Strange Quark Mass from the Invariant Mass Distribution of Cabibbo-Suppressed Tau Decays

Quark mass corrections to the tau hadronic width play a significant role only for the strange quark, hence providing a method for determining its mass. The experimental input is the vector plus axial-vector strange spectral function derived from a complete study of tau decays into strange hadronic final states performed by ALEPH. New results on strange decay modes from other experiments are also incorporated. The present analysis determines the strange quark mass at the Mtau mass scale using moments of the spectral function. Justified theoretical constraints are applied to the nonperturbative components and careful attention is paid to the treatment of the perturbative expansions of the moments which exhibit convergence problems. The result obtained, m_s(Mtau^2) = (120 +- 11_exp +- 8_Vus +- 19_th) MeV = (120^+21_-26) MeV, is stable over the scale from Mtau down to about 1.4 GeV. Evolving this result to customary scales yields m_s(1 GeV^2) = (160^+28_-35) MeV and m_s(4 GeV^2) = (116^+20_-25) MeV.Comment: LaTex, 8 pages, 4 figures (EPS

Multiscale Decompositions and Optimization

In this paper, the following type Tikhonov regularization problem will be systematically studied: [(u_t,v_t):=\argmin_{u+v=f} {|v|_X+t|u|_Y},] where $Y$ is a smooth space such as a \BV space or a Sobolev space and $X$ is the pace in which we measure distortion. Examples of the above problem occur in denoising in image processing, in numerically treating inverse problems, and in the sparse recovery problem of compressed sensing. It is also at the heart of interpolation of linear operators by the real method of interpolation. We shall characterize of the minimizing pair $(u_t,v_t)$ for (X,Y)=(L_2(\Omega),\BV(\Omega)) as a primary example and generalize Yves Meyer's result in [11] and Antonin Chambolle's result in [6]. After that, the following multiscale decomposition scheme will be studied: [u_{k+1}:=\argmin_{u\in \BV(\Omega)\cap L_2(\Omega)} {1/2|f-u|^2_{L_2}+t_{k}|u-u_k|_{\BV}},] where $u_0=0$ and $\Omega$ is a bounded Lipschitz domain in $\R^d$. This method was introduced by Eitan Tadmor et al. and we will improve the $L_2$ convergence result in \cite{Tadmor}. Other pairs such as $(X,Y)=(L_p,W^{1}(L_\tau))$ and $(X,Y)=(\ell_2,\ell_p)$ will also be mentioned. In the end, the numerical implementation for (X,Y)=(L_2(\Omega),\BV(\Omega)) and the corresponding convergence results will be given.Comment: 33 page

Limited-Memory Greedy Quasi-Newton Method with Non-asymptotic Superlinear Convergence Rate

Non-asymptotic convergence analysis of quasi-Newton methods has gained attention with a landmark result establishing an explicit superlinear rate of O$((1/\sqrt{t})^t)$. The methods that obtain this rate, however, exhibit a well-known drawback: they require the storage of the previous Hessian approximation matrix or instead storing all past curvature information to form the current Hessian inverse approximation. Limited-memory variants of quasi-Newton methods such as the celebrated L-BFGS alleviate this issue by leveraging a limited window of past curvature information to construct the Hessian inverse approximation. As a result, their per iteration complexity and storage requirement is O$(\tau d)$ where $\tau \le d$ is the size of the window and $d$ is the problem dimension reducing the O$(d^2)$ computational cost and memory requirement of standard quasi-Newton methods. However, to the best of our knowledge, there is no result showing a non-asymptotic superlinear convergence rate for any limited-memory quasi-Newton method. In this work, we close this gap by presenting a limited-memory greedy BFGS (LG-BFGS) method that achieves an explicit non-asymptotic superlinear rate. We incorporate displacement aggregation, i.e., decorrelating projection, in post-processing gradient variations, together with a basis vector selection scheme on variable variations, which greedily maximizes a progress measure of the Hessian estimate to the true Hessian. Their combination allows past curvature information to remain in a sparse subspace while yielding a valid representation of the full history. Interestingly, our established non-asymptotic superlinear convergence rate demonstrates a trade-off between the convergence speed and memory requirement, which to our knowledge, is the first of its kind. Numerical results corroborate our theoretical findings and demonstrate the effectiveness of our method