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Let $R$ be a commutative ring with identity. The
co-annihilating-ideal graph of $R$, denoted by $\mathcal{A}_R$,
is
a graph whose vertex set is the set of all non-zero proper ideals
of $R$ and two distinct vertices $I$ and $J$ are adjacent
whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we
initiate the study of the co-annihilating ideal graph of a
commutative ring and we investigate its properties.

Let $H$ be a group. The co-maximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are non-trivial and proper subgroups of
$H$ and two distinct vertices $L$
and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In
this paper, we study the connectivity, diameter, clique number
and vertex
chromatic number of $\Gamma(H)$. For instance, we show that
if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$
is connected with diameter at most $3$. Also, we characterize
all finite groups whose co-maximal graphs are connected.
Among other results, we show that if $H$ is a finitely generated
solvable group and $\Gamma(H)$ is connected and moreover the
degree of a maximal subgroup is finite, then $H$ is finite.
Furthermore, we show that the degree of each vertex in the
co-maximal graph of a general linear group over an algebraically
closed field is zero or infinite.

All finite simple self $2$-distance graphs with no square, diamond,
or triangles with a common vertex as subgraph are determined.
Utilizing these results, it is shown that there is no cubic self
$2$-distance graph.

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$
a non-nilpotent ideal of $R$. The purpose of this article is
to investigate Faltings' finiteness
dimension $f_I(R)$ and equidimensionalness of certain homomorphic
image of $R$. As a consequence
we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$
and if $\operatorname{mAss}_R(R/I)$
is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq
1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$.
Moreover, we will obtain a lower bound for injective dimension
of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case
$(R, \frak m)$ is a complete equidimensional local ring.

We prove the existence of
a (in fact many)
holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq
0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$.

Let $P$ be a finite N-free poset. We consider the hypergraph
$\mathcal{H}(P)$ whose vertices are the elements of $P$ and whose
edges are the maximal intervals of $P$. We study the dual
König property of $\mathcal{H}(P)$ in two subclasses of N-free class.

We identify when a tubular group (the fundamental group of a
finite
graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free
by
cyclic and show, using Wise's equitable sets criterion, that
every
tubular free by
cyclic group acts freely on a CAT(0) cube complex.

Let $D$ be an integral domain, $X^1(D)$ be the set of height-one
prime ideals of $D$,
$\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be
two disjoint nonempty sets of indeterminates over $D$,
$D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and
$D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type
power series ring over $D[\{X_{\beta}\}]$.
Assume that $D$ is a Prüfer $v$-multiplication domain (P$v$MD)
in which each proper integral $t$-ideal has only finitely many
minimal prime ideals
(e.g., $t$-SFT P$v$MDs, valuation domains, rings of Krull type).
Among other things, we show that if
$X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$,
then
${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D - \{0\}}$ is a
Krull domain.
We also prove that if $D$ is a $t$-SFT P$v$MD, then the complete
integral closure of $D$ is a Krull domain and
ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every
height-one maximal $t$-ideal $M$ of $D$.

Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give
a sufficient and necessary condition
for weighted translation operators on the Lebesgue space $\ell^p(G)$
to be densely disjoint hypercyclic.
The characterization for the dual of a weighted translation to
be densely disjoint hypercyclic is also obtained.

In this
paper,
double commutativity and the reverse order law for the core inverse
are considered. Then, new characterizations of the Moore-Penrose
inverse of a regular element are given by one-sided invertibilities
in a ring. Furthermore, the characterizations and representations
of
the core and dual core inverses of a regular element are considered.

Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*-algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak-$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all word-length functions
on finitely generated nilpotent-by-finite groups.

Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to
Y$.
We prove the Ulam-Hyers stability theorem
for the cubic functional equation
\begin{align*}
f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)=0
\end{align*}
in restricted domains. As an application we consider a measure
zero stability problem
of the inequality
\begin{align*}
\|f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon
\end{align*}
for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure
0.

Let $G$ be a connected graph with vertex set $V(G)$. The degree
Kirchhoff index of $G$ is defined as $S'(G) =\sum_{\{u,v\}\subseteq
V(G)}d(u)d(v)R(u,v)$, where $d(u)$ is the degree of vertex $u$,
and
$R(u, v)$ denotes the resistance distance between vertices $u$
and
$v$. In this paper, we characterize the graphs having maximum
and
minimum degree Kirchhoff index among all $n$-vertex bicyclic
graphs
with exactly two cycles.

