D. Hawkins [1] propose un modèle stochastique simple de la distribution des nombres premiers. II construit un crible aléatoire analogue à celui d' Eratosthène. La règle pour obtenir une suite de nombres premiers aléatoires est la suivante: 2 est un nombre premier aléatoire, que nous désignerons par P1; chaque entier plus grand que P1 est ensuite criblé avec probabilité 1/P1 d' être éliminé par le crible; si P2 est le premier entier plus grand que P1 qui n' a pas été éliminé par le crible, on crible chaque nombre plus grand que p2, qui n' a pas été éliminé par le crible à la première opération, avec probabilité 1/P2 d'être élimine; à la niéme étape, si Pn est le premier nombre plus grand que Pn, qui n' a pas été éliminé par les n - i premières opérations du crible, on crible chaque nombre plus grand que Pn, qui n' a pas été éliminé par les n - 1 premières opérations du crible, avec probabilité l / Pn d' être éliminé. On obtient ainsi une suite { Pn} n = 1, 2, 3, … de nombres premiers aléatoires.

It is a well-known fact that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G$. We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraϊssé theory and Ramsey theory. This work generalizes some of the known results about countable structures

We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.

Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.

In this work we consider the equation

1

where K(x, y) is singular in the sense that it does not properly belong to L2 and f(x) is an arbitrary L2 function.

A Lebesgue measurable function K(x, y) of two variables, having real values on [0.1] × [0.1] is called a singular normal kernel of

1. (i)

   There exists approximating kernels Km(x, y) satisfying

2. (ii)

3. (iii)

4. (iv)

A function f(x) is said to be completely monotonic on (0, ∞) if

1

Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].

Let a1 < a2 < … be any sequence of integers. Assume that the infinite sequence of numbers un satisfies the following condition: To every ɛ > 0 there is an no = no (ɛ) such that for all n > no and all k

1

Obreanu asked (Problem P. 35 Can. Math. Bull.) under what conditions on the sequence a1 < a2 < … does (1) imply that the sequence u is convergent. N. G. de Bruijn and P. Erdos proved that a necessary and sufficient condition for (1) to imply the convergence of un is that the sequence {an} be infinite and that the greatest common divisor of the a1 should be 1.

This paper deals with the convergence of iterations of the equation

1.1

with u0 ∊ Rn, both as a root and a point of repulsion. Here u and ϕ are real n-vectors and ϕ is continuously differentiate with respect to u. By taking u0 as an initial approximation to u0 in a neighbourhood of u0 we define the sequence of iterates {uk+1} by

1.1

For the sake of convenience we shall, sometimes, call the equation (1.1) an iteration.

It is well known [1] that the variational problem of minimizing

1

where F is positive homogeneous of degree one in (henceforth abbreviated to "plus-one" in ) leads to a Hamiltonian H{xi, pi} and corresponding Hamilton-Jacobi equation

2

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions of) the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\unicode[STIX]{x1D70B}$.

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

Consider three externally tangent circles A, B, C, with radii a, b, c, respectively. Let D be the circle which is externally tangent to all three and enclosed by them. Pack circles so that Z1 = D and Zi is externally tangent to A, B, Zi-1.

The inversion and the characterization of the convolution transform is derived via the concept of unimodality introduced by Khintchine (1938). This method yields simple and intuitively appealing proofs.

The kernels of the transforms in the class that we shall treat satisfy

1.1

where Re ak = ak Re ck = ck, 0 ≤ ak/ck and Σa-2k (see also [1], [2], [3], [6], [7] and [8]).

In this paper we obtain the bv0-norms for several well-known classes of matrix operators.

We study the semilinear Schrödinger equation

$$\left\{ _{u\,\,\in \,\,{{H}^{1}}({{\mathbf{R}}^{N}}),}^{-\Delta \,u+V(x)u=f(x,u),\,\,\,\,\,x\in \,\,{{\mathbf{R}}^{N}},} \right.$$

where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)={{V}_{0}}(x)+{{V}_{1}}(x)$, ${{V}_{0}}\in C({{\mathbf{R}}^{N}}),\,{{V}_{0}}(x)$ is 1-periodic in each of ${{x}_{1}},{{x}_{2}},...,{{x}_{N}}$, $\sup [\sigma (-\Delta +{{V}_{0}})\,\cap \,(-\infty ,0)]<0<$$\inf [\sigma (-\Delta +{{V}_{0}})\cap (0,\infty )],\,{{V}_{1}}\in C({{\mathbf{R}}^{N}})$, and ${{\lim }_{|x|\to \infty }}\,{{V}_{1}}(x)=0$. A new super-quadratic condition is obtained that is weaker than some well-known results.

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral.

The article studies the class of abelian groups G such that in every direct sum decomposition G = A ⊕ B, A is 5-projective. Such groups are called pds groups and they properly include the quasi-projective groups.

The pds torsion groups are fully determined.

The torsion-free case depends on a lemma that establishes freedom in the non-indecomposable case for several classes of groups. There is evidence suggesting freedom in the general reduced torsion-free case but this is not established and prompts a logical discussion. It is shown, for example, that pds torsion-free groups must be Whitehead if they are not indecomposable, but that there exists Whitehead groups that are not pds if there exist non-free Whitehead groups.

The mixed case is characterized and examples are given.