. Math. China. 2017, 12 (5): 1139-1162.   DOI: 10.1007/s11464-017-0631-6
Abstract   PDF (236KB)

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.

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Contact process on regular tree with random vertex weights
Yu PAN, Dayue CHEN, Xiaofeng XUE
Front. Math. China. 2017, 12 (5): 1163-1181.   DOI: 10.1007/s11464-017-0633-4
Abstract   PDF (212KB)

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), xTd} are independent and identically distributed (i.i.d.) vertex weights. We show that when dis large enough, there is a phase transition at λc(d) ∈ (0,) such that for λ<λc (d),the contact process dies out, and for λ>λc(d),the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at λe(d) such that for λ<λe(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.

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Spectral square moments of a resonance sum for Maass forms
Nathan SALAZAR, Yangbo YE
Front. Math. China. 2017, 12 (5): 1183-1200.   DOI: 10.1007/s11464-016-0621-0
Abstract   PDF (210KB)

Let fbe a Maass cusp form for Γ0(N) with Fourier coefficients λf (n)and Laplace eigenvalue 14+k2.For real α0 and β>0,consider the sum SX(f; α, β) =nλf(n)e(αnβ)ϕ(n/X),where φis a smooth function of compact support. We prove bounds for the second spectral moment of SX(f; α,?β),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 X12+ε,the standard resonance main term for SX(f;±2q, 1/2), q?+,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 KεLK1ε. The same bounds can be proved in a similar way for holomorphic cusp forms.

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Harmonic moments and large deviations for supercritical branching processes with immigration
Front. Math. China. 2017, 12 (5): 1201-1220.   DOI: 10.1007/s11464-017-0642-3
Abstract   PDF (190KB)

We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Zn, extending the previous results for non-immigration cases in literature. As a by-product, the large deviations for Zn+1/Zn 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.

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g-Good-neighbor conditional diagnosability of star graph networks under PMC model and MM* model
Shiying WANG, Zhenhua WANG, Mujiangshan WANG, Weiping HAN
Front. Math. China. 2017, 12 (5): 1221-1234.   DOI: 10.1007/s11464-017-0657-9
Abstract   PDF (172KB)

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 Sn has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of Sn under the PMC model and MM∗ model.

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Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces
Wei WANG, Jingshi XU
Front. Math. China. 2017, 12 (5): 1235-1246.   DOI: 10.1007/s11464-017-0653-0
Abstract   PDF (168KB)

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

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Generating series of intersection numbers on Hilbert schemes of points
Zhilan WANG, Jian ZHOU
Front. Math. China. 2017, 12 (5): 1247-1264.   DOI: 10.1007/s11464-017-0686-4
Abstract   PDF (211KB)

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_yTS[n]. We also propose some related conjectures, including an equivariant version of Lehn’s conjecture.

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Finite groups with permutable Hall subgroups
Xia YIN, Nanying YANG
Front. Math. China. 2017, 12 (5): 1265-1275.   DOI: 10.1007/s11464-017-0641-4
Abstract   PDF (157KB)

Let σ={σi|iI} be a partition of the set of all primes P, and let G be a finite group. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member 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 σiπ(G)φ. In this paper, we study the structure of G under the assuming that some subgroups of G permutes with all members of H .

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