Each permutation representation of a finite group $G$ can be used to pull cohomology classes back from a symmetric group to $G$. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of $G$.

We also investigate the effect of restricting to special types of permutation representations, such as $\mathrm{GL}_n(\mathbb{F}_p)$ acting on flags of subspaces.

AMS 2000 Mathematics subject classification: Primary 20J06

Given an integer $n$, we show that $\mathcal{I}_{n}$ embeds in a 2-generated subsemigroup of $\mathcal{I}_{n+2}$, which is an inverse semigroup. An immediate consequence of this result is the following, which is analogous to the case for groups and semigroups: every finite inverse semigroup may be embedded in a finite 2-generated semigroup which is an inverse semigroup.

AMS 2000 Mathematics subject classification: Primary 20M18. Secondary 20M20

The explicit defining equations of a new family of curves whose members have a trivial automorphism group are given. Each member is defined for characteristic zero and all but a finite number of characteristics greater than zero. This family, in conjunction with a previously appearing family of the author’s, provides explicit examples of algebraic curves which possess only the trivial automorphism for each genus greater than three. The family is then used to construct Riemann surfaces without anticonformal automorphisms and Klein surfaces with no non-trivial automorphisms.

AMS 2000 Mathematics subject classification: Primary 14H37; 30F50; 30F99

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.

Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.

In this paper, we give a positive answer to this question.

This paper is devoted to the study of the elliptic problems with a critical potential,where N ≥ 3, λ ≥ 0 and 0 < q < 1 < p ≤ (N + 2)/(N − 2). Existence, multiplicity, behaviour in x = 0 and bifurcation are considered under some hypotheses in h and g.

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second-order elliptic operators, these estimates hold true on bounded C1,1 domains. For higher-order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to non-cooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

This paper deals with the Riemann problem for a partial differential equation's model arising in phase-transition dynamics and consisting of an hyperbolic–elliptic system of two conservation laws. First of all, we provide a complete description of all solutions of the Riemann problem that are consistent with the mathematical entropy inequality associated with the total energy of the system. Second, following Abeyaratne and Knowles, we impose a kinetic relation to determine the dynamics of subsonic phase boundaries. Based on the requirement that subsonic phase boundaries are preferred whenever available, we determine the corresponding wave curves associated with composite waves (shocks, rarefaction fans, phase boundaries). It turns out that even after the kinetic relation is imposed, the Riemann problem may admit up to two solutions. A nucleation criterion is necessary to select between a solution remaining in a single phase and a solution containing two phase boundaries. Alternatively, a strong assumption on the kinetic relation ensures that the Riemann solution is unique and depends continuously upon its initial data.

Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.

The class of masked factorable matrices is introduced and simple necessary and sufficient conditions are given for matrices in the class to represent bounded transformations between Lebesgue sequence spaces.

A method for explicit Wiener–Hopf factorization of 2 × 2 matrix-valued functions is presented together with an abstract definition of a class of functions, C(Q1, Q2), to which it applies. The method involves the reduction of the original factorization problem to certain nonlinear scalar Riemann–Hilbert problems, which are easier to solve. The class C(Q1, Q2) contains a wide range of classes dealt with in the literature, including the well-known Daniele–Khrapkov class. The structure of the factors in the factorization of any element of the class C(Q1, Q2) is studied and a relation between the two columns of the factors, which gives one of the columns in terms of the other through a linear transformation, is established. This greatly simplifies the complete determination of the factors and gives relevant information on the nature of the factorization. Two examples suggested by applications are completely worked out.

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det Fp(x) ≥ μ+ > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequalitywhich is just our result with Fp = 11. With slight modifications, we are also able to treat forms of the type

A recent paper of Ockendon et al. discusses the Fanno model for quasi-one-dimensional flow of gas in a tube, in situations where the flow is turbulent and the tube is long enough for wall drag to be important. Based on appropriate scalings and with associated boundary conditions they derive equations for similarity solutions and make predictions about travelling and evolving waves. In this paper the existence, uniqueness and asymptotic behaviour of these wave forms is proved rigorously. Techniques include shooting methods (both one- and two-parameter), appropriate changes of variables, and comparison techniques.

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

A leader is to be elected from n people using the following algorithm. Each person flips a coin. Those people who wind up with tails (which occurs with probability p, 0 < p < 1) move on to the next stage. Those with heads are eliminated. Let Hn denote the number of stages needed until there is a single winner. We analyze the moments and the probability distribution of Hn. In the symmetric model we have an unbiased coin with p = 1/2; in the asymmetric model p ≠ 1/2. We analyze these models asymptotically, for n → ∞, using a variety of analytical and numerical approaches. This leads to simple derivations of some existing results, as well as some new results for the asymmetric case. Our analysis makes some assumptions about the forms of various asymptotic expansions as well as their asymptotic matching.

We consider the Lyapunov functional,of the rescaled Extended Fisher-Kolmogorov equationThis is a fourth order generalization of the Fisher–Kolmogorov or Allen–Cahn equation. We prove that if ε → 0, then tends to the area functional in the sense of Γ-limits, where the transition energy is given by the one-dimensional kink of the Extended Fisher–Kolmogorov equation.

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment emanating from the vertex, so we have liquid at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundary, the region occupied by the fluid is the image of a semidisc (a domain bounded by a semicircle and its diameter) in the ζ-plane under a conformal map given by a function of the form ζα times a rational function of ζ. The form of this rational function can be written down, and the parameters appearing in it then determined as the solution to a set of algebraic equations. Examples of such flows are given (including one which shows that, in a certain sense, injection can produce a cusp), and the limiting situation in the wedge configuration as one injection point is moved to infinity is also considered.

Solutions to a boundary-value problem involving a second-order linear functional differential equation with an advanced argument are investigated in this paper. The boundary conditions imposed on the differential equation are analogous to conditions defining various singular Sturm-Liouville problems, and if an eigenvalue parameter is introduced certain properties of the spectrum can be deduced having analogues with the classical problem. Dirichlet series solutions are constructed for the problem and it is established that the spectrum contains an infinite number of real positive eigenvalues. A Laplace transform analysis of the problem then reveals that the spectrum does not generically consist of isolated points and that there may be an infinite number of eigenfunctions corresponding to a given eigenvalue. In contrast, it is also shown that there is a subset of eigenvalues that correspond to the zeros of an entire function for which the corresponding eigenfunctions are unique.

Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment that does not adjoin the vertex, so we have air at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundaries, the region occupied by the fluid is the image of a rectangle under a conformal map that can be expressed in terms of elliptic functions if α = 1 or α = 2, and in terms of theta functions if 0 < α < 1 or 1 < α < 2. The form of the function giving the map can be written down, and the parameters appearing in it then determined as the solution to a set of transcendental equations. The procedure is illustrated by a number of examples.

This paper presents a new class of solutions for steady nonlinear capillary waves on a curved sheet of fluid in the plane. The solutions are exact in that the free surfaces of the sheet and the associated flow field can be found in closed form. The solutions are generalizations of the classic solutions for finite amplitude waves on fluid sheets [5] to the case where the fluid sheets are curved.