We provide a simple extension of Bolthausen's Morita type proof the replica symmetric formula [E. Bolthausen, Stat. Mech. Classical and Disordered Systems, pp. 63-93 (2018)] for Sherrington-Kirkpatrick model prove symmetry all $(\beta,h)$ that satisfy $\beta^2 E \operatorname{sech}^2(\beta\sqrt{q}Z+h) \leq 1$, where $q = E\tanh^2(\beta\sqrt{q}Z+h)$. Compared to (2018)], key argument is apply co...

Let $ f: {\mathcal H} \rightarrow \mathbb{R} be a convex differentiable function whose solution set {{\rm{argmin}}}\; f is nonempty. To attain of the problem \min_{\mathcal H}f $, we consider second order dynamic system \;\ddot{x}(t) + \alpha \, \dot{x}(t) \beta (t) \nabla f(x(t)) c x(t) = 0 where positive such that \lim_{t\rightarrow +\infty}\beta(t) +\infty $. By imposing adequate hypothesis ...

Neutrinoless double beta decay ($0\nu\beta\beta$) nuclear matrix elements (NME) are the object of many theoretical calculation methods, and very important for analysis guidance a large number experimental efforts. However, there discrepancies between NME values provided by different methods. In this paper we propose statistical $^{48}$Ca $0\nu\beta\beta$ using interacting shell model, emphasizi...

$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles are related to $\beta$-Hermite, $\beta$-Laguerre, $\beta$-Jacobi ensembles. For fixed there exist associated weak limit theorems (WLTs) in the freezing regime $\beta\to\infty$ $\beta$-Hermite $\beta$-Laguerre case by Dumitriu Edelman (2005) explicit formulas for covariance matrices $\Sigma_N$ te...

Consider $\beta > 1$ and $\lfloor \beta \rfloor$ its integer part. It is widely known that any real number $\alpha \in \Bigl[0, \frac{\lfloor \rfloor}{\beta - 1}\Bigr]$ can be represented in base $\beta$ using a development series of the form = \sum_{n 1}^\infty x_n\beta^{-n}$, where $x (x_n)_{n \geq 1}$ sequence taking values into alphabet $\{0,\; ...\; ,\; \lfloor \rfloor\}$. The so called $\...

– Recently an extension of beta function is defined by introducing an extra parameter is proved to be useful earlier (Aslam Chaudhry 1997 [8] and A. R. Miller 1998 [1]). In this research note, we generalize the incomplete beta function and obtained the various integral representations and properties. Furthermore, we obtained the beta distribution for generalized beta function. Keywords––Incompl...

A bstract Extending earlier work, we find the two-loop term in beta-function for scalar coupling ? a generalized Wilson loop operator of $$ \mathcal{N} N = 4 SYM theory, working planar weak-coupling expansion. The has fixed points at ±1 and 0, corresponding respectively to supersymmetric Wilson-Maldacena stan...