The middle coefficient of a polynomial of an even degree $d$ is the coefficient of $x^{{d}/{2}}$. In this note we compute the middle coefficient of the cyclotomic polynomial $\Phi _{3p_2p_3}(x)$ when $p_3\equiv \pm 2,\pm 4, \pm 5\bmod 3p_2$.

We study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and $p$-gonal surfaces defined by divisors supported on their branch points. Moreover, we study invariant spin divisors under automorphisms and antiholomorphic involutions of Riemann surfaces and count them. We generalize a formula that gives $2$-spin divisors, proved by Mumford, to the case of $m$-spin divisors for an even $m$, supported on branch points of a hyperelliptic surface.

The $\sigma ^+$-game is played on a directed graph where the vertices (lights) are assigned an off or on state. The player can press any light, thereby toggling it along with its neighbors. The objective of the game is to turn all lights off given an initial configuration. Our particular interest are rectangular grid graphs with all lights initially on. We explore different classes of grid sizes and dimensions with unique solutions, and we construct the solutions to grids of size $(2^n-1)\times (2^n-1)$ with all lights initially on.

Motivated by the way numerical semigroups arise from primitive digraphs, we initiate the study of the graphic structure of a numerical semigroup. We lay some general groundwork for determining this structure and apply our results to a minimization problem.

For Banach spaces $E_1, \dots ,E_m$, $E$ and $F$ with their bases, we show that a particular monomial sequence forms a basis of $\mathcal {P}(^mE; F)$, the space of continuous $m$-homogeneous polynomials from $E$ to $F$ (resp.\ a basis of $\mathcal {L}(E_1,\dots ,E_m;F)$, the space of continuous $m$-linear operators from $E_1\times \cdots \times E_m$ to $F$) if and only if the basis of $E$ (resp. the basis of $E_1,\dots ,E_m$) is a shrinking basis and every $P \in \mathcal {P}(^mE; F)$ (resp.\ every $T \in \mathcal {L}(E_1,\dots ,E_m;F)$) is weakly continuous on bounded sets.

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this paper, we first determine a condition under which a given integer $t$ is a lower bound for the cohomological dimension $\operatorname {cd}(I,M)$, and use this to conclude that non-catenary Noetherian domains contain prime ideals that are not set-theoretic complete intersection. We also show the existence of a descending chain of ideals with successive diminishing cohomological dimensions. We then resolve the Artinianness of top local cohomology modules over local unique factorization domains of Krull dimension at most three, and obtain several related results on the top local cohomology modules for much more general cases.

Let $A$ be a an infinite dimensional stably finite unital simple separable exact ${\rm C^*}$-algebra. Let $B\subset A$ be a stably large subalgebra in $A$ such that $B$ has $m$-comparison of positive elements property (strong tracial $m$-comparison of positive elements property). Then $A$ has $m$-comparison of positive elements property (strong tracial $m$-comparison of positive elements property).

We give a new proof and minor refinements, in the polynomial case, of Jack's lemma. This proof is essentially based on the Bernstein's inequality for polynomials on the unit circle. We also discuss the cases of equality of \[ \frac {d^2}{d \theta ^2} \ln | {f(e^{i\theta })}| \leq 0, \] where $f$ is analytic in the unit disc and in a neighborhood of $e^{i\theta }$ where $|{f(e^{i\theta })}|=\sup _{|z|\lt 1}|f(z)|$.

In this paper, we study the existence of at least two non-trivial solution for a class of $p(x)$-curl systems with a small perturbation. We provide one new criterion to ensure the existence of two solutions. Recent results in the literature are extended and significantly improved.

For a group $G$ generated by $S\doteq \{g_1,\ldots ,g_n\}$, one can construct the Cayley graph $\mathrm {Cayley}({G},{S})$. Given a distance set $D\subset \mathbb Z _{\geq 0}$ and $\mathrm{Cayley}{G}{S}$, one can construct a $D$-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. Group $G$ acts on this chain complex, and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of $G$. This paper uses tools from group theory, representation theory and homological algebra to further our understanding of the interplay between generated groups, corresponding representations on their associated $D$-neighborhood complexes and the homology of the $D$-neighborhood complexes. This paper is an exposition of the results in my dissertation focusing on the case of two generators.

