In this paper, we study a class of nonhomogeneous sublinear-superlinear $p$-Laplacian equations in $\mathbb {R}^N$. By applying a minimization method on the Nehari manifold $\mathcal {N}^{\alpha }$, the existence of positive solutions and the continuity in the perturbation term are obtained.

Let $L/\mathbb {Q}$ be a cyclic extension of degree~$p$, where $p$ is an odd unramified prime in $L/\mathbb {Q}$. An explicit description of the integral trace form $Tr _{L/\mathbb {Q}}(x^2)|_{\mathfrak O_L}$, where~$\mathfrak O_L$ is the ring of algebraic integers of $L$, is given, and an application to finding the minima of certain algebraic lattices is presented.

The objective of the present work is to present information on the set of periodic points of a continuous self-map on a closed surface which can be obtained using the action of this map on homological groups of the closed surface.

We classify the centers of polynomial differential systems in $\mathbb {R}^2$ of odd degree $d \ge 5$, in complex notation, as $ \cdot z = i z + (z \overline z)^{({d-5})/{2}} (A z^5 + B z^4 \overline z + C z^3 \overline z^2+ D z^2 \overline z^3+ E z \overline z^4 + F \overline z^5)$, where $A, B, C, D, E, F \in \mathbb {C}$ and either $A=\Re (D)=0$, $A=\Im (D)=0$, $\Re (A)=D=0$ or $\Im (A)=D=0$.

Let $G \subsetneq \mathbb {R}^n$ be a domain, and let $d_1$ and $d_2$ be two metrics on $G$. We compare the geometries defined by the two metrics to each other for several pairs of metrics. The metrics we study include the distance ratio metric, the triangular ratio metric and the visual angle metric. Finally, we apply our results to study Lipschitz maps with respect to these metrics.

Characterizations of bounded and finite factorization domains are given using topological notions. Using our characterizations, the almost Dedekind domain and Pr\"ufer domain constructed by Grams \cite {Grams} are shown to be a BFD and an FFD, respectively. For a class of almost Dedekind (not Dedekind) domains it is shown that satisfying the ascending chain condition for principal ideals implies BFD.

In this short note, we introduce the notions of ``factorable ring" and ``fully factorable ring" for commutative rings based upon the notion of ``factorable domain" advanced by Anderson, Kim and Park~\cite {AKP}. Using a novel sufficient condition for an ideal to be a product of nonfactorable ideals, we classify the Artinian rings that are (fully) factorable. We also explore the intersection of the class of factorable rings with the class of Noetherian rings. An analogue for multiplication rings of a characterization result due to Butts~\cite {HSB} concerning when such a unique factorization occurs is provided.

Let $K$ be an algebraically closed field of characteristic~$0$, and let $H^{\mu }_{\mathbb {A}^n_{K }}$ denote the Hilbert scheme of $\mu $ points of $\mathbb {A}^n_{K}$. An \textit {elementary component} $E$ of $H^{\mu }_{\mathbb {A}^n_{K }}$ is an irreducible component such that every $K$-point $[I]\in E$ represents a length-$\mu $ closed subscheme $Spec (K [x_1,\ldots ,x_n]/I)\subseteq \mathbb {A}^n_{K}$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $\mu > 1$) of elementary components \cite {Iarrob-Emsalem}; in their examples, the ideals $I$ were homogeneous (up to a change of coordinates corresponding to a translation of $\mathbb {A}^n_{K}$). We generalize their construction to obtain new examples of elementary components.

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2=x^3+\alpha x^2-n^2 x$. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with the $\alpha =0$ case of congruent numbers. Congruent numbers are positive integers equal to the area of a right triangle with rational side lengths.

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under the convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first and second kind. Hence, we call them Stirling operators. In order to construct matrix representations of the roots of GFPs, we use Stirling operators of the first kind. We give explicit examples to show how Stirling operators of the second kind appear in low degree cases for the WIP-roots. As a consequence of the matrix construction we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.

Inspired by Stark's analytic proof of class number finiteness of a ring of integers in an algebraic number field, we give a new proof of the finiteness of the number of classes of ideals in a maximal order of a totally definite quaternion algebra over a totally real number field. Our proof makes use of Epstein zeta function properties. This approach leads to alternative proofs of Eichler's mass formula and even parity of the number of ramified primes in the quaternion algebra.

We describe the $C^*$-algebra generated by an irreducible Toeplitz operator~$T_{\psi }$, with continuous symbol~$\psi $ on the unit circle $\mathbb {T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For specific automorphism-induced composition operators and certain types of irreducible Toeplitz operators, we show that the above $C^*$-al\-ge\-bra is not isomorphic to that generated by the shift and composition operators.

Let $R$ be a commutative ring with identity, and let $R[x]$ be the collection of polynomials with coefficients in~$R$. We observe that there are many multiplications in $R[x]$ such that, together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications belong to a class of functions $\lambda $ from $\mathbb {N}_0$ to $\mathbb {N}$. The trivial case when $\lambda (i) = 1$ for all $i$ gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when $\lambda (i) = i!$ for all $i$. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we study Krull dimension and unique factorization in $R_H[x]$. We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$. When the ring $R$ is Noetherian we prove that $\dim R \leq \dim R_H[x] \leq \dim R+1$. A condition for the ring $R$ is also given in order to determine whether $\dim R_H[x] = \dim R$ or $\dim R_H[x] = \dim R +1$ in this case. We show that $R_H[x]$ is a unique factorization domain, respectively, a Krull domain, if and only if $R$ is a unique factorization domain, respectively, a Krull domain, containing all of the rational numbers.

In this paper, we establish two regularity criteria for the two-dimensional (2D) incompressible generalized magnetohydrodynamic (GMHD) equations in terms of only one quantity, namely, the current density $j=\nabla \times b$ or the vorticity $\omega =\nabla \times u$. More precisely, it is proved that, if one of the following holds true: \[ \int _{0}^{T}{\|j(t)\|_{\dot {B}_{\infty ,\,\infty }^{0}(\mathbb {R}^{2})}\,dt}\lt \infty , \] \[ \int _{0}^{T}{\|\omega (t)\|_{\dot {B}_{\infty ,\,\infty }^{0}(\mathbb {R}^{2})}\,dt}\lt \infty , \] then the solution $(u,\,b)$ actually remains regular on $[0, T ]$.

This paper is concerned with front-like entire solutions of a delayed nonlocal dispersal equation with convolution type bistable nonlinearity. Here, a solution defined for all $(x, t)\in \mathbb {R}^2$ is an entire solution. It is known that the equation has an increasing traveling wavefront with nonzero wave speed under some reasonable conditions. We first give the asymptotic behavior of traveling wavefronts at infinity. Then, by the comparison argument and sub-super-solutions method, we construct new types of entire solutions other than traveling wavefronts and equilibrium solutions of the equation, which behave like two increasing traveling wavefronts propagating from both sides of the $x$-axis and annihilate at a finite time. Finally, various qualitative properties of the entire solutions are also investigated.