中国科学院数学与系统科学研究院期刊网
AMS

15 January 2011, Volume 27 Issue 2
    

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  • Heat Equation with Memory in Anisotropic and Non-Homogeneous Media
    Jiongmin YONG, Xu ZHANG
    Acta Mathematica Sinica. 2011, 27(2): 219-254. https://doi.org/10.1007/s10114-010-0077-1
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    A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.  
  • Behavioral Portfolio Selection with Loss Control
    Song ZHANG, Han Qing JIN, Xun Yu ZHOU
    Acta Mathematica Sinica. 2011, 27(2): 255-274. https://doi.org/10.1007/s10114-011-0380-5
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    In this paper we formulate a continuous-time behavioral (à la cumulative prospect theory) portfolio selection model where the losses are constrained by a pre-specified upper bound. Economically the model is motivated by the previously proved fact that the losses occurring in a bad state of the world can be catastrophic for an unconstrained model. Mathematically solving the model boils down to solving a concave Choquet minimization problem with an additional upper bound. We derive the optimal solution explicitly for such a loss control model. The optimal terminal wealth profile is in general characterized by three pieces: the agent has gains in the good states of the world, gets a moderate, endogenously constant loss in the intermediate states, and suffers the maximal loss (which is the given bound for losses) in the bad states. Examples are given to illustrate the general results.  
  • The Coherent-Constructible Correspondence and Fourier-Mukai Transforms
    Bohan FANG, Chiu-Chu Melissa LIU, David TREUMANN, Eric ZASLOW
    Acta Mathematica Sinica. 2011, 27(2): 275-308. https://doi.org/10.1007/s10114-011-0462-4
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    As evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In a previous paper, the authors showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on Rn. In this paper we investigate and reprove some of Kawamata's results from this perspective.  
  • Global Solutions of the Evolutionary Faddeev Model with Small Initial Data
    Zhen LEI, Fang Hua LIN, Yi ZHOU
    Acta Mathematica Sinica. 2011, 27(2): 309-328. https://doi.org/10.1007/s10114-011-0465-1
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    We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space R1+n to the unit sphere S2, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.  
  • On the Longest Length of Consecutive Integers
    Min Zhi ZHAO, Qi-Man SHAO
    Acta Mathematica Sinica. 2011, 27(2): 329-338. https://doi.org/10.1007/s10114-011-0496-7
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    Choose m numbers from the set {1, 2, . . . , n} at random without replacement. In this paper we first establish the limiting distribution of the longest length of consecutive integers and then apply the result to test randomness of selecting numbers without replacement.  
  • Effective Maxwell Equations from Time-dependent Density Functional Theory
    Weinan E, Jianfeng LU, Xu YANG
    Acta Mathematica Sinica. 2011, 27(2): 339-368. https://doi.org/10.1007/s10114-011-0555-0
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    The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained.  
  • h?-transforms of Positivity Preserving Semigroups and Associated Markov Processes
    Xin Fang HAN, Zhi-Ming MA
    Acta Mathematica Sinica. 2011, 27(2): 369-376. https://doi.org/10.1007/s10114-011-0597-3
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    The h?-transforms of positivity preserving semigroups and their associated Markov processes are investigated in this paper. In particular, it is shown that any quasi-regular positivity preserving coercive form is h?-associated with a pair of special standard processes which are in weak duality.  
  • Towards a Unified Recurrent Neural Network Theory:The Uniformly Pseudo-Projection-Anti-Monotone Net
    Zong Ben XU, Chen QIAO
    Acta Mathematica Sinica. 2011, 27(2): 377-396. https://doi.org/10.1007/s10114-011-0598-2
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    In the past decades, various neural network models have been developed for modeling the behavior of human brain or performing problem-solving through simulating the behavior of human brain. The recurrent neural networks are the type of neural networks to model or simulate associative memory behavior of human being. A recurrent neural network (RNN) can be generally formalized as a dynamic system associated with two fundamental operators: one is the nonlinear activation operator deduced from the input-output properties of the involved neurons, and the other is the synaptic connections (a matrix) among the neurons. Through carefully examining properties of various activation functions used, we introduce a novel type of monotone operators, the uniformly pseudo-projectionanti- monotone (UPPAM) operators, to unify the various RNN models appeared in the literature. We develop a unified encoding and stability theory for the UPPAM network model when the time is discrete. The established model and theory not only unify but also jointly generalize the most known results of RNNs. The approach has lunched a visible step towards establishment of a unified mathematical theory of recurrent neural networks.  
  • Non-existence of KAM Torus
    Chong Qing CHENG
    Acta Mathematica Sinica. 2011, 27(2): 397-404. https://doi.org/10.1007/s10114-011-0631-5
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    Given an integrable Hamiltonian h0 with n-degrees of freedom and a Diophantine frequency ω, then, arbitrarily close to h0 in the Cr topology with r < 2n, there exists an analytical Hamiltonian h∈ with no KAM torus of rotation vector ω. In contrast with it, KAM tori exist if perturbations are small in Cr topology with r > 2n.  
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