On the Generalized Volume Conjecture and Regulator

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL_2(C) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding $\mathbb{C}^{*}$-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over $\mathbb{C}^{*}\times \mathbb{C}^{*}$. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.Comment: 9 pages, revised version of section 3 of math.GT/0604057, section 3.4 is ne

The equivalent classical metrics on the Cartan-Hartogs Domains

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, we prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the Bergman metric is equivalent to the Einstein-K\"ahler metric. That means the Yau's conjecture is true on Cartan-Hartogs domain.Comment: 19 page

Volume Conjecture, Regulator and SL_2(C)-Character Variety of a Knot

In this paper, by using the regulator map of Beilinson-Deligne, we show that the quantization condition posed by Gukov is true for the SL_2(\mathbb{C}) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding \mathbb{C}^{*}-valued 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over \mathbb{C}^{*}\times \mathbb{C}^{*}. The second part of the paper is to define an algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in $S^3$. We establish a Casson type invariant for these 3-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications.Comment: 19 pages, this is the revised and corrected versio

Explicit formulas of Euler sums via multiple zeta values

Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein--Borwein--Girgensohn theorems, and the Flajolet--Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of $S_{r^m,q}$, $S_{r^m,\bar{q}}$, $S_{\bar{r}^m,q}$ and $S_{\bar{r}^m,\bar{q}}$, are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight $w\leq 11$ for non-alternating case and of weight $w\leq 6$ for alternating case

An SL(2,C) Algebro-Geometric Invariant of Knots

In this paper, we define a new algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in S^3. We establish a Casson type invariant for these 3-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some 3-manifolds resulting from the Dehn surgery along the figure-eight knot.Comment: 17 pages, revised version of sections 4 and 5 of math.GT/0604057,some computations of the invariant are adde

The Levels of Conceptual Interoperability Model: Applying Systems Engineering Principles to M&S

This paper describes the use of the Levels of Conceptual Interoperability Model (LCIM) as a framework for conceptual modeling and its descriptive and prescriptive uses. LCIM is applied to show its potential and shortcomings in the current simulation interoperability approaches, in particular the High Level Architecture (HLA) and Base Object Models (BOM). It emphasizes the need to apply rigorous engineering methods and principles and replace ad-hoc approaches.Comment: 9 pages, 2 figures, 4 tables, Proceedings of Spring Simulation Multiconference (SpringSim'09). San Diego, CA, US

On a spectral sequence for twisted cohomologies

Let ($\Omega^{\ast}(M), d$) be the de Rham cochain complex for a smooth compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$, there are a twisted de Rham cochain complex $(\Omega^{\ast}(M), d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the filtration $F_p(\Omega^{\ast}(M))=\bigoplus_{i\geq p}\Omega^i(M)$ of $\Omega^{\ast}(M)$, which converges to the twisted de Rham cohomology $H^*(M,H)$. We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.Comment: 25 page

Max-Diversity Distributed Learning: Theory and Algorithms

We study the risk performance of distributed learning for the regularization empirical risk minimization with fast convergence rate, substantially improving the error analysis of the existing divide-and-conquer based distributed learning. An interesting theoretical finding is that the larger the diversity of each local estimate is, the tighter the risk bound is. This theoretical analysis motivates us to devise an effective maxdiversity distributed learning algorithm (MDD). Experimental results show that MDD can outperform the existing divide-andconquer methods but with a bit more time. Theoretical analysis and empirical results demonstrate that our proposed MDD is sound and effective

Cavity-meidated collisionless sympathetic cooling of molecules with atoms

Cooling a range of molecules to ultracold temperatures (<1 mK) is a difficult but important challenge in molecular physics and chemistry. Collective cavity cooling of molecules is a promising method that does not rely on molecular energy level and thus can be applied to all molecules in principle. However, the initial lack of cold molecules leads to the difficulty in its experimental implementation. We show that efficient collective sympathetic cooling of molecules to sub-mK temperatures using a large ensemble of atoms within a cavity is feasible. This approach is a new type of sympathetic cooling which does not rely on direct collisions between atoms and molecules, but utilizes thermalization via their mutual interaction with a cavity field. Two important mechanisms are identified. This include: (1) giant enhancement of cavity optical field from the efficient scattering of the pump light by the atoms; (2) cavity-mediated collective interaction between the atoms and the molecules. We show an optimal cavity detuning for maximizing cooling, which is dependent on the atom and molecule numbers. We determine a threshold for the molecular pump strength and show that it is independent of molecule number when the number of atoms is much greater than the molecules. This can be reduced by orders of magnitude when compared to cavity cooling of single molecular species only. Using this new sympathetic cavity cooling technique, cooling molecules to sub-mK within a high-Q cavity could be within reach of experimental demonstration

Service-oriented high level architecture

Service-oriented High Level Architecture (SOHLA) refers to the high level architecture (HLA) enabled by Service-Oriented Architecture (SOA) and Web Services etc. techniques which supports distributed interoperating services. The detailed comparisons between HLA and SOA are made to illustrate the importance of their combination. Then several key enhancements and changes of HLA Evolved Web Service API are introduced in comparison with native APIs, such as Federation Development and Execution Process, communication mechanisms, data encoding, session handling, testing environment and performance analysis. Some approaches are summarized including Web-Enabling HLA at the communication layer, HLA interface specification layer, federate interface layer and application layer. Finally the problems of current research are discussed, and the future directions are pointed out.Comment: 12 pages, 3 figures, 2 tables, Proceedings of 2008 European Simulation Interoperability Workshop, 08E-SIW-02