The aim of this article is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of certain known transcendental differential functions, as of $\Gamma(x)$. Furthermore, it also determines differential transcendence of solution of some functional equations.

Some necessary and/or sufficient conditions for the uniform exponential dichotomy are pointed out. Thus are extended known results due to Barbashin, Datko, Lovelady, Pazy, Preda.

We obtain Gagliardo-Nirenberg interpolation inequalities of the form $\| \nabla u\|_{X}\leq C_1\sqrt{\| u|\|_{Y}\|\nabla^{(2)}u\|_Z}+C_2\| u\|_Y$, where $X,Y,Z$ are Orlicz spaces related to a single measure which may not satisfy the doubling condition. Some examples among homogeneous, logarithmic and exponential spaces are given.

Let $\phi $, $\theta $ be odd increasing homeomorphisms from $\mathbb{R}$ onto $\mathbb{R}$ satisfying $\phi (0)=\theta (0)=0$, $f:[0,1]\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}$ be a function satisfying Carathéodory conditions and $e:[0,1]\rightarrow \mathbb{R}$ be a function in $L^{1}[0,1]$. Let $\xi _{i}$,$\tau _{j}\in (0,1)$, $a_{i}$, $ b_{j}\in \mathbb{R}$, $i=1$, $2$, $\cdot \cdot \cdot $, $m-2$, $j$ $=$ $1$, $ 2$, $\cdot $ $\cdot $ $\cdot $, $n-2$, $0<\xi _{1}<\xi _{2}<\cdot \cdot \cdot <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\cdot \cdot \cdot <\tau_{n-2}<1 $ be given. We study the problem of existence of solutions for the generalized multi-point boundary value problem \begin{gather} (\phi (x^{\prime }))^{\prime }=f(t,x,x^{\prime })+e\text{, }0<t<1, \notag \\ x(0)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \theta (x^{\prime }(1))=\sum_{j=1}^{n-2}b_{j}\theta (x^{\prime }(\tau _{j})), \end{gather} in the non-resonance case. We say that this problem is non-resonant if the associated problem: \begin{gather} (\phi (x^{\prime }))^{\prime }=0\text{, }0<t<1, \notag \\ x(0)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \theta (x^{\prime }(1))=\sum_{j=1}^{n-2}b_{j}\theta (x^{\prime }(\tau _{j})), \end{gather} has the trivial solution as its only solution. This is the case if \begin{equation*} (1-\sum_{j=1}^{n-2}b_{j})(1-\sum_{i=1}^{m-2}a_{i})\neq 0. \end{equation*} Our methods consist in using topological degree and some a priori estimates.

In this paper we study a class of quasilinear elliptic problems involving critical exponents and non-standard growth condition. We establish the existence of at least one nontrivial solution using as main tool Ekeland's variational principle.

Given two harmonic functions $u_{+}(x,y)$, $u_{-}(x,y)$ defined on opposite sides of the $y$-axis in $\mathbb{R}^2$ and periodic in $y$, we consider the problem of constructing a {\it family of gluing elliptic functions}, i.e. a family of functions $u_{\epsilon}(x,y)$ of class ${\mathcal C}^{1,1}$ that coincide with $u_+$ and $u_-$ outside neighborhoods of the $y$-axis of width less than $\epsilon$ and are solutions to linear, uniformly elliptic equations without zero order terms. We first show that not always there is such a family and we give a necessary condition for its existence. Then we give a sufficient condition for the existence of a family of gluing elliptic functions and a way for its construction.

Let $M$ be an $n$-dimensional orientable compact hypersurface in an $(n+1)$-dimensional real space form $\overline{M}(c)$, $n\geq 2$. If the lengths $\left\| R\right\| $, $\left\| A\right\| $ and $\left\| \nabla \alpha \right\| $ of the curvature tensor field $R$, the shape operator $A$, the gradient $\nabla \alpha $ of the mean curvature $\alpha $ and the scalar curvature $S$ of the hypersurface $M$ satisfy the inequality \begin{equation*} \frac{1}{2}\left\| R\right\| ^{2}\leq cS+\delta \left\| A\right\| ^{2}-n(n-1)\left\| \nabla \alpha \right\| ^{2} \end{equation*} where $\delta =\min Ric=\underset{p\in M\quad v\in T_{p}M\quad \left\| v\right\| =1}{\min }Ric_{p}(v)$, $Ric$ is Ricci curvature of the hypersurface, then it is shown that $M$ is an extrinsic sphere in $\overline{M}(c)$. In particular we deduce that the condition $\frac{1}{2}\left\| R\right\| ^{2}\leq \delta \left\| A\right\| ^{2}-n(n-1)\left\| \nabla \alpha \right\| ^{2}$ characterizes spheres in the Euclidean space $R^{n+1}$ among the compact orientable hypersurfaces whose Ricci curvatures are bounded below by a constant $\delta >0$.

