Let H be a subgroup of a group G. Then, we call H weakly s-supplemented in G if G has a subgroup T such that HT = G and H ∩ T ≤ HsG, where HsG is the largest s-permutable subgroup of G contained in H. In this paper, we use the weakly s-supplemented subgroups to characterize the structure of groups. A series of known results in the literature are unified and generalized.

Let G be a finite nilpotent group, χ and ψ be irreducible complex characters of G with prime degree. Assume that χ(1) = p. Then, either the product χψ is a multiple of an irreducible character or χψ is the linear combination of at least distinct irreducible characters.

We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this.

By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes.

In an earlier paper, for ‘large’ (but otherwise unspecified) subsets , , , of q, Sárközy showed the solvability of the equations a + b = cd with a ∈ , b ∈ , c ∈ , d ∈ . This equation has been studied recently by many other authors. In this paper, we study the solvability of systems of equations of this type using additive character sums.

In this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems—fixed initial data and fixed asymptote, respectively—we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections.

We study moduli spaces of framed perverse instantons on ℙ3. As an open subset, it contains the (set-theoretical) moduli space of framed instantons studied by I. Frenkel and M. Jardim in [9]. We also construct a few counter-examples to earlier conjectures and results concerning these moduli spaces.

Regular maps, that is, graph embeddings with the ‘highest level’ of orientation-preserving symmetry, can be identified with two-generator presentations of groups G of the form 〈x, y; xm = y2 = (xy)n = … = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Zm* is an exponent of such a map if the assignments x ↦ xj and y ↦ y extend to an automorphism of G. The element −1 is an exponent if and only if the map is reflexible, that is, isomorphic to its mirror image. Non-trivial exponents, j ≠ ±1, induce automorphisms of the underlying graph, which are not map automorphisms but can nevertheless be considered as ‘external symmetries’ of the map. In this paper we show with the help of residual finiteness of triangle groups that for any given m, n≥3 such that 1/m + 1/n ≤ 1/2, there exist infinitely many finite, reflexible and regular maps of valence m and covalence n with no non-trivial exponent.

We consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

We compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

Following Craw, Maclagan, Thomas and Nakamura's works[2, 7] on Hilbert schemes for abelian groups, we give an explicit description of theHilbGℂ3 scheme for G = 〈diag(ϵ, ϵa, ϵr−a)〉 by a classification of all G-sets. We describe how the combinatorial properties of the fan of HilbGℂ3 relates to the Euclidean algorithm.

This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra ←p(F), with a transparent combinatorial structure, which controls the rational representation theory of GL2(F).

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

In this paper, we establish an extended Loomis–Whitney inequality for positive double John bases, which generalises Ball's result [1]. Moreover, a different extension of the Loomis–Whitney inequality is deduced.

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.

Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

We start here the study of some algebraic varieties related to cluster algebras. These varieties are defined as the fibres of the projection map from the cluster variety to the affine space of coefficients. We compute the number of points over finite fields on these varieties, for all simply laced Dynkin diagrams. We also compute the cohomology with compact support in some cases.

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

We investigate eigenvalue intervals for the Dirichlet problem when the nonlinearity may be singular at t = 0 or t = 1. Our approach is based on variational methods and cover both sublinear and superlinear cases. We also study the continuous dependence of solutions on functional parameters.