Existence and duality theorems for convex problems of Bolza
R. T.
Rockafellar
1-40
Abstract: The theory of conjugate convex functions is applied to a fundamental class of ``convex'' problems in the calculus of variations and optimal control. This class has many special properties which have not previously been exploited and for which the standard methods of approach are inadequate. Duality theorems are established which yield new results on the existence of optimal arcs, as well as necessary and sufficient conditions for optimality. These results have some relevance also to the study of ``nonconvex'' problems.
Extensions in certain topological algebraic categories
Robert C.
Busby
41-56
Abstract: Categories which we call ``sufficiently algebraic'' are defined, and for certain objects $A$ (called faithful) in such categories, and arbitrary objects $C$, we partially order the sets $\operatorname{Ext} (C,A)$ of extensions of $ A$ by $C$. We prove that the maximal elements in $\operatorname{Ext} (C,A)$ (with respect to this ordering) are in bijective correspondence with the morphisms from $C$ to a canonical object $O(A)$. If the short five lemma holds in the category, all extensions are maximal and therefore obtained in this way. As an application we compute extensions in certain categories of topological rings. In particular we investigate the possible extensions of one group algebra (of a locally compact group) by another in the category of Banach algebras with norm decreasing homomorphisms, and using some analytic tools we give conditions for the splitting of such extensions. Previous results of the author on extensions of ${C^ \ast }$-algebras are also included in this theory as a special case.
Integration theory on infinite-dimensional manifolds
Hui Hsiung
Kuo
57-78
Abstract: The purpose of this paper is to develop a natural integration theory over a suitable kind of infinite-dimensional manifold. The type of manifold we study is a curved analogue of an abstract Wiener space. Let $H$ be a real separable Hilbert space, $ B$ the completion of $ H$ with respect to a measurable norm and $i$ the inclusion map from $H$ into $B$. The triple $(i,H,B)$ is an abstract Wiener space. $B$ carries a family of Wiener measures. We will define a Riemann-Wiener manifold to be a triple $ (\mathcal{W},\tau ,g)$ satisfying specific conditions, $ \mathcal{W}$ is a $ {C^j}$-differentiable manifold $(j \geqq 3)$ modelled on $B$ and, for each $x$ in $\mathcal{W},\tau (x)$ is a norm on the tangent space $ {T_x}(\mathcal{W})$ of $\mathcal{W}$ at $x$ while $g(x)$ is a densely defined inner product on $ {T_x}(\mathcal{W})$. We show that each tangent space is an abstract Wiener space and there exists a spray on $\mathcal{W}$ associated with $g$. For each point $x$ in $ \mathcal{W}$ the exponential map, defined by this spray, is a ${C^{j - 2}}$-homeomorphism from a $ \tau (x)$-neighborhood of the origin in $ {T_x}(\mathcal{W})$ onto a neighborhood of $x$ in $ \mathcal{W}$. We thereby induce from Wiener measures of ${T_x}(\mathcal{W})$ a family of Borel measures ${q_t}(x, \cdot ),t > 0$, in a neighborhood of $x$. We prove that $ {q_t}(x, \cdot )$ and ${q_s}(y, \cdot )$, as measures in their common domain, are equivalent if and only if $t = s$ and $ {d_g}(x,y)$ is finite. Otherwise they are mutually singular. Here $ {d_g}$ is the almost-metric (in the sense that two points may have infinite distance) on $ \mathcal{W}$ determined by $ g$. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals.
Conjugacy separability of the groups of hose knots
Peter F.
Stebe
79-90
Abstract: Let $G$ be a group. An element $g$ of $G$ is c.d. in $G$ if and only if, given any element $h$ of $G$, either $h$ is conjugate to $g$ or there is a homomorphism $\xi$ from $G$ onto a finite group such that $\xi (g)$ is not conjugate to $ \xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. In this paper we show that the groups of hose knots are c.s.
On zonotopes
P.
