Let $G$ be a graph. The minimum number of colors needed to color the edges of $G$ is called the chromatic index of $G$ and is denoted by $X'\left( G \right)$. It is well known that $\Delta \left( G \right)\,\le \,\mathcal{X}'\left( G \right)\,\le \Delta \left( G \right)\,+\,1$, for any graph $G$, where $\Delta \left( G \right)$ denotes the maximum degree of $G$. A graph $G$ is said to be class 1 if ${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)$ and class 2 if ${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)\,+\,1$. Also, ${{G}_{\Delta }}$ is the induced subgraph on all vertices of degree $\Delta \left( G \right)$. Let $f:\,V\left( G \right)\,\to \mathbb{N}$ be a function. An $f$ -coloring of a graph $G$ is a coloring of the edges of $E\left( G \right)$ such that each color appears at each vertex $v\,\in \,V\left( G \right)$ at most $f\left( v \right)$ times. The minimum number of colors needed to $f$ -color $G$ is called the $f$ -chromatic index of $G$ and is denoted by ${{{\mathcal{X}}'}_{f}}\left( G \right)$. It was shown that for every graph $G,\,{{\Delta }_{f}}\,\left( G \right)\,\le \,{{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,\le \,{{\Delta }_{f}}\,\left( G \right)\,+\,1$, where ${{\Delta }_{f}}\left( G \right)\,=\,{{\max }_{v\in \left( G \right)}}\,\left\lceil {{{d}_{G}}\left( v \right)}/{f\left( v \right)}\; \right\rceil $. A graph $G$ is said to be $f$ -class 1 if ${{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,=\,{{\Delta }_{f}}\left( G \right)$, and $f$ -class 2, otherwise. Also, ${{G}_{{{\Delta }_{f}}}}$ is the induced subgraph of $G$ on $\left\{ v\,\in \,V\left( G \right)\,:\,{{{d}_{G}}\left( V \right)}/{f\left( v \right)}\;\,=\,{{\Delta }_{f}}\left( G \right) \right\}$. Hilton and Zhao showed that if ${{G}_{\Delta }}$ has maximum degree two and $G$ is class 2, then $G$ is critical, ${{G}_{\Delta }}$ is a disjoint union of cycles and $\delta \left( G \right)\,=\,\Delta \left( G \right)-1$, where $\delta \left( G \right)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$ -coloring of graphs. Also, we determine the $f$ -chromatic index of a connected graph $G$ with $\left| {{G}_{{{\Delta }_{f}}}} \right|\,\le \,4$.

For every subnormal $m$-variable weighted shift $S$ (with bounded positive weights), there is a corresponding positive Reinhardt measure $\mu $ supported on a compact Reinhardt subset of ${{\mathbb{C}}^{m}}$. We show that, for $m\,\ge \,2$, the dimensions of the 1-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual $\widetilde{S}$ are different if a certain radial function happens to be integrable with respect to μ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to $\widetilde{S}$. We next prove that, for $m\,\ge \,2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual.

We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, $\text{BMOA}$, or the recently defined ${{\text{Q}}_{p}}$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on ${{\mathbb{C}}^{n}}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on ${{\mathbb{C}}^{n}}$ has a dense orbit.

Let $\left( R,\,m \right)$ be a non-zero commutative Noetherian local ring (with identity) and let $M$ be a non-zero finitely generated $R$-module. In this paper for any $\mathfrak{p}\,\in \,\text{Spec}\left( R \right)$ we show that

1

$$\text{injdi}{{\text{m}}_{{{R}_{\mathfrak{p}}}}}\,H_{\mathfrak{p}{{R}_{\mathfrak{p}}}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

1

$$\text{f}{{\text{d}}_{{{R}_{\mathfrak{p}}}}}H_{\mathfrak{p}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

are bounded from above by $\text{injdi}{{\text{m}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ and $\text{f}{{\text{d}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ respectively, for all integers $i\,\ge \,\dim\left( {R}/{\mathfrak{p}}\; \right)$.

