We study a problem from nonlinear optics that leads to an integro-differential equation which can be written as the abstract bifurcation problem Au = λLu + ∇φ(u). For such a type of equation, a general theorem on the bifurcation of solutions from the essential spectrum is proved. The hypothesis that φ is non-negative (as is usually assumed) could be omitted and replaced by other conditions that allow us to use the results for the particular example. We also indicate applications to semilinear elliptic equations.

We define a new class of admissible solutions for the parabolic problems where the theory of viscosity solutions does not apply. A typical example is the porous medium equation complemented by lower-order terms, nonlinear with respect to the gradient. In the case, the nonlinearity in the equation fails to be proper in the sense of the theory of viscosity solutions. The admissible solutions satisfy a very general comparison principle and, consequently, the corresponding initial-boundary value problems are well-posed in this class. They coincide with the standard viscosity solutions provided the nonlinearity in the equation is proper.

Charles Darwin in his Autobiography, written in 1876, gives in half a dozen pages (Life and Letters, vol. i, pp. 36–42) an account of his two academic sessions as a medical student in Edinburgh. This account includes brief references to his naturalist friends and acquaintances, a statement that he collected specimens in the tidal pools on the shore of the Firth of Forth and by going out with the “trawlers,” and that he made new observations on the “so-called ova of Flustra,” which “were in fact larvæ,” and on the egg-cases of Pontobdella muricata.

The apparent prodigality with which Nature provides for the reproduction of plants and animals has a marked exception in the case of sterile organisms, often produced on a large scale, in which every function except that of propagation may be carried on in a quite perfect manner. The cases of the sterilisation of an organism are best seen in insects, but I wish at present to consider its occurrence high up in the animal kingdom in the well-known case of the Free-Martin, an apparent sterile cow born co-twin with a potent bull.

In July 1913, while collecting the larvæ of the large Pine Weevil (Hylobius abietis), I discovered a number of tiny larvæ feeding on some of them. These I succeeded in rearing, and they proved to be the larvæ of a Bracon which seemed to be B. hylobii. In July 1914 I recorded my experiments and observations.* In this record I stated that my identification of the species was based on Ratzeburg's description.† All the specimens I examined—and I have examined many—agree, for the most part, with his description of B. hylobii. In April 1914, specimens were sent for determination to Dr Szeplegeti, the authority for the family, in Budapest, but so far no reply has been received.

Let H be the Hopf line bundle over the complex projective space of complex dimension k – 1. We determine the codegree of the virtual bundle –nH in the range l ≦ n ≦ k. This codegree has geometric significance as a stable James number.

We consider convolution operators arising in the study of abstract initial boundary value problems. These operators are of the form

where {S(t)}t ≧0 is a C0-semigroup in a Banach space X,, with infinitesimal generator A0,: D(A0), ⊂ X, → X, and K(z): Y → X is a linxear, continuous mapping defined in another Banach space Y., We study the continuity of T between the spaces Lp([0, + ∞), Y), and Lq([0, + ∞), X), 1 ≦ p, q, ≦ + ∞. We give several examples of the applicability of the results to some familiar initial boundary value problems, including both parabolic and hyperbolic cases.

Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.

Let K be the class of all operators T in a Banach space × which have the property that, for any pair of integers (n, k) with n ≧2 and l≦ k ≦ n – l, there exists a constant Cnk such that

for all fϵdom Tn. If T ϵ K, then the best possible constant for the norm inequality (*) is the smallest non-negative value of the constant Cnk in (*). Any operator T which is the adjoint of a maximal symmetric operator in a Hilbert space belongs to the class K, as was shown by Ljubič [Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 825–864].

This article is concerned with the computation of the best possible constant for the differentiation operator Tf=if′ on the maximal domain in L2(0, ∞). Three algorithms, proposed by Ljubič [ibid.] and Kupcov [Trudy Mat. Inst. Steklov. 138 (1975)], are discussed and related to one another, asymptotic expressions (valid for large n) and numerical values of the best possible constant are presented, and the extremals (i.e. the elements / ∈ dom Tn for which equality holds in (*) with the best possible constant) are given.

For a sign-changing function a(x) we consider the solutions of the following semilinear elliptic problem in ℝn with n ≥ 3:

where γ > 0 and 0 < q < 1 < p < (n + 2)/(n − 2). Under an appropriate growth assumption on a− at infinity, we show that all solutions are compactly supported. When Ω+ = {x ∈ ℝn | a(x) > 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a(x) = a(|x|), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω+.

In this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.

By an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.

Similar results are obtained for 0-simple semigroups without nonzero idempotents.

The following conjecture of I. D. Macdonald is confirmed in this paper. If G is a finite ρ-group generated by elements of breadth at most n, then the nilpotency class of G is bounded in terms of nonly.

We consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

Professor Andrews, in the “Philosophical Magazine” for 1852, recalled the attention of physicists to the method originally devised by Davy of making a vacuum so perfect, that the residual gas exercised no appreciable pressure as registered by the depression of a barometric column. This he effected by filling the vessel to be exhausted with carbonic acid gas, having previously inserted a cup containing a concentrated solution of caustic potash. On rapidly exhausting with an air-pump, and leaving time for the absorption of the residual carbonic acid by the caustic potash, he obtained a vacuum as perfect as a Torricellian. Andrews' method was afterwards employed by Gassiot in his well-known investigations on the passage of electricity through attenuated media.

This paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equation

The key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problem

No condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

Measurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

By use of the theory of asymptotic expansions for first-order linear systems of ordinary differential equations, asymptotic formulas are obtained for the solutions of an nth order linear homogeneous ordinary differential equation with complex coefficients having asymptotic expansions in a sector of the complex plane. These asymptotic formulas involve the roots of certain polynomials whose coefficients are obtained from the asymptotic expansions of the coefficients of the differential operator.

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

This paper is concerned with the limit of the vanishing ratio of the electron mass to the ion mass in the hydrodynamic models for plasmas in critical Besov spaces. We give a new construction of approximation solutions and show that periodic initial-value problems of certain scaled hydrodynamic models have smooth solutions in a (finite) time interval where the Euler solution is known to exist. Furthermore, it is justified that, as the electron mass tends to zero, the smooth solutions converge rigorously to solutions of the incompressible Euler equations, and the definite convergence orders are also obtained.