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The dimensions of the graded quotients of the
cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$
with respect to the Hodge filtration are described in terms of
simple geometrical invariants. The case of curves with ordinary
singularities is discussed in detail. We also give a precise
numerical estimate for the difference between the Hodge filtration
and the pole order filtration on $H^2(U,\mathbb C)$.

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.

We use George Bergman's recent normal form for universally adjoining
an inner inverse to show that, for general rings, a nilpotent
regular element $x$ need not be unit-regular.
This contrasts sharply with the situation for nilpotent regular
elements in exchange rings (a large class of rings), and for
general rings when all powers of the nilpotent element $x$ are
regular.

Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the one-dimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.

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.

Suppose that $G$ is a finite group and $k$ is a field of characteristic
$p\gt 0$. A ghost map is a map in the stable category of
finitely generated $kG$-modules which induces the zero map
in Tate cohomology in all degrees. In an earlier paper we showed
that the
thick subcategory generated by the trivial module
has no nonzero ghost maps if and only if
the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3.
In this paper we introduce and study variations of ghost
maps.
In particular, we consider the behavior of ghost maps under
restriction
and induction functors. We find all groups satisfying a strong
form
of Freyd's generating hypothesis and show that ghosts can
be detected on a finite range of degrees of Tate cohomology.
We also
consider maps which mimic ghosts in high degrees.

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 $D$.

In this note, we study the recurrence and topologically multiple
recurrence of a sequence of operators on Banach spaces.
In particular, we give a sufficient and necessary condition for
a cosine operator function,
induced by a sequence of operators on the Lebesgue space of a
locally compact group, to be topologically multiply recurrent.

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.

The thickness of a graph $G$ is the minimum number
of planar subgraphs whose union is $G.$ A
$t$-minimal graph is a graph of thickness $t$ which contains
no proper subgraph of thickness $t.$ In this paper, upper and
lower bounds are obtained for the thickness, $t(G\Box H)$, of
the Cartesian
product of two graphs $G$ and $H$, in terms of the thickness
$t(G)$ and $t(H)$.
Furthermore, the thickness of the Cartesian product of two planar
graphs and of a $t$-minimal graph and a planar graph are determined.
By using a new planar decomposition of the complete bipartite
graph $K_{4k,4k},$ the thickness of the Cartesian product of
two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also
given, for $n\neq 4k+1$.

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.

We investigate the bi-orderability of two-bridge knot groups
and the groups of knots with 12 or fewer crossings by applying
recent theorems of Chiswell, Glass and Wilson.
Amongst all knots with 12 or fewer crossings (of which there
are 2977), previous theorems were only able to determine bi-orderability
of 499 of the corresponding knot groups. With our methods we
are able to deal with 191 more.

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant
cohomology theory, we provide a systematic framework for computing
$E_T^*$ in the context of equivariantly stratified smooth complex
projective varieties. This allows us to explicitly compute $E_T^*(X)$
as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of
smooth complex projective $T_{\mathbb{C}}$-varieties with finitely
many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$,
$K_T^*$, and $MU_T^*$. We perform this computation on the affine
Grassmannian of a complex semisimple group.

We show that the discrete Hilbert transform
and the discrete Kak-Hilbert transform
are infinitesimal generator of one-parameter groups of
operators in $\ell^2$.

There are several kinds of classification problems for real hypersurfaces
in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$.
Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb
C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection.
In this paper, we introduce the notion of generalized Tanaka-Webster
(in shortly, GTW) Reeb parallel Ricci tensor for Hopf hypersurface
$M$ in $G_2({\mathbb C}^{m+2})$. Next, we give a complete classification
of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ with GTW Reeb
parallel Ricci tensor.

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.

