Let $T$ be a complete local (Noetherian) ring. Let $C$ be a countable set of pairwise incomparable nonmaximal prime ideals of $T$. We find necessary and sufficient conditions for $T$ to be the completion of a local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. Furthermore, if the characteristic of $T$ is zero, we provide necessary and sufficient conditions for $T$ to be the completion of an \textit{excellent} local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. In addition, for a positive integer $k$, we construct local integral domains that contain a prime ideal of height~$k$ whose formal fiber has countably many maximal elements.

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements. Let $f(x)=ax^{n} + x^{m}$ be a binomial with coefficients in $\mathbb{F}_{q}$ and $d = \mbox{gcd\,}(n-m,q-1)$. In this paper, we prove that there does not exist any permutation binomial such that $d$ satisfies certain congruence conditions, and we do some computations to list all non permutation binomials for $n-m=3$ and $q\leq 100$.

We study a McKendrick-von Foerster type equation with renewal. This model is represented by a single equation which describes one species which produces young individuals. The renewal condition is linear but takes into account some history of the population. This model addresses nonlocal interactions between individuals structured by age. The vast majority of size-structured models are also treatable. Our model generalizes a number of earlier models with delays and integrals. The existence and uniqueness is proved through a fixed-point approach to an equivalent integral problem in $L^{\infty}\cap L^1$.

We introduce new concepts of asymptotically anti-periodic function and semi-Lipschitz continuity. The former is a natural generalization of the well-known anti-periodic function. Then, sufficient conditions, ensuring the existence of asymptotically anti-periodic mild solutions to a Cauchy problem of nonlinear evolution equation with nonlocal initial condition, are established. It is mentioned that one of our main results is proved in the absence of the compactness and Lipschitz continuity of nonlocal item and of the Lipschitz continuity of nonlinearity. Finally, an example is presented as an application.

Assuming that the linear equation $x'' + h(t)x' + a(t)x = 0$ has a positive Green's function, we study the existence of nontrivial periodic solutions of second order damped dynamical systems \[ x'' + h(t)x' + a(t)x = f(t, x) + e(t), \] where $h$, $a\in \C(\!(\R/T\Z),\R)$, $e\! =\! (e_1,\ldots, e_N\!)^T\!\! \in \C(\!(\R/T\Z),\R^N\!)$, $N \ge 1$, and the nonlinearity $f = (f_1,\ldots, f_N)^T\in\C((\R=T\Z)\times\R^N\setminus\{0\},\R^N)$ has a repulsive singularity at the origin. We consider a very general singularity and do not need any kind of strong force condition. The proof is based on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and improved.

By employing Kranoselskii's fixed point theorem, we obtain sufficient conditions for the existence of nonoscillatory solutions of the forced higher-order nonlinear neutral dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^m }+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] on a time scale, where $p_{i}(t)$, $f_{i}(t)$ and $q(t)$ may be oscillatory. Then we establish sufficient and necessary conditions for the existence of nonoscillatory solutions to the equation $[x(t)+p(t)x(\tau(t))]^{\nabla^m }+F(t,x(\delta(t)))=q(t)$. Finally, we deal with dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^{m-1} \Delta}+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] with mixed $\nabla$ and $\Delta$ derivatives. In particular, some interesting examples are included to illustrate the versatility of our results.

Here, we find all the solutions of the title Diophantine equation in positive integer variables $(m,n,x,y)$, where $F_k$ is the $k$-th term of the Fibonacci sequence.

In this note, we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures, we show that the tile counting group associated to a set $T$ of tiles and a set of regions tileable by $T$ is isomorphic to a quotient of the second homology group of a 2-complex built from $T$. In this topological setting, we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.

We study the class of groups $G$ satisfying the condition that, for every ordered pair $x,y \in G$, one of the following is true: (1)~$xy=yx$; (2)~$x$ and $y$ are conjugate; (3)~$x^y=x^{-1}$; (4)~$y^x=y^{-1}$. We describe all such groups completely and give a further condition that characterizes these groups in terms of their $3$-$\text{S}$-rings.

Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. We study the relationship between $SU(2)$ and $\sldos$-representations of this group, looking at their characters. Using the description of the character variety of $G$, $X(G)$, we give a geometric description of $Y(G)\subset X(G)$, the set of characters arising from $SU(2)$-representations.

Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, according to their moduli. We refine one of those regions and replace it with the closed interior of a lemniscate that provides more precise information on the location of the zeros. Moreover, Pellet's theorem is considered the generalization of a zero inclusion region due to Cauchy. Using linear algebra tools, we derive a different generalization that leads to a sequence of smaller inclusion regions, which are also the closed interiors of lemniscates.

We generalize the character sum formulas of McLeman-Rasmussen attached to isogenies of elliptic curves in positive characteristic. Two improvements are given: sums are evaluated for isogenies of degree greater than two, and over arbitrary finite fields. We prove a transfer formula to evaluate such sums quickly over the domain curve and use this to evaluate the character sums attached to several standard families of isogenies of low degree.

Recently, substantial progress has been made on generalized factorization techniques in integral domains, in particular, $\tau$-factorization. There have also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by %C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing $\tau$-factorization to rings with zero-divisors by using the notion of U-factorization.

In this paper, we discuss the question of when the substructures, the singular sub-bimodule $\Delta[M,N]$ and the cosingular bi-submodule $\nabla[M,N]$ of $\mbox{Hom\,}(M,N)$, are equal to zero. Some well-known results of regular rings are obtained. Moreover, the substructures $\Delta[M,N]$ and $\nabla[M,N]$ with $M$ and $N$ that are direct sums of submodules are studied.

In this paper, we introduce the more general g-frame which is called a $K$-g-frame by combining a g-frame with a bounded linear operator $K$ in a Hilbert space. We give several equivalent characterizations for $K$-g-frames and discuss the stability of perturbation for $K$-g-frames. We also investigate the relationship between a $K$-g-frame and the range of the bounded linear operator $K$. In the end, we give two sufficient conditions for the remainder of a $K$-g-frame after an erasure to still be a $K$-g-frame. It turns out that although $K$-g-frames share some properties similar to g-frames, a large part of $K$-g-frames behaves completely different from g-frames.

In this paper, we provide some necessary and sufficient conditions to check when a (semi-)perfect graph is well-covered. We show that the checking process of these conditions can be achieved in a polynomial time. Since the comparability graph of a simplicial complex is a perfect graph, our result carries over to simplicial complexes.