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Frontiers of Mathematics in China


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, Volume 13 Issue 3 Previous Issue   
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Efficient algorithm for principal eigenpair of discrete p-Laplacian
Front. Math. China. 2018, 13 (3): 509-524.
Abstract   PDF (198KB)

This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379–1418; 2017, 12(5): 1023–1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.

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Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation
Shou-Ting CHEN, Wen-Xiu MA
Front. Math. China. 2018, 13 (3): 525-534.
Abstract   PDF (779KB)

A (2+ 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specific presented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2+ 1)-dimensional nonlinear partial differential equations which possess lump solutions.

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A reduced-form model with default intensities containing contagion and regime-switching Vasicek processes
Jie GUO, Guojing WANG
Front. Math. China. 2018, 13 (3): 535-554.
Abstract   PDF (322KB)

The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps (CDSs). Numerical results are presented to show the impacts of the model parameters on the fair spreads.

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Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming
Yongguang HE, Huiyun LI, Xinwei LIU
Front. Math. China. 2018, 13 (3): 555-578.
Abstract   PDF (514KB)

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman-Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman-Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.

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Global geometrical optics method for vector-valued Schrödinger problems
Jiashun HU, Xiang MA, Chunxiong ZHENG
Front. Math. China. 2018, 13 (3): 579-606.
Abstract   PDF (2496KB)

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11(1): 105-140], to the more general vectorvalued Schrödinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian ow. The classical Wentzel-Kramers-Brillouin (WKB) analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level Schrödinger system and the other is the periodic Schrödinger equation. Numerical tests validate the theoretical results.

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Existence of anti-periodic solutions for hemivariational inequalities
Xiaoyou LIU
Front. Math. China. 2018, 13 (3): 607-618.
Abstract   PDF (154KB)

J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747–767; 2009, 71: 3203–3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.

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Integral-type functionals of first hitting times for continuous-time Markov chains
Yuanyuan LIU, Yanhong SONG
Front. Math. China. 2018, 13 (3): 619-632.
Abstract   PDF (174KB)

We investigate integral-type functionals of the first hitting times for continuous-time Markov chains. Recursive formulas and drift conditions for calculating or bounding integral-type functionals are obtained. The connection between the subexponential integral-type functionals and the subexponential ergodicity is established. Moreover, these results are applied to the birth-death processes. Polynomial integral-type functionals and polynomial ergodicity are studied, and a sufficient criterion for a central limit theorem is also presented.

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Commutator of Riesz potential in p-adic generalized Morrey spaces
Huixia MO, Xiaojuan WANG, Ruiqing MA
Front. Math. China. 2018, 13 (3): 633-645.
Abstract   PDF (166KB)

Suppose that Ipα is the p-adic Riesz potential. In this paper, we established the boundedness of Ipα on the p-adic generalized Morrey spaces, as well as the boundedness of the commutators generated by the p-adic Riesz potential Ipα and p-adic generalized Campanato functions.

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Oscillatory hyper-Hilbert transform along curves on modulation spaces
Xiaomei WU, Dashan FAN
Front. Math. China. 2018, 13 (3): 647-666.
Abstract   PDF (338KB)

We consider the boundedness of the n-dimension oscillatory hyper-Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems signicantly improve some known results.

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Phase transition for SIR model with random transition rates on complete graphs
Xiaofeng XUE
Front. Math. China. 2018, 13 (3): 667-690.
Abstract   PDF (248KB)

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs Cn with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λc0 such that when λλc, the proportion r of vertices which have ever been infective converges to 0 weakly as n+ while when λλc, there exist c(λ)0 and b(λ)0 such that for each n1 with probability pb(λ), the proportion rc(λ). Furthermore, we prove that λc is the inverse of the production of the mean of ρ and the mean of the inverse of ξ.

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A recollement construction of Gorenstein derived categories
Peng YU
Front. Math. China. 2018, 13 (3): 691-713.
Abstract   PDF (248KB)

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.

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Tilting sheaves for weighted projective lines of weight type (p)
Xiaofeng ZHANG
Front. Math. China. 2018, 13 (3): 715-724.
Abstract   PDF (169KB)

We give a complete classification of tilting sheaves over weighted projective lines of weight type (p).

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Hörmander index in finite-dimensional case
Yuting ZHOU, Li WU, Chaofeng ZHU
Front. Math. China. 2018, 13 (3): 725-761.
Abstract   PDF (326KB)

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

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