By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C*-algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroupℑℐt.

In a previous communication (Proc. Roy. Soc. Edin., vol. lii, 1932, p. 337) a visual method of measuring the diffusion coefficients of Br2 : H2, Br2 : N2, Br2 : O2, and Br2 : CO2 was described. The investigation has been extended to include similar measurements for bromine diffusing into argon, nitrous oxide, methane, and hydrogen chloride. Preliminary experiments in which a photoelectric cell replaces the human eye have also been made and confirmatory results have been obtained.

Depuis Murray et Eenard deux travaux importants sont venus enrichir la bibliographie de la Glauconie: le mémoire de MM. Calderon et Cliaves, “Contribuciones al estudio de la Glauconita,” et l'œuvre magistrale de M. L. Cayeux, Etude micrographique des terrains sédimentaires.

Murray et Eenard avaient étudié spécialement la Glauconie actuelle; MM. Calderon et Cliaves baserent leur synthese de la Glauconie actuelle sur une analyse de Glauconie des roches sédimentaires; M. L. Cayeux limita strictement ses belles recherches à la Glauconie des roches sédimentaires et n'attaquapas le problème de sa genèse.

When a Mach number dependent parameter σ ≠ 0 is accounted forin an expansive free interaction analogous to that of Stewartson [7], the solution terminates in a singularity with a finite pressure, but with a pressure gradient varying as the inverse three-fifths power of the distance from the singularity, apart from a subtle logarithmic modification.

Levels in the ground state bands of 158Dy, 160Dy and 162Dy have been populated by (α, 2n) reactions on metallic targets of separated 156Gd, 158Gd and 160Gd isotopes. Two Ge(Li) detectors were used to study singles gamma-ray spectra, gamma-ray angular distributions, gamma-gamma coincidence spectra and relative yields of gamma-rays for bombarding energies from 20 to 27 MeV. Transitions from all levels in the ground state bands up to the 12+ member were identified and the following level energies in keV established: 158Dy: 0 (0+), 99·0 (2+), 317·4 (4+), 637·9 (6+), 1044·1 (8+), 1520·1 (10+) and 2049·4 (12+); 160Dy: 0 (0+), 86·7 (2+), 283·7 (4+), 581·4 (6+), 967·4 (8+), 1429·0 (10+), 1951·7 (12+); 162Dy: 0 (0+), 80·7 (2+), 265·7 (4+), 548·5 (6+), 921·1 (8+), 1374·8 (10+) and 1901·0 (12+). There is weak evidence for the 14+ levels in 158Dy and 160Dy. The level energies are compared with calculated values using the variable moment of inertia model.

Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).

When a rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity field and the pressure field of the fluid motion and the angular velocity of the rigid body.

In this paper we obtain existence and uniqueness results for the steady state problem in which a rigid body rotates about an axis of symmetry in a viscous incompressible fluid.

This paper considers existence of periodic solutions for vector Liénard differential equations

In our main result we write

where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation

the assumptions are weakened.

Our approach is based on Leray-Schauder's degree theory and a priori estimates.

We consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

Several criteria are given for some or all polynomials in an expression M(y) = –(py′)′+qy which is non-oscillatory on (0, ∞) to be limit-point. One of these states that if M is non-oscillatory and , then every polynomial in M is limit-point.

In this paper the author first sketches the history of Rhizodus, from its discovery by Ure of Rutherglen to the most recent papers on the subject.

Placed by Agassiz among the “Coelacanthi” (i.e., cycliferous Crossopterygia of modern nomenclature) it was classified by Professor Huxley in the cycliferous division of his family Glyptodipterini, along with Holoptychius, Glyptolepis, Dendrodus. The discovery by the author, in 1875, of its subacutely lobate pectoral fin revealed the fact that it was much more closely allied to Rhizodopsis than to Holoptychius, and that it ought, along with the former genus, to be classed in a family (Cyclodipterini) distinct from the acutely lobate Holoptychiidae. The author does not consider the identity of Leidy's genus Apepodus with Rhizodus as proved.

In a paper read in this Society on the 3rd of March last, “On the Law of Inertia,” &c., I had occasion to adduce for consideration a problem to the following effect:—

The aim of this paper is to study the asymptotic behaviour of the solutions of the linearized elasticity system, posed on thin reticulated structures involving several small parameters. We show that this behaviour depends on the relative size of the parameters. In each case, we obtain a limit system where the microstructure and macrostructure appear simultaneously. From it, we get a suitable approximation in L2 of the displacements and the linearized strain tensor.

We give a new characterization, in the equicharacteristic case, of Teter rings by using Macaulay inverse systems. We extend the previous characterizations due to Teter, to Huneke and Vraciu and to Ananthnarayan et al., to any characteristic of the ground field and remove the hypothesis on the socle ideal. We construct and describe the variety parametrizing Teter covers and we show how to check if an Artin ring is Teter. If this is the case, we show how to compute a Teter cover.

We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).

We study Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3. The close connections of these algebras with Lie algebras graded over the non-reduced root system BC1, with J-ternary algebras and with Freudenthal–Kantor triple systems are explored.

We consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.