Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

In this paper, we prove the existence of at least one solution to the problem

where ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

As is well known, the deficiency indices of the minimal symmetric differential operator of even order 2m with real coefficients on the half-line lie within the limits m and 2m, and are equal. I. M. Glazman [6] first showed, by examples, that for such operators the deficiency indices can actually take all the intermediate values m ≦ P ≦ 2m. Subsequently fresh examples of operators with real coefficients, whose deficiency indices take all possible values, were constructed by various authors (S. A. Orlov [11], M. V. Fedoryuk [4], and others).

The main goal of this work is to study the existence and uniqueness of a positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

A variational problem for a functional with slowly growing principal part and involving critical Orlicz–Sobolev lower term with respect to the principal part is discussed. The principal part of the functional is not Fréchet differentiable. The lack of differentiability and the critical growth rate of the lower term demand a precise compactness argument in the variational approach. A non-negative solution for the Euler equation is given.

Several definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.

It is recognised that infecundity and infertility are among the major problems which confront the breeder of livestock, and in this country, as elsewhere, investigation into their causes, prevention, and cure is actively encouraged. A number of pathological and nutritional factors have been shown to lie at the root of these conditions, and it has become abundantly clear that in many instances genetic factors are undoubtedly responsible for the inability on the part of the individual to elaborate functional gametes and/or of these gametes in their union to give rise to fully viable zygotes. These factors may be discovered in intergeneric crosses.

We investigate the existence of positive principal eigenvalues of the problem - ∆u(x) = λg(x)u(x) for x ∈ ℝk where the weight function g changes sign in ℝk and is negative for |x| sufficiently large.

Dr Anderson's paper (Proc. Roy. Soc. Edin., vol. Ivi, 1936, pp. 128–157) is of considerable interest as a contribution both to geology and to the theory of elasticity. One of his assumptions about fracture, however, has little support, and I am inclined to think that some difficulties that he mentions towards the end would be lessened if it were removed. This assumption is that the complete form of a fracture surface can be inferred directly from the distribution of stress before fracture, intersecting the surfaces of principal stress at some constant angle. It is often to be found in both geological and mechanical literature; but it rests on no theoretical grounds and is, I think, untrue.

The calming effect of oil on troubled waters is so well known and so often referred to, that we might have expected there would, by this time, have been a considerable amount of written information on the subject. This calming effect of oil is not only well known, but it has also become so stereotyped by constant repetition, that it seems to have acquired the stamp of classical antiquity upon it, yet one is surprised to find that reference to it in early writers is strangely conspicuous by its absence. Our knowledge of the subject seems to have lived almost entirely in tradition.

The object of this paper is to record certain temperatures which were measured in the deep Boreland Bore, near Dysart, in Fife, and to discuss the bearing of these and some other observations on the amount of heat which escapes from the crust, by conduction, in Britain.

The elastic properties of plant cell walls are major factors controlling the swelling and shrinking of plant cells under conditions of varying water stress. The importance of these properties for the adaptation of plants to the changing water stresses they encounter is discussed.

In a previous paper an expression was given by one of us for the Doppler effect in a de Sitter universe. In the present communication an attempt is made to obtain a more general expression for the Doppler effect, and for this purpose certain results have been derived relating to the properties of null geodesics in a general Riemannian space. In the paper referred to, the problem of defining spatial distance in general space was discussed and certain formulæ suggested which satisfy the essential conditions associated with the idea of spatial distance. This problem has now been approached from a different standpoint, and a new formula of more general form is now put forward. In the final section of the present paper a physical interpretation is given of the theorem relating to null geodesics which is proved in the first section. This involves a discussion of the concept of energy in general relativity, in order that a decision may be arrived at as to what law governs the transference of energy by radiation from one point to another. It appears that when the simplest and most natural assumptions are made for this purpose, then these are precisely the assumptions which are necessary to ensure the general consistency of relativity with the principles of quantum theory. This is illustrated by an application to a statical universe, and to the reddening of light in an expanding universe.

Extensions of the integral version of Hardy's Inequality were given by Kadlec and Kufner (1967) and by Copson (1976). This paper provides several levels of further generalization of their results, obtained mostly by specializing four main inequalities. Most of the inequalities have the form ∥Kf∥ ≤ C ∥f∥, where K is an integral transform and ∥.∥ is a generalized Lp-norm; some have the inequality sign reversed. Best possible constants C are obtained in several cases, under mild extra hypotheses.

We use a modified form of the Stieltjes inversion formula due to Atkinson together with some recent results on the asymptotic form of Titchmarch–Weyl m-functions to obtain an asymptotic expression for the spectral functions τ(t) associated with the differential equation

A unified description and treatment is given for a large and important class of homogeneous Siegel domains in finite and infinite dimensions. These domains are shown to be linearly equivalent to generalized upper half-planes in spaces of operators having a kind of triple product structure.

We study the stability of periodic solutions of the scalar delay differential equation where f(t) is a periodic forcing term and δ,p∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.

Vector-borne diseases, such as chikungunya, dengue, malaria, West Nile virus, yellow fever and Zika, pose a major global public health problem worldwide. In this paper we investigate the propagation dynamics of diffusive vector-borne disease models in the whole space, which characterize the spatial expansion of the infected hosts and infected vectors. Due to the lack of monotonicity, the comparison principle cannot be applied directly to this system. We determine the spreading speed and minimal wave speed when the basic reproduction number of the corresponding kinetic system is larger than one. The spreading speed is mainly estimated by the uniform persistence argument and generalized principal eigenvalue. We also show that solutions converge locally uniformly to the positive equilibrium by employing two auxiliary monotone systems. Moreover, it is proven that the spreading speed is the minimal wave speed of travelling wave solutions. In particular, the uniqueness and monotonicity of travelling waves are obtained. When the basic reproduction number of the corresponding kinetic system is not larger than one, it is shown that solutions approach to the disease-free equilibrium uniformly and there is no travelling wave solutions. Finally, numerical simulations are presented to illustrate the analytical results.

Routine observations on the growth of chickens from a flock of Brown Leghorns maintained at this Institute indicated that factors, apparently closely associated with season of the year, caused considerable variations in the rate of growth. It is possible to conceive of two distinct means by which seasonal factors may produce their effect (1) directly on the growing chick, or (2) indirectly in that their influence may be exerted on the hen and transmitted to the egg from which the chick subsequently develops; either or both may contribute to the final variations obtained.