A Continuous Opinion Dynamic Model in Co-evolving Networks--A Novel Group Decision Approach

Opinion polarization is a ubiquitous phenomenon in opinion dynamics. In contrast to the traditional consensus oriented group decision making (GDM) framework, this paper proposes a framework with the co-evolution of both opinions and relationship networks to improve the potential consensus level of a group and help the group reach a stable state. Taking the bound of confidence and the degree of individual's persistence into consideration, the evolution of the opinion is driven by the relationship among the group. Meanwhile, the antagonism or cooperation of individuals presented by the network topology also evolve according to the dynamic opinion distances. Opinions are convergent and the stable state will be reached in this co-evolution mechanism. We further explored this framework through simulation experiments. The simulation results verify the influence of the level of persistence on the time cost and indicate the influence of group size, the initial topology of networks and the bound of confidence on the number of opinion clusters.Comment: 24 pages, 3 figure

Quantum state complexity and the thermodynamic arrow of time

Why time is a one-way corridor? What's the origin of the arrow of time? We attribute the thermodynamic arrow of time as the direction of increasing quantum state complexity. Inspired by the work of Nielsen, Susskind and Micadei, we checked this hypothesis on both a simple two qubit and a three qubit quantum system. The result shows that in the two qubit system, the thermodynamic arrow of time always points in the direction of increasing quantum state complexity. For the three qubit system, the heat flow pattern among subsystems is closely correlated with the quantum state complexity of the subsystems. We propose that besides its impact on macroscopic spatial geometry, quantum state complexity might also generate the thermodynamic arrow of time.Comment: 4 pages, 4 figure

The Sup-norm Perturbation of HOSVD and Low Rank Tensor Denoising

The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces and the low rank tensors are attainable in the Euclidean norm. In this article, we analyze the sup-norm perturbation bounds of HOSVD and introduce estimators of the singular subspaces with sharp deviation bounds in the sup-norm. We also investigate a low rank tensor denoising estimator and demonstrate its fast convergence rate with respect to the entry-wise errors. The sup-norm perturbation bounds reveal unconventional phase transitions for statistical learning applications such as the exact clustering in high dimensional Gaussian mixture model and the exact support recovery in sub-tensor localizations. In addition, the bounds established for HOSVD also elaborate the one-sided sup-norm perturbation bounds for the singular subspaces of unbalanced (or fat) matrices

Understanding over-parameterized deep networks by geometrization

A complete understanding of the widely used over-parameterized deep networks is a key step for AI. In this work we try to give a geometric picture of over-parameterized deep networks using our geometrization scheme. We show that the Riemannian geometry of network complexity plays a key role in understanding the basic properties of over-parameterizaed deep networks, including the generalization, convergence and parameter sensitivity. We also point out deep networks share lots of similarities with quantum computation systems. This can be regarded as a strong support of our proposal that geometrization is not only the bible for physics, it is also the key idea to understand deep learning systems.Comment: 6 page

On Quantum de Rham Cohomology Theory

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the correspoding quantum Stokes theorem is proved. We also derive quantum hard Lefschetz theorem. By replacing d by d_h and wedge by wedge_h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of classical Chern-Weil theory. Quantum equivariant de Rham cohomology is defined in the similar fashion.Comment: 8 pages, AMSLaTe

DGBV Algebras and Mirror Symmetry

We describe some recent development on the theory of formal Frobenius manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk (DGBV) algebras and formulate a version of mirror symmetry conjecture: the extended deformation problems of the complex structure and the Poisson structure are described by two DGBV algebras; mirror symmetry is interpreted in term of the invariance of the formal Frobenius manifold structures under quasi-isomorphism.Comment: 11 pages, to appear in Proceedings of ICCM9

Formal Frobenius manifold structure on equivariant cohomology

For a closed K\"{a}hler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.Comment: AMS-LaTex, 14 page

Degenerate Chern-Weil Theory and Equivariant Cohomology

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a very simple proof of a localization formula due to Jeffrey-Kirwan in the case of G = SU(2) or SO(3).Comment: 23 pages, AMSLaTe

Frobenius Manifold Structure on Dolbeault Cohomology and Mirror Symmetry

We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.Comment: 10 pages, AMS LaTe

Identification of Two Frobenius Manifolds In Mirror Symmetry

We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology. Our result can be considered as a generalization of the identification of the Dolbeault cohomology ring with the complexified de Rham cohomology ring on a Kaehler manifold.Comment: 12 pages, AMS LaTe