Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups.
J. L.
Alperin;
Richard
Brauer;
Daniel
Gorenstein
1-261
Abstract: The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups ${L_3}(q)$ with $q \equiv - 1 \pmod 4,{U_3}(q)$ with $q \equiv 1 \pmod 4$, or ${M_{11}}$. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of $ {Z_{{2^n}}}$ and ${Z_2},n \geqq 2$.
Strong renewal theorems with infinite mean
K. Bruce
Erickson
263-291
Abstract: Let F be a nonarithmetic probability distribution on $(0,\infty )$ and suppose $1 - F(t)$ is regularly varying at $\infty$ with exponent $\alpha ,0 < \alpha \leqq 1$. Let $U(t) = \Sigma {F^{{n^ \ast }}}(t)$ be the renewal function. In this paper we first derive various asymptotic expressions for the quantity $U(t + h) - U(t)$ as $t \to \infty ,h > 0$ fixed. Next we derive asymptotic relations for the convolution ${U^ \ast }z(t),t \to \infty $, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function $m(t) = \smallint _0^t[1 - F(x)]dx$, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case $ \alpha = 1$ to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.
Meromorphic functions of elements of a commutative Banach algebra
Barnett W.
Glickfeld
293-307
Quadratic functionals of second order
Walter
Leighton
309-322
Abstract: In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions $y(x)$ with fixed endpoint conditions on $y(x)$ and its derivative at $x = a$ and at $x = b$. Necessary conditions and sufficient conditions are obtained. These lead, in turn, to natural extensions of the Sturm comparison theorem to fourth-order selfadjoint equations. These extensions include and are more general than previously stated comparison theorems. Finally, it is found that the present variational theory provides an orderly approach to second-order Wirtinger-like inequalities, and the results include as special cases many published results of this type.
Dual spaces of weighted spaces
W. H.
Summers
323-333
Abstract: The topological duals of a large class of weighted spaces of continuous functions are characterized as spaces of Radon measures which can be factored into a product of a weight function and a bounded Radon measure. We next obtain a representation for a base for the equicontinuous subsets of these dual spaces and for the extremal points of the members of this base. Finally, among other applications, these ideas make possible an extension of the representation theorem for biequicontinuous completed tensor products of weighted spaces obtained by the author in an earlier paper.
Global dimension of orders
Richard B.
Tarsy
335-340
Abstract: We prove that the finitistic global dimension (fGD) of an order in a quaternion algebra over the quotient field of a Dedekind domain is one. Examples are given of orders of global dimension $n - 1$ in $n \times n$ matrices over the quotient field of a discrete valuation ring.
Spaces of countable and point-countable type
J. E.
Vaughan
341-351
Abstract: These spaces were introduced by M. Henriksen and J. R. Isbell, and A. V. Arhangel'skiĭ, who proved results about the placement of such spaces in their compactifications. In the present paper, these results are consolidated using new terminology. In addition, further results concerning the heredity of these spaces are obtained. Generalizations of these spaces are introduced, and an analogous treatment is given for them. Finally, some examples are given of which one gives a solution to a problem raised by Arhangel'skiĭ by showing that the perfect image of a first countable space need not be of point-countable type.