arXiv subjects

Mohamed Bouali

Publications and source records attributed to Mohamed Bouali.

16 recordsLinked to original sources

arXiv2025

Statistical density of particles in one dimensional interaction and Jellium Model

We study a one-dimensional gas of $n$ charged particles confined by a potential and interacting through the Riesz potential or a more general potential. In equilibrium, and for symmetric potential the particles arrange themselves symmetrically around the origin within a finite region. Various models will be studied by modifying both the confining potential and the interaction potential. Focusing on the statistical properties of the system, we analyze the position of the rightmost particle, $x_{\text{max}}$, and show that its typical fluctuations are described by a limiting distribution different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also derive the large deviation functions governing the atypical fluctuations of $x_{\text{max}}$ far from its mean.

cond-mat.stat-mech↗

arXiv2024

Convexity and concavity of a class of functions related to the elliptic functions

We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{{\cal K}{(\sqrt x)}}{a-(1/2)\log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\geq a_c$ and $1/f_a$ is strictly convex on $(0,1)$ if and only if $a\leq\log 4$, where $a_c$ is some critical value. The second main result of the paper is to study the log-convexity and log-concavity of the function $$h_p(x)=(1-x)^p{\cal K}(\sqrt x).$$ We prove that $h_p$ is strictly log-concave on $(0,1)$ if and only if $p\geq 7/32$ and strictly log-convex if and only if $p\leq 0$. This solves some problems posed by Yang and Tian and complete their result and a result of Alzer and Richards that $f_a$ is strictly concave on $(0,1)$ if and only if $a=4/3$ and $1/f_a$ is strictly concave on $(0,1)$ if and only if $a\geq 8/5$. As applications of the convexity and concavity, we establish among other inequalities, that for $a\geq a_c$ and all $r\in(0,1)$ $$\frac{2π\sqrtπ}{(2a+\log 2)Γ(3/4)^2}\leq \frac{{\cal K}(\sqrt r)}{a-\frac12\log (r)}+\frac{{\cal K}(\sqrt{1-r})}{a-\frac12\log (1-r)}<1+\fracπ{2a},$$ and for $p\geq 3(2+\sqrt 2)/8$ and all $r\in(0,1)$ $$\sqrt{(r-r^2)^p{\cal K}(\sqrt{1-r}){\cal K}(\sqrt r)}< \frac{π\sqrtπ}{2^{p+1}Γ(3/4)^2}<\frac{r^p{\cal K}(\sqrt{1-r})+(1-r)^p{\cal K}(\sqrt r)}{2}.$$

math.GM↗

arXiv2024

A mean value inequalities for the polygamma and zeta functions

A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-γ<-γH(x,1/x)<\frac{γ^2}{ψ\big(H(x,1/x)\big)}<ψ\Big(1/H(x,1/x)\Big)<H\Big(ψ(x), ψ(1/x)\Big),$$ $$H\Big(ζ(x),ζ(1/x)\Big)<-2,$$ and for all $x\in(0,1)$ $$ζ(1/2)<H\Big(ζ(x),ζ(1-x)\Big)<-1,$$ $$\frac{\log 4}{1+\log 4}<H\Big(η(x),η(1-x)\Big)<(1-\sqrt 2)ζ(1/2).$$ Here, $ψ=Γ'/Γ$ denotes the digamma function, $γ$ is Euler's constant, $ζ$ is the Riemann's zeta function and $η$ is the Dirichlet's eta function.

math.GM↗

arXiv2024

Convexity properties related to Gauss hypergeometric function

We investigate the convexity property on $(0,1)$ of the functions $φ_{a,b,c}$ and $1/φ_{a,b,c}$, where $$φ_{a,b,c}(x)= \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\leq -2γ-ψ(a)-ψ(b),$ (respectively $c\geqα_0$) and $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\geq c(a,b)$ (respectively $c\in[δ_-,δ_+]$), where $ψ$ is the Polygamma function. This generalizes some problems posed by Yang and Tian and complete the study of convexity properties of functions studied by the author in [bouali]. As applications of the convexity and concavity, we establish among other inequalities, that for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\geq c(a,b)$ $$c+\frac{Γ(a)Γ(b)}{Γ(a+b)}\leq \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)}+\frac{c-\log(x)}{\,_2F_1(a,b,a+b,1-x)}\leq\frac{(2c+2\log 2)}{\,_2{F}_1(a,b;a+b;1/2)},$$ and for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\in [δ_-,δ_+]$ $$\frac1c+\frac{Γ(a+b)}{Γ(a)Γ(b)}\leq \frac{\,_2F_1(a,b,a+b,x)}{c-\log(1-x)}+\frac{\,_2F_1(a,b,a+b,1-x)}{c-\log(x)}\leq\frac{\,_2{F}_1(a,b;a+b;1/2)}{(2c+2\log 2)}.$$

math.GM↗

arXiv2022

On some complete monotonic functions

Motivated by open questions in the papers " Refinements and sharpenings of some double inequalities for bounding the gamma function" and "Complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function",we confirm among other results and disprove other one.

