The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.

Groups having two generators and two relations are studied. The Reidemeister-Schreier algorithm is used to determine presentations for their derived groups. This enables the orders of the groups to be found. Necessary and sufficient conditions are given for the groups to be metabelian. Certain classes closely related to the class are also discussed.

A new construction of derived equivalences is given, which relates different endomorphism rings and, more generally, cohomological endomorphism rings, including higher extensions, of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable to a wide variety of situations, covering finite-dimensional algebras as well as certain infinite-dimensional algebras, Frobenius categories and n-Calabi–Yau categories.

In this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix

$$ \begin{pmatrix} -\dfrac{\mathrm{d}}{\mathrm{d} x}\rho(x) \dfrac{\mathrm{d}}{\mathrm{d} x}+q(x) & \dfrac{\mathrm{d}}{\mathrm{d} x}\dfrac{\beta(x)}{x} \\[12pt] -\dfrac{\beta(x)}{x}\dfrac{\mathrm{d}}{\mathrm{d} x} & \dfrac{m(x)}{x^2} \end{pmatrix} $$

is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range $\text{range}((m\rho-\beta^2)/\rho x^2)$ in the case where the quasi-regularity conditions are not satisfied.

The structure of non-trivial non-negative solutions to singularly perturbed semilinear Dirichlet problems of the form −ε2Δu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ε → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and . It is shown that there are many non-trivial non-negative solutions and they are spike-layer solutions. Moreover, the measure of each spike layer is estimated as ε → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0, ∞). Uniqueness of a large positive solution and many positive intermediate spike-layer solutions are obtained for ε sufficiently small.

We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).

This paper is concerned with the theory of the probability distribution of the total number of electrons in the avalanche produced by the release of a single electron in the gas of a proportional counter. The disagreement between existing theory and experimental results is discussed and a new theory is proposed, based on the fact that fluctuations in the number of electrons at a given point in the avalanche are accompanied by fluctuations in the average electron energy. This aspect of the problem is incorporated directly into a simple one-dimensional model of the multiplication process, and the resulting distribution function has a mathematical foim in agreement with that observed experimentally.

The fluctuation in the number of electrons predicted by this theory is not constant, but is determined by a parameter which, for large mean values, is essentially the mean fraction of electrons in the avalanche having energies above the ionization energy of the counter gas. Limits on the variation of this parameter are obtained by calculations of the mean values using a particular two-dimensional model, in which electrons are divided into two classes according as they have energies above, or below, the ionization energy. The experimentally observed fluctuation lies within the predicted range and close to the lower limit; it is concluded that there is little scope for improvement in the resolution to be obtained from the conventional type of proportional counter.

In this paper, I prove that on every fractal structure with three vertices, for suitable weights, the renormalization operator has an eigenform.

A Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.

It is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.

Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

Sufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

In 1932, Hardy and Littlewood [1] proved the inequality

The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where

y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)

and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

We classify multiplicative white noise perturbations k(·) dw, of generalised KPP equations and their effects on deterministic approximate travelling wave solutions by the behaviour of , the solutions of the stochastic generalised KPP equations converge to deterministic approximate travelling waves and if

being an associated potential energy, Фs a solution of the corresponding classical mechanical equations of Newton, D being a certain domain in R1 × Rr then the white noise perturbations essentially destroy the wave structure and force the solutions to die down.

For the case

(suppose the existence of the limit) we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations. Numerical solutions are included and give good agreement with theoretical results.

Although the action of potassium persulphate on silver nitrate solution was one of the first persulphate reactions observed (vol. xviii. p. 64), I had not until lately paid any special attention to the behaviour of the ammonium salt in this respect. It appears, however, that in the latter case there are additional actions of great interest, not possible with the potassium salt. A general description of these will be given now, but there are still some points deserving of further investigation.