In this article, we study localizations of hearts of cotorsion pairs ($\mathcal{U}, \mathcal{V}$) where $\mathcal{U}$ is rigid on an extriangulated category $\mathcal{B}$. The hearts of such cotorsion pairs are equivalent to the functor categories over the stable category of $\mathcal{U}$ ($\bmod \underline{\mathcal{U}}$). Inspired by Marsh and Palu (Nagoya Math. J. 225(2017), 64–99), we consider the mutation (in the sense of Iyama and Yoshino, Invent. Math. 172(1) (2008), 117–168) of $\mathcal{U}$ that induces a cotorsion pair ($\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$). Generally speaking, the hearts of ($\mathcal{U}, \mathcal{V}$) and ($\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$) are not equivalent to each other, but we will give a generalized pseudo-Morita equivalence between certain localizations of their hearts.

The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley–Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley–Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology, we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the grand crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin–Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in ${\mathcal{E}}_{n}$-algebra.

We study the temporal decay estimate of the Oseen semigroup in a two-dimensional exterior domain. We establish the local energy decay estimate with a suitable dependence on the small translation speed, which is a significant improvement of Hishida’s result in 2016. As an application, we prove the $L^{q}$-$L^{r}$ estimates of the Oseen semigroup uniformly in the small translation speed.

Dolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and $q\notin \{7,11,29\}$, we obtain a full classification of all pairs of non-trivial coprime subdegrees.

Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/( log N)ω(1). We consider the random greedy algorithm for forming a matching in $ {\cal H} $. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\cal H} $ that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1). This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

Let k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8.

A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang–Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

Let $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$. The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$-sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.

In this note it is shown that the class of all multipliers from the d-parameter Hardy space $H_{{\rm prod}}^1 ({\open T}^d)$ to L2 (𝕋d) is properly contained in the class of all multipliers from L logd/2L (𝕋d) to L2(𝕋d).

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

There does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group.

Recently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 <F0) ( [9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞).

The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.