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We study the existence of fixed points for contraction multivalued
mappings in modular metric spaces endowed with a graph. The
notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. This paper can
be seen as a generalization of Nadler's and Edelstein's fixed
point theorems to modular metric spaces endowed with a graph.

In this paper, we introduce the definition of a convex real
valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We
prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f
\geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept,
we state and prove discrete Hermite-Hadamard inequality using
the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$).
Second, we state and prove the discrete fractional Hermite-Hadamard
inequality using the basics of discrete fractional calculus.
We close the paper by defining the convexity of a real valued
function on any time scale.

The $Q_p$ spaces of holomorphic functions on
the disk, hyperbolic Riemann surfaces or complex unit ball have
been studied deeply.
Meanwhile, there are a lot of papers devoted to the $Q^\#_p$
classes of meromorphic functions on the disk or hyperbolic Riemann
surfaces. In this paper, we prove the nesting property (inclusion
relations) of $Q^\#_p$ classes on hyperbolic Riemann surfaces.
The same property for $Q_p$ spaces was also established systematically
and precisely in
earlier work
by the authors of this paper.

The classification of Euclidean frieze groups into seven conjugacy
classes is well known, and many articles on recreational mathematics
contain frieze patterns that illustrate these classes. However,
it is
only possible to draw these patterns because the subgroup of
translations that leave the pattern invariant is (by definition)
cyclic, and hence discrete. In this paper we classify the conjugacy
classes of frieze groups that contain a non-discrete subgroup of
translations, and clearly these groups cannot be represented
pictorially in any practical way. In addition, this discussion
sheds
light on why there are only seven conjugacy classes in the classical
case.

Let $f$ be a holomorphic function of the unit
disc $\mathbb{D},$ preserving the origin. According to Schwarz's
Lemma, $|f'(0)|\leq1,$ provided that $f(\mathbb{D})\subset\mathbb{D}.$
We prove that this bound still holds, assuming only that $f(\mathbb{D})$
does not contain any closed rectilinear segment
$[0,e^{i\phi}],\;\phi\in[0,2\pi],$ i.e. does not contain any
entire radius of the closed unit disc. Furthermore, we apply
this result to the hyperbolic density and we give a covering
theorem.

An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.

We prove that the existence spectrum of Mendelsohn triple systems
whose associated quasigroups satisfy distributivity corresponds
to the Loeschian numbers, and provide some enumeration results.
We do this by considering a description of the quasigroups in
terms of commutative Moufang loops.

In addition we provide constructions of Mendelsohn quasigroups
that fail distributivity for as many combinations of elements
as possible.

These systems are analogues of Hall triple systems and anti-mitre
Steiner triple systems respectively.

In this note we present a simple alternative proof
for the Bernstein problem in the three-dimensional Heisenberg
group $\operatorname{Nil}_3$
by using the loop group technique. We clarify the geometric
meaning of the two-parameter ambiguity of entire minimal graphs
with prescribed Abresch-Rosenberg differential.

An incomplete pairwise balanced design is equivalent to a pairwise
balanced design with a distinguished block, viewed as a `hole'.
If there are $v$ points, a hole of size $w$, and all (other)
block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition
to congruence restrictions on $v$ and $w$, there is also a necessary
inequality: $v \gt (k-1)w$. This article establishes two main existence
results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$
is large, and the other in the case $v \gt (k-1+\epsilon) w$ when
$w$ is large (depending on $\epsilon$). Several possible generalizations
of the problem are also discussed.

This paper studies the uncertainty principle for spherical
$h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated
with a weight function invariant under a general finite reflection
group, which
is in full analogy with the classical Heisenberg inequality.
Our proof is motivated by a new decomposition of the Dunkl-Laplace-Beltrami
operator on the weighted sphere.

