We study the existence of extremal periodic solutions for nonlinear evolution inclusions defined on an evolution triple of spaces and with the nonlinear operator establish A being time-dependent and pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence of extremal periodic solutions. Subsequently, by assuming that A(t, ·) is monotone, we prove a strong relaxation theorem for the periodic problem. Two examples of nonlinear distributed parameter systems illustrate the applicability of our results.

We show that if a periodic residually-finite group G has a 2-element with finite centralizer then G is locally finite.

Felix and Murillo introduced the group $\mathrm{Aut}_\varOmega(X)$ of self-maps $f$ of $X$, which satisfy $\sOm f=1_{\varOmega X}$, and proved that the group is nilpotent with the order of nilpotency bounded by the Lusternik–Schnirelmann category of $X$. In this paper we construct a spectral sequence converging to the group $\mathrm{Aut}_\varOmega(X)$ and derive several interesting consequences.

AMS 2000 Mathematics subject classification: Primary 55P42

Various solutions of Mathieu's equation, or the equation of the elliptic cylinder functions

have recently been discussed in an elegant series of papers in these Proceedings. These papers have dealt with the periodic and quasi-periodic solutions, but the present paper merely considers determinants which give the infinite series of relations between a and q, so that the solutions are purely periodic, i.e. the solutions denoted by Professor Whittaker

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.

Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.

Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of $\mathbb D^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where $\mathbb D:=\mathbb R, \mathbb C$ or $ \mathbb H $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $.

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficients

Parseval's formula gives

We show that if G is a compact abelian group and U is a weakly continuous representation of G by means of isometries on a Banach space X, then holds for each measure µ in reg(M(G)), where π(µ) denotes the generalized convolution operator in B(X) defined by , σ the usual spectrum in B(X), sp(U) the Arveson spectrum of U, the Fourier-Stieltjes transform of µ and reg(M(G)) the largest closed regular subalgebra of the convolution measure algebra M(G) of G. reg(M(G)) contains all the absolutely continuous measures and discrete measures.

The present paper deals with the problem of asymptotic equivalence of the system with impulse effect

and

where x, y:I→Rn; g:I→Rn; f:I × Rn→Rn; bk:Rn→Rn; I = [0, ∞); Rn is the n-dimensional Euclidean space with a norm |·|; A and B are constant matrices; the moments {tk} constitute an increasing sequence 0 < t1 < … < tk < …, limk→∞tk = ∞.

Many problems involving the solution of partial differential equations require the solution over a finite region with fixed boundaries on which conditions are prescribed. It is a well known fact that the numerical solution of many such problems requires additional conditions on these boundaries and these conditions must be chosen to ensure stability. This problem has been considered by, amongst others, Kreiss [11, 12, 13], Osher [16, 17], Gustafsson et al. [9] Gottlieb and Tarkel [7] and Burns [1]

A particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.

In this paper we shall deal with radicals γ of near-rings such that for every near-ring N its radical γ(N) contains all left ideals (left invariant subgroups, respectively) I of N with I∈γ. At first, examples of such radicals will be given. Then we shall prove that these radicals are hypersolvable and have hereditary semisimple classes.

We develop here the recurrence relations for the generalised C.F.'s introduced in Part 3 (Shenton 1956). In the main the discussion will be limited to second order C.F.'s, but results for higher orders will be given when these are not complicated.

In a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.

This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

Let H be any group. We call a cardinal number r the rank r(H) of H if H can be generated by a generating system X with cardinal number r but not by a generating system Y with cardinal number s less than r. Let r(H) be the rank of H.

We call a generating system X of H a minimal generating system (M.G.S.) of H if X has the cardinal number r(H).

In this note we prove the following.

Throughout this paper X will be a finite connected CW-complex of dimension m, and ξ will be a real (n + l)-plane bundle over X(n >0) equipped with a Riemannian metric. We aim to give a systematic account of the space ГSξ of sections of the sphere-bundle Sξ.

We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.