Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. We prove basic properties of these expectations and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.Comment: Published in at http://dx.doi.org/10.1214/09-AAP619 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces

Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.Comment: 32 pages, 6 figure

Recent developments in the shell model Monte Carlo approach to nuclei

The shell model Monte Carlo (SMMC) approach provides a powerful method for the microscopic calculation of statistical and collective nuclear properties in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods. We discuss recent applications of the method to describe the emergence of collectivity in the framework of the configuration-interaction shell model and the crossover from vibrational to rotational collectivity in families of rare-earth nuclei. We have calculated state densities of these rare-earth nuclei and find their collective enhancement factors to be correlated with the pairing and shape phase transitions. We also discuss an accurate method to calculate the ground-state energy of odd-even and odd-odd nuclei, circumventing the sign problem that originates in the projection on an odd number of particles. We have applied this method to calculate pairing gaps in families of isotopes in the iron region.Comment: 7 pages, 5 figures, Proceedings of Horizons of Innovative Theories, Experiments, and Supercomputing in Nuclear Physics (HITES 2012

The Differential Scheme and Quantum Computation

It is well-known that standard models of computation are representable as simple dynamical systems that evolve in discrete time, and that systems that evolve in continuous time are often representable by dynamical systems governed by ordinary differential equations. In many applications, e.g., molecular networks and hybrid Fermi-Pasta-Ulam systems, one must work with dynamical systems comprising both discrete and continuous components. Reasoning about and verifying the properties of the evolving state of such systems is currently a piecemeal affair that depends on the nature of major components of a system: e.g., discrete vs. continuous components of state, discrete vs. continuous time, local vs. distributed clocks, classical vs. quantum states and state evolution. We present the Differential Scheme as a unifying framework for reasoning about and verifying the properties of the evolving state of a system, whether the system in question evolves in discrete time, as for standard models of computation, or continuous time, or a combination of both. We show how instances of the differential scheme can accommodate classical computation. We also generalize a relatively new model of quantum computation, the quantum cellular automaton, with an eye towards extending the differential scheme to accommodate quantum computation and hybrid classical/quantum computation. All the components of a specific instance of the differential scheme are Convergence Spaces. Convergence spaces generalize notions of continuity and convergence. The category of convergence spaces, Conv, subsumes both simple discrete structures (e.g., digraphs), and complex continuous structures (e.g., topological spaces, domains, and the standard fields of analysis: R and C). We present novel uses for convergence spaces, and extend their theory by defining differential calculi on Conv. It is to the use of convergence spaces that the differential scheme owes its generality and flexibility