We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm {RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm {RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

This paper is focused on spreading dynamics for a discrete Nicholson's blowflies model with time convolution kernel. This problem arises in the invasive activity of blowflies scattered in discrete spatial environment and has distributed maturated age. We found that for a general convolution kernel, the model can exhibit travelling wave phenomena in a discrete spatial habitat. In particular, we determine the minimal wave speed of travelling waves by deriving the non-existence of travelling waves, and we demonstrate that the minimal wave speed can determine the long time behaviour of solutions with compact initial function. Moreover, we prove that all travelling waves are strictly increasing, which implies that the waveforms remain monotone in the propagation process. Some numerical simulations are also presented to confirm the analytical results.

In this paper, we study the quadratic perturbations of a one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. By the criterion function for determining the lowest upper bound of the number of zeros of Abelian integrals, we obtain that the cyclicity of either period annulus is two. To the best of our knowledge, this is the first result for the cyclicity of period annulus of the one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. Moreover, the simultaneous bifurcation and distribution of limit cycles from two-period annuli are considered.

We determine all three-dimensional homogeneous and $1$-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.

In a paper, entitled “On the Action of Carbonic Acid Water on Minerals and Kocks,” read before the Edinburgh Geological Society on the 18th February 1886, and subsequently published in their Transactions for that year (vol. v. part ii. p. 282), I gave some account of the simple experiments made with a view to elucidate the action of carbonated water on various minerals, which up to that period I had been able to carry through.

Obs. Diagnoses specierum novarum ampliores et observationes variæ, enumeratio completa omnium Lichenum a participibus expeditionum oculatiss. Balfourio (B.C.S.) et Schweinfurthio (Schweinf.) in hac neglectissima insula lectorum mihique benevole submissorum, nec non indicatio locorum accurata in Balfourii opere integro currente anno edendæ sunt.

The methods and the apparatus were the same as in the two previous articles of the series and hence need not be described, the only change being in connection with the galvanometer. Previously it had rested on the table, but when half the results in the infra-red recorded in this paper were obtained, the arrangement was changed; it was suspended by three iron wires each about one and a half metres long from a bracket in the wall and hung with its three levelling screws clearing the table by about one centimetre. Between the table and levelling screws were placed loose wads of cotton wool for the purpose of damping any vibrations that might arise. With this arrangement the zero is very much less sensitive to vibration. With the lamp and scale at one and a half metres—the distance at which they are ordinarily used now–vibrations of the laboratory rarely cause the zero to move mm. either way.

We may write the electro-magnetic equations of Clerk-Maxwell as

For plane waves, running with normal velocity vα= −μ−1, we have

whence at once

[For the moment, we assume that φ and ψ are self-conjugate, so that a linear function of them is also self-conjugate. And we employ the method sketched in Tait's Quaternions, §§ 438–9.]

The various factors which are supposed to influence the composition of milk have already been the subject of numerous investigations. The general result of these has been to show that within very wide limits the composition of milk is very little affected by the nature of the food supplied. As regards the mineral constituents the results of some of the investigations are rather conflicting, and while there is a general consensus of opinion that the composition and amount of the mineral constituents are independent of the food-supply, certain investigators claim to have been able to increase both the calcium and phosphoric acid in the milk by slight amounts.

The increased interest taken in Orthogonants during the period 1880–1900 is more than maintained in the period which we have now reached. Out of considerably more than 100 writings sought out and examined, as many as 65 have been found to deal directly with the theory of orthogonal substitution, and 35 to be more closely concerned with the practical applications of it. These numbers, 65 and 35, are what have to be set over against 38 and 12 in the earlier period referred to, note, however, at the same time being taken that the contrast is unfairly heightened by several papers of the first period, chiefly Magyar, having been received only in time to be joined with those of the second. These Magyar papers, it may be added, make also an appreciable contribution in the matter of quality.

By the kindness of M. de Boisbaudran I have been enabled to experiment upon a specimen of nickel, very nearly pure. Its thermo-electric relations are exceedingly interesting, and are easily observed by employing palladium as the second metal in the circuit. The nickel line in the thermo-electric diagram presents nearly the same appearance as that of iron, but its peculiarities occur at much lower temperatures.

§ 1. In my previous papers on this subject I used the relationship terms, not in a representative but in a class sense; for instance, cA was employed to denote the children of the man A, and U was employed to denote the total assemblage of mankind, or a limited portion of that assemblage. I have found it useful for the purpose on hand to analyse these symbols into their component elements. Let U denote a man representatively, that is, any man, then mankind is appropriately represented by ΣU, where Σ has its ordinary mathematical meaning of taking the sum. Also UA is the appropriate mathematical expression for the man who has the name A, and A standing by itself is to be regarded as a contraction for UA. This notation is useful where, as in the present case, the universe of the investigation is composed of individuals; but since the universe may be continuous in its nature, by taking U to denote the whole, a more general basis is given to the Algebra of Logic, and accordingly I adopted that notation in my work on the subject.

Most of the work which has been done directly on the cathode fall of potential refers to discharges where the potential difference applied was less than 1000 volts. In view of the large amount of work which has been done with discharge tubes worked from an induction coil at voltages of the order of 30,000, it seems of interest to consider how far the results obtained at the lower voltages can be applied to this case.

The mouse placenta implanted into immature female mice evokes pregnancy conditions in the vaginal epithelium, growth of the follicles in the ovary, and enlargement of the interstitial tissue. It is demonstrated, therefore, that mouse placenta contains an active kyogenic substance.

L'existence des Macruridæ à l'intérieur du Cercle Polaire Arctique est connue depuis longtemps.

C'est Sven Lovén qui en rapporta le premier exemplaire (Macrurus berglax), de Hammerfest (Finmark), lors de la première Expédition suédoise au Spitzberg, sur l'Enigheten, en 1837.

The Deputation appointed by the Council to represent the Society on the occasion of the 400th anniversary of the founding of the University of Upsala consisted of Mr Sprague and myself. Professor Balfour and Professor Sir Wyville Thomson, who represented the Edinburgh University, also joined the Deputation.

The transformations of volume integrals into surface integrals, and of surface integrals into line integrals, are usually established first for a small parallelepiped and parallelogram. These elements are not, however, very convenient, or indeed altogether satisfactory, for the extension of the theorems over finite domains. If arranged in straight rows and plates, the elements cannot be fitted to a general boundary; and if they are arranged in curved rows and shells like the stones of a vault, there will be, in general, between various elements of the first order, spaces of the second not represented. This inconvenience disappears if tetrahedra and triangles are taken as elements. In Quaternions, the theorems are easily proved for such elements.

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.