We study the C*-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C*-algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C*-algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C*-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C*-algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C*-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

The integral inequalities with which this paper is concerned are

and

where K(μ) and K1(μ) are positive numbers which depend on μ.

It is shown that each of these inequalities is valid only when μ = 0. Results relating to the corresponding inequalities on the extended interval (−∞, ∞) are also given.

Let R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.

In this paper we show that for the whole class of differential expressions in the limit-point case considered by Kauffman in [2], the perturbation theory yields a limit-point criterion for a much wider class of ordinary differential expressions. More general coefficients are admitted which may be eventually negative provided they are “dominated” by some other positive coefficients. This generalises results in [4], [5] and [6].

Given a non-negative integer n and a complete hereditary cotorsion triple , the notion of subcategories in an abelian category is introduced. It is proved that a virtually Gorenstein ring R is n-Gorenstein if and only if the subcategory of Gorenstein injective R-modules is with respect to the cotorsion triple , where stands for the subcategory of Gorenstein projectives. In the case when a subcategory of is closed under direct summands such that each object in admits a right -approximation, a Bazzoni characterization is given for to be . Finally, an Auslander–Reiten correspondence is established between the class of subcategories and that of certain subcategories of which are -coresolving covariantly finite and closed under direct summands.

The following formula

is given by Laplace (Theorie Analytique des Probabilités, 2 ed. Paris, 1814, p. 195) as an approximate value of Δn0i/ni, when n and i are very large numbers, and is applied immediately afterwards to the case where i is of the order n log n.

In a paper read to the Society in June last, “Proceedings,” June 1879, on the fungus disease affecting salmon and other fish, I discussed the various theories which were advocated, as to the cause of the disease; the effects of the fungus upon the fish, the vegetative and reproductive aspects of the fungus, and the belief that salt water had a curative effect upon salmon affected with fungus disease, on their reaching and remaining for some time in that element.

IN our communication to the Royal Society of Victoria of the 11th March, 1909 (1), describing our recent discovery of forty-two Tasmanian crania hitherto quite unknown to the world of science, we stated that “one of the earliest purposes to which it is proposed to utilise the present material is the determination of the relationship of the Tasmanian to the anthropoids and primitive man on the one hand, and to the Australian aboriginal on the other hand. Schwalbe's study of Pithecanthropus erectus (2) may serve as a basis for the former purpose, and Klaatsch's recent work (3) for the latter, though it must be remembered that innumerable authors have contributed to both subjects.” The present work is the fulfilment of the first part of this undertaking, namely, the determination of the relationship of the Tasmanian to the anthropoids and primitive man.

The portable dust-counting apparatus described in a previous communication to this Society was designed with a view of making observations on the air at situations where it would be inconvenient to work with the larger laboratory apparatus; and also to enable these observations to be made under conditions more favourable for avoiding local impurities than is possible when working in a house. As the construction of this portable apparatus was completed just as I was about to start for the Continent, the opportunity seemed a favourable one for continuing the investigation into the number of dust particles in the atmosphere, and extending our knowledge on the subject in other countries. The portable apparatus is reduced to such dimensions that it adds but little to one's personal luggage, and can be easily carried to’ the place where the observations are to be made.

The nephridia of Hirudo medicinalis, as is well known to biologists, are in pairs extending from the second to the eighteenth segments (somites). Each nephridium consists of a much convoluted cellular tube. The cells of the tube are perforated by small ducts. The nephridia (“segmental organs”) open externally on the ventral side of the body.

The composition of the ice produced in saline solutions, and more particularly in sea-water, has frequently been the object of investigation and of dispute. It might be thought that to a question of whether ice so formed does or does not contain salt, experiment would at once give a decisive answer. Yet, relying on experiment alone, competent authorities have given contradictory answers. All agree that ice, whether formed artificially in the laboratory by freezing sea-water, or found in nature as one of the varieties of sea-water ice, retains, in one form or another, and with great tenacity, some of the salt existing in solution in the water. The question at issue is whether this salt is to be attributed to the solid matter of the ice or to the liquor mechanically adhering to it, from which it is impossible to free it. Most bodies, and especially those which take a crystalline form, are easily purified and freed from all suspected foreign matter, with a view to analysis, by the simple operation of washing and drying. It is impossible to wash the crystals, formed by freezing a saline solution, with distilled water, because they melt at a temperature below that at which distilled water freezes.

The problem of defining the concept of spatial distance in a general riemannian space-time has been discussed by E. T. Whittaker, who showed how to translate into the terms of four-dimensional geometry the method adopted by astronomers in determining the distance of a star from the earth by means of a comparison of its absolute and apparent luminosities. The problem was further considered by the present writer, who derived a different formula for spatial distance by a method which was essentially one of partitioning the whole of space-time into space and time relative to the given observer. It was suggested that the new spatial distance might be that determined practically by parallax-measurements, though the evidence in support of this suggestion was perhaps hardly sufficient to carry conviction.

This note discusses primeness for semigroup rings of semigroups satisfying a certain condition, weaker than the u.p. property. These semigroups include free products of semigroups, semigroups presented by a single relator and at least three generators, and free inverse semigroups of infinite rank.

The position of this new species of Ganoid, under our commonly accepted classification, the author gave as follows:—

After referring to the Polyodon folium of Laépède (the P. reticulata of Shaw, the Planirostra spatula of Owen), the paddle-fish or spoon-bill sturgeon of the Ohio and Mississippi and their tributaries, as a well-known species of the genus in question, Dr Handyside went on to state that the new species now to be described was first observed on a Chinese fishmonger's stall at Woosung, 12 miles from Shanghai, and had since been found in the Yang-tsze-Kiang, and, as was alleged, in the northern Japanese sea. He then sketched the history of the Polyodontidoæ family, and narrated the researches of Lacépède, Von Martens, Blakiston, Kaup, and Duméril.

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delays

is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.

We describe products of arbitrary L-, R- and H-classes with the set of idempotents of full transformation semigroups.

Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

A system of partial differential equations modelling a van der Waals fluid or an elastic medium with nonmonotone pressure-density relation is studied. As the system changes type, regularisations are considered. The existence of one-dimensional periodic travelling waves, with prescribed average density in a certain range, average velocity and wavelength, is proved. They exhibit layer structure when the regularisation parameter is small. Similarities with the Cahn–Hilliard equation are explored.

Schaefer's fixed-point theorem is used to obtain sufficient conditions for the existence of a periodic solution of the non-linear differential equation f(D)x+BMg(D)x = p. The most significant feature of these conditions is a geometrical restriction on the range of the matrix M which is the same as the elliptic ball criterion encountered in stability theory. The extension of the results to delay-differential equations with constant time lags is also discussed.