A Nikishin-Maurey characterization is given for bounded subsets
of weak-type Lebesgue spaces. New factorizations for linear and
multilinear operators are shown to follow.

We prove Lawton's conjecture about the upper bound on the measure
of the set on the unit circle on which a complex polynomial with
a bounded number of coefficients takes small values. Namely,
we prove that Lawton's bound holds for polynomials that are not
necessarily monic. We also provide an analogous bound for polynomials
in several variables. Finally, we investigate the dependence
of the bound on the multiplicity of zeros for polynomials in
one variable.

Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not central-valued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:

(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;

(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on
$R$;

(III) $char(R)=2$ and one of the following holds:

(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;

(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is central-valued on $R$;

(iii) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;

(iv) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued
on $R$.

Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.

It is known that if $E$ is a closed subset of an open Riemann
surface $R$ and $f$ is a holomorphic function on a neighbourhood
of $E,$ then it is ``usually" not possible to approximate $f$
uniformly by functions holomorphic on all of $R.$ We show, however,
that for every open Riemann surface $R$ and every closed subset
$E\subset R,$ there is closed subset $F\subset E,$ which approximates
$E$ extremely well, such that every function holomorphic on $F$
can be approximated much better than uniformly by functions holomorphic
on $R$.

This paper introduces a unified operator theory approach to the
abstract Plancherel (trace) formulas over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be
the normalized $G$-invariant measure on $G/H$ associated to the
Weil's formula.
Then, we present a generalized abstract notion of Plancherel
(trace) formula for the Hilbert space $L^2(G/H,\mu)$.

We characterize two important notions of amenability and compactness
of
a locally compact quantum group ${\mathbb G}$ in terms of certain
homological
properties. For this, we show that ${\mathbb G}$ is character
amenable if and only if it is both amenable and co-amenable.
We finally apply our results to
Arens regularity problems of the quantum group algebra
$L^1({\mathbb G})$; in particular, we improve an interesting result
by Hu, Neufang and Ruan.

In this paper, we establish $BMO$ estimates for generalized commutators
of
rough fractional maximal and integral operators on generalized
weighted
Morrey spaces, respectively.

In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$-convex-normal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$- and $(k+1)$-convex-normality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.

We previously obtained a congruence modulo four for the number
of real solutions to many Schubert problems on
a square Grassmannian given by osculating flags.
Here, we consider Schubert problems given by more general isotropic
flags, and prove this
congruence modulo four for the largest class of Schubert problems
that could be expected to exhibit this
congruence.

In the Simpson moduli space $M$ of semi-stable sheaves with
Hilbert polynomial $dm-1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).

Fix an irreducible (finite) root system $R$ and a choice
of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the
Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$.
Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such
that
$\mu$ is maximal with this property. Garibaldi, Guralnick, and
Nakano
have asked under which condition there exists $k$ such that $L
(\mu)_k$
is a composition factor of $V (\lambda)_k$, and they exhibit an
example
in type $E_8$ where this is not the case. The purpose of this
paper
is to to show that their example is the only one. It contains
two proofs
for this fact, one that uses a classification of the possible
pairs $(\lambda, \mu)$,
and another one that relies only on the classification
of root systems.

Alfred Schild has established conditions
that Lorentz transformations map world-vectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. These transformations
are called integral Lorentz transformations.

The present paper contains supplements to
our earlier work
with a new focus on group theory. To relate the results to the
familiar matrix group nomenclature we associate Lorentz transformations
with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the
lattice of subgroups of the group originated in Schild's paper
and obtain generating sets for the full group and its subgroups.

It is shown that the Dirichlet problem for the slab $(a,b) \times
\mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof
is based
on a generalized Schwarz reflection principle. Moreover, it is
shown that
for a given entire harmonic function $g$
the inhomogeneous difference equation $h
( t+1,y) -h (t,y) =g ( t,y)$
has an entire harmonic solution $h$.

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.

We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic
(or rational) field of discriminant $D$ and Dirichlet character
$\chi$, if a prime $p$ is large enough compared to $D$, there
is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with
respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f
\otimes \chi,1) \neq 0$. This result is obtained through an estimate
of a weighted sum of twists of $L$-functions which generalises
a result of Ellenberg. It relies on the approximate functional
equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and
a Petersson trace formula restricted to Atkin-Lehner eigenspaces.
An application of this nonvanishing theorem will be given in
terms of existence of rank zero quotients of some twisted jacobians,
which generalises a result of Darmon and Merel.

