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3  The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph Alfuraidan, Monther Rashed
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.


13  On classes $Q_p^\#$ for Hyperbolic Riemann surfaces Aulaskari, Rauno; Chen, Huaihui
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.


30  A Geometric Extension of Schwarz's Lemma and Applications Cleanthous, Galatia
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.


36  Distributive and Antidistributive Mendelsohn Triple Systems Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David
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.


50  On the Bernstein Problem in the Threedimensional Heisenberg Group Dorfmeister, Josef F.; Inoguchi, Junichi; Kobayashi, Shimpei
In this note we present a simple alternative proof
for the Bernstein problem in the threedimensional Heisenberg
group $\operatorname{Nil}_3$
by using the loop group technique. We clarify the geometric
meaning of the twoparameter ambiguity of entire minimal graphs
with prescribed AbreschRosenberg differential.


62  Uncertainty Principles on Weighted Spheres, Balls and Simplexes Feng, Han
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 DunklLaplaceBeltrami
operator on the weighted sphere.


73  Positive Solutions for the Generalized Nonlinear Logistic Equations Gasiński, Leszek; Papageorgiou, Nikolaos S.
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.


87  Approximation of a Function and its Derivatives by Entire Functions Gauthier, Paul M.; Kienzle, Julie
A simple proof is given for the fact that, for $m$ a nonnegative
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.


95  Faithful Representations of Graph Algebras via Branching Systems Gonçalves, Daniel; Li, Hui; Royer, Danilo
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 CuntzKrieger
uniqueness theorem for graph algebras by means of branching systems.


104  LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls He, Ziyi; Yang, Dachun; Yuan, Wen
In this paper, the authors characterize secondorder 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 LittlewoodPaley $g_\lambda^\ast$function in
terms of ball means.


119  A Simple Proof and Strengthening of a Uniqueness Theorem for Lfunctions Hu, PeiChu; Li, Bao Qin
We give a simple proof and strengthening of a uniqueness theorem
for functions in the extended Selberg class.


123  Discrete Spacetime and Lorentz Transformations Jensen, Gerd; Pommerenke, Christian
Alfred Schild has established conditions
that Lorentz transformations map worldvectors $(ct,x,y,z)$ 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 numbertheoretic arguments, the subject is also
interesting when isolated from its physical background.


136  Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series Kajihara, Yasushi
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 WhippleSears 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.


144  A Brief Note Concerning Hard Lefschetz for Chow Groups Laterveer, Robert
We formulate a conjectural hard Lefschetz property
for Chow groups, and prove this in some special cases: roughly
speaking, for varieties with finitedimensional motive, and
for varieties whose selfproduct has vanishing middledimensional
Griffiths group. An appendix includes related statements that
follow from results of Vial.


159  Rotors in Khovanov Homology MacColl, Joseph
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.


170  A Note on Fine Graphs and Homological Isoperimetric Inequalities MartínezPedroza, Eduardo
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.


182  Generalized Torsion in Knot Groups Naylor, Geoff; Rolfsen, Dale
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 biorderable. Examples include all torus
knots, the (hyperbolic) knot $5_2$ and algebraic knots in the
sense of Milnor.


190  Ramsey Number of Wheels Versus Cycles and Trees Raeisi, Ghaffar; Zaghian, Ali
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 $m$, $n\geq m1$, where $T_n$ is
either a caterpillar, a tree with diameter at most four or a
tree
with a vertex adjacent to at least $\lceil
\frac{n}{2}\rceil2$ 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.


197  Quasicopure Submodules Rajaee, Saeed
All rings are commutative with identity and all modules are unital.
In this paper we introduce the concept of quasicopure 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.


204  Restricted Khinchine Inequality Spektor, Susanna
We prove a Khintchine type inequality under the assumption that
the sum of
Rademacher random variables equals zero. We also show a new
tailbound for a hypergeometric random variable.


211  Universality Under Szegő's Condition Totik, Vilmos
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 Lebesguepoints both
for the weight function $w$ and for $\log w$.