Let $R$ be a prime ring of characteristic different from
$2$, $Q_r$ be its right Martindale quotient ring and
$C$ be its extended centroid. Suppose that $F$ is
a generalized skew derivation of $R$, $L$ a non-central Lie ideal
of $R$, $0 \neq a\in R$,
$m\geq 0$ and $n,s\geq 1$ fixed integers. If
\[
a\biggl(u^mF(u)u^n\biggr)^s=0
\]
for all $u\in L$, then either $R\subseteq M_2(C)$, the ring of
$2\times 2$ matrices over $C$, or $m=0$ and there exists $b\in
Q_r$ such that
$F(x)=bx$, for any $x\in R$, with $ab=0$.

We consider a nonlinear parametric elliptic equation driven
by a nonhomogeneous differential
operator with a logistic reaction of the superdiffusive type.
Using variational methods coupled with suitable truncation
and comparison techniques,
we prove a bifurcation type result describing the set of positive
solutions
as the parameter varies.

A simple proof is given for the fact that, for $m$ a non-negative
integer, a function $f\in C^{(m)}(\mathbb{R}),$ and an arbitrary positive
continuous function $\epsilon,$ there is an entire function $g,$
such that $|g^{(i)}(x)-f^{(i)}(x)|\lt \epsilon(x),$ for all $x\in\mathbb{R}$
and for each $i=0,1\dots,m.$ We also consider the situation,
where $\mathbb{R}$ is replaced by an open interval.

In this paper we study the zero sets of harmonic functions on
open sets in $\mathbb{R}^N$ and holomorphic functions on open sets in
$\mathbb{C}^N.$
We show that the non-extendability of such zero sets is a generic
phenomenon.

We continue to investigate branching systems of directed graphs
and their connections with graph algebras. We give a sufficient
condition under which the representation induced from a branching
system of a directed graph is faithful and construct a large
class of branching systems that satisfy this condition. We finish
the paper by providing a proof of the converse of the Cuntz-Krieger
uniqueness theorem for graph algebras by means of branching systems.

In this paper, the authors characterize second-order Sobolev
spaces $W^{2,p}({\mathbb R}^n)$,
with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and
$n\in\{1,2,3\}$, via the Lusin area
function and the Littlewood-Paley $g_\lambda^\ast$-function in
terms of ball means.

We show that the product rank of the $3 \times 3$ determinant
$\det_3$ is $5$,
and the product rank of the $3 \times 3$ permanent
$\operatorname{perm}_3$
is $4$.
As a corollary, we obtain that the tensor rank of $\det_3$ is
$5$ and the tensor rank of $\operatorname{perm}_3$ is $4$.
We show moreover that the border product rank of $\operatorname{perm}_n$ is
larger than $n$ for any $n\geq 3$.

Alfred Schild has established conditions
that Lorentz transformations map world-vectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. The problem
was dealt with in the context of tensor and spinor calculus.
Due to Schild's number-theoretic arguments, the subject is also
interesting when isolated from its physical background.

The paper of Schild is not easy to understand. Therefore we first
present a streamlined version of his proof which is based on
the use of null vectors. Then we present a purely algebraic proof
that is somewhat shorter. Both proofs rely on the properties
of Gaussian integers.

In this paper, we develop a generalized Jordan canonical form
theorem for a certain class of operators in $\mathcal
{L}(\mathcal {H})$. A complete criterion for similarity for this
class of operators in terms of $K$-theory for Banach
algebras is given.

A master formula of transformation formulas for bilinear sums
of basic hypergeometric series
is proposed.
It is obtained from the author's previous results on
a transformation formula for Milne's multivariate generalization
of basic hypergeometric
series of type $A$ with different dimensions and it can be considered
as a
generalization of the Whipple-Sears transformation formula for
terminating balanced ${}_4 \phi_3$
series.
As an application of the master formula, the one variable cases
of some transformation formulas
for bilinear sums of basic hypergeometric series are given as
examples.
The bilinear transformation formulas seem to be new in the literature,
even in one variable case.

We study and generalize a classical theorem of L. Bers that classifies
domains up to biholomorphic equivalence in terms of the algebras
of
holomorphic functions on those domains. Then we develop applications
of these results to the study of domains with noncompact automorphism
group.