In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
Hardy-Littlewood maximal operator. We obtain some new bounds
for the derivative of the one-dimensional multisublinear
fractional maximal operators acting on vector-valued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

We study a population model with nonlocal diffusion, which
is a delayed integro-differential equation with double nonlinearity
and two integrable kernels. By comparison method and analytical
technique, we obtain globally asymptotic stability of the zero
solution and the positive equilibrium. The results obtained
reveal that the globally asymptotic stability only depends on
the property of nonlinearity. As application, an example for
a population model with age structure is discussed at the end
of the article.

The chromatic functor of a simple graph is a functorization of
the chromatic polynomial. M. Yoshinaga showed
that two finite graphs have isomorphic chromatic functors if
and only if they have the same chromatic polynomial. The key
ingredient in the proof is the use of stable partitions of graphs.
The latter is shown to be closely related to chromatic functors.
In this note, we further investigate some interesting properties
of chromatic functors associated to simple graphs using stable
partitions. Our first result is the determination of the group
of natural automorphisms of the chromatic functor, which is in
general a larger group than the automorphism group of the graph.
The second result is that the composition of the chromatic functor
associated to a finite graph restricted to the category $\mathrm{FI}$
of finite sets and injections with the free functor into the
category of complex vector spaces yields a consistent sequence
of representations of symmetric groups which is representation
stable in the sense of Church-Farb.

It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=C-S$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the Gauss-Seidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
Gauss-Seidel iteration, no matter whether the CSCS itself is
convergent or not.

We prove Wise's $W$-cycles conjecture: Consider a compact graph
$\Gamma'$ immersing into another graph $\Gamma$. For any immersed
cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$
from
the circular components $\mathbb{S}$ of the pullback to $\Gamma'$.
Unless
$\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to
S^1$ is bounded above by minus the Euler characteristic of
$\Gamma'$. As a corollary, any finitely generated subgroup
of a
one-relator group has finitely generated Schur multiplier.

Let $G$ be a finite group and
let $A(G)$ denote the Burnside ring of $G$.
Then an inverse limit $L(G)$ of the groups $A(H)$ for
proper subgroups $H$ of $G$ and a homomorphism
${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural
way.
Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$.
For a prime $p$,
let $N(p)$ be the minimal
normal subgroup of $G$ such that the order of $G/N(p)$ is
a power of $p$, possibly $1$.
In this paper we prove that $Q(G)$ is isomorphic to
the cartesian product of the groups $Q(G/N(p))$, where $p$
ranges over the primes dividing the order of $G$.

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 $R[x]$.

Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping.
For any rational number $r \ne 2$, C. Baak, D. H.
Boo, and Th. M. Rassias have proved the Hyers-Ulam stability
of the following functional equation:
\begin{align*}
r f
\left(\frac{\sum_{j=1}^d x_j}{r}
\right)
& + \sum_{\substack{i(j) \in \{0,1\}
\\ \sum_{j=1}^d i(j)=\ell}} r f
\left(
\frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r}
\right)
= (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d
f(x_j)
\end{align*}
where $d$ and $\ell$ are positive integers so that $1 \lt \ell
\lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$,
$p, q \in \mathbb{N}$ with $p \le q$.

In this note we solve this equation for arbitrary nonzero scalar
$r$ and show that it is actually Hyers-Ulam stable.
We thus extend and generalize Baak et al.'s result.
Different questions concerning the *-homomorphisms and the
multipliers between C*-algebras are also
considered.

In an attempt to resolve a folklore conjecture of Erdös regarding
the non-vanishing at $s=1$ of the $L$-series
attached to a periodic arithmetical function with period $q$
and values in $\{ -1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$
- linear independence of logarithms of certain algebraic numbers.
In this paper, we disprove Livingston's conjecture for composite
$q \geq 4$, highlighting that a new approach is required to settle
Erdös's conjecture. We also prove that the conjecture is
true for prime $q \geq 3$, and indicate that more ingredients
will be needed to settle Erdös's conjecture for prime $q$.

