Let ${\mathcal{R}}$ be a small preadditive category, viewed as a “ring with several objects.” A right ${\mathcal{R}}$ -module is an additive functor from ${\mathcal{R}}^{\text{op}}$ to the category $Ab$ of abelian groups. We show that every hereditary torsion theory on the category $({\mathcal{R}}^{\text{op}},Ab)$ of right ${\mathcal{R}}$ -modules must be differential.

In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$ -Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$ -well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$ , ( $t\in \mathbb{R}$ ), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$ , $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$ .

We construct a family of self-affine tiles in $\mathbb{R}^{d}$ ( $d\geqslant 2$ ) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$ , and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and  $Q$ , we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with  $Q$ .

We consider the unital Banach algebra $\ell ^{1}(\mathbb{Z}_{+})$ and prove directly, without using cyclic cohomology, that the simplicial cohomology groups ${\mathcal{H}}^{n}(\ell ^{1}(\mathbb{Z}_{+}),\ell ^{1}(\mathbb{Z}_{+})^{\ast })$ vanish for all $n\geqslant 2$ . This proceeds via the introduction of an explicit bounded linear operator which produces a contracting homotopy for $n\geqslant 2$ . This construction is generalised to unital Banach algebras $\ell ^{1}({\mathcal{S}})$ , where ${\mathcal{S}}={\mathcal{G}}\cap \mathbb{R}_{+}$ and ${\mathcal{G}}$ is a subgroup of  $\mathbb{R}_{+}$ .

In 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism that is the restriction of a universal entire function.

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$ -sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$ , and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$ -term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$ -term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally “special” Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$ -degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.

This paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$ . These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

We classify up to automorphisms all left-invariant non-Einstein solutions to the Einstein–Maxwell equations on four-dimensional Lie algebras.

We construct a shifted version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted $r$ -cycles. These are the first concrete results which count the number of cycles over “all tournaments”.

It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length ${<}\mathfrak{c}^{+}$ that is increasing with respect to the Rudin–Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from B. Kuzeljevic and D. Raghavan. It is also proved that Jensen’s diamond principle implies the existence of an unbounded strictly increasing sequence of P-points of length $\unicode[STIX]{x1D714}_{1}$ in the Rudin–Keisler ordering. This shows that restricting to the class of rapid P-points is essential for the first result.

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in  $C$ , but badly approximable by rationals outside of  $C$ . More precisely, we construct them so that all but finitely many of their convergents lie in  $C$ .

Given a prime $p\geqslant 5$ and an integer $s\geqslant 1$ , we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$ , such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$ . We also provide some explicit formulas for the optimal $M$ .

We say that two elements of a group or semigroup are $\Bbbk$ -linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$ -linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field  $\Bbbk$ .

We study the structure of the stable category $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen–Macaulay module over $S/(f)$ where $S$ is a graded ( $\pm 1$ )-skew polynomial algebra in $n$ variables of degree 1, and $f=x_{1}^{2}+\cdots +x_{n}^{2}$ . If $S$ is commutative, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if $n\leqslant 5$ , then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to $\mathbb{P}^{1}$ .

In this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$ -Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$ , there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$ -dimensional Euclidean space.