In this paper we prove a useful formula for the graded commutator
of the Hodge
codifferential with the left wedge multiplication by a fixed
$p$-form acting on
the de Rham algebra of a Riemannian manifold. Our formula generalizes
a formula
stated by Samuel I. Goldberg for the case of 1-forms. As first
examples of
application we obtain new identities on locally conformally Kähler
manifolds
and quasi-Sasakian manifolds. Moreover, we prove that under suitable
conditions
a certain subalgebra of differential forms in a compact manifold
is quasi-isomorphic as a CDGA to the full de Rham algebra.

We study the distribution of the discrete spectrum of the Schrödinger
operator perturbed by a fast oscillating decaying potential depending
on a small parameter $h$.

We study the total graph of a finite commutative ring. We calculate
its metric dimension in the case when the Jacobson radical of
the ring is nontrivial and we examine the metric dimension of
the total graph of a product of at most two fields, obtaining
either exact values in some cases or bounds in other, depending
on the number of elements in the respective fields.

A dynamical approximation of a stochastic wave
equation with large interaction is derived.
A random invariant manifold is discussed. By a key linear transformation,
the random invariant manifold is shown to be close to the random
invariant manifold
of a second-order stochastic ordinary differential equation.

We propose a version of the classical Artin
approximation
which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a
Nash equation by a Nash solution in a
compatible way with a given Nash change of variables.
This result is closely related to the so-called nested Artin
approximation and becomes false in the analytic setting. We provide
local and global versions of this approximation in real and complex
geometry together with an application to the Right-Left equivalence
of Nash maps.

We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.

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 $L^2(G/H,\mu)$.

A subset $E$ of a discrete abelian group is called $\epsilon
$-Kronecker if
all $E$-functions of modulus one can be approximated to within
$\epsilon $
by characters. $E$ is called a Sidon set if all bounded $E$-functions
can be
interpolated by the Fourier transform of measures on the dual
group. As $%
\epsilon $-Kronecker sets with $\epsilon \lt 2$ possess the same
arithmetic
properties as Sidon sets, it is natural to ask if they are Sidon.
We use the
Pisier net characterization of Sidonicity to prove this is true.

Let $R$ be an associative ring with identity.
First we prove some results about zero-divisor graphs of reversible
rings. Then we study the zero-divisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$-compatible.

In this article, we give necessary and sufficient conditions
on a function to be a low-pass filter on a local field $K$ of
positive characteristic associated to the scaling function for
multiresolution analysis of $L^2(K)$. We use probability and
martingale methods to provide such a characterization.

The energy of a type II superconductor submitted to an external
magnetic field of intensity close to the second critical field
is given by the celebrated Abrikosov energy. If the external
magnetic field is comparable to and below the second critical
field, the energy is given by a reference function obtained as
a special (thermodynamic) limit of a non-linear energy. In this
note, we give a new formula for this reference energy. In particular,
we obtain it as a special limit of a linear energy defined
over configurations normalized in the $L^4$-norm.

On a real hypersurface $M$ in a non-flat complex
space form there exist the Levi-Civita and the k-th generalized
Tanaka-Webster connections. The aim of the present paper is to
study three dimensional real hypersurfaces in non-flat complex
space forms, whose Lie derivative of the structure Jacobi operator
with respect to the Levi-Civita connections coincides with the
Lie derivative of it with respect to the k-th generalized Tanaka-Webster
connection. The Lie derivatives are considered in direction of
the structure vector field and in directions of any vecro field
orthogonal to the structure vector field.

We show that the conjectural criterion of $p$-incompressibility
for products of projective homogeneous varieties in terms of
the factors, previously known in a few special cases only, holds
in general.
Actually, the proof goes through for a wider class of varieties
which includes the norm varieties associated to symbols in Galois
cohomology of arbitrary degree.

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 $h$.

A commuting family of subnormal operators need
not have a commuting normal extension. We study when a representation
on an abelian semigroup can be extended to a normal representation,
and show that it suffices to extend the set of generators to
commuting normals. We also extend a result due to Athavale to
representations on abelian lattice ordered semigroups.

