With each Fell bundle over a discrete group~$G$ we associate a partial action of $G$ on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional $C^*$-algebras in terms of the dynamics of this partial action.

In this paper, we study the notion of strong differential superordination as a dual concept of strong differential subordination, introduced in~\cite {1.a}. The notion of strong differential superordination has recently been studied by many authors, see, for example, \cite {2.a, 3.a, 5.a}. Let $q(z)$ be an analytic function in $\mathbb {D}$ that satisfies the first order differential equation $$\theta (q(z))+F(z)q'(z)\varphi (q(z))=h(z).$$ \smallskip Suppose that $p(z)$ is analytic and univalent in the closure of the open unit disk $\overline {\mathbb {D}}$ with $p(0)=q(0)$. We shall find conditions on $h(z),G(z),\theta (z)$ and $\varphi (z)$ such that $$ h(z)\prec \prec \theta (p(z))+\frac {G(\xi )}{\xi }zp'(z)\varphi (p(z))\Longrightarrow q(z)\prec p(z). $$ Applications and examples of the main results are also considered.

We make a connection between the braid group and signed permutations. Using this link, we describe a commutative diagram which contains the fundamental sequence for the braid group.

An augmented generalized happy function, ${S_{[c,b]}} $ maps a positive integer to the sum of the squares of its base $b$ digits and a non-negative integer~$c$. A positive integer $u$ is in a \textit {cycle} of ${S_{[c,b]}} $ if, for some positive integer~$k$, ${S_{[c,b]}}^k(u) = u$, and, for positive integers $v$ and $w$, $v$ is $w$-\textit {attracted} for ${S_{[c,b]}} $ if, for some non-negative integer~$\ell $, ${S_{[c,b]}} ^\ell (v) = w$. In this paper, we prove that, for each $c\geq 0$ and $b \geq 2$, and for any $u$ in a cycle of ${S_{[c,b]}} $: (1)~if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted integers, and (2)~if $b$ is odd, then there exist arbitrarily long sequences of 2-consecutive $u$-attracted integers.

All groups considered in this paper are finite. A subgroup~$H$ of a group~$G$ is said to \textit {seminormal} in $G$ if $H$ is normalized by all subgroups~$K$ of~$G$ such that $\gcd (\lvert H\rvert , \lvert K\rvert )=1$. We call a group $G$ an MSN-\textit {group} if the maximal subgroups of all the Sylow subgroups of~$G$ are seminormal in~$G$. In this paper, we classify all MSN-groups.

We prove that, if $\mathcal F $ is the class of torsion free discrete modules over a profinite group $G$, that is, the class of discrete $G$-modules which are torsion free as abelian groups, then $({\mathcal F},{\mathcal F}^\bot )$ is a complete cotorsion pair. Moreover, we find a structure theorem for torsion free and cotorsion discrete $G$-modules and for finitely generated cotorsion discrete $G$-modules.

In this paper, among other things, we show that there is a close connection between the existence of a bounded projection on some Banach algebras associated to a locally compact group~$G$ and the existence of a left invariant mean on $L^\infty (G)$. A necessary and sufficient condition is found for a locally compact group to possess a left invariant mean.

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language $L_{\infty \omega }$, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in $L_{\infty \omega }$.

This paper is concerned with the asymptotic behavior of a doubly degenerate parabolic equation in non-divergence form. The proofs are divided into three cases according to exponent values of the source, and, by using different methods, we prove the stability of the steady states. We also expand the discussion of asymptotic stability for equations with the periodic source.

Let $R$ be a noncommutative prime ring with extended centroid~$C$ and with $Q_{mr}(R)$ its maximal right ring of quotients. From the viewpoint of functional identities, we give a complete characterization of Jordan $\sigma $-derivations of $R$ with $\sigma $ an epimorphism. Precisely, given such a Jordan $\sigma $-derivation $\de \colon R\to Q_{mr}(R)$, it is proved that either $\delta $ is a $\sigma $-derivation or a derivation $d\colon R\to Q_{mr}(R)$ and a unit $u\in Q_{mr}(R)$ exist such that $\delta (x)=ud(x)+\mu (x)u$ for all $x\in R$, where $\mu \colon R\to C$ is an additive map satisfying $\mu (x^2)=0$ for all $x\in R$. In addition, if $\sigma $ is an X-outer automorphism, then $\delta $ is always a $\sigma $-derivation.

