Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.

A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.

In this paper we consider the Sturm–Liouville operator d2/dx2 − 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L2(a, 0) and L2(0, b) of the space L2(a, b) = L2(a, 0) ⊕ L2(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L2(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition limx→0+(f′(x)−f′(−x)) = γf(0) with γ∈C+∪R or by the Dirichlet condition f(0+) = f(0−) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L2(a, b).

Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$. We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$. We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

It is well known that the Dirichlet problem for hyperbolic equations is a classical “not well posed” problem. Here we consider the Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for the hyperbolic equation uxy = 0 in all positions of the square and a class of rectangles. We also get a partial answer to the problem which deals with a ray that moves from any point on the boundary of a rectangle and is reflected on the boundary such that the angle between every ray and its reflection is π/2.

We derive the one-dimensional bending–torsion equilibrium model for the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin, rod-like bodies. By taking the limit as the thickness of the three-dimensional rods tends to zero, and by using ideas from the theory of Γ-convergence, we find that the resulting model consists of the union of the usual one-dimensional nonlinear bending–torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections; balance of contact forces and contact couples.

It is known that the condition ‘either ∂L (F) ≠ Ø or there exist υ1,…,υq ∈ Rnsuch thatF ∈ int co {υ1,…,υq} characterizes solvability of the problemwith f(·) = 〈F,·〉.

We extend this result to the case of lower semicontinuous integrands L : Rn → R.

We also show that validity of this condition for all F ∈ Rn is both a necessary and sufficient requirement for solvability of all minimization problems with sufficiently regular Ω and f. Moreover, the assumptions on Ω and f can be completely dropped if L has sufficiently fast growth at infinity.

We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.

Two races or physiological species (A and B) were found in Drosophila pseudo-obscura by Lancefield (1922), who described later (1925, 1929) the genetical behaviour of the interracial hybrids. The present writer (1931, 1932, a, b) has reported further investigations concerning the sterility of the male and the regional reduction of crossing-over in the female of the interracial hybrids. The results obtained suggested that there are genetic fertility factors, located in the sex chromosome complex, which are primarily responsible for the sterility of the F1 males. A similar conclusion has been reached more recently by Dobzhansky (1933 b) from his cytological studies of hybrid males.

In a former jjaper to this Society, we pointed out the important role played by carbonic acid in modern seas, with special reference to the vast deposits of carbonate of lime now taking place in coral reefs and those other calcareous deposits known as Globigerina and Pteropod Oozes. It was pointed out that carbonic acid was the chief agent in the disintegration of felspars and other silicates of the earth's surface, that it was concerned in all the changes that result in the secretion of carbonate of lime by marine organisms from any of the lime salts in sea-water, that a vast amount of Carbonic acid was being locked up in the calcareous deposits now in process of formation on the sea-bed, and that there was an accumulation of these calcareous deposits chiefly towards the equatorial regions of the ocean basins. In the present paper we propose to deal with the great antagonistic power to carbonic acid, viz., silica, and with the siliceous organic remains in the ocean.

The “Doctrine of Uniformity” in Geology, as held by many of the most eminent of British Geologists, assumes that the earth's surface and upper crust have been nearly as they areat present in temperature, and other physical qualities, during millions of millions of years. But the heat which we know, by observation, to be now conducted out of the earth yearly is so great, that if this action had been going on with any approach to uniformity for 20,000 million years, the amount of heat lost out of the earth would have been about as much as would heat, by 100° Cent., a quantity of ordinary surface rock of 100 times the earth's bulk.

A well-known theorem proved by Lazer and Solimini claims that the singular equation

has a periodic solution if and only if the mean value of the continuous external force is positive. In this paper, we show that this result cannot be extended to the case when h is an integrable function, unless additional assumptions are introduced. In addition, for each p ≥ 1 and h-integrable function in the pth power, we give a sharp condition guaranteeing the existence of periodic solutions to the above-mentioned equation, showing that there is a close relation between p and the order of the singularity λ.

We study the resonance set ∑ of pairs (α,β) ∊ ℝ2 for which the problem ∆u + αu+ − βu− = 0 has a nontrivial solution . We show that if λ0, is an eigenvalue of multiplicity two of −Δ, then has measure zero, where are the neighbouring eigenvalues of λ0. Moreover, we have that, if the operator Δ + αIu<0 + βIu < 0 has a kernel of dimension one for(α, β) ∊ ∑ and u ≠ 0 such that Δu + αu+ − βu− = 0, then (α, β) is an isolated point on ∑ ∩ L, where L is the straight line parallel to the diagonal of ℝ+ × ℝ+ through (α, β).

In this paper, a sufficient condition (H) is given on initial values for which there is a unique smooth global in time solution of the initial value problem for a special nonisentropic gas dynamics system.

We study decay estimates for the solutions to the initial value problem for a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. The method is integral estimation.

We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.

I discuss implications of the rank-one convexity condition in a particular case.

In this paper, we consider non-negative solutions of , We prove that if pq ≤ 1, every solution is global while if pq > 1, all solutions blow up in finite time. We also show that if p, q ≥ 1, then blow-up can occur only on the boundary.

An improperly posed problem is studied for a linear partial integro-differential equation of convolution type on the semi-axis. The problem originates from a generalised process of heat conduction in materials with fading memory, where the temperature of the material has to be determined for prescribed homogeneous boundary conditions and for a given final temperature distribution. By using eigenfunctions of the n-dimensional Laplacian, the problem is reduced to a family of equivalent initial-value problems for ordinary integro-differential equations; the latter are treated by the method of factorisation in a suitable function algebra. Sufficient conditions for the existence and uniqueness of solutions to the original problem are obtained in terms of the solvability conditions of the reduced problems. The whole analysis is performed simultaneously in a broad variety of spaces consisting of functions with an exponential growth rate (in the time variable) at infinity. One of the main advantages in the present approach is that solutions, if they exist, can always be computed explicitly.

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.