A theorem of Gil', relating the zeros of entire functions of finite order to traces of powers of matrices, is generalized to entire functions of finite rank and then analyzed from the point of view of spectral theory. Plenty of relevant examples are given, including a generalization of Viete's relations for the elementary symmetric functions of the roots of a polynomial.

We propose a generalization of a concept of bounded index for analytic functions in the unit ball. Use of directional derivatives gives us a possibility to deduce the necessary and sufficient conditions of boundedness of $L$-index in a direction for analytic functions of several variables, namely, we obtain an analog of Hayman's theorem and a logarithmic criteria for this class. The criteria describe the behavior of the directional logarithmic derivative outside the zero set and a uniform distribution of zeros in some sense. The criteria are useful for studying analytic solutions of partial differential equations and estimating their growth. We present a scheme of this application.

We prove an explicit formula for the conductor of an irreducible, admissible representation of $\operatorname{GL} _{n}(F)$ twisted by a character of $F^{\times} $ where the field $F$ is local and non-archimedean. As a consequence, we quantify the number of character twists of such a representation of fixed conductor.

We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations over distinguished order-ideals of the signed partition lattice. We also define the quotient of a signed graph by a signed permutation, and show that its signed graphic arrangement is closely related to an induced arrangement on a distinguished subspace. Lastly, a formula for the number of unlabeled acyclic orientations of a signed graph is presented which recalls classical reciprocity theorems of Stanley and Zaslavsky.

We show that in the extended modular group $\overline \Gamma = \rm {PGL}(2, \mathbb {Z})$ there are exactly seven finite subgroups up to conjugacy; three subgroups of size 2, one subgroup each of size 3, 4, and 6, and the trivial subgroup of size 1.

Let $\varphi $ be an analytic self-map of the open unit disk $\mathbb {D}$ and let $\psi $ be an analytic function on $\mathbb {D}$ such that the weighted composition operator $C_{\psi ,\varphi }$ defined by $C_{\psi ,\varphi }(f)=\psi f\circ \varphi $ is bounded on the Hardy and weighted Bergman spaces. We characterize those weighted composition operators $C_{\psi ,\varphi }$ on $H^{2}$ and $A_{\alpha }^{2}$ that are essentially hypo-normal, when $\varphi $ is a linear-fractional non-automorphism.

In this note, we discuss open book embeddings of closed non-orientable $3$-manifolds in $5$-manifolds. We also show that a huge class of closed orientable $3$-manifolds, namely the orientation double covers of certain closed non-orientable $3$-manifolds, open book embeds in the $5$-sphere $S^5$. Finally, we give a new proof of a well-known theorem which states that every closed non-orientable $3$-manifold smoothly embeds in $S^5$.

In this paper, we study a $(p,q)$-Laplace equation with singular and critical terms and establish the existence of multiple positive solutions via use of variational methods.

In this paper, we study the solvability of a mixed problem of a partial differential equation of high order with fractional derivatives with respect to time and with Sturm-Liouville operators with spatial variables and non-local boundary conditions; the solution is found as a series of eigenfunctions of the Sturm-Liouville operator with non-local boundary conditions.

We introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the $5$-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo $5$. Finally, we pose some problems for future work.

Let $D \subseteq E$ be an extension of commutative rings with identity, $I$ a nonzero proper ideal of $D$, $(\Gamma , \leq )$ a strictly totally ordered monoid such that $0 \leq \alpha $ for all $\alpha \in \Gamma $, and $\Gamma ^*=\Gamma \setminus \{0\}$. Let $D+[\![E^{\Gamma ^*, \leq }]\!]=\{f \in [\![E^{\Gamma , \leq }]\!] \mid f(0) \in D\}$ and $D+[\![I^{\Gamma ^*, \leq }]\!] =\{f \in [\![D^{\Gamma , \leq }]\!] \mid f(\alpha ) \in I$ for all $\alpha \in \Gamma ^*\}$. In this paper, we give some conditions for the rings $D+[\![E^{\Gamma ^*, \leq }]\!]$ and $D+[\![I^{\Gamma ^*, \leq }]\!]$ to satisfy the ascending chain condition on principal ideals.

This paper is devoted to the study of a nonhomogeneous discrete fractional initial value problem. Using the Laplace transform, we present the existence and uniqueness of the solution. We illustrate the applicability of results by an example.

Using associated trees, we construct a spectral triple for the $\mathrm {C}^*$-algebra of continuous functions on the ring of integers $R$ of a nonarchimedean local field $F$ of characteristic zero, and investigate its properties. Remarkably, the spectrum of the spectral triple operator is closely related to the roots of a $q$-hypergeometric function. We also study a noncompact version of this construction for the $\mathrm {C}^*$-algebra of continuous functions on $F$, vanishing at infinity.

For the symmetric group $S_4$ we determine all the integer values taken by its group determinant when the matrix entries are integers.

We examine the well-posedness of a strongly damped wave equation equipped with fractional diffusion operators. Ranges on the orders of the diffusion operators are determined in connection with global well-posedness of mild solutions or the global existence of weak solutions. Local existence proofs employ either semigroup methods or a Faedo-Galerkin scheme, depending on the type of solution sought. Mild solutions arising from semigroup methods are either analytic or of Gevrey class; the former produce a gradient system. We also determine the critical exponent for the nonlinear term depending on the orders of the fractional diffusion operators. Thanks to the nonlocal presentation of the fractional diffusion operators, we are able to work on arbitrary bounded domains. The nonlinear potential is only assumed to be continuous while satisfying a suitable growth condition.

We reconsider intermediate holomorphic convexity introduced by Barlet and Silva in the more general case of convexity with respect to hermitian holomorphic line bundles whose weight functions are (weakly) $q$-convex. Also we give completeness and Kahler properties of the canonical reduction.

In this paper, we study existence and stability of solutions to a partial differential equation with $\psi $-fractional derivative. Existence results are established by means of the fixed point and upper and lower solution methods. In addition, Ulam-type stability results are derived by using the Gronwall inequality method. Finally, an example is provided to illustrate the theoretical results.

In this paper, we consider the existence of infinitely many sign-changing solutions for the following Hardy-Sobolev-Maz'ya equation \[ \left \{\begin{aligned} &-\triangle u-\frac {\mu u}{|y|^2}=\lambda u+\frac {|u|^{2^{\ast }(t)-2}u}{|y|^t} \quad \mbox {in } \Omega , \\ &u=0 \quad \qquad \qquad \qquad \qquad \qquad \qquad \,\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega $ is an open bounded domain in $\mathbb {R}^N, \mathbb {R}^N=\mathbb {R}^k\times \mathbb {R}^{N-k}$, $\lambda >0$, $0\leq \mu \lt {(k-2)^2}/{4}$ when $k>2$, $\mu =0$ when $k=2$ and $2^{\ast }(t)={2(N-t)}/({N-2})$. A point $x\in \mathbb {R}^N$ is denoted as $x=(y,z)\in \mathbb {R}^k\times \mathbb {R}^{N-k}$, and the points $x^0=(0,z^0)$ are contained in $\Omega $. By using a compactness result obtained previously, we prove the existence of infinitely many sign changing solutions by a combination of the invariant sets method and the Ljusternik-Schnirelman type minimax method.

We provide formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. We derive from that exact formulas for the depth of those ideals. We also give some examples to show that the assumptions cannot be dropped.