We formulate a conjectural hard Lefschetz property
for Chow groups, and prove this in some special cases: roughly
speaking, for varieties with finite-dimensional motive, and
for varieties whose self-product has vanishing middle-dimensional
Griffiths group. An appendix includes related statements that
follow from results of Vial.

Let $T$ be a quadratic operator on a complex Hilbert space $H$.
We show that $T$ can be written as a product of two positive
contractions if and only if $T$ is of the form
\begin{equation*}
aI \oplus bI \oplus
\begin{pmatrix} aI & P \cr 0 & bI \cr
\end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus
H_3)
\end{equation*}
for some $a, b\in [0,1]$ and strictly positive operator $P$ with
$\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we
give a necessary condition for a bounded linear operator $T$
with operator matrix
$
\big(
\begin{smallmatrix} T_1 & T_3
\\ 0 & T_2\cr
\end{smallmatrix}
\big)
$ on $H\oplus K$ that can be written as a product
of two positive contractions.

In this paper, a
nonlinear stage-structured model for Lyme disease is considered.
The model is a system of differential equations with two time
delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is
derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium
is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$,
then there exists
a unique positive equilibrium whose local asymptotical stability
and the existence of
Hopf bifurcations are established by analyzing the distribution
of the characteristic values.
An explicit algorithm for determining the direction of Hopf bifurcations
and the
stability of the bifurcating periodic solutions is derived by
using the normal form and
the center manifold theory. Some numerical simulations are performed
to confirm the correctness
of theoretical analysis. At last, some conclusions are given.

Let $a_1, \cdots, a_9$ be non-zero integers and $n$ any integer.
Suppose
that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$
for $1 \leq i \lt j \leq 9$.
In this paper we prove that

(i) if $a_j$ are not all of the same sign, then the cubic
equation $a_1p_1^3+\cdots +a_9p_9^3=n$ has prime solutions satisfying
$p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{8+\varepsilon};$

(ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{25+\varepsilon}$,
then
$a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$.

This results improve our previous results (Canad. Math. Bull.,
56 (2013), 785-794)
with the bounds $\textrm{max}\{|a_j|\}^{14+\varepsilon}$ and
$\textrm{max}\{|a_j|\}^{43+\varepsilon}$
in place of $\textrm{max}\{|a_j|\}^{8+\varepsilon}$ and $\textrm{max}\{|a_j|\}^{25+\varepsilon}$
above, respectively.

Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.

In the framework of homological characterizations of relative
hyperbolicity, Groves and Manning posed the question of whether
a simply connected $2$-complex $X$ with a linear homological
isoperimetric inequality, a bound on the length of attaching
maps of $2$-cells and finitely many $2$-cells adjacent to any
edge must have a fine $1$-skeleton. We provide a positive answer
to this question. We revisit a homological characterization
of relative hyperbolicity, and show that a group $G$ is hyperbolic
relative to a collection of subgroups $\mathcal P$ if and only if
$G$ acts cocompactly with finite edge stabilizers on an connected
$2$-dimensional cell complex with a linear homological isoperimetric
inequality and $\mathcal P$ is a collection of representatives of
conjugacy classes of vertex stabilizers.

A doubly stochastic measure on the unit square is a Borel probability
measure whose horizontal and vertical marginals both coincide
with the Lebesgue measure. The set of doubly stochastic measures
is convex and compact so its
extremal points are of particular interest. The problem number 111
of
Birkhoff (Lattice Theory 1948) is to provide a necessary and
sufficient condition on the support of a doubly stochastic measure
to guarantee extremality. It was proved by
Beneš and Štėpán that an extremal doubly stochastic measure is concentrated
on a set which admits an aperiodic decomposition.
Hestir and Williams later found a necessary condition which
is nearly sufficient by
further refining the aperiodic structure of the support of extremal
doubly stochastic measures.
Our objective in this work is to
provide a more practical necessary and nearly sufficient
condition for a set to support an extremal doubly stochastic
measure.

