We study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented.

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.

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

It is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

The equational complexity function $\beta \nu \,:\,{\open N} \to {\open N}$ of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting $O(\log_{2}^{3}(n))$ growth among varieties of semilattice-ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.

Let G be a linear group such that for every g ∈ G there is a finite set ${\cal R}(g)$ with the property that for every x ∈ G all sufficiently long commutators [g, x, x, …, x] belong to ${\cal R}(g)$. We prove that G is finite-by-hypercentral.

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good for G.

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.

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.

A polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

Let A and G be finite groups and suppose that A acts via automorphisms on G with $(|A|, |G|)=1$. We study how certain conditions on the Sylow 2-subgroups of the fixed point subgroup of the action $C_G(A)$ may imply the non-simplicity or solubility of G.

We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized 4n2-inequality for complete intersection singularities and the technique of hypertangent divisors.

For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.