We consider a steady flow of an incompressible viscous fluid in a two-dimensional unbounded domain with unbounded boundaries. The domain has two outlets. The part of the domain upstream is a cylinder and the part of the domain downstream is a wedge. In the part of the domain upstream, the velocity is required to approach the Poiseuille flow. In the part of the domain downstream, the velocity is required to approach Jeffery-Hamel's flow. This problem has been treated by C.J. Amick and L.E. Frankel, V.A. Solonnikov and many others. Recently, T. Kobayashi obtained the unique solution of Jeffery-Hamel's flows in a wedge. Therefore we reconsider this problem. In this paper, we succeed in proving the existence of such a steady flow under the restricted flux condition which depends only on the part of the domain upstream and downstream.

We study both canonical reproducing kernels and constructive reproducing kernels for holomorphic functions in $\mathbb C^1$ and $\mathbb C^n$. We compare and contrast the two, and also develop important relations between the two types of kernels. We prove a new result about the relationship between these two kernels on certain domains of finite type.

We show that the set of analytic functions from $\mathbb C^2$ into $\mathbb C^2$, which are not Lorch-analytic is spaceable and strongly $\mathfrak {c}$-algebrable, but is not residual in the space of entire functions from $\mathbb C^2$ into $\mathbb C^2$. We also show that the set of functions which belongs to the disk algebra but not a Dales-Davie algebra is strongly $\mathfrak {c}$-algebrable and is residual in the disk algebra.

Given an integer $N\geq 2$ and a self-orthogonal subcategory $\mathcal {W}$ of an abelian category $\mathscr {A}$, we investigate the $\mathcal {W}$-Gorenstein $N$-complexes. We show that an $N$-complex $G$ is $\mathcal {W}$-Gorenstein if and only if $G$ is an $N$-complex consisting of $\mathcal {W}$-Gorenstein objects in $\mathscr {A}$. As an application, we improve a result of Estrada.

We establish a correspondence between the limit distribution and the asymptotic entropy of random walks on $\mathbb {Z}$, which have a sequence of step length that is exponentially increasing up to some error, and Bernoulli convolutions.

In this paper we extend the continuous wavelet transform to Schwartz distributions and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. But the uniqueness theorem for our wavelet inversion formula is valid for the space $\cal D '_F$ obtained by filtering (deleting) (i) all non-zero constant distributions from the space $\cal D '$, (ii) all non-zero constants that appear with a distribution as a union, as for example, for $ {x^2}/(1+x^2) = 1-1/(1+x^2)$, 1 is deleted and $-1/(1+x^2)$ is retained. The kernel of our wavelet transform is an element of $\cal D $ which when integrated along the real line vanishes, but none of its moments of order $m\ge 1$ along the real line is zero. The set of such kernels will be denoted by $D_m$.

We deal with positive solutions for the equation $ -\Delta _p u +\mu \alpha (x)|\nabla u|^q = a(x)f(u) + \lambda b(x)g(u)$ in $ \Omega $, where $\Omega \subset \mathbb {R}^N$ is a smooth bounded domain; $\Delta _p$ is the $p$-Laplacian operator with $1\lt p\lt N$; $0 \leq q \leq p$; $\lambda , \mu >0$ are real parameters; $ f: (0, \infty )\rightarrow [0,\infty )$ may be singular at $0$; $g: (0, \infty )\rightarrow [0,\infty )$ may be superlinear and $\alpha , a, b:{\Omega } \to (0,\infty )$ are suitable functions. The main novelties in this paper are the presence of singular and superlinear nonlinearities linked with the gradient term, which make it impossible to apply standard comparison principles and variational techniques. We overcome these difficulties by improving a technique of regularization-monotonicity of nonlinearities $f$ and $g$, using sub- and supersolution methods and avoiding the direct use of comparison principles to the operator $ -\Delta _p u +\mu \alpha (x)|\nabla u|^q$.

We study a plate equation with presumed nonuniqueness for the associated Cauchy problem. We establish the existence of global weak solutions by the Faedo--Galerkin method, and our main result refers to the existence of a global attractor using the method of generalized semiflows.

Let $F$, $G$ be monoidal comonads on a monoidal category $\mathcal {C}$. The aim of this paper is to discuss the smash coproducts of $F$ and $G$. As an application, the smash coproduct of Hom-bialgebras is discussed. Further, the Hom-entwining structure and Hom-entwined modules are investigated.