Laplace Contour Integrals and Linear Differential Equations

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ L [ w ] ( z ) : = w ( n ) + ∑ j = 0 n - 1 ( a j + b j z ) w ( j ) = 0 . This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.

On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height $${\mathcal {H}}$$ H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to $${\mathcal {H}}$$ H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $$k \in \{0,d\}$$ k ∈ { 0 , d } or $$\gcd (k,d) = 1$$ gcd ( k , d ) = 1 . We therefore study the behaviour in the case where $$0< k < d$$ 0 < k < d and $$\gcd (k,d) > 1$$ gcd ( k , d ) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

On mean convergence of Fourier-Jacobi series

We establish the order of growth of modified Lebesgue constants of Fourier-Jacobi sums in $L_{p,w}$ spaces.

Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type="bibr" rid="b12">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id="M1">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id="M2">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id="M3">\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M4">\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.</p>

On [p, q]-order of growth of solutions of linear differential equations in the unit disc

<abstract><p>The $ [p, q] $-order of growth of solutions of the following linear differential equations $ (**) $ is investigated,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{'}+A_{0}(z)f = 0, (**) $\end{document} </tex-math></disp-formula></p> <p>where $ A_{i}(z) $ are analytic functions in the unit disc, $ i = 0, 1, ..., k-1 $. Some estimations of $ [p, q] $-order of growth of solutions of the equation $ (\ast*) $ are obtained when $ A_{j}(z) $ dominate the others coefficients near a point on the boundary of the unit disc, which is generalization of previous results from S. Hamouda.</p></abstract>

On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

$(p,q)$th order oriented growth measurement of composite $p$-adic entire functions

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and let $\mathcal{A}\left(\mathbb{K}\right)$ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire function $f\in \mathcal{A}\left( \mathbb{K}\right) $ and $r>0$, we denote by $|f|\left(r\right)$ the number $\sup \left\{ |f\left( x\right) |:|x|=r\right\}$, where $\left\vert \cdot \right\vert (r)$ is a multiplicative norm on $\mathcal{A}\left( \mathbb{K}\right)$. For any two entire functions $f\in \mathcal{A}\left(\mathbb{K}\right)$ and $g\in \mathcal{A}\left(\mathbb{K}\right)$ the ratio $\frac{|f|(r)}{|g|(r)}$ as $r\rightarrow \infty $ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norms. Likewise to complex analysis, in this paper we define the concept of $(p,q)$th order (respectively $(p,q)$th lower order) of growth as $\rho ^{\left( p,q\right) }\left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$ (respectively $\lambda ^{\left( p,q\right) }\left( f\right) =\underset{ r\rightarrow +\infty }{\lim \inf }\frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$), where $p$ and $q$ are any two positive integers. We study some growth properties of composite $p$-adic entire functions on the basis of their $\left(p,q\right)$th order and $(p,q)$th lower order.

A Comparison Between AHP and Hybrid AHP for Mobile Based Culinary Recommendation System

Mobile based culinary recommendation system has become critical topic in mobile application. Some methods presented in the literature propose the use of the AHP (Analytic Hierarchy Process), AHP TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and fuzzy AHP for mobile based culinary recommendation system. However, there are no comparative studies of these three methods when applied to mobile based culinary recommendation system. Thus, this research presents a comparative analysis of these three methods in the context of culinary recommendation system in mobile environment. The comparison was made based on accuracy and time complexity because mobile application environment needs low time complexity. The results have shown that all of these methods are suitable for culinary recommendation system in mobile environment. Fuzzy AHP have the highest accuracy between all of these methods, it have 66,67 % accuracy. But, AHP TOPSIS shows the best performance in time complexity, with order of growth in quadratic class (n2)