The object of this note is to derive a form of Poisson's equation from general relativistic mechanics, without assuming the field to be either static or “weak”. The problem is essentially a “local” problem, all observations being made by one observer; this observer determines the apparent gravitational field in his vicinity by observing the motions of free (isolated) particles. Defining gravitational mass by means of Poisson's equation, we find the relation between the densities of gravitational and inertial mass relative to any observer. We also find what may be called the non-rotating frame of reference belonging to any observer.

This paper is a continuation of (4). The main aim of this paper is the introduction of the concept of sex-linked duplication. In addition, we shall give several equivalent definitions for the concept of a genetic algebra and make several remarks on overlapping of generations.

Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

The boundary value problem

is studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.

This boundary value problem arises in the search of positive radially symmetric solutions to

where Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

The notion of Arens regularity of a bilinear form on a Banach space E is extended to continuous m-linear forms, in such a way that the natural associated linear mappings, E→L (m−1E) and (m – l)-linear mappings E × … × E → E', are all weakly compact. Among other applications, polynomials whose first derivative is weakly compact are characterized.

The classical Hamburger moment problem can be formulated as follows: Given a sequence {cn:n=0,1,2,…} of real numbers, find necessary and sufficient conditions for the existence of a distribution function ψ (i.e. a bounded, real-valued, non-decreasing function) on (– ∞,∞) with infinitely many points of increase, such that , n = 0,1,2, … This problem was posed and solved by Hamburger [5] in 1921. The corresponding problem for functions ψ on the interval [0,∞) had already been treated by Stieltjes [15] in 1894. The characterizations were in terms of positivity of Hankel determinants associated with the sequence {cn}, and the original proofs rested on the theory of continued fractions. Much work has since been done on questions connected with these problems, using orthogonal functions and extension of positive definite functionals associated with the sequence. Accounts of the classical moment problems with later developments can be found in [1,4,14]. Good modern accounts of the theory of orthogonal polynomials can be found in [2,3].

Consider the Darboux problem

where φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:

(A1) f(.,.,z) is measurable for every z ∈ Rd;

(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;

(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

In this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

We consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.

AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber–Pink conjecture for curves in a mixed Shimura variety of dimension 4, as well as the study of polynomial Pell equations with non-separable discriminants.

In this paper we generalize techniques used by Klyachko and the authors to prove some tessellation results about S2. These results are applied to prove the solvability of certain equations with torsion-free coefficients.

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.

The conditions under which the solution of the non-local thermistor problem

\begin{gather*} u_t=\Delta u+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^{2}},\quad x\in\varOmega\subset\mathbb{R}^N,\ N\geq2,\ t>0, \\ \frac{\partial u(x,t)}{\partial\nu}+\beta(x)u(x,t)=0,\quad x\in\partial\varOmega,\ t>0, \\ u(x,0)=u_0(x),\quad x\in\varOmega, \end{gather*}

blows up are investigated. We assume that $f(s)$ is a decreasing function and that it is integrable in $(0,\infty)$. Considering a suitable functional we prove that for all $\lambda\gt0$ the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that $\lambda$ is sufficiently large $(\lambda\gg 1)$. In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either $\lambda\gt\lambda^*$ or $0\lt\lambda\lt\lambda^*$, for some critical value $\lambda^*$, and that the initial condition is sufficiently large provided also that $f(s)$ satisfies the decay condition $\int_0^\infty[sf(s)-s^2f'(s)]\,\mathrm{d} s\lt\infty$.

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.

AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theory and a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg), we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study a quasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinct solutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25

The integral equations we consider are

and

where 0 < α < 1. The asymptotic behaviour of the eigen-values of the latter equation is already known (see (1) and (4)). The former equation has been studied by many authors but as yet no explicit statement seems to have been made about the behaviour of its eigen-values.

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentation

the subgroup PSL(2, ℤ) being generated by UV and VW.

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.

We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

An analysis is made of the motion of a simple undamped pendulum which performs small oscillations in one plane while its string is raised or lowered through the point of support at a constant rate. The behaviour of the system and the variation of its angular and linear amplitudes are expressible in terms of Bessel functions of orders 0, 1 and 2.