McMullen
91-109
Abstract: In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with $n$ zones of dimensions $d$ and $n - d$, and enables one to enumerate all the combinatorial types of $d$-zonotopes with $n \leqq d + 2$ zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements.
Representation of vector valued nonlinear functions
Victor J.
Mizel;
K.
Sundaresan
111-127
Abstract: A representation theorem for ``additive'' nonlinear functional on spaces ${L^p}(\mu )$ is here extended to ``additive'' nonlinear functions from Lebesgue-Bochner function spaces $ L_E^p(\mu )$ ($ E$ a separable Banach space) into Banach spaces $B$. A counterexample is provided to show that the restriction to separable $E$ is essential.
Estimates for best approximation to rational functions
S. J.
Poreda
129-135
Abstract: Estimates for the deviation of certain rational functions and their polynomials of best uniform approximation on various sets are given. As a result, in some cases these deviation and polynomials are explicitly calculated. For example, the polynomials of best uniform approximation to the function $(\alpha z + \beta )/(z - a)(1 - \bar az),\vert a\vert \ne 1$, on the unit circle are given.
Value distribution of harmonic polynomials in several real variables.
Morris
Marden
137-154
Abstract: Using Bergman's integral operator method, the author studies an arbitrary axisymmetric harmonic polynomial $H(x,\rho )$ in ${R^3}$ and ${R^N}$ in relation to its associate polynomial $ h(\zeta )$ in $ C$. His results pertain to the value distributions and critical circles of $H(x,\rho )$ in certain cones; bounds on the gradient of an $H(x,\rho )$ assumed bounded in sphere ${x^2} + {\rho ^2} \leqq 1$; axisymmetric harmonic vectors. Corresponding results are also obtained for axisymmetric harmonic functions $F(x,\rho )$ with rational associate $f(\zeta )$.
A pointwise convergence theorem for sequences of continuous functions.
K.
Schrader
155-163
Abstract: Let $\{ {f_k}\}$ be a sequence of continuous real valued functions defined on an interval $I$ and $N$ a fixed nonnegative integer such that if ${f_k}(x) = {f_i}(x)$ for more than $N$ distinct values of $x \in I$ then ${f_{k}}(x) \equiv {f_i}(x)$ for $ x \in I$. It follows that there is a subsequence $ \{ {g_j}\}$ of $\{ {f_k}\}$ such that for each $x$ the subsequence $\{ {g_j}(x)\}$ is eventually monotone. Thus ${\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists for all $x$, where $f$ is an extended real valued function. If $\vert{f_k}(x)\vert$ is bounded for each $ x \in I$ then $ {\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists as a finite limit for all $ x \in I$. For $ N = 0$ this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.
Computability by probabilistic Turing machines
Eugene S.
Santos
165-184
Abstract: In the present paper, the definition of probabilistic Turing machines is extended to allow the introduction of relative computability. Relative computable functions, predicates and sets are discussed and their operations investigated. It is shown that, despite the fact that randomness is involved, most of the conventional results hold in the probabilistic case. Various classes of ordinary functions characterizable by computable random functions are introduced, and their relations are examined. Perhaps somewhat unexpectedly, it is shown that, in some sense, probabilistic Turing machines are capable of computing any given function. Finally, a necessary and sufficient condition for an ordinary function to be partially recursive is established via computable probabilistic Turing machines.
Limit theorems for measures on nonmetrizable locally compact abelian groups
David C.
Bossard
185-205
Abstract: In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assumption. This paper consists of a reworking of certain results contained in Parthasarathy to see what can be done without the metric restriction. Among the topics considered are: necessary and sufficient conditions for a limit law to have an idempotent factor; the relationship between limits of compound Poisson laws and limits of sums of independent random variables; and a representation theorem for certain limit laws.
Interpolation theorems for the pairs of spaces $(L\sp{p},\,L\sp{\infty })$ and $(L\sp{1},\,L\sp{q})$
George G.
