The monotonicity and convexity of a function involving digamma one and their applications

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left( 4/15,\infty \right)$ or $\left( 0,\infty \right) \times \left( 1/15,\infty \right)$ by the formula% \begin{equation*} \mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}% \right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{% 15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and convexity of the function $x\rightarrow F_{a}\left( x\right) =\psi \left( x+1\right) -\mathcal{L}(x,a)$, where $\psi$ denotes the Psi function. And, we determine the best parameter $a$ such that the inequality \psi \left( x+1\right) \right) \mathcal{L}% (x,a) holds for $x\in \left( -1,\infty \right)$ or $\left( 0,\infty \right)$, and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence $\left( l_{n}\left( a\right) \right)$ defined by $l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right)$, which gives extremely accurate values for $\gamma$.Comment: 20 page

Solutions of fractional logistic equations by Euler's numbers

In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler's numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler's numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set, The idea of the proof consists to explore some analogies with logistic function and Euler's numbers, and then to generalize them in the fractional case.Comment: Euler's numbers, Biological Application, Fractional logistic equatio

Geometric Phase Integrals and Irrationality Tests

Let $F(x)$ be an analytical, real valued function defined on a compact domain $\mathcal {B}\subset\mathbb{R}$. We prove that the problem of establishing the irrationality of $F(x)$ evaluated at $x_0\in \mathcal{B}$ can be stated with respect to the convergence of the phase of a suitable integral $I(h)$, defined on an open, bounded domain, for $h$ that goes to infinity. This is derived as a consequence of a similar equivalence, that establishes the existence of isolated solutions of systems equations of analytical functions on compact real domains in $\mathbb{R}^p$, if and only if the phase of a suitable ``geometric'' complex phase integral $I(h)$ converges for $h\rightarrow \infty$. We finally highlight how the method can be easily adapted to be relevant for the study of the existence of rational or integer points on curves in bounded domains, and we sketch some potential theoretical developments of the method

Field-free nucleation of antivortices and giant vortices in non-superconducting materials

Giant vortices with higher phase-winding than $2\pi$ are usually energetically unfavorable, but geometric symmetry constraints on a superconductor in a magnetic field are known to stabilize such objects. Here, we show via microscopic calculations that giant vortices can appear in intrinsically non-superconducting materials, even without any applied magnetic field. The enabling mechanism is the proximity effect to a host superconductor where a current flows, and we also demonstrate that antivortices can appear in this setup. Our results open the possibility to study electrically controllable topological defects in unusual environments, which do not have to be exposed to magnetic fields or intrinsically superconducting, but instead display other types of order.Comment: Revised version; 4 pages manuscript, 4 pages supplemental, 6 figure