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 $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent
operators
(i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$).
In the dynamical systems theory the above quasinilpotency property
is equivalent
to a very strong concept of stability for the solutions of autonomous
systems.
This concept is frequently called superstability and weakens
the classical finite time extinction property
(roughly speaking, disappearing solutions).
We show that under some assumptions, the quasinilpotency, or
equivalently, the superstability property
of a C$_0$-semigroup is preserved under the perturbations of
its infinitesimal generator.

The notion of coaxers is introduced in a pseudo-complemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudo-complemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.

The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position,
then up to conjugacy, there are only finitely many $r$-tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r-1)n^2 + 1$.

Let $K$ be an ultraspherical hypergroup associated to a locally
compact group $G$ and a spherical projector $\pi$ and let $VN(K)$
denote the dual of the Fourier algebra $A(K)$ corresponding to
$K.$ In this note, we show that the set of invariant means on
$VN(K)$ is singleton if and only if $K$ is discrete. Here $K$
need not be second countable. We also study invariant means on
the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$
in the $cb$-multiplier norm. Finally, we consider generalized
translations and generalized invariant means.

Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besov-type spaces, Triebel-Lizorkin-type
spaces and Besov-Morrey spaces via the sequence
$\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{-k}}f$. These results give a way
to introduce Besov-type spaces,
Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of Morrey-Sobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.

We prove a comprehensive version of the Ruelle-Perron-Frobenius
Theorem
with explicit estimates of the spectral radius of the Ruelle
transfer operator and various other
quantities related to spectral properties of this operator. The
novelty here is that the Hölder
constant of the function generating the operator appears only
polynomially, not exponentially as
in previous known estimates.

This paper is dedicated to studying the
semilinear Schrödinger equation
$$
\left\{
\begin{array}{ll}
-\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N},
\\
u\in H^{1}({\mathbf{R}}^{N}),
\end{array}
\right.
$$
where $f$ is a superlinear, subcritical nonlinearity. It focuses
on the case where
$V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic
in each of $x_1, x_2, \ldots, x_N$ and
$\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle
+V_0)\cap (0, \infty)]$,
$V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic
condition is obtained,
which is weaker than some well known results.

A subset $W$ of the vertex set of a graph $G$ is called a resolving
set of $G$ if for every pair of distinct vertices $u, v$ of $G$,
there is $w \in W$ such that the~distance of $w$ and $u$ is different
from the distance of $w$ and $v$. The~cardinality of a~smallest
resolving set is called the metric dimension of $G$, denoted
by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists
of the vertices $v_0, v_1, \dots , v_{n-1}$ and the~edges $v_i
v_{i+j}$, where $0 \le i \le n-1$, $1 \le j \le t$ $(2 \le t
\le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo
$n$. Grigorious et al. [On the metric dimension of circulant
and Harary graphs, Applied Mathematics and Computation 248 (2014),
47--54] proved that $dim(C_n (1,2, \dots , t))
\ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they
presented a~conjecture saying that $dim(C_n (1,2, \dots , t))
= t+p-1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove
both statements. We show that if $t \ge 4$ is even, there exists
an infinite set of values of $n$ such that $dim(C_n (1,2, \dots
, t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le
t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even,
$t \ge 4$, $2 \le p \le t$ and $k \ge 1$.

Let $R$
be a ring and $b, c\in R$.
In this paper, we give some characterizations of the $(b,c)$-inverse,
in terms of the direct sum decomposition, the annihilator and
the invertible elements.
Moreover, elements with equal $(b,c)$-idempotents related to
their $(b, c)$-inverses are characterized, and the reverse order
rule for the $(b,c)$-inverse is considered.

In this paper, we study condition $C'_{\wedge}$ which is a
projective tensor product analogue of condition $C'$. We show
that
the finite-dimensional OLLP operator spaces have condition
$C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.

In this paper, the concept of the classical $f$-divergence for
a pair of measures is extended to the mixed $f$-divergence for
multiple pairs of measures. The mixed $f$-divergence provides
a way to measure the difference between multiple pairs of (probability)
measures. Properties for the mixed $f$-divergence are established,
such as permutation invariance and symmetry in distributions.
An
Alexandrov-Fenchel type inequality and an isoperimetric inequality
for the
mixed $f$-divergence are proved.

In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.

This paper generalizes the Aleksandrov problem: the Mazur-Ulam
theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance
preserving mapping is an $n$-isometry if and only if it has the
zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries
on $n$-G-quasi normed space are equivalent; we generalize the
Benz theorem to n-normed spaces with no restrictions on the dimension
of spaces.