In this paper, we study the existence of infinitely many solutions for a class of Kirchhoff-type problems with critical growth in $\mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable positive parameters $\alpha , \beta $. The proofs are based on variational methods and the concentration-compactness principle.

In this paper, we investigate the average behavior of coefficients of the triple product $L$-function $L(f \otimes f \otimes f,s)$ attached to a primitive holomorphic cusp form $f(z)$ of weight~$k$ for the full modular group $SL(2, \Z )$. Here we call $f(z)$ a primitive cusp form if it is an eigenfunction of all Hecke operators simultaneously.

In this paper, we study the existence and uniqueness of positive solutions for some nonlinear fractional partial differential equations via given boundary value problems by using recent fixed point results for a class of mixed monotone operators with convexity.

We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature and derive its algebra structure. Our construction is based on Riemannian tractor calculus, allowing us to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry.

Let $K$ be a field, $S=K[x_1,\ldots ,x_m, y_1,\ldots , y_n]$ a standard bigraded polynomial ring, and $M$ a finitely generated bigraded $S$-module. In this paper, we study the sequentially Cohen-Macaulayness of~$M$ with respect to $Q=(y_1,\ldots ,y_n)$. We characterize the sequentially Cohen-Macaulayness of $L\otimes _KN$ with respect to $Q$ as an $S$-~module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \ldots , x_m]$ and $K[y_1, \ldots , y_n]$, respectively. All hypersurface rings that are sequentially Cohen-Macaulay with respect to $Q$ are classified.

A number of recent papers have been devoted to generalizations of the classical AM-GM inequality. Those generalizations which incorporate \textit {variance} have been the most useful in applications to economics and finance. In this paper, we prove an inequality which yields the best possible upper and lower bounds for the geometric mean of a sequence solely in terms of its arithmetic mean and its variance. A particular consequence is the following: among all positive sequences having given length, arithmetic mean and nonzero variance, the geometric mean is maximal when all terms in the sequence except one are equal to each other and are less than the arithmetic mean.

Recently, the concept of g-Riesz dual sequences for g-Bessel sequences has been introduced. In this paper, we investigate under what conditions a g-Riesz sequence $\Phi =\{\Phi _j\in L(H,H_j):j\in \mathcal I\}$ is the g-Riesz dual sequence of a given g-frame $\Lambda =\{\Lambda _i\in L(H,H_i):i\in \mathcal I\}$.

In this article, we consider nonlinear elliptic fourth-order equations with the monotone principal part satisfying the common growth and coerciveness conditions for Sobolev space $W^{2,p}(\Omega )$, $\Omega \subset \mathbb {R}^{n}$. It is supposed that the lower-order term of the equations admits arbitrary growth with respect to an unknown function and is arbitrarily close to the growth limit with respect to the derivatives of this function. We assume that the lower-order term satisfies the sign condition with respect to the unknown function. We prove the existence of continuous generalized solutions for the Dirichlet problem in the case $n=2p$.

In this paper, we are concerned with elliptic equations with Hardy potential and critical Sobolev exponents where $2^{*}={2N}/({N-2})$ is the critical Sobolev exponent, $N\geq 3$, $0\leq \mu \lt \overline {\mu }={(N-2)^2}/{4}$, $\mathbf {\Omega }\subset \mathbb {R}^{N}$ an open bounded set. For $\lambda \in [0,\lambda _{1})$ with $\lambda _{1}$ being the first eigenvalue of the operator $-\Delta -{\mu }/{|x|^{2}}$ with zero Dirichlet boundary condition, and for $f\in H_{0}^{1}(\mathbf {\Omega })^{-1}=H^{-1}$, $f\neq 0$, we show that (\ref {eq1}) admits at least two distinct nontrivial solutions $u_{0}$ and $u_{1}$ in $H_{0}^{1}(\mathbf {\Omega })$. Furthermore, $u_{0}\geq 0$ and $u_{1}\geq 0$ whenever $f\geq 0$.