The existence, non-existence and multiplicity of solutions to periodic boundary value problems of Liénard type \begin{eqnarray*} (|u'|^{p-2}u')'+ f(u)u'+ g(u) = e(t) + s,\quad u(0)-u(T)=0=u'(0)-u'(T), \end{eqnarray*} is discussed, where $p>1,$ $f$ is arbitrary and $g$ is assumed to be bounded, positive and $g(\pm\infty)=0.$ The function $e$ is continuous on $[0,T]$ with mean value $0$ and $s$ is a parameter.

In the paper we consider the spaces of entire functions $f(z), z\in C^n$, satisfying the condition $$ \int_{R^n}\left(\int_{R^n}|f(x+iy)|^p dx \right)^s |y|^{\alpha}e^{-\sigma |y|^{\rho}}dy <+\infty . $$ For these classes the following integral representation is obtained: $$ f(z)=\int_{C^n}f(u+iv)\Phi(z,u+iv)|v|^{\alpha}e^{-\sigma |v|^{\rho}}dudv ,\quad z\in C^n ,$$ where the reproducing kernel $\Phi(z,u+iv)$ is written in an explicit form as a Fourier type integral. Also, an estimate for $\Phi$ is obtained.

In this note we will show how one can describe the representation varieties of certain Fuchsian groups using the Luna Slice theorem and the technique of local quivers. We will illustrate with one specific example: the Fuchsian group $\mathbb{Z}_2 * \mathbb{Z}_4$.

We consider the problem: given a regular form (linear functional) $v$, find all the regular forms $u$ which satisfy the relation $xu=\lambda x^2v,\lambda \in\C -\{0\}$. We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to $u$. Some examples are studied.

We introduce a new formalization of Higher-Order-Logic (abbreviated Hol), which we baptized Formath, an acronym for FORMAl MATHematics. We discuss the syntax, semantics, deduction-rules, axioms and principles of extension, after which we prove soundness and consistency. The semantics are comparable to other systems for Hol, such as Hol-4 and Hol-Light, but other parts differ from the traditional way of working, for example the deduction-rules and axioms. We discuss these differences in large extent. We also talk about porting theorems to the Formath library, provide examples and discuss the applications.

In this paper, we prove $L^\infty$-regularity for solutions of some nonlinear elliptic equations with degenerate coercivity whose prototype is $$ \left\{\begin{array}{lll} {\rm-div}({\frac{1}{(1+|u|)^{\theta(p-1)}}}|\nabla u|^{p-2}{\nabla u})=f&{\rm in}&\Omega, \\ u=0&{\rm on}& \partial{\Omega}, \end{array} \right. $$ where $\Omega$ is a bounded open set in ${\rm \mathbb{R}^N}$, $N\geq 2$, $1<p<N$, $\theta$ is a real such that $0\leq\theta\leq1$ and $f\in L^{\frac{N}{p}}log^{\alpha}L$ with some $\alpha>0.$

The approximate orthogonalization method for a finite set of linearly independent vectors was introduced by Z. Kovarik in which it is necessity to compute explicitly the inverse of a matrix in every iteration. It is proved that Kovarik's method converges quadratically. Several modifications have been proposed for Kovarik's method, all of which try to eliminate the necessity of explicit computation of the inverse. Most of these methods are linear convergent. The best modification, with a good convergent behaviour, is Petcu and Popa's, although they did not express any satisfactory reason for the origin of this modification. In this paper, we present a class of modifications for Kovarik's method which consists of Petcu and Popa's method. We prove that the methods from this class are, generally, linear convergent, while, only for the special case of the Petcu and Popa's method, it is quadratic convergent. Therefore, we show that Petcu and Popa's method, in contrast with their claim, is not linear but quadratic convergent, turning it into an optimal method in this class.