James constructed a functorial homotopy decomposition for path-connected, p ointed CW-complexes X. We generalize this to a p-local functorial decomposition of ΣA, where A is any functorial retract of a looped co-H-space. This is used to construct Hopf invariants in a more general context. In addition, when A = ΩY is the loops space of a co-H-space, we show that the wedge summands of ΣΩY further functorially decompose by using an action of an appropriate symmetric group. As a valuable example, we give an application to the theory of quasi-symmetric functions.

A mixed-order system of singular partial differential equations occurring in the study of rotating stars is considered. It is shown that the associated operator L0 is closable and that the essential spectrum of its closure L coincides with the essential spectrum of a bounded operator. Finally, some parts of the essential spectrum of L are determined explicitly.

A real polynomial $P$ of degree $n$ in one real variable is hyperbolic if its roots are all real. A real-valued function $P$ is called a hyperbolic polynomial-like function of degree $n$ if it has $n$ real zeros and $P^{(n)}$ vanishes nowhere. Denote by $x_k^{(i)}$ the roots of $P^{(i)}$, $k=1,\dots,n-i$, $i=0,\dots,n-1$. Then, in the absence of any equality of the form

\begin{equation} x_i^{(j)}=x_k^{(l)} \tag{*} \label{*} \end{equation}

for all $i<j$ we have

\begin{equation} x_k^{(i)}<x_k^{(j)}<x_{k+j-i}^{(i)} \tag{**} \label{**} \end{equation}

(the Rolle theorem). For $n\geq4$ (respectively, for $n\geq5$) not all arrangements without equalities \eqref{*} of $\tfrac12n(n+1)$ real numbers $x_k^{(i)}$ and compatible with \eqref{**} are realizable by the roots of hyperbolic polynomials (respectively, of hyperbolic polynomial-like functions) of degree $n$ and of their derivatives. For $n=5$ and when

$$ x_1^{(1)}<x_1^{(3)}<x_2^{(1)}<x_3^{(1)}< x_2^{(3)}<x_4^{(1)} $$

we show that all such 102 arrangements are realizable by hyperbolic polynomial-like functions (of which 66 are obtained by hyperbolic polynomials and another 8 by perturbations of such).

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time-dependent heat source and time-dependent boundary conditions.

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem

on a general bounded domain Ω in ℝN, with Navier boundary condition u = Δu on ∂Ω. Firstly, we prove the extremal solution is smooth for any p > 1 and N ⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst. A 34 (2014), 2561–2580). Secondly, if p = 3, N = 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case Ω = 𝔹, which completes the result of Dávila et al. (Math. Annalen348 (2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u = –l/up in ℝN with u > 0.

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

We study the inhomogeneous semilinear parabolic equation

with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—

Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

In August 1901 Mr Thomas Anderson, merchant, Hillswick, to whom and to other members of whose family I have on several occasions been indebted for specimens to illustrate the Zoology of Shetland, wrote to tell me that a dead whale had been found floating near Hillswick and had been towed into Roeness Voe. It was within three miles of land when seen by the fishermen and was claimed by the Crown. From the appearance of the animal Mr Anderson thought that it was a sperm whale, and he sent me the following description, from which it was clear that he had correctly recognised it.

Monthly observations, extending over one year, were made on the rectal temperature of 114 domestic fowls (Gallus gallus, ♀) and records from forty-one of these were obtained for two years. Six different breeds were used, each located in a separate pen, all under similar conditions, and the mean temperatures for each group were plotted out to form an annual temperature curve. It was found that—

1. The lowest temperatures occur in December, January, and February, and the highest in June, July, and August, corresponding in a general way with the temperature of the external air.

2. Barometric pressure does not appear to have any influence on the body temperature of the hen.

3. The curve of egg-production does not coincide with the annual temperature curve, the former reaching its highest level in April and May, the latter in June, July, and August.

If we compare the mean rectal temperature at two periods of the year when the external or weather conditions are approximately the same (April-May and September-October), but when the vitality of the birds, as indicated by the curve of egg-production, moulting, etc., is at a maximum and minimum respectively, we find that the figures are practically identical. This would seem to show that cyclical bodily changes have little effect on body temperature as compared with outside influences.

Of the more recent researches on the subject of this paper, mention may be made of the work of H. Zahn* on the galvanomagnetic and thermomagnetic effects in various metals. Zahn has measured these effects in many different metals, and has used his results to test the electron theory of the properties of metals as developed by P. Drude. He has also determined in some cases the temperature variation of the effects.

