Crystal of affine sl^ℓ\widehat{\mathfrak{sl}}_{\ell} and Hecke algebras at a primitive 2ℓ2\ellth root of unity

Let $\ell\in\mathbb{N}$ with $\ell>2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke algebras $H_n$ of type $A$ and their level two cyclotomic quotients with Hecke parameter being a primitive $2\ell$th root of unity. We categorify the Kashiwara operators for the crystal as the functors of taking socle of certain two-steps restriction and of taking head of certain two-steps induction. For any finite dimensional irreducible $H_n$-module $M$, we prove that the irreducible submodules of $\rm{res}_{H_{n-2}}^{H_n}M$ which belong to $\widehat{B}(\infty)$ (Definition 6.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and the affine Hecke algebras of type $A$ at a primitive $\ell$th root of unity

Variational principle for contact Tonelli Hamiltonian systems

We establish an implicit variational principle for the equations of the contact flow generated by the Hamiltonian $H(x,u,p)$ with respect to the contact 1-form $\alpha=du-pdx$ under Tonelli and Osgood growth assumptions. It is the first step to generalize Mather's global variational method from the Hamiltonian dynamics to the contact Hamiltonian dynamics.Comment: arXiv admin note: text overlap with arXiv:1408.379

Weak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases} \end{equation*} where $\phi(x)\in C(M,\mathbb{R})$. Under some assumptions on the convexity of $H(x,u,p)$ with respect to $p$ and the Osgood growth of $H(x,u,p)$ with respect to $u$, we establish an implicitly variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. Moreover, we obtain a representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation

High energy tau neutrinos: production, propagation and prospects of observations

High energy tau neutrinos with energy greater than several thousands of GeV may be produced in some astrophysical sites. A summary of the intrinsic high energy tau neutrino flux estimates from some representative astrophysical sites is presented including the effects of neutrino flavor oscillations. The presently envisaged prospects of observations of the oscillated high energy tau neutrino flux are mentioned. In particular, a recently suggested possibility of future observations of Earth-skimming high energy tau neutrinos is briefly discussed.Comment: 4 pages, 2 figs, talk given at 28th International Cosmic Ray Conference (ICRC 2003), Tsukuba, Japan, 31 July-7 Aug, 2003, appeared in its proceedings edited by T. Kajita et al., HE, pp. 1431-143

Aubry-Mather and weak KAM theories for contact Hamiltonian systems. Part 1: Strictly increasing case

This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $00$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry set for the contact Hamiltonian system showing that it is the intersection of two Legendrian pseudographs $G_{u_-}$ and $G_{u_+}$, and that the projection $\pi:T^*M\times \mathbb{R}\to M$ induces a bi-Lipschitz homeomorphism $\pi|_{\tilde{\mathcal{A}}}$ from Aubry set $\tilde{\mathcal{A}}$ onto the projected Aubry set $\mathcal{A}$. At last, we introduce the notion of barrier functions and study their interesting properties along calibrated curves. Our analysis is based on a recent method by [43,44].Comment: 34 page

Rigorous Effective Field Theory Study on Pion Form Factor

We study $e^+e^-\to\pi^+\pi^-$ cross section and phase shift of $I=l=1$ $\pi-\pi$ scattering below 1GeV in framework of chiral constituent quark model. The results including all order contribution of the chiral perturbation expansion and all one-loop effects of pseudoscalar mesons, but without any adjust parameters. Width of $\rho$ predicted by the model strongly depends on transition momentum-square $q^2$. We show that the mass pamameter of $\rho$-meson in its propagator is very different from its physical mass due to momentum-dependent width of $\rho$. The mass difference between $\rho^0$ and $\omega$ are predicted successfully. The rigorous theoretical prediction on $e^+e^-\to\pi^+\pi^-$ cross section and the phase shift in $I=l=1$ $\pi-\pi$ scattering agree with data excellentlly.Comment: revtex file, 10 pages, 4 eps figure