math.CA↗

arXiv2022

Double Inequalities for Complete Monotonicity Degrees of Remainders of Asymptotic Expansions of the Gamma and Digamma Functions

Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and the digamma function and we give two bounded for this degrees.

math.CA↗

arXiv2020

A harmonic mean inequality for the $q-$gamma and $q-$digamma functions

We prove amongs others results that the harmonic mean of $Γ_q(x)$ and $Γ_q(1/x)$ is greater than or equal to $1$ for arbitrary $x > 0$ and $q\in J$ where $J$ is a subset of $[0,+\infty)$. Also, we prove that for there is $p_0\in(1,9/2)$, such that for $q\in(0,p_0)$, $ψ_q(1)$ is the minimum of the harmonic mean of $ψ_q(x)$ and $ψ_q(1/x)$ for $x > 0$ and for $q\in(p_0,+\infty)$, $ψ_q(1)$ is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.

math.CA↗

arXiv2018

Short note on an open problem

In this work, we investigate a problem posed by Feng Qi and Bai-Ni Guo in their paper Complete monotonicities of functions involving the gamma and digamma functions.

math.PR↗

arXiv2016

Product of Non-Hermitian Random Matrices

We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N\times N$ in the class of elliptic matrices, with independent identically distributed entries. The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process.

math.PR↗

arXiv2016

Product of Independent Cauchy-Lorentz Random Matrices

We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N_i\times N_{i+1}$ $(i=1,...,n)$, with independent identically distributed Cauchy entries (Cauchy-Lorentz matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Cauchy-Lorentz matrix, but with weight given by a Meijer G-function depending on $n$ and $N_i$.

math.PR↗

arXiv2015

Constrained Eigenvalues Density of Invariant Random Matrices Ensembles

We compute exact asymptotic of the statistical density of random matrices belonging to invariant random matrices ensemble (RMT) orthogonal, unitary and symplectic ensembles, where all its eigenvalues lie within the interval $[σ, +\infty[$ or $]-\infty,τ]$ or $[σ,τ]$. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied by Dean-Majumdar.

math.PR↗

arXiv2015

Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[σ, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n\times n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-βθ(α)n^2]$ where the Dyson index $β$ characterizes the ensemble, $α$ is some extra parameter and the exponent $θ(α)$ is a function of $α$ which will be given explicitly. For $α=0$, $θ(0)= (\log 3)/4 = 0.274653...$ is universal. We compute the probability that the eigenvalues lie in the interval $[σ,+\infty[$ with $(σ>0,\; {\rm if}\;α>0)$ and $(σ\in\mathbb R,\; {\rm if }\;α=0)$. This generalizing the celebrated Wigner semicircle law to these restricted ensembles. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied in \cite{D}, \cite{S}.

math.PR↗

arXiv2015

Sobolev Freud polynomials

We investigate the uniform asymptotic of some Sobolev orthogonal polynomials. Three term recurrence relation is given, moreover we give a recurrence relation between the so-called Sobolev orthogonal polynomials and Freud orthogonal polynomials.

math.CA↗

arXiv2014

Asymptotic density of zeros of half range generalized Hermite polynomials

We investigate the global density of zeros of generalized Hermite orthogonal polynomials, subject to certain truncated conditions on its weight. We shall given explicitly the global density of zeros under some asymptotic conditions on the weight. Moreover we compute the asymptotic of the total energy of the equilibrium position of the system of $n$ movable unit charges in an external field determined by the weight of the generalized Hermite polynomials. We will see that for finite $n$ the energy is in direct relationship with the zeros of the orthogonal polynomials.

math.PR↗

arXiv2014

Generalized Gaussian Random Unitary Matrices Ensemble

We describe Generalized Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We will calculate a Laplace transform of such a density for finite $n$, which will be expressed through an hypergeometric function. When the dimensional of the hermitian matrix begin large enough, we will prove that the statistical density of eigenvalues converge in the tight topology to some probability measure, which generalize the Wigner semi-circle law.

math.PR↗

arXiv2014

Generalized $β$-Gaussian Ensemble Equilibrium measure method

We investigate $β$-Generalized random Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We investigate general method names as equilibrium measure method. When taking $n$ large limit we will see that the asymptotic density of eigenvalues generalize the Wigner semi-circle law.

math.PR↗