We find a sign-changing solution for a class of Schr\"odinger-Poisson systems in $\mathbb {R}^3$ as an existence result by minimization in a closed subset containing all the sign-changing solutions of the equation. The proof is based on variational methods in association with the deformation lemma and Miranda's theorem.

Let $T$ be a unital, continuous linear functional defined on complex Banach algebra $A$. First, we prove an approximate version of the Gleason-Kahane-\.Zelazko theorem: given $\epsilon >0$, there exists an $M>0$ such that, if $$ T(\exp x)\neq 0,\quad x\in A,\ \|x\|\leq M, $$ then $T$ is $\epsilon $-almost multiplicative. Then, we show that this result remains true if the exponential function is replaced by a nonsurjective entire function~$F$ with $F'(0)\neq 0$.

For an abelian number field $K$, the discriminant can be obtained from the conductor~$m$ of~$K$, the degree of~$K$ over $\mathbb {Q}$, and the degrees of extensions $K\cdot \mathbb {Q}(\zeta _{m/p^{\alpha }})/\mathbb {Q}(\zeta _{m/p^{\alpha }})$, where $p$ runs through the set of primes that divide $m$, and $p^{\alpha }$ is the greatest power that divides~$m$. In this paper, we give a formula for computing the discriminant of any abelian number field.

We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as actions on $C^*$\nobreakdash -algebras by Hilbert bimodules and describe the section algebras of these Fell bundles.

Our constructions give saturated Fell bundles over non-Hausdorff \'etale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms, that is, the Packer-Raeburn stabilization trick does not generalize to non-Hausdorff groupoids.

We characterize the closed sets in Euclidean space that satisfy a two-sided rolling ball property. As an application we show that certain conformal metric ratios have boundary value~1.

Since the nonlinear integral transforms $$ J_{\alpha }[f](z) = \int _{0}^{z}(f'(u))^{\alpha } du $$ and $$ \ \ I_{\alpha }[f](z) =\int _0^z (f(u)/u)^{\alpha } du $$ with a complex number $\alpha $ were introduced, a great number of studies have been dedicated to deriving sufficient conditions for univalence on the unit disk. However, little is known about the conditions where $J_{\alpha }[f]$ or $I_{\alpha }[f]$ produce a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper, we discuss quasiconformal extendability of the integral transforms $J_{\alpha }[f]$ and $I_{\alpha }[f]$ for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordination and Loewner theory are applied to this problem.

We study Tate cohomology of modules over a commutative Noetherian ring with respect to semidualizing modules. First, we show that the class of modules admitting a Tate $\mathcal{F}_C $-resolution is exactly the class of modules in $\mathcal{B}_{C} $ with finite $\mathcal{GF}_{C} $-projective dimension. Then, the interaction between the corresponding relative and Tate cohomologies of modules is given. Finally, we give some new characterizations of modules with finite $\mathcal{F}_C $-projective dimension.

We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with modulus squared weight of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng problem.

Let $G$ be an undirected simple, connected graph with $n \geq 3$ vertices and $m$ edges, with vertex degree sequence $d_1\ge d_2 \ge \cdots \ge d_n$. The general Randi\'c index is defined by \[ R_{-1}=\sum _{(i,j)\in E}\frac {1}{d_id_j}. \] Lower and upper bounds for $R_{-1}$ are obtained in this paper.

The discrete part of the discrete-continuous orthogonality \[ \mathscr {B}(f,g)=\mathscr {B}_d( f,g)+\mathscr {B}_c(f^{(N)},g^{(N)}), \] is studied for families of classical orthogonal polynomials such that the associated three-term recurrence relation \[ xp_n=p_{n+1}+\beta _np_n+ \gamma _n p_{n-1}, \] presents one vanishing coefficient $\gamma _n$, as in the case of Laguerre polynomials $L_n^{(-N)}$, Jacobi polynomials $P_n^{(-N,\beta )}$ and Gegenbauer polynomials $C_n^{(-N+1/2)}$ with $N\in \mathbb {N}$. It is shown that the discrete bilinear functional $\mathscr {B}_d$ can be replaced by a linear functional, $\mathscr {L}$, or by another bilinear functional related with $\mathscr {L}$, which allows us to reformulate the orthogonality in a much simpler way in the case of Laguerre polynomials and in a totally explicit form in the case of Jacobi and Gegenbauer polynomials.

We prove that the underlying set of an orbifold equipped with the ring of smooth real-valued functions completely determines the orbifold atlas. Consequently, we obtain an essentially injective functor from orbifolds to differential spaces.

In this paper, we study the existence of positive ground state solutions for non-autonomous Kirchhoff type problems: $$ -\Big (1+b\int _{\mathbb R^3} |\nabla u|^2\Big ) \Delta u+u=a(x)|u|^{p-1}u \quad \mbox {in } \mathbb {R}^3, $$ where $b>0$, $3\lt p\lt 5$ and $a:\mathbb R^3\rightarrow \mathbb R$ is such that $$ \lim _{|x|\rightarrow \infty } a(x)=a_\infty >0, $$ but no symmetry property on $a(x)$ is required.