In this paper a combination is given of the two methods mentioned in the title with the aim of optimising the definition of the absolute temperature with respect to physical transparency and mathematical simplicity. The mathematical tools used are the elements of vector analysis, i.e. “usual” mathematics (“normale” Mathematik) in the sense of Born [1, p. 162, cf. also 22, p. 579].

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

Schrödinger operators of the form T = (i grad + b(x))2 + a(x) · grad + q(x) in Rm are considered, where a, b ate real vector-valued functions and q is a scalar complex-valued function. It is shown that T is essentially quasi-m-accretive in L2(Rm) if (1 + #x2223;∣)−1a ∈ L4 + L∞, div a ∈ L∞, , and Re q ≧ 0. The proof is elementary.

The principal concern here is with conditions on f or on special solutions of the equation

which ensure that the higher differences of the zeros and related quantities of solutions of (1) are regular in sign. In particular, by choosing f(x)= 2v−2x1/v−2, it is shown that if ⅓ ≦|v|<½, then

where cvk denotes the kth positive zero of a Bessel function of order v and Δµk = Δk+1 − µk. Lorch and Szego [15] conjectured that (2) should hold for the larger range | v | < ½ but the methods used here do not apply to the range | v <| ⅓.

The inequality considered is

where p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L2(a, b).

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.

We prove the global existence of non-negative weak solutions for a strongly coupled, fourth-order degenerate parabolic system governing the motion of two thin fluid layers in a porous medium when capillarity is the sole driving mechanism.

We consider a non-homogeneous generalized Burgers equation

Here, ν is small and positive, f is strongly convex and satisfies a growth assumption, while ηω is a space-smooth random ‘kicked’ forcing term. For any solution u of this equation, we consider the quasi-stationary regime, corresponding to t ⩾ 2. After taking the ensemble average, we obtain upper estimates and time-averaged lower estimates for a class of Sobolev norms of u. These estimates are of the form Cν-β with the same values of β for bounds from above and from below. They depend on η and f, but do not depend on the time t or the initial condition.

We consider the eigenfunction expansions associated with a symmetric differential operator M[·] of order 2n with coefficients defined on an open interval (a, b). Each singular endpoint of (a, b) is assumed to be of limit-n type. A direct convergence theory is established for the eigenfunction series expansion of a function y in a set Termwise differentiation of the series is established for the derivatives of order up to n. For O ≤ i ≤ n − 1, the i-fold differentiated series converges absolutely and uniformly to y(i) on compact intervals; the n−fold differentiated series converges to yn in the mean. The expansion theory is valid also when an essential spectrum is present. An explicit formula is given for the calculation of the spectral matrix.

This paper is concerned with integral inequalities of the form

where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

The scalar nonlinear convection-diffusion equation

is considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.

This paper considers the Cauchy problem for an extended model of combustion (u + qz)t + f(u)x = 0, zt + kg(u)z = 0 with Lp bounded initial data, where g(u) is a piecewise Lipschitz continuous function and its discontinuous points have no finite limit point. The integral representation gives a definition of a weak solution in Lp space. Some existence results are obtained based on a simplified method of compensated compactness in which the weak continuity theorem of 2 * 2 determinants plays a more important role, but the idea of Young measures has been avoided.

We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

The law which is sometimes associated with the name of Talbot is generally stated thus:—Once complete fusion has been reached, no alteration in the intensity of the resultant impression produced by a series of flashes takes place, however short the time during which each flash acts may be, provided that the number of flashes in a given time and the length of each stimulation be always kept inversely proportional. Complete fusion of stimuli here is analogous to complete tetanus of muscle. And, as in the latter case, it has been discussed as to where the upper limit of frequency lies, or whether there be an upper limit, so in the former case are like questions in place. With the various answers which have been given in regard to muscle tetanus we are not here concerned; except that it may be noticed that the later investigators, where they have at all admitted the probability of a limit, have had a tendency to shorten the time between each stimulus which they regarded as the minimum.

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

The effective conductivity tensor σ* of a two-dimensional polycrystalline material depends on the conductivity tensor σ0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation field R(x) giving the orientation of the crystal at each point x. Here it is established that the dependence of σ* on σ0 in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.

We consider operator-valued boundary-value problems in (0, 2π)n with periodic or, more generally, ν-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary-value problem. As an application, we study vector-valued parabolic initial boundary-value problems in cylindrical domains (0, 2π)n × V with ν-periodic boundary conditions in the cylindrical directions. We show that, under suitable assumptions on the coefficients, we obtain maximal Lq-regularity for such problems. For symmetric operators such as the Laplacian, related results for mixed Dirichlet-Neumann boundary conditions on (0, 2π)n × V are deduced.

While observations were being carried out on the fecundity of certain experimental Drosophila pseudo-obscura females, it became apparent that there was a regular alternation of periods of rapid laying with periods of no laying. Since an irregularity in rate of oviposition in this species had been noticed by other investigators, and since, under similar circumstances, Drosophila melanogaster lays approximately the same number of eggs each day, it was decided to investigate more closely the possibility that differences in laying habit existed among the various species.

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problem and which models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A ∈ C([0, 1]) with A(s) > 0 for s ∈ (0, 1] and there is a constant L ∈ (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problems.

This paper is concerned with the existence of maximizers for a certain non-convex energy functional, relative to a class of rearrangements of a given function. Physically, the solution represents a uniform shear flow containing a bounded vortex anomaly, in R2. The prescribed data are the rearrangement class of the vorticity field.