Lorentz;
Tetsuya
Shimogaki
207-221
Abstract: A Banach space $ Z$ has the interpolation property with respect to the pair $(X,Y)$ if each $T$, which is a bounded linear operator from $ X$ to $X$ and from $Y$ to $Y$, can be extended to a bounded linear operator from $ Z$ to $Z$. If $X = {L^p},Y = {L^\infty }$ we give a necessary and sufficient condition for a Banach function space $ Z$ on $(0,l),0 < l \leqq + \infty$, to have this property. The condition is that $g \prec {}^pf$ and $f \in Z$ should imply $g \in Z$; here $ g \prec {}^pf$ means that $ {g^{ \ast p}} \prec {f^{ \ast p}}$ in the Hardy-Littlewood-Pólya sense, while ${h^ \ast }$ denotes the decreasing rearrangement of the function $\vert h\vert$. If the norms $\vert\vert T\vert{\vert _X},\vert\vert T\vert{\vert _Y}$ are given, we can estimate $\vert\vert T\vert{\vert _Z}$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding ${\lambda _p},{\lambda _p} \leqq {2^{1/q}},1/p + 1/q = 1$. Similar results hold if $X = {L^1},Y = {L^q}$. For all these theorems, the complete continuity of $T$ on $Z$ is assured if $T$ has this property on $X$ or on $Y$, and if $Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $\vert\vert{\sigma _a}\vert{\vert _Z},a > 0$, where ${\sigma _a}$ is the compression operator $ {\sigma _a}f(t) = f(at),0 \leqq t < l$.
How changing $D[[x]]$ changes its quotient field
Philip B.
Sheldon
223-244
Abstract: Let $D[[x]]$ be the ring of formal power series over the commutative integral domain $ D$. It is shown that changing $D[[x]]$ to $D[[x/a]]$ changes (i.e. increases) the quotient field by an infinite transcendence degree over the original field whenever $\cap _{i = 1}^\infty {a^i}D = 0$. From this it follows that if ${D_1}$ and ${D_2}$ are two distinct rings between the integers and the rational numbers, with ${D_1}$ contained in ${D_2}$, then the change in the ring of coefficients from $ {D_1}[[x]]$ to $D_{2}[[x]]$ again yields a change in the quotient fields by an infinite transcendence degree. More generally, it is shown that $D$ is completely integrally closed iff any increase in the ring of coefficients yields an increase in the quotient field of $D[[x]]$. Moreover, $D$ is a one-dimensional Prüfer domain iff any change in the ring of coefficients from one overring of $D$ to another overring of $D$ yields a change in the quotient field of the respective power series rings. Finally it is shown that many of the domain properties of interest are really properties of their divisibility groups, and some examples are constructed by first constructing the required divisibility groups.
Generally $p\sp{\alpha }$-torsion complete abelian groups
Paul F.
Dubois
245-255
Abstract: A generalized $ p$-primary cotorsion abelian group $G$ is a $ {p^\alpha }$-injective, that is satisfies ${p^\alpha }\operatorname{Ext} ( - ,G) = 0$, iff ${G_t}$ is $ {p^\alpha }$-injective in the category of torsion abelian groups. Such a torsion group is generally $ {p^\alpha }$-torsion complete, but an example shows that all its Ulm factors need not be complete. The injective properties of generally ${p^\alpha }$-torsion complete groups are investigated. They are an injectively closed class, and the corresponding class of sequences is the class of ${p^\alpha }$-pure sequences with split com-c pletion when $\alpha$ is ``accessible". Also, these groups are the $ {p^\alpha }$-high injectives.
Ramsey's theorem for $n$-parameter sets
R. L.
Graham;
B. L.
Rothschild
257-292
Abstract: Classes of objects called $n$-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into $ r$ classes of the $ k$-parameter subsets of any sufficiently large $n$-parameter set must result in some $ l$-parameter subset with all its $k$-parameter subsets in one class. Among the immediate corollaries are the lower dimensional cases of a Ramsey theorem for finite vector spaces (a conjecture of Rota), the theorem of van der Waerden on arithmetic progressions, a set theoretic generalization of a theorem of Schur, and Ramsey's Theorem itself.