An improved estimate is provided for the number of ${{\text{F}}_{q}}$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

For Banach spaces $X$ and $Y$, we show that if ${{X}^{*}}$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact, then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either ${{X}^{*}}$ or $Y$ has the bounded compact approximation property.

We prove a necessary and sufficient condition on the list of nonzero integers ${{u}_{1}},...,{{u}_{k}}$, $k\,\ge \,2$, under which a monic polynomial $f\,\in \,\mathbb{Z}\left| x \right|$ is expressible by a linear form ${{u}_{1}}\,{{f}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{u}_{k}}\,{{f}_{k}}$ in monic polynomials ${{f}_{1}},...,\,{{f}_{k}}\,\in \,\mathbb{Z}\left| x \right|$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials ${{f}_{1}},...,\,{{f}_{k}}$ can be chosen to be irreducible in $\mathbb{Z}\left[ x \right]$.

In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\text{S}{{\text{L}}_{2\left( \mathbb{Z} \right)}}$ by means of the cross-ratio, weight 2 modular forms, quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

Let $A$ be a Banach algebra and let $\pi :\,A\,\to \,\mathcal{L}\left( H \right)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with dim $H\,=\,\text{m}$. Let ${{\pi }_{ij}}$ be the coordinate functions of $\pi $ with respect to an orthonormal basis and suppose that for each $1\,\le \,j\,\le \,\text{m,}\,{{C}_{j}}\,=\,\sum\nolimits_{i=1}^{\text{m}}{\left\| {{\pi }_{ij}} \right\|}{{A}^{*}}\,<\,\infty $ and ${{\sup }_{j}}\,{{C}_{j}}\,<\,\infty $. Under these conditions, we call an element $\overline{\Phi }\,\in \,{{\iota }^{\infty }}\,\left( \mathfrak{m},\,{{A}^{**}} \right)$ left $\pi $-invariant if $a\,\cdot \overline{\Phi }\,={{\,}^{^{t}\pi }}\left( a \right)\overline{\Phi }$ for all $a\in A$ In this paper we prove a link between the existence of left $\pi $-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where $\pi :\,A\,\to \text{C}$ is a non-zero character on $A$.

The Riemann zeta-function is employed to generate universally overconvergent power series.

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., $\sum $-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module $_{R}U$ to have an indecomposable local dual $U_{R}^{\#}.$

In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms.

Let $R$ be a semiprime ring with center $Z\left( R \right)$. For $x,\,y\,\in \,R$, we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$. If $\sigma $ is a non-identity automorphism of $R$ such that

1

$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$

for all $x\,\in \,R$, where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$. In particular, when $R$ is a prime ring, $R$ is commutative.

We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes, and V. Trevisan.

We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.

Randers metrics are a special class of Finsler metrics. Every Randers metric can be expressed in terms of a Riemannian metric and a vector field via Zermelo navigation. In this paper, we show that a Randers metric has constant scalar curvature if the Riemannian metric has constant scalar curvature and the vector field is homothetic

In this paper, we study the spread of virus/worm in computer networks with a view to addressing cyber security problems. Epidemic models have been applied extensively to model the propagation of computer viruses, which characterize the fact that infected machines may spread malware to other hosts connected to the network. In our framework, the dynamics of hosts evolves according to a modified inhomogeneous Susceptible-Infectious-Susceptible $\left( \text{SIS} \right)$ epidemic model with time-varying transmission rate and recovery rate. The infection of computers is subject to direct attack as well as propagation among hosts. Based on optimal control theory, optimal attack strategies are provided by minimizing the cost (equivalently maximizing the profit) of the attacker. We present a threshold function of the fraction of infectious hosts, which captures the dynamically evolving strategies of the attacker and reflects the persistence of virus spreading. Moreover, our results indicate that if the infectivity of a computer worm is low and the computers are installed with antivirus software with high reliability, the intensity of attacks incurred will likely be low. This agrees with our intuition.