We show under some conditions that a Gorenstein ring $R$ satisfies the
Generalized Auslander-Reiten Conjecture if and only if so does
$R[x]$. When $R$ is a local ring we prove the same result for some
localizations of $R[x]$.

In a group, a nonidentity element is called
a generalized torsion element if some product of its conjugates
equals the identity. We show that for many classical knots one
can find generalized torsion in the fundamental group of its
complement, commonly called the knot group. It follows that
such a group is not bi-orderable. Examples include all torus
knots, the (hyperbolic) knot $5_2$ and algebraic knots in the
sense of Milnor.

Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

We investigate whether the total character of a finite group $G$
is a polynomial in a suitable irreducible character of $G$. When
$(G,Z(G))$ is a generalized Camina
pair, we show that the total character is a polynomial in a faithful
irreducible character of $G$
if and only if $Z(G)$ is cyclic.

Let $G_1, G_2, \dots , G_t$ be arbitrary graphs. The
Ramsey number $R(G_1, G_2, \dots, G_t)$ is the smallest positive
integer $n$ such that if the edges of the complete graph $K_n$
are
partitioned into $t$ disjoint color classes giving $t$ graphs
$H_1,H_2,\dots,H_t$, then at least one $H_i$ has a subgraph
isomorphic to $G_i$. In this paper, we provide the exact value
of
the $R(T_n,W_m)$ for odd $m$, $n\geq m-1$, where $T_n$ is
either a caterpillar, a tree with diameter at most four or a
tree
with a vertex adjacent to at least $\lceil
\frac{n}{2}\rceil-2$ leaves. Also, we
determine $R(C_n,W_m)$ for even integers $n$ and $m$, $n\geq
m+500$, which improves a result of Shi and confirms a
conjecture of Surahmat et al. In addition, the multicolor Ramsey
number of trees
versus an odd wheel is discussed in this paper.

All rings are commutative with identity and all modules are unital.
In this paper we introduce the concept of quasi-copure submodule
of
a multiplication $R$-module $M$ and will give some results of
them.
We give some properties of tensor product of finitely generated
faithful multiplication modules.

In this paper, we consider the quasi-linear elliptic
problem
\[
\left\{
\begin{aligned}
&
-M
\left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx
\right){\rm
div}
\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u
\right)
\\
&
\qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda
h_{1}(x)|u|^{q-2}u,
\\
&
-M
\left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx
\right){\rm
div}
\left(|x|^{-ap}|\nabla v|^{p-2}\nabla v
\right)
\\
&
\qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu
h_{2}(x)|v|^{q-2}v,
\\
&u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N}
\end{aligned}
\right.
\]
where $\lambda, \mu\gt 0$, $1\lt p\lt N$,
$1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq
a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$,
$k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$
are
continuous functions which change sign in $\mathbb{R}^{N}$. We
will prove that the problem has at least two positive solutions
by
using the Nehari manifold and the fibering maps associated with
the Euler functional for this problem.

We prove a Khintchine type inequality under the assumption that
the sum of
Rademacher random variables equals zero. We also show a new
tail-bound for a hypergeometric random variable.

This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesgue-points both
for the weight function $w$ and for $\log w$.

For any $C^*$-algebra $A$ with an approximate
unit of projections, there is a smallest ideal $I$ of $A$ such
that the quotient $A/I$ is stably finite.
In this paper, a sufficient and necessary condition is obtained
for an ideal of a $C^*$-algebra with real rank zero is this smallest
ideal by $K$-theory.

A graph $G=(V,E)$ is $L$-colorable if for a given list
assignment $L=\{L(v):v\in V(G)\}$, there exists a proper coloring
$c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V$. If $G$ is
$L$-colorable for every list assignment $L$ with $|L(v)|\geq
k$ for
all $v\in V$, then $G$ is said to be $k$-choosable. Montassier
(Inform. Process. Lett. 99 (2006) 68-71) conjectured that every
planar
graph without cycles of length 4, 5, 6, is 3-choosable. In this
paper,
we prove that every planar graph without 5-, 6- and 10-cycles,
and
without two triangles at distance less than 3 is 3-choosable.