Soliton equations whose solutions are expressed by Pfaffians are briefly discussed. Included are a discrete-time Toda equation of BKP type, a modified Toda equation of BKP type, a coupled modified KdV equation and a coupled modified KdV equation of derivative type.

In a recent paper we presented a truncation-type method of deriving Bäcklund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painlevé equations. For the third Painlevé equation we are able to obtain all fundamental Bäcklund transformations for the case where the parameters satisfy \gamma \delta \neq 0. For the fifth Painlevé equation our approach yields what appears to be all known Bäcklund transformations.

It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k).

Hirota representations of soliton equations have proved veryuseful. They produced many of the known families of multisoliton solutions, andhave often led to a disclosure of the underlying Lax systems and infinite sets ofconserved quantities.

A striking feature is the ease with which direct insight can be gained into thenature of the eigenvalue problem associated with soliton equations derivable from aquadratic Hirota equation (for a single Hirota function), such as the KdV equationor the Boussinesq equation. A key element is the bilinear Bäcklund transformation(BT) which can be obtained straight away from the Hirota representation of theseequations, through decoupling of a related “two field condition” by means of anappropriate constraint of minimal weight. Details of this procedure have beenreported elsewhere. The main point is that bilinear BT's are obtained systematically,without the need of tricky “exchange formulas”. They arise in the formof “Y-systems”, each equation of which belongs to a linear space spanned by a basisof binary Bell polynomials (Y-polynomials).

It is proved that the subalgebra of constants of a derivation on a free associative algebra in prime characteristic is free provided that some constraining conditions are satisfied. As a particular case, it follows that the constants of the partial derivatives on a free algebra form a free subalgebra. The main result is also applied in order to provide a simplified proof of a previous result by the author on extensions of tensor rings.

A Banach space X is said to be an {\cal M}-space if every continuous multilinear form on X is weakly sequentially continuous. We study in this paper the stability properties of the class of {\cal M}-spaces.

In this paper BP-theory is used to give a proof that there exists a stable homotopy element in \pi _{2^{n+1} - 2}^{S}( {\tf="times-b"R}P^{\infty }) with non-zero Hurewicz image in ju-theory if and only if there exists an element of \pi _{2^{n+1} - 2}^{S}( S{\hskip1}^{0}) that is represented by a framed manifold of Arf invariant one.

We give an example of a finitely generated commutative semigroup that is not automatic.

A group G is called co-Dedekindian if every subgroup of G is invariant under all central automorphisms of G. In this paper we give some necessary conditions for certain finite p-groups with non-cyclic abelian second centre to be co-Dedekindian. We also classify 3-generator co-Dedekindian finite p-groups which are of class 3, having non-cyclic abelian second centre with |\Omega_1(G^p)|=p.

In this paper, we obtain an exact formula for the Hausdorff and box dimensions of a class of self-affine sets in two dimensions, namely those with disjoint projections. We prove, in particular, that fractals in this class have a Hausdorff and box dimension that is equal to the maximum Hausdorff and box dimension of one of their projections.

We construct three p-adic L-functions attached to the symmetric square of a modular elliptic curve. Following a calculation of Perrin-Riou for one of these functions, we compute the derivative of the p-adic L-function associated to the square of the non-unit root of Frobenius at p. This generalises Greenberg's notion of [Lscr]-invariant to these three-dimensional Galois representions.

It is shown that the creation operator is the only (up to a multiplicative constant) injective weighted shift all of whose translations (or at least one) are still injective weighted shifts regardless of what the weight sequences and the bases are. A similar result is true for the annihilation operator as well as for the Heisenberg and Schrödinger couples.

This paper discusses the fractional chromatic number of the direct product of graphs. It is proved that if H is a circulant graph G^k_d, or a Kneser graph, or a direct sum of such graphs, then for any graph G, \chi_f{\hskip1}(G\times H{\hskip1}) = {\text min}\{\chi_f{\hskip1}(G), \chi_f{\hskip1}(H{\hskip1})\}.

A unit speed curve \gamma =\gamma (s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along \gamma and a positive constant k such that \nabla _s \gamma '(s)=k Y(s),\, \nabla _s Y(s)=-k \gamma '(s). The main purpose of this article is to investigate the fundamental relationships between circles, maximal tori in compact symmetric spaces, and immersions of finite type.

In this paper, we consider the dependence of the Dirichlet eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain of definition. We prove that the dependence of a certain eigenvalue and of the corresponding eigenspace is analytic on the set of perturbations that leave the multiplicity constant.

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f(u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution andthe solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

Equilibrium solutions to the one-dimensional Gierer–Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity ε. For a pattern with k spikes, the construction yields k1 spikes that have a common small amplitude and k2 = k−k1 spikes that have a common large amplitude. A k-spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value Dm that depends upon k1, on k2, and on other parameters associated with the Gierer–Meinhardt model. It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch sk, for which k spikes have equal height. These asymmetric solutions provide connections between the branch sk and the other symmetric branches sj , for j = 1,…, k−1. The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(ε2) eigenvalues is also analyzed. It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > De, where De depends on k1 and k2, on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence. Numerical values for De are obtained from numerical solutions of a matrix eigenvalue problem. Another matrix eigenvalue problem that determines the small eigenvalues is derived. For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues.

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.

A three-phase ensemble-averaged model is developed for the flow of water and air through a deformable porous matrix. The model predicts a separation of the flow into saturated and unsaturated regions. The model is closed by proposing an experimentally-motivated heuristic elastic law which allows large-strain nonlinear behaviour to be treated in a relatively straightforward manner. The equations are applied to flow in the ‘nip’ area of a roll press machine whose function is to squeeze water out of wet paper as part of the manufacturing process. By exploiting the thin-layer limit suggested by the geometry of the nip, the problem is reduced to a nonlinear convection-diffusion equation with one free boundary. A numerical method is proposed for determining the flow and sample simulations are presented.

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.