A representation theorem for large and small analytic solutions of algebraic differential equations in sectors
Steven
Bank
293-305
Abstract: In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field. In the particular case where the coefficients are rational functions, our main result states that for any given sector $S$ in the plane, there exists a positive real number $N$, depending only on the equation and the angle opening of $S$, such that any solution $y(z)$, which is meromorphic in $ S$ and satisfies the condition $ {z^{ - N}}y \to \infty$ as $z \to \infty$ in $S$, must be of the form $\exp \int {c{z^m}(1 + o(1))}$ in subsectors, where $c$ and $m$ are constants. (From this, we easily obtain a similar representation for analytic solutions in $ S$, which are not identically zero, and for which $ {z^K}y \to 0$ as $z \to \infty$ in $S$, where the positive real number $K$ again depends only on the equation and the angle opening of $S$fs
Theorems on Cartan subalgebras like some on Carter subgroups
Ernest L.
Stitzinger
307-315
Abstract: We consider some results on the Cartan subalgebras of a solvable Lie algebra which are analogues to some results on Carter subgroups of a finite solvable group. Only solvable Lie algebras are considered here.
Grunsky-Nehari inequalities for a subclass of bounded univalent functions
D. W.
DeTemple
317-328
Abstract: Let ${D_1}$ be the class of regular analytic functions $F(z)$ in the disc $U = \{ z:\vert z\vert < 1\}$ for which $F(0) > 0,\vert F(z)\vert < 1$, and $F(z) + F(\zeta ) \ne 0$ for all $z,\zeta \in U$. Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in $ {D_1}$, the proof being based on the area method. By subordination it is shown univalency is unnecessary for certain special cases of the inequalities. Employing a correspondence between $ {D_1}$ and the class $ {S_1}$ of bounded univalent functions, the results can be reinterpreted to apply to this latter class.
Some characterizations of $n$-dimensional $F$-spaces
M. J.
Canfell
329-334
Abstract: In this paper we obtain characterizations of an $n$-dimensional $F$-space in terms of the rings of continuous real-valued and complex-valued functions defined on the space. Motivation for these results is the work of Gillman and Henriksen on $U$-spaces ($F$-spaces of dimension 0) and $T$-spaces ($F$-spaces of dimension 0 or 1).
On isomorphisms between ideals in rings of continuous functions
David
Rudd
335-353
Abstract: A ring of continuous functions is a ring of the form $C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space $X$. For an arbitrary ideal $I$ in $C(X)$, the author shows that the maximal ideals of $ I$ are precisely the ideals of the form $I \cap M$, for some maximal ideal $M$ in $C(X)$ not containing $I$. The author shows that any ring isomorphism between ideals in any two rings of continuous functions preserves order, boundedness, and lattice structure; and he uses these results to obtain one of the main theorems: An isomorphism of a maximal ideal in $C(X)$ onto a maximal idea in $C(Y)$ can be extended to an isomorphism of $C(X)$ onto $C(Y)$. Another of the main theorems characterizes those isomorphisms between ${C^ \ast }(X)$ and $ {C^ \ast }(Y)$ (the subrings of bounded functions in $C(X)$ and $C(Y)$ respectively) which can be extended to isomorphisms between $C(X)$ and $C(Y)$. The author proves that, given any ideal $ I$ in $C(X)$, there exists a space $ X(I)$ so that the uniform closure of $I$ is isomorphic to a maximal ideal in $ C(X(I))$.
