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Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution
considerations, we define, for each prime $\ell$, a local factor
that
measures the relative frequency with which $f\bmod \ell$ occurs
as the
characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$.
For a certain
class of polynomials, we show that the resulting infinite product
calculates the number of principally polarized abelian surfaces
over $\mathbb{F}_q$
with Weil polynomial $f$.

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.

Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first Chevalley-Eilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.

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 Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
square-free integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, non-squares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.

Motivated by Almgren's work on the isoperimetric inequality,
we prove a sharp inequality relating the length and maximum curvature
of a closed curve in a complete, simply connected manifold of
sectional curvature at most $-1$. Moreover, if equality holds,
then the norm of the geodesic curvature is constant and the torsion
vanishes. The proof involves an application of the maximum principle
to a function defined on pairs of points.

We show that a class of semilinear Laplace-Beltrami equations
on the unit sphere
in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions.
The solutions to
these equations are the solutions to a two point boundary value
problem for a
singular ordinary differential equation. We prove the existence
of such solutions
using energy and phase plane analysis. We derive a
Pohozaev-type
identity
in
order to prove that the energy to an associated initial value
problem tends
to infinity as the energy at the singularity tends to infinity.
The nonlinearity is allowed to grow as fast as $|s|^{p-1}s$ for
$|s|$ large
with $1 \lt p \lt (n+5)/(n-3)$.

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.

We prove the non-existence of Hopf real hypersurfaces in complex
two-plane Grassmannians with harmonic curvature with respect
to the generalized Tanaka-Webster connection if they satisfy
some further conditions.

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.

Let $p$ be a prime number and $F$ a field containing a root of
unity of order $p$.
We relate recent results on vanishing of triple Massey products
in the mod-$p$ Galois cohomology of $F$,
due to Hopkins, Wickelgren, Mináċ, and Tân, to classical
results in the theory of central simple algebras.
For global fields, we prove a stronger form of the vanishing
property.

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 $\mathcal{E}$ be an injectively resolving subcategory of
left $R$-modules. A left $R$-module $M$
(resp. right $R$-module $N$) is called $\mathcal{E}$-injective
(resp. $\mathcal{E}$-flat)
if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$)
for any $G\in\mathcal{E}$.
Let $\mathcal{E}$ be a covering subcategory.
We prove that a left $R$-module $M$ is $\mathcal{E}$-injective
if and only if $M$ is a direct sum
of an injective left $R$-module and a reduced $\mathcal{E}$-injective
left $R$-module.
Suppose $\mathcal{F}$ is a preenveloping subcategory of right
$R$-modules such that
$\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$.
It is shown that a finitely presented right $R$-module $M$ is
$\mathcal{E}$-flat if and only if
$M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right
$R$-module.
In addition, we introduce and investigate the
$\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of
modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary
rings and $\mathcal{E}$-von Neumann regular rings and characterize
them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat
modules.

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 article we prove the embedding theorem for inhomogeneous
Besov and Triebel-Lizorkin spaces on RD-spaces.
The crucial idea is to use the geometric density condition
on the measure.

We prove character sum estimates for additive Bohr subsets modulo
a prime.
These estimates are analogous to classical character sum bounds
of
Pólya-Vinogradov and Burgess. These estimates are applied to
obtain results on
recurrence mod $p$ by special elements.

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.

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.

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.

In this paper, we generalize the finite generation result of
Sormani
to non-branching $RCD(0,N)$
geodesic spaces (and in particular, Alexandrov spaces) with full
support measures. This is a special case of the Milnor's Conjecture
for complete non-compact $RCD(0,N)$ spaces. One of the key tools
we use is the Abresch-Gromoll type excess estimates for non-smooth
spaces obtained by Gigli-Mosconi.

Suppose that $G$ is a
finite group and $H$ is a subgroup of $G$. $H$ is said to be
$s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow
$p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is said to be
$s$-quasinormally embedded in $G$ if for each prime $p$ dividing the
order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow
$p$-subgroup of some $s$-quasinormal subgroup of $G$. We fix in
every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$
satisfying $1\lt |D|\lt |P|$ and study the structure of $G$ under the
assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either
$s$-semipermutable or $s$-quasinormally embedded in $G$.
Some recent results are generalized and unified.

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.

This paper is concerned with the study of
the regularity for the multisublinear maximal operator. It is
proved that the multisublinear maximal operator is bounded on
first-order Sobolev spaces. Moreover, two key point-wise
inequalities for the partial derivatives of the multisublinear
maximal functions are established. As an application, the
quasi-continuity on the multisublinear maximal function is also
obtained.

We consider the class of polynomial differential systems of the
form
$\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
this
class of differential systems we summarize the known results
for the
existence of limit cycles, and we provide new results for their
nonexistence and existence.

Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated
by all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. The
category of all finite-dimensional representations of $(Q, J^2)$ over
$A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we
introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a
subcategory of
$\operatorname{rep}{}(Q,J^2,A)$ of all exact representations.
The main result of this paper explicitly describes the Gorenstein-projective representations in $\operatorname{rep}{}(Q,J^2,A)$,
via the exact representations plus an extra condition.
As a corollary, $A$ is a self-injective algebra, if
and only if the Gorenstein-projective representations are exactly the
exact representations of $(Q, J^2)$ over $A$.

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.

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 $0.235711131719\ldots$.

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.

Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider
the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where
the action of $\mathbb{Z}$ on $\mathbb{R}^{n}$ is induced by
the left multiplication by $A$. Let $(\alpha,\tau)$ be a strongly
continuous action of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a
$C^{*}$-algebra $B$ where $\alpha$ is a strongly continuous action
of $\mathbb{R}^{n}$ and $\tau$ is an automorphism. The map $\tau$
induces a map $\widetilde{\tau}$ on $B \rtimes_{\alpha} \mathbb{R}^{n}$.
We show that, at the $K$-theory level, $\tau$ commutes with the
Connes-Thom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$.
As an application, we recompute the $K$-groups of the Cuntz-Li
algebra associated to an integer dilation matrix.

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 $\log w$.

Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$
denote the Dedekind eta function. We use a recent product-to-sum
formula in conjunction with conditions for the non-representability
of integers by certain ternary quadratic forms to give explicitly
10 eta quotients
\[
f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi
i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0,
\]
such that the Fourier coefficients $c(n)$ vanish for all positive
integers $n$ in each of infinitely many non-overlapping arithmetic
progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$
we have $c(n)=0$ for all $n$ in each of the arithmetic progressions
$\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k
\geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.

Under sufficiently strong assumptions about the first term in
an arithmetic progression, we prove that for any integer $a$,
there are infinitely many $n\in \mathbb N$ such that for each
prime factor $p|n$, we have $p-a|n-a$. This can be seen as a
generalization of Carmichael numbers, which are integers $n$
such that $p-1|n-1$ for every $p|n$.

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.

We show that if $v$ is a regular semi-classical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = -\lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semi-classical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the three-term recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.

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.