A multi-mode area-efficient SCL polar decoder

Polar codes are of great interest since they are the first provably capacity-achieving forward error correction codes. To improve throughput and to reduce decoding latency of polar decoders, maximum likelihood (ML) decoding units are used by successive cancellation list (SCL) decoders as well as successive cancellation (SC) decoders. This paper proposes an approximate ML (AML) decoding unit for SCL decoders first. In particular, we investigate the distribution of frozen bits of polar codes designed for both the binary erasure and additive white Gaussian noise channels, and take advantage of the distribution to reduce the complexity of the AML decoding unit, improving the area efficiency of SCL decoders. Furthermore, a multi-mode SCL decoder with variable list sizes and parallelism is proposed. If high throughput or small latency is required, the decoder decodes multiple received codewords in parallel with a small list size. However, if error performance is of higher priority, the multi-mode decoder switches to a serial mode with a bigger list size. Therefore, the multi-mode SCL decoder provides a flexible tradeoff between latency, throughput and error performance, and adapts to different throughput and latency requirements at the expense of small overhead. Hardware implementation and synthesis results show that our polar decoders not only have a better area efficiency but also easily adapt to different communication channels and applications.Comment: 13 pages, 9 figures, submitted to TVLS

Noncommutative QED and Muon Anomalous Magnetic Moment

The muon anomalous $g$ value, $a_\mu=(g-2)/2$, is calculated up to one-loop level in noncommutative QED. We argue that relativistic muon in E821 experiment nearly always stays at the lowest Landau level. So that spatial coordinates of muon do not commute each other. Using parameters of E821 experiment, $B=14.5$KG and muon energy 3.09GeV/c, we obtain the noncommutativity correction to $a_\mu$ is about $1.57\times 10^{-9}$, which significantly makes standard model prediction close to experiment.Comment: revtex, 6 page, 5 figure

Symbol-Decision Successive Cancellation List Decoder for Polar Codes

Polar codes are of great interests because they provably achieve the capacity of both discrete and continuous memoryless channels while having an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. In this paper, we propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. First, we propose to use a recursive channel combination to calculate symbol-wise channel transition probabilities, which lead to symbol decisions. Our proposed recursive channel combination also has a lower complexity than simply combining bit-wise channel transition probabilities. The similarity between our proposed method and Arikan's channel transformations also helps to share hardware resources between calculating bit- and symbol-wise channel transition probabilities. Second, a two-stage list pruning network is proposed to provide a trade-off between the error performance and the complexity of the symbol-decision SCL decoder. Third, since memory is a significant part of SCL decoders, we propose a pre-computation memory-saving technique to reduce memory requirement of an SCL decoder. Finally, to evaluate the throughput advantage of our symbol-decision decoders, we design an architecture based on a semi-parallel successive cancellation list decoder. In this architecture, different symbol sizes, sorting implementations, and message scheduling schemes are considered. Our synthesis results show that in terms of area efficiency, our symbol-decision SCL decoders outperform both bit- and symbol-decision SCL decoders.Comment: 13 pages, 17 figure

Weak KAM theory for general Hamilton-Jacobi equations I: the solution semigroup under proper conditions

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x). \end{cases} \end{equation*} Under some assumptions on $H(x,u,p)$ with respect to $p$ and $u$, we provide a variational principle on the evolutionary Hamilton-Jacobi equation. By introducing an implicitly defined solution semigroup, we extend Fathi's weak KAM theory to certain more general cases, in which $H$ explicitly depends on the unknown function $u$. As an application, we show the viscosity solution of the evolutionary Hamilton-Jacobi equation with initial condition tends asymptotically to the weak KAM solution of the following stationary Hamilton-Jacobi equation: \begin{equation*} H(x,u(x),\partial_xu(x))=0. \end{equation*}.Comment: This is a revised version of arXiv:1312.160