We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.

The Fulcher Bands of Hydrogen have recently been arranged by Richardson. An examination of Richardson's arrangement in the light of recent accurate measurements of the hydrogen spectrum by Gale, Monk, and Lee, while strongly confirming Richardson's allocation of the Q branches, has failed to yield confirmation of the remaining branches which he gives.

The present paper gives a number of combination relations which have been found to hold between the lines of the Fulcher Bands. These indicate that there are three main branches in each band, designated the R', Q, and P' branches, the Q branch being identical with that of Richardson. The R' and P' branches have a common initial level differing from that of the Q branch, while all three branches have a common final level. The arrangement yields term differences consistent throughout the bands.

As far as these term differences have been investigated, they have been found to be in agreement with the new mechanics.

The present investigation was suggested to me by Prof. H. Stanley Allen of St Andrews, and my warmest thanks are due to him for constant encouragement and helpful criticisms and suggestions.

1. Degeneration of the germinal epithelium of the testis invariably follows the operation of vasoligation or of vasectomy in the mouse, but does not become evident until a considerable period of time has elapsed.

2. Similar degeneration follows the operation of epididymo-deferentectomy, but after a much shorter interval of time.

3. This degeneration is caused perhaps by the pressure within the seminiferous tubules exerted by the accumulation of seminal products, or more probably by the action of toxic substances liberated from the disintegrating cell and sperm debris.

This paper is the continuation of the papers by Born (1939, Part I) and by Born and myself (1940, Parts II and III). Here the theory is extended to the case of spinor wave functions satisfying the Dirac equation.

The only observations, so far as I am aware, that have been made on the pitchers of Nepenthes seedlings, are those by Sir J. D. Hooker, who, in 1859, described their external configuration in his paper “On the Origin and Development of the Pitchers of Nepenthes, &c,” in vol. xxii. of the Linnean Society's Transactions, and afterwards added a few details of anatomical structure, in his admirable address on Insectivorous Plants, delivered at the Belfast meeting of the British Association in 1874.

In a previous paper the relation between the illumination I on a panchromatic plate, the duration t of the exposure, the wave-length λ of the light, and the resulting density D of the deposit on the developed plate was investigated.

1. Experiments made at 15° C. with various adsorbents and gases (chiefly nitrogen and hydrogen) show that a cylinder filled with adsorbent granules has a capacity for dry gas under a given pressure which is generally greater than its capacity when containing no adsorbent. For example, a cylinder charged with nitrogen at 35 atmospheres has its capacity increased by 66 per cent. by filling it with cocoanut charcoal.

2. Sudden outbursts of firedamp in coal-mines are the result of releasing immense quantities of gas adsorbed under pressure in coal.

3. The logarithmic relation derived by Williams is shown to apply to the adsorption isotherms of gases above their critical temperature up to pressures of 100 atmospheres, providing a correction be applied for the gas in the capillaries which is not adsorbed, but which exists under simple compression.

In a previous communication the electrosynthesis of n-dotriacontanedi-carboxylic acid was described. This work has been continued, and three more acids of this series have been prepared, namely, eicosanedicarboxylic acid, HOOC. (CH2)20. COOH; tetracosanedicarboxylic acid, HOOC. (CH2)24. COOH; and octacosanedicarboxylic acid, HOOC. (CH2)28. COOH.

We are concerned with a Cauchy problem for the semilinear heat equation

then u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

This paper deals with the large-time behaviour of solutions to the exterior problem of the non-Newtonian filtration equation with first-order term and nonlinear boundary source. In particular, the critical global exponent and the critical Fujita exponent are determined or estimated. An interesting phenomenon is shown: there exists a threshold value for the coefficient of the first-order term such that the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this value.

We consider the bifurcation of positive solutions of the two-point boundary value problem

where λ> 0 is a real bifurcation parameter, and f ∊ C2 satisfies (fl) f(0) < 0, (f2) f′(s) > 0 for s > 0, (f3) f″(s) < 0 for s > 0 and (f4) limS→+∞f(s) = M where 0 < M ≦+∞. This problem has been studied by Casto and Shivaji under two additional hypotheses (f5) lims→+∞sf′(s) = 0, and (f6)f(θ)/θ < f′(θ), where θ is a positive number satisfying Assuming (fl)−(f6), Castro and Shivaji obtain some existence and nonexistence results and hence partial information on the bifurcation diagram, and they conjecture that this problem has at most two positive solutions. We prove this conjecture. Furthermore, we are able to generalise and improve their results under hypotheses (fl)−(f4). As a corollary, we show that there exists μ1 > 0 such that there exist no positive solutions for 0 <λ <μ1 and at most two positive solutions for μ1≦λ< + ∞, which improves a result of Brunovsky and Chow.

In any discussion of the zeros and turning points of the elliptic-cylinder function it is not necessary to define them more precisely than as functions of period π or 2π which satisfy the Mathieu equation

for the corresponding characteristic values of a. To simplify the following discussion, it is convenient merely to impose the condition that y(0)>0 if y(x) is even, and y′(0)>0 if y(x) is odd.

We prove uniform Morrey–Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted L2-estimates for the solution. We also prove a uniform L2-estimate without weight for the trace of the solution on the interface.

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

Let D be an open connected subset of the open unit ball in Ṟd, d ≧ 2. We give an estimate of the harmonic measure of ∂D∩{|x| = 1} with respect to D. This estimate depends in a simple way on the geometry of D. An essential tool is a rearrangement theorem for differential inequalities. When d = 2, examples are given which illustrate the precision of the results.

Theories regarding the chemistry of the contraction process in striped muscle have been revolutionised during the last year. The Hill-Meyerhof theory was based on the assumption that the contraction process was due to the liberation of lactic acid by the break-down of glycogen, a non-oxidative reaction, and that the recovery process consisted in the oxidation of part of the lactic acid and the resynthesis of glycogen from the remainder. Meyerhof now believes (1, p. 305) that contraction is produced by the break-down of creatin-phosphoric acid (phosphagen), and that the energy produced by carbohydrate break-down is utilised to resynthesise phosphagen.

Cyclic irrotational motion, [§ 60 (z)] once established through an aperture or apertures, in a movable solid immersed in a liquid, continues for ever after with circulation or circulations unchanged, [§60 (a)] however the solid be moved, or bent, and whatever influences experienced from other bodies. The solid, if rigid and left at rest, must clearly continue at rest relatively to the fluid surrounding it to an infinite distance, provided there be no other solid within an infinite distance from it. But if there be any other solid or solids at rest within any finite distance from the first, there will be mutual forces between them, which, if not balanced by proper application of force, will cause them to move.

We consider the set S(f) of λ-values for which there exists a function g ∊ L(t0, ∞) such that

has a solution not tending to 0; here f is a fixed function which is positive, non-decreasing and tends to ∞ with t. It is shown that if the jumps of logf(t) at its discontinuities are uniformly bounded, then S(f) is an additive group. This group is determined in some cases; some related groups are noted, which may coincide with S(f).

This paper studies a linked system of second order ordinary differential equations

where xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

We prove the existence of weak solutions for the strongly nonlinear parabolic problem

in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.