Area measure and Rad\'o's lower area
Togo
Nishiura
355-367
Abstract: The theory of Geöcze area for two-dimensional surfaces in threedimensional space had been essentially completed by the mid 1950's. The only hypothesis needed for all theorems in this case is the finiteness of the area. See [2] for an account of this theory. In the early 1960's, H. Federer established, in his paper [6], fundamental facts concerning his integral geometric area for higher dimensional area theory by employing the theory of normal and integral currents. These facts employ not only the finiteness of area as a basic hypothesis but certain other hypotheses as well. The extensions of Geöcze type area to higher dimensions also employ not only the finiteness of area but certain added hypotheses. These hypotheses are of such a nature as to allow the use of the theory of quasi-additivity [3], [11]. The present paper concerns these added hypotheses which play such an important part of higher-dimensional area theory of today. It is shown that Radó's lower area is the best Geöcze type area to describe these added hypotheses. That is, it is shown that the quasi-additivity hypotheses of Geöcze area in [11] imply the quasi-additivity hypotheses of lower area. Second, it is shown that the quasi-additivity hypotheses for lower area imply that the surface has the essential cylindrical property defined by J. Breckenridge in [5]. This essential cylindrical property is proved to be equivalent to the existence of area measures on the middle space of the mapping representing the surface. Finally, it is shown that the essential cylindrical property of a surface is equivalent to the quasi-additivity condition for lower area. Thus, an intrinsic property of the surface characterizes the quasi-additivity condition for the lower area of a surface.
Removable sets for pointwise subharmonic functions
Victor L.
Shapiro
369-380
Abstract: Pointwise subharmonic is defined in terms of the pointwise ${L^1}$ total derivative of order 2. The class $\mathcal{A}({x^ \ast },{r_ \ast })$ is introduced for the ball $ B({x^ \ast },{r_ \ast })$, and the following theorem is established: Let $ Q$ be a Borel set of Lebesgue measure zero contained in $B({x^ \ast },{r_ \ast })$. Then a necessary and sufficient condition that $Q$ be removable for pointwise subharmonic functions with respect to the class $\mathcal{A}({x^ \ast },{r_ \ast })$ is that $ Q$ be countable. It is also shown that the class $\mathcal{A}({x^ \ast },{r_ \ast })$ is in a certain sense best possible for the sufficiency of the above theorem.
Bounds on the nilpotency class of certain semidirect products
Larry
Morley
381-390
Abstract: Gilbert Baumslag has shown that the standard wreath product of $ A$ by $B$ is nilpotent if and only if $ A$ and $B$ are $p$-groups for the same prime $p, A$ is nilpotent of bounded exponent and $B$ is finite. L. Kaloujnine and Marc Krasner have shown that the standard (unrestricted) wreath product of $A$ by $B$ contains an isomorphic copy of every group $G$ which is an extension of $A$ by $B$. Thus it follows that any extension subject to the above condition on $A$ and $B$ is nilpotent. In this paper, the author gives an explicit characterization of the terms of the lower central series of a semidirect product $W$ of an abelian group by an arbitrary group. He then establishes a formula for an upper bound on the nilpotency class of $W$ when $W$ is a semidirect product of an abelian $ p$-group $X$ of bounded exponent by a finite $ p$-group $B$. This new bound is given in terms of the exponent of $X$ and the cycle structure of the factor groups of the lower central series of $B$.
Teichm\"uller inequalities without coefficient normalization
Arthur E.
Obrock
391-416
Abstract: Teichmüller's relation between the coefficients of extremal schlicht functions and quadratic differentials is extended. The coefficient normalization hypothesis in his theorem is dropped with the result that the new coefficient relations become more complex. This completes the partial result in this direction which is contained in Jenkins' General Coefficient Theorem. A modification of the version of the length-area method used by Teichmüller and Jenkins is introduced in our proof.
Equivariant bordism and Smith theory
R. E.
Stong
417-426
Abstract: The relationship between equivariant bordism and Smith homology theory on the category of pairs with involution is studied.
Some examples in degree of approximation by rational functions
D.
Aharonov;
J. L.
Walsh
427-444
Abstract: We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of $E$, (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.
