• Volume 90, Issue 3

July 1981,   pages  167-286

• EOL and ETOL array languages

In this paper we study the families of ETOL and EOL array languages. Standard forms for ETOL and EOL array systems are defined and closure properties of the families are studied. Relations of these families with other developmental array languages and other array languages are studied.

• On the relation of generalized Valiron summability to Cesàro summability

A family (Vak) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (Bα,γ), (Eρ, (Tα), (Sβ) and (Va) are members of this family. In §3 some properties of the (Bα,γ) and (Vak) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (Vak) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If sn (n ≥ 0) isa real sequence satisfying$$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$, and if sns (Vak) thensn → s (C, 2ρ).

• On the mean square value of Hurwitz zeta function

R Balasubramanian has shown that$$\mathop \smallint \limits_1^{\rm T} |\zeta (\tfrac{1}{2} + it)|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by$$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\| {2a} \right\|}}} \right)$$ for 0 &lt;a &lt; 1.

• On a generalization of the class of functions with bounded Mocanu variation

The object of this paper is to generalise the well-known class of functions analytic in the unit disc having bounded Mocanu variation. Certain properties of this more general class are investigated using convolution techniques.

• Analyse asymptotique des équations de Transport dans le cas d’évolution

Nous démontrons la convergence de la solution d’une équation de Transport vers la solution d’une équation de Diffusion quand Je libre parcours moyen tend vers zéro, pour des équations d’évolution et un domaine borné dans deux cas: flux incident nul et réflexion spéculaire.

• Application of Newton’s method to a homogenization problem

The homogenization of a family (Pε) of uniformly elliptic semilinear partial differential equations of second order is studied. The main result is that any non-singular solutionu of the homogenized problem (P) is the limit of non-singular solutions of (Pε). The method consists of specifying a functionwε starting from which the Newton iterates converge to a solutionuε ofPε. These solutionsuε converge to the given solutionu of (P).

• Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λε = ε−2 λ + λ0 +O (ε), Stekloff: λε = ελ1 +O2), Neumann: λε = λ0 + ελ1 +O2).

Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.

• A note on the mean value of L-series

Using Hilbert’s inequality, we give a new asymptotic formula (uniform inq andT) for$$\mathop \Sigma \limits_{\begin{array}{*{20}c} {\chi (mod q)} \hfill \\ {\chi primitive} \hfill \\ \end{array} } \smallint _T^{2T} |L(\tfrac{1}{2} + it,\chi )^4 |dt$$