Let $\mathcal{A}$ be a unital $C^*$-algebra and let $M_1$ and $M_2$ be Banach left $\mathcal{A}$-modules. In this paper, we prove the generalized Hyers-Ulam-Rassias stability for a generalized form, \begin{eqnarray} g\Big( \sum_{i=1}^{n}r_i x_i \Big)= \sum_{i=1}^{n} s_i g(x_i) \end{eqnarray} of a Cauchy-Jensen functional equation $2g(\frac{x+y}{2})=g(x)+g(y)$ for a mapping $g : M_1 \rightarrow M_2.$ As an application, we show that every approximate $C^*$-algebra isomorphism $h:\mathcal{A} \rightarrow \mathcal{B}$ between unital $C^*$-algebras is a $C^*$-algebra isomorphism when $h$ satisfies some regular conditions.

In this paper we consider the existence of positive solutions for the following class of singular elliptic nonlocal problems of Kirchhoff-type $$ \left\{\begin{array}{rclcc} -M(\|u\|^{2})\Delta u = \frac{h(x)}{u^{\gamma}}+k(x)u^{\alpha} \mbox{in} \Omega ,\\ u > 0 \mbox{in} \Omega ,\\ u = 0 \mbox{on} \partial\Omega ,\\ \end{array} \right. $$ where $\Omega \subset \mathbb R^{N}, N \geq 2,$ is a bounded smooth domain, $M:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $\|u\|^{2}=\int_{\Omega}|\nabla u|^{2}$ is the usual norm in $H^{1}_{0}(\Omega )$. The main tools used are the Galerkin method and a Hardy-Sobolev inequality.

We show that the set ${\mathcal L}$ of complex-valued everywhere surjective functions on $\mathbb{C}$ is algebrable. Specifically, ${\mathcal L}$ contains an infinitely generated algebra every non-zero element of which is everywhere surjective. We also give a technique to construct, for every $n \in \mathbb N$, $n$ algebraically independent everywhere surjective functions, $f_1, f_2, \dots, f_n$, so that for every non-constant polynomial $P \in \mathbb{C}[z_1, z_2, \dots, z_n]$, $P(f_1, f_2, \dots, f_n)$ is also everywhere surjective.

In this paper we investigate the $p$-adic validity of the classical Dieudonné-Schwartz Theorem proved in [5] and of its main improvements given in the archimedean literature. We show that the situation differs substantially from the real or complex one.

We show the existence of forced periodic solutions to certain symmetric ordinary differential equations. First and second order systems of ordinary differential equations are investigated with and without damping with periodic and symmetric forcings. We study both resonance and nonresonance cases.

By constructing suitable kernel functions, a higher order Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra is obtained, leading to a higher order Cauchy integral formula.

A straightforward generalization of the classical inverse of a real function based on reflections leads to several insuperable difficulties. We introduce a new type of inverse w.r.t. monotone bijections $\phi$ that is determined by the direction of the base vectors of the real Euclidean plane. Inverting a monotone function in the real plane does not necessarily result in a function. Given an increasing real function $f$, Schweizer and Sklar geometrically construct a set of inverse functions. We will largely extend their construction to our new concept of $\phi$-inverses, also incorporating decreasing functions $f$. Furthermore, the geometrical and algebraical aspects of our approach are elaborated comprehensively. Special attention goes to the symmetry of a monotone function $f$ w.r.t. some monotone bijection $\phi$.

Projective planes $\Pi $ of order up to $q^{3}$ with a collineation group $G$ acting $2$-transitively on a subplane of order $q$ are investigated.

We establish the generalized Hyers-Ulam-Rassias stability of derivations in $C^*$-ternary rings associated to the Cauchy functional equation. We also show that any so-called almost derivation on a $C^*$-ternary ring is a true derivation.

A decision-maker observes sequentially a given permutation of $n$ uniquely rankable options. He has to invest capital into these opportunities at the moment when they appear. At each step only relative ranks are known. At the end the true rank of the option, at which the investment has been made, is known. [Bruss and Ferguson] have considered such problems under the assumption that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return. Uninvested capital keeps its risk-free value. Wrong investments lose their value. In this paper we partially extend results by [Bruss and Ferguson]. We confine our study to linear utility but a wider range of payoffs is taken into account. Two cases are considered. The first-type payoff gives a positive rate of return if the investment is made on the best or the second best option. The second-type payoff pays when the investment is at the second best option. We motivate these payoff choices. A few examples are explicitly solved.

We obtain maximum modulus principles for solutions to some quasilinear and fully nonlinear ODEs and discuss their applications to quasilinear PDEs involving $p$-Laplacian. Our approach is convenient to deal with singular PDEs. Its idea can be tracked back to the old theory by Szegö on orthogonal polynomials.

We prove that the index of every $m$-dimensional vector bundle over $B$ is equal to $m$ if $m \geq 2 dim{B}$. We also determine the smallest integer $k$ for which every $m$-dimensional vector bundle with $m\geq k$ is I-stable in the cases $B=FP^n$ and $B=S^n$.