The author of the present paper has confined his attention to the thermomagnetic transverse effects and the Hall effect. These have been measured in magnetic fields of various strengths and at temperatures varying over a range of about 100 Centigrade degrees.

The experiments were carried out with a view to obtaining some light on the electron theory, and the ratios of the effects are discussed in relation to this theory.

Professor Allman described a new genus of Polyzoa, obtained by the Rev. A. M. Norman and Mr J. Gwyn Jeffreys, from deep-sea dredgings in Shetland.

Its cœnœcium consists of a branched tube, partly adherent and partly free, the free portion forming tubes of egress, through which the polypides move in the acts of exsertion and retraction. In the walls of the adherent portion a rigid chitinous rod is developed along their attached side, and to this rod the polypides are connected from distance to distance each by a flexible cord or funiculus.

Among many geological questions which wait solution, there is probably none more interesting or perplexing than the agency by which Boulders or “blocs erratiques,” as the French term them, have come to their present sites. I allude, of course, not to blocks lying at the foot of some mountain crag from which they have fallen by the decay or weathering of the overhanging rocks, but to blocks which have manifestly been transported great distances, after being detached from the rocks of which they originally formed part.

I propose making a few brief observations on the relations in which we stand to the aborigines of the Andaman Islands, their place in the economy of this convict settlement, and their more noticeable habits and customs. In the consideration of the latter, I shall only state such information as I have obtained myself from conversation with, and personal observation on the manners of the natives. Detailed monographs on their ethnology, &c., have been written already by several observers, and are to be found in the “Transactions of the Anthropological Society,” &c.

Since the discovery by Heusler in 1903 of a magnetic alloy of copper, manganese, and aluminium, considerable interest has been aroused by the peculiarities of its properties and the difficulty of their explanation. In the summer of 1905 a preliminary investigation of a sample of the alloy was made in the Physical Laboratory of Glasgow University, and the results obtained have been published in a paper by Professor A. Gray.

The genus Branchiomaldane, represented by a single species vincenti, was founded by Langerhans to contain certain small Polychætes, which he discovered living in sand-covered tubes, among algæ, on the northern shore of Teneriffe. Professor Mesnil has since found specimens, inhabiting mucous tubes situated on the lower side of the calcareous alga Lithothamnion, in rock-pools at St Martin, near Cape la Hague, and has given an account of some of the structural features of these worms, which he showed to be adult and hermaphrodite. Professor Fauvel considered B. vincenti to be a dwarf Arenicola, arrested in development; he therefore merged Branchiomaldane with Arenicola, and designated the worm Arenicola vincenti. The writer has investigated the anatomy of several specimens of this worm, kindly sent to him by Professor Mesnil, and has concluded that the genus Branchiomaldane should be retained; the original name of the worm—B. vincenti—is therefore employed in the following account.

Several additional members of the series of normal saturated dicarboxylic acids have been investigated by X-rays, and the results are in agreement with the conclusions arrived at in an earlier paper. In the series of mono- and di-alkyl malonic acids, successive reflection planes are separated by the length of one molecule. The series of fatty acid amides closely resembles that of the fatty acids themselves. The cleavage planes are separated by the length of two molecules oriented in opposite directions.

My thanks are due to Mr N. K. Adam for the specimens of mono- and dioctadecyl malonic acid, and to Sir James Walker for his advice and helpful criticism.

Numerous authors have been concerned with phenomena and problems of senescence, duration of life, order of dying-out, and related subjects. With the exception of studies on Drosophila (Pearl and Parker, 1921; Pearl, 1922), other insects (for literature see Korschelt, 1924) and, in some respects, human subjects (see actuarial literature), most of the research of previous authors was embarrassed by the lack of sufficiently large and sufficiently homogeneous material; many conclusions rest on an insecure basis, since they refer only to isolated specimens or to experiments carried out on small numbers of animals.

Mr Clerk Maxwell ingeniously suggests the extension of the common theory of the foci of the conic sections to curves of a higher degree of complication in the following manner:—

(1.) As in the ellipse and hyperbola, any point in the curve has the sum or difference of two lines drawn from two points or foci = a constant quantity, so the author infers, that curves to a certain degree analogous, may be described and determined by the condition that the simple distance from one focus plus a multiple distance from the other, may be = a constant quantity; or more generally, m times the one distance + n times the other = constant.