In this paper, using Calderón's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the Triebel-Lizorkin
and Besov spaces associated with Zygmund dilations.

Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$-norms:
\[
1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]

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 design an elementary method to study the problem, getting
an asymptotic formula which is better than Hooley's and Heath-Brown's
results for certain cases.

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.

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.

The
eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the well-known case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.

We prove that for the linear scalar delay differential
equation
$$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-1) $$
with non-negative periodic coefficients of period $P\gt 0$, the
stability threshold for the trivial solution is
$r:=\int_{0}^{P}
\left(b(t)-a(t)
\right)\mathrm{d}t=0,$
assuming that $b(t+1)-a(t)$ does not change its sign. By constructing
a class of explicit examples, we show the counter-intuitive result
that in general, $r=0$ is not a stability threshold.

It has been known that there exists a canonical system for every
finite real reflection group. The first and the third authors
obtained
an explicit formula for a canonical system in the previous paper.
In this article, we first define canonical systems for the finite
unitary reflection groups, and then prove their existence.
Our proof does not depend on the classification of unitary reflection
groups.
Furthermore, we give an explicit formula for a canonical system
for every unitary reflection group.

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 $R[x]$.

We introduce a weaker notion of (semi)stability for vector bundles
on
reducible curves which does not depend on a choice of polarization,
and
which suffices for many applications of degeneration techniques.
We explore the basic
properties of this alternate notion of (semi)stability. In a
complementary
direction, we record a proof of the existence of semistable extensions
of vector bundles in suitable degenerations.

We give a precise description of the homology group of the Fermat
curve as a cyclic module over a group ring.
As an application, we prove the freeness of the profinite homology
of the Fermat tower.
This allows us to define measures, an equivalent of Anderson's
adelic beta functions,
in a similar manner to Ihara's definition of $\ell$-adic universal
power series for Jacobi sums.
We give a simple proof of the interpolation property using a
motivic decomposition of the Fermat curve.

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 $0.235711131719\ldots$.

The annihilating-ideal graph
of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a
graph whose vertex set consists of all non-zero annihilating
ideals and two distinct
vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,
we show that if $R$ is a reduced ring and the independence
number of $\mathbb{AG}(R)$ is finite, then the edge chromatic
number of $\mathbb{AG}(R)$ equals its maximum degree
and this number equals $2^{|{\rm Min}(R)|-1}-1$; also, it is
proved that the independence number of $\mathbb{AG}(R)$ equals
$2^{|{\rm Min}(R)|-1}$, where ${\rm Min}(R)$ denotes the set
of minimal prime ideals of $R$.
Then we give some criteria for a graph to be isomorphic with
an annihilating-ideal graph of a ring.
For example, it is shown that every bipartite annihilating-ideal
graph is a complete bipartite graph with at most two horns. Among
other results, it is shown that a finite graph $\mathbb{AG}(R)$
is not Eulerian, and it is Hamiltonian if and only if $R$ contains
no Gorenstain ring as its direct summand.

The unitary Cayley graph of a ring $R$, denoted
$\Gamma(R)$, is the simple graph
defined on all elements of $R$, and where two vertices $x$ and
$y$
are adjacent if and only if $x-y$ is a unit in $R$. The largest
distance between all pairs of vertices of a graph $G$ is called
the
diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved
that for each integer $n\geq1$, there exists a ring $R$ such
that
${\rm diam}(\Gamma(R))=n$. We also show that ${\rm
diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$
self-injective and classify all those rings with ${\rm
diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.

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 $k \ge 1$.

The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.

Let $R$ be a ring. The following results are proved: $(1)$ every
element of $R$ is a sum of an idempotent and a tripotent that
commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times
R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of
exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb
Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference
of two commuting idempotents iff $R\cong R_1\times R_2$, where
$R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$,
and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s;
$(3)$ every element of $R$ is a sum of two commuting tripotents
iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean
with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect
product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect
product of $\mathbb Z_5$'s.