Symmetrized separable convex programming
L.
McLinden
1-44
Abstract: The duality model for convex programming studied recently by E. L. Peterson is analyzed from the viewpoint of perturbational duality theory. Relationships with the traditional Lagrangian model for ordinary programming are explored in detail, with particular emphasis placed on the respective dual problems, Kuhn-Tucker vectors, and extremality conditions. The case of homogeneous constraints is discussed by way of illustration. The Slater existence criterion for optimal Lagrange multipliers in ordinary programming is sharpened for the case in which some of the functions are polyhedral. The analysis generally covers nonclosed functions on general spaces and includes refinements to exploit polyhedrality in the finite-dimensional case. Underlying the whole development are basic technical facts which are developed concerning the Fenchel conjugate and preconjugate of the indicator function of an epigraph set.
Distribution of eigenvalues of a two-parameter system of differential equations
M.
Faierman
45-86
Abstract: In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval $0\, \leqslant \,{x_1}\, \leqslant \,1$, the second for the interval $0\, \leqslant \,{x_{2\,}}\, \leqslant \,1$, and each containing the parameters $\lambda$ and $\mu$. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $({\lambda _{j,k}},{\mu _{j,k}})$ and $ {\psi _{j,k}}({x_{1,}}{x_2})$, respectively, $j,\,k\, = \,0,\,1,\, \ldots \,$, asymptotic methods are employed to derive asymptotic formulae for these expressions, as $j + k \to \infty$ when $ (j,\,k)$ is restricted to lie in a certain sector of the $(x,\,y)$ -plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the ${\psi _{j,k}}\,({x_1},\,{x_2})$.
On the construction of branched coverings of low-dimensional manifolds
Israel
Berstein;
Allan L.
Edmonds
87-124
Abstract: Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of ${S^3}$, $ {S^2}\, \times \,{S^1}$, ${P^2}\, \times \,{S^1}$, or the nontrivial ${S^3}$-bundle over ${S^1}$, and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of ${P^2}\, \times \,{S^1}$ of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of ${P^2}\, \times \,{S^1}$.
Examples of noncatenary rings
Raymond C.
Heitmann
125-136
Abstract: A technique is developed for constructing a new family of noetherian integral domains. To each domain, there naturally corresponds its poset (partially ordered set) of prime ideals. The resulting family of posets has the following property: every finite poset is isomorphic to a saturated subset of some poset in the family. In the process, it is determined when certain power series may be adjoined to noetherian rings without destroying the noetherian property.
Irrational connected sums and the topology of algebraic surfaces
Richard
Mandelbaum
137-156
Abstract: Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by ${V_m}$ a nonsingular element of $\left\vert {mV} \right\vert$. Let $ {V_1}$, ${V_m}$, $ {V_{m\, + \,1}}$ be generic elements of $ \left\vert V \right\vert$, $\left\vert {mV} \right\vert$, $\left\vert {(m\, + \,1)V} \right\vert$ respectively such that they have normal crossing in W. Let $ {S_{1m}}\, = \,{V_1}\, \cap \,{V_m}$ and $C\, = \,{V_1}\, \cap \,{V_m}\, \cap \,{V_{m + 1}}$. Then ${S_{1m}}$ is a nonsingular curve of genus $ {g_m}$ and C is a collection of $N\, = \,m\left( {m + 1} \right)V_1^3$ points on $ {S_{1m}}$. By [MM2] we find that $ ( \ast )\,{V_{m\, + \,1}}$ is diffeomorphic to $\overline {{V_m}\, - \,T({S_{1m}})} \,{ \cup _\eta }\,\overline {{V_1}'\, - \,T({S_{1m}}')}$, where $T\left( {{S_{1m}}} \right)$ is a tubular neighborhood of ${S_{1m}}$ in ${V_m}$, ${V_1}'$ is ${V_1}$ blown up along C, ${S_{1m}}'$ is the strict image of $ {S_{1m}}$ in $ {V_1}'$,
On the existence of good Markov strategies
Theodore Preston
Hill
157-176
Abstract: In contrast to the known fact that there are gambling problems based on a finite state space for which no stationary family of strategies is at all good, in every such problem there always exist $ \varepsilon$-optimal Markov families (in which the strategy depends only on the current state and time) and also $\varepsilon$-optimal tracking families (in which the strategy depends only on the current state and the number of times that state has been previously visited). More generally, this result holds for all finite state gambling problems with a payoff which is shift and permutation invariant.
Thickenings of CW complexes of the form $S\sp{m}\cup \sb{\alpha }e\sp{n}$
George
Cooke
177-209
Abstract: Necessary conditions are given for the existence of a thickening of $ {S^m}\,{ \cup _\alpha }\,{e^n}$ in codimension k. I give examples of such complexes requiring arbitrarily large codimension in order to thicken. Sufficient conditions are given for the existence of a tractable thickening in codimension $k\, + \,1$. The methods used include the study of the reduced product space of a pair of CW complexes.
A consistent consequence of AD
E. L.
Bull;
E. M.
Kleinberg
211-226
Abstract: We are concerned with the existence of a normal measure concentrating on the $ \omega$-closed unbounded sets. We strengthen the known result that the axiom of choice implies no such measure exists. It is shown that the existence of these measures is consistent (relative to a large cardinal). In particular, ${\aleph _2}$ may admit exactly two normal measures: one which contains the $\omega$-closed unbounded sets and the other, the ${\aleph _1}$-closed unbounded sets. This property of ${\aleph _2}$ is a well-known consequence of AD.
Partitions and sums and products of integers
Neil
Hindman
227-245
Abstract: The principal result of the paper is that, if $r\, < \,\omega$ and ${\{ {A_i}\} _{i < r}}$ is a partition of $ \omega$, then there exist $ i\, < \,r$ and infinite subsets B and C of $\omega$ such that $\sum F\, \in \,{A_i}$ and $\prod {G\, \in \,{A_i}}$ whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.
Differential algebraic Lie algebras
Phyllis Joan
Cassidy
247-273
Abstract: A class of infinite-dimensional Lie algebras over the field $\mathcal{K}$ of constants of a universal differential field $ \mathcal{U}$ is studied. The simplest case, defined by homogeneous linear differential equations, is analyzed in detail, and those with underlying set $\mathcal{U}\, \times \,\mathcal{U}$ are classified.
Frattini subgroups of $3$-manifold groups
R. B. J. T.
Allenby;
J.
Boler;
B.
Evans;
L. E.
Moser;
C. Y.
Tang
275-300
Abstract: In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.
Semi-algebraic groups and the local closure of an orbit in a homogeneous space
Morikuni
Goto
301-315
Abstract: Let L be a topological group acting on a locally compact Hausdorff space M as a transformation group. Let m be in M. A subset Q of M is called the local closure of the orbit Lm if Q is the smallest locally compact invariant subset of M with $m\, \in \,Q$. A partition $\displaystyle M = \,\bigcup\limits_{\lambda \in \wedge } \,{Q_\lambda },\,\,\,\... ...,}}\, \cap \,\,{Q_\mu } = \,\emptyset \,\,\,\,\left( {\lambda \ne \mu } \right)$ is called an LC-partition of M with respect to the L action if each ${Q_\lambda }$ is the local closure of Lm for any m in $ {Q_\lambda }$. Theorem. Let G be a connected Lie group, and let A and B be subgroups of G with only finitely many connected components. Suppose that B is closed. Then the factor space $G/B$ has an LC-partition with respect to the A action.
Erratum to: ``Generalized super-solutions of parabolic equations'' (Trans. Amer. Math. Soc. {\bf 220} (1976), 235--242)
Neil
Eklund
317-318