Weak KAM theory for general Hamilton-Jacobi equations II: the fundamental solution under Lipschitz 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 uniform Lipschitz of $H(x,u,p)$ with respect to $u$, we establish a variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. By introducing an implicitly defined {\it fundamental solution}, we obtain a variational representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss the large time behavior of the viscosity solution of the evolutionary Hamilton-Jacobi equation and provide a dynamical representation formula of the viscosity solution of the stationary Hamilton-Jacobi equation with strictly increasing $H(x,u,p)$ with respect to $u$

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

A Dynamical Approach to Viscosity Solutions of Hamilton-Jacobi Equations

In this paper, we consider the following Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,t,u(x,t),\partial_xu(x,t))=0, u(x,0)=\phi(x). \end{cases} \end{equation*} Under some assumptions on the convexity of $H(x,t,u,p)$ w.r.t. $p$, we develop a dynamical approach to viscosity solutions and show that there exists an intrinsic connection between viscosity solutions and certain minimal characteristics.Comment: This paper has been withdrawn by the author due to a crucial error in Lemma 3.

A hybrid partial sum computation unit architecture for list decoders of polar codes

Although the successive cancelation (SC) algorithm works well for very long polar codes, its error performance for shorter polar codes is much worse. Several SC based list decoding algorithms have been proposed to improve the error performances of both long and short polar codes. A significant step of SC based list decoding algorithms is the updating of partial sums for all decoding paths. In this paper, we first proposed a lazy copy partial sum computation algorithm for SC based list decoding algorithms. Instead of copying partial sums directly, our lazy copy algorithm copies indices of partial sums. Based on our lazy copy algorithm, we propose a hybrid partial sum computation unit architecture, which employs both registers and memories so that the overall area efficiency is improved. Compared with a recent partial sum computation unit for list decoders, when the list size $L=4$, our partial sum computation unit achieves an area saving of 23\% and 63\% for block length $2^{13}$ and $2^{15}$, respectively.Comment: 5 pages, presented at the 2015 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP

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

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

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

Large N_c Expansion in Chiral Quark Model of Mesons

We study SU(3)_L\timesSU(3)_R chiral quark model of mesons up to the next to leading order of $1/N_c$ expansion. Composite vector and axial-vector mesons resonances are introduced via non-linear realization of chiral SU(3) and vector meson dominant. Effects of one-loop graphs of pseudoscalar, vector and axial-vector mesons is calculated systematically and the significant results are obtained. We also investigate correction of quark-gluon coupling and relationship between chiral quark model and QCD sum rules. Up to powers four of derivatives, chiral effective lagrangian of mesons is derived and evaluated to the next to leading order of $1/N_c$. Low energy limit of the model is examined. Ten low energy coupling constants $L_i(i=1,2,...,10)$ in ChPT are obtained and agree with ChPT well.Comment: 49 pages, latex file, 6 eps figure

Solutions to the ABS lattice equations via generalized Cauchy matrix approach

The usual Cauchy matrix approach starts from a known plain wave factor vector $r$ and known dressed Cauchy matrix $M$. In this paper we start from a matrix equation set with undetermined $r$ and $M$. From the starting equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The starting matrix equation set admits more choices for $r$ and $M$ and in the paper we give explicit formulae for all possible $r$ and $M$. As applications, we get more solutions than usual multi-soliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation and some lattice equations in ABS list.Comment: 24 page

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