We call a semigroup right perfect if every object in the category of unitary right acts over that semigroup has a projective cover. In this paper, we generalize results about right perfect monoids to the case of semigroups. In our main theorem, we will give nine conditions equivalent to right perfectness of a factorizable semigroup. We also prove that right perfectness is a Morita invariant for factorizable semigroups.

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

Given a graph $G$ and an integer $\ell \ge 2$, we denote by $\alpha _{\ell }(G)$ the maximum size of a $K_{\ell }$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $\alpha _{\ell }(G) = o(n)$, which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $\alpha _\ell (G)=n^{1-o(1)}$ for all $r\ge \ell \ge 2$.

A random two-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log (n)$. While there are many families of graphs whose expected number of faces is $\Theta (n)$, none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any $n$-vertex multigraph, the expected number of faces in a random two-cell embedding is at most $2n\log (2\mu )$, where $\mu$ is the maximum edge-multiplicity. This bound is best possible up to a constant factor.

The time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type $\rm I\!I$ functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.

John Rognes developed a notion of Galois extension of commutative ring spectra, and this includes a criterion for identifying an extension as unramified. Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André–Quillen homology. In the classical algebraic context, it is important to distinguish between tame and wild ramification. Noether’s theorem characterizes tame ramification in terms of a normal basis, and tame ramification can also be detected via the surjectivity of the trace map. For commutative ring spectra, we suggest to study the Tate construction as a suitable analog. It tells us at which integral primes there is tame or wild ramification, and we determine its homotopy type in examples in the context of topological K-theory and topological modular forms.

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.

Using skew polynomials $f\in R$, we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

For a $k$-uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$-uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega (1/n^{k - 1})$, ${\mathrm{coker}}(M(\mathcal{H}))$ is torsion-free.

This paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

For linear stochastic differential equations with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and the whole ${\Bbb R}$ separately, in the sense that such an NMS-ED persists under a sufficiently small linear perturbation. The result for the nonuniform mean-square exponential contraction is also discussed. Moreover, in the process of proving the existence of NMS-ED, we use the observation that the projections of the ‘exponential growing solutions’ and the ‘exponential decaying solutions’ on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are different but related. Thus, the relations of three types of projections on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are discussed.

Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$, in agreement with experiments.

The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of sequences of nonnegative functions with $p>2$. Furthermore, we show the blow-up phenomena for solutions to the extended Nirenberg problem modelled by fractional $p$-Laplacian with the prescribed negative functions.

This work studies the asymptotic behavior of a waves coupled system with a boundary dissipation of the fractional derivative type. We prove well-posedness and polynomial stability based on the semigroup approach, the energy method, and the result of stability.

We investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup on the total space, and the Kähler property of special equivariant compactifications. We provide many examples illustrating that the main result is optimal.

As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

Let G denote a possibly discrete topological group admitting an open subgroup I which is pro-p. If H denotes the corresponding Hecke algebra over a field k of characteristic p, then we study the adjunction between H-modules and k-linear smooth G-representations in terms of various model structures. If H is a Gorenstein ring, we single out a full subcategory of smooth G-representations which is equivalent to the category of all Gorenstein projective H-modules via the functor of I-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-Archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of H-modules admits a right transfer. On the homotopy level, the right derived functor of I-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.

We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$:(*)

\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align}

$b\geq0$

$\partial\Omega\neq\emptyset$

$\mathbb{R}^{3}\backslash\Omega$

$u\in H_{0}^{1}(\Omega)$

$f\in C^1(\mathbb{R},\mathbb{R})$

\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}

$H^1(\mathbb R^3)$

$\mathbb R^3\setminus \Omega$