Inner ideals in quadratic Jordan algebras
Kevin
McCrimmon
445-468
Abstract: The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the theory of associative algebras. In particular, the Jordan algebras with descending chain condition on inner ideals are intimately related to the Artinian associative algebras. In this paper we will completely characterize all inner ideals in the semisimple Jordan algebras with descending chain condition. It is well known that any left or right ideal $\mathfrak{B}$ in a semisimple Artinian $\mathfrak{A}$ is determined by an idempotent, $ \mathfrak{B} = \mathfrak{A}f$ or $\mathfrak{B} = e\mathfrak{A}$. We show that any inner ideal in the quadratic Jordan algebra ${\mathfrak{A}^ + }$ has the form $ \mathfrak{B} = e\mathfrak{A}f$, and if $ \mathfrak{A}$ has involution $^\ast$ the inner ideals of the Jordan algebra $\mathfrak{H}(\mathfrak{A}, ^ \ast )$ of $ ^ \ast$-symmetric elements are ``usually'' of the form $\mathfrak{B} = {e^ \ast }\mathfrak{H}e$. We also characterize the inner ideals in the Jordan algebras $ \mathfrak{J}(Q,c)$ or $ \mathfrak{J}(N,c)$ determined by a quadratic or cubic form.
Modules over Pr\"ufer domains
Syed M.
Fakhruddin
469-487
Abstract: We prove that the category of torsion-free modules over a Prüfer domain is equivalent to a certain category of diagrams. As applications we derive (i) a strong version of Baer-Kulikov-Kaplansky theorem for modules over a valuation ring, (ii) the determination of a complete set of invariants for modules of almost countable rank over an almost-maximal valuation ring, (iii) a new nonhomological proof of the characterization of a maximal valuation ring due to Matlis.
A separable postliminal $C\sp{\ast} $-algebra without maximal closed ideals
H.
Leptin
489-496
Abstract: Let $G$ be the free abelian group with a countable number of generators. We construct a separable locally compact $G$-transformation space $X$ without closed minimal invariant subsets, such that the corresponding $ {C^ \ast }$-algebra ${C^ \ast }(G,X)$ has the properties mentioned in the title. Using $X$ we also give an example of a transformation space $(G,Z)$ without closed minimal invariant subset, on which $G$ acts freely.
Rearrangements
Henryk
Minc
497-504
Abstract: It is shown that if ${a^{(t)}} = (a_1^{(t)},a_2^{(t)}, \ldots ,a_n^{(t)}),t = 1, \ldots ,m$, are nonnegative $ n$-tuples, then the maxima of $ \sum\nolimits_{i = 1}^n {a_i^{(1)}a_i^{(2)} \cdots a_i^{(m)}}$ of $\prod\nolimits_{i = 1}^n {{{\min }_t}(a_i^{(t)})}$ and of $ \Sigma _{i = 1}^n$ min $(a_i^{(t)})$, and the minima of $\prod\nolimits_{i = 1}^n {(a_i^{(1)} + a_i^{(2)} + } \cdots + a_i^{(m)})$, of $ \prod\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ and of $ \sum\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ are attained when the $ n$-tuples $ {a^{(1)}},{a^{(2)}}, \ldots ,{a^{(m)}}$ are similarly ordered. Necessary and sufficient conditions for equality are obtained in each case. An application to bounds for permanents of $ (0,1)$-matrices is given.
Analytic continuation of the series $\sum \,(m+nz)\sp{-s}$
Joseph
Lewittes
505-509
Abstract: The series $\Sigma {(m + nz)^{ - s}},m,n$ ranging over all integers except both zero, for $s$ an integer greater than two is well known from the theory of elliptic functions and modular forms. In this paper, we show that this series defines an analytic function $(z,s)$ for $\operatorname{Im} z > 0$ and $\operatorname{Re} s > 2$ which has an analytic continuation to all values of $s$. It is then shown that $G$ satisfies a functional equation under the transformation $z \to - 1/z$, and finally as an application some numerical results are obtained.
Errata to ``Hypergeometric operator series and related partial differential equations''
Louis R.
Bragg
511