Related papers: Densities, Laplace Transforms and Analytic Number …

We generalize the derivation of the Wallis formula for $\pi$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational…

Let $(X_1, \dots, X_n)$ be multivariate normal, with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$, and $S_n=\mathrm{e}^{X_1}+\cdots+\mathrm{e}^{X_n}$. The Laplace transform ${\cal…

By using the Lie's invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schr\"{o}dinger equations with variable coefficients. Starting from the equivalence generators we construct…

Non-alternating Hamiltonian Lie algebras in three variables over a perfect field of characteristic 2 are considered. A classification of non-alternating Hamiltonian forms over an algebra of divided powers in three variables and of the…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

In this paper, we make use of Robin and Lagarias' criteria to prove Riemann hypothesis. The goal is, using Lagarias criterion for $n\geq 1$ since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality…

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous…

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and K\"ahler manifolds under weak…

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish…

A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this…

We prove the following noncommutative version of Lewis's classical result. Every n-dimensional subspace E of Lp(M) (1<p<\infty) for a von Neumann algebra M satisfies d_{cb}(E, RC^n_{p'}) \leq c_p n^{\abs{1/2-1/p}} for some constant c_p…

The concept of weak Lie motion (weak Lie symmetry) is introduced through ${\cal{L}}_{\xi}{\cal{L}}_{\xi}g_{ab}=0,$ (${\cal{L}}_{\xi}{\cal{L}}_{\xi}f=0$). Applications are given which exhibit a reduction of the usual symmetry, e.g., in the…

We establish a correspondence between general relativity with diffeomorphism invariance and scalar field theories with Galilean invariance: notions such as the Levi-Civita connection and the Riemann tensor have a Galilean counterpart. This…

In this paper we continue our research on the concept of liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in $\mathbb{R}$. The most important examples of…

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra,…

In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010)…

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…

We define the notion of L^2 homology and L^2 Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L^2 homology…

We revisit the proof of the de Moivre--Laplace theorem, which is the ancestor of the central limit theorem for the binomial distribution. Our goal is to provide a proof that can be reasonably presented to undergraduate students within a…

We study the Wiener-Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener-Ikehara…