Nous voulons montrer que le théorème de Bernstein est une conséquence simple de la formule de Taylor et d' un théoréme de Helly. Rappelons une définition: une fonction réelle f définie sur (0, ∞) est dite complètement monotone si f est indéfiniment derivable, si f ainsi que toutes ses dérivées paires sont positive set si toutes les dérivées impaires sont négatives.

Constructions are made of a T1 space which does not have a T1 completion and of a quasi-uniform space which is complete, but not strongly complete. An example relating to a completion due to Popa is given. An alternate definition for Cauchy filter, called C-filter, is examined and a construction of a C-completion is given. We discuss quasi-pseudometrics over a Tikhonov semifield RΔ. Every topological space is quasi-pseudometrizable over a suitable RΔ. It is shown that if a quasi-pseudometric space over RΔ is complete, the corresponding quasi-uniform structure is C-complete. A general method for constructing compatible quasiuniform structures is given.

In this paper, we study the behaviour of singular isometries in orthogonal groups. These are isometries whose path is a singular subspace. We shall prove that the path of such a singular isometry is always even-dimensional. We shall use this result to show that the subgroup of the orthogonal group On(K, Q) which is generated by the singular isometries is the commutator subgroup Ωn(K, Q).

An asymptotic approximation is obtained, as k → ∞, for the integral

where Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), where

Hence S-1R is a topological ring for all S if and only if M ⊂ p*(R), where

Nous donnons, dans ce qui suit, la preuve de la proposition suivante.

We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

For a given list assignment $L\,=\,\{L\left( v \right)\,:\,v\,\in \,V\left( G \right)\}$, a graph $G\,=\,\left( V,\,E \right)$ is $L$-colorable if there exists a proper coloring $c$ of $G$ such that $c\left( v \right)\,\in \,L\left( v \right)$ for all $v\,\in \,V$. If $G$ is $L$-colorable for every list assignment $L$ having $\left| L\left( v \right) \right|\,\ge \,k$ for all $v\,\in \,V$, then $G$ is said to be $k$-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.

A recent publication entitled "Magic Squares and Cubes", by W. S. Andrews, published by the Dover Publications, New York, is an excellent authoritative book on magic squares, and magic cubes. In fact, chapter XIV entitled "Magic Octrahedroids", has shown examples of the extension into four-dimensional space. Andrews' method seems to be that of extending symmetrical considerations, which he calls reversions, in forming higher dimensional forms of magic squares. Neither he, nor any other author, to the best of my knowledge, has extended magic squares to higher than four dimensions.

La Hire must be given full credit for his method of breaking down magic squares into component squares, and conversely constructing magic squares from component squares. The approach of this author is to extend La Hire's method for n-dimensional space by means of a relation:

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

A theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

We construct the quantum $s$-tuple subfactors for an AFD $\text{I}{{\text{I}}_{1}}$ subfactor with finite index and depth, for an arbitrary natural number $s$. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

Let (𝓜,g) be a Riemannian manifold and let 1, 2, 3 be mutually orthogonal distributions on 𝓜 of dimensions p1, p2,P3 such that p1 + p2 + p3 = dim 𝓜. We assume that , and 1, ⊕ 2 are integrable and that all the geodesies of 𝓜 with initial tangent vector in 2 remain tangent to 2. Then, we prove that Pontk(2, ⊕ 3) = 0 for k > p2 + 2p3, where Pontk(2, ⊕ 3) is the subspace of the Pontrjagin algebra of 2 ⊕ 3 generated by forms of degree k.

Let $R$ be a ring with 1 and $P(R)$ the periodic radical of $R$. We obtain necessary and sufficient conditions for $P(\text{RG})=0$ when $\text{FG}$ is the group ring of an $\text{FC}$ group $G$ and $R$ is commutative. We also obtain a complete description of $P\left( I(X,R) \right)$ when $I(X,R)$ is the incidence algebra of a locally finite partially ordered set $X$ and $R$ is commutative.

We consider Dirichlet series R(s), constructed by taking scalar products of Hecke L-series with ray-class characters. Using a theorem of G. W. Mackey on tensor products of representations of finite groups we show that R(s) has a meromorphic continuation into Re(s) > 1/2 (obtained by more sophisticated methods in [l]-[5]); we then obtain estimates for the growth of R(s) on vertical lines. Via the Mellin transformation we deduce asymptotics for various weighted moment sums involving ideals of given ray-class and norm, in one or several fields simultaneously.