Gelfond's method for algebraic independence
W. Dale
Brownawell
1-26
Abstract: This paper extends Gelfond's method for algebraic independence to fields $K$ with transcendence type $\leqslant \tau$. The main results show that the elements of a transcendence basis for $K$ and at least two more numbers from a prescribed set are algebraically independent over $Q$. The theorems have a common hypothesis: $\{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\}$ are sets of complex numbers, each of which is $Q$-linearly independent. THEOREM A. If $(2\tau - 1) < MN$, then at least two of the numbers ${\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$. THEOREM B. If $ 2\tau (M + N) \leqslant MN + M$, then at least two of the numbers $ {\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$. THEOREM C. If $ 2\tau (M + N) \leqslant MN$, then at least two of the numbers $ 1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $K$.
The subgroups of a tree product of groups
J.
Fischer
27-50
Abstract: Let $G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})$ be a tree product with $H$ a subgroup of $G$. By extending the technique of using a rewriting process we show that $H$ is an HNN group whose base is a tree product with vertices of the form $x{A_i}{x^{ - 1}} \cap H$. The associated subgroups are contained in vertices of the base, and both the associated subgroups of $H$ and the edges of its base are of the form $y{U_{jk}}{y^{ - 1}} \cap H$. The $ x$ and $y$ are certain double coset representatives for $ G\bmod (H,{A_i})$ and $G\bmod (H,{U_{jk}})$, respectively, and the elements defined by the free part of $ H$ are specified. More precise information about $H$ is given when $H$ is either indecomposable or $ H$ satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of $G$ is contained in the center of its two vertices then the cartesian subgoup of $ G$ is a free group. We also use our subgroup theorem in proving that if each edge of $G$ is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then $G$ is a finite extension of a free group iff the orders of the ${A_i}$ are uniformly bounded.
Hypercontractive semigroups and Sobolev's inequality
George F.
Feissner
51-62
Abstract: If $H \geqslant 0$ is the generator of a hypercontractive semigroup (HCSG), it is known that ${(H + 1)^{ - 1/2}}$ is a bounded operator from $ {L^p}$ to ${L^p},1 \leqslant p \leqslant \infty$. We prove that ${(H + 1)^{ - 1/2}}$ is bounded from ${L^2}$ to the Orlicz space ${L^2}{\text{ I}}{{\text{n}}^ + }L$, basing the proof on the uniform semiboundedness of the operator $H + V$, for suitable $V$. We also prove by an interpolation argument, that ${(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}$ to ${L^p}{\text{ I}}{{\text{n}}^ + }L,2 \leqslant p < \infty$. Another interpolation argument shows that $ {(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^m}$ to $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^{m + 1}}$ and $m$ a positive integer. Finally, we identify the topological duals of the spaces mentioned above.
Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc
Stephan
Ruscheweyh
63-74
Abstract: Let $A$ be the set of functions regular in the unit disc $ \mathcal{U}$ and $ {A_0}$ the set of all functions $f \in A$ which satisfy $f(0) = 1$. For $V \subset {A_0}$ define the dual set $ {V^ \ast } = \{ f \in {A_0}\vert f \ast g \ne 0{\text{ for all }}g \in V,z \in \mathcal{U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }$. Here $f \ast g$ denotes the Hadamard product. THEOREM. Let $V \subset {A_0}$ have the following properties: (i) $V$ is compact, (ii) $f \in V$ implies $f(xz) \in V$ for all $\vert x\vert \leqslant 1$. Then $ \lambda (V) = \lambda ({V^{ \ast \ast }})$ for all continuous linear functionals $\lambda$ on $A$. This theorem has many applications to functions in $A$ which are defined by properties like bounded real part, close-to-convexity, univalence etc.
On minimal immersions of $S\sp{2}$ into $S\sp{2m}$
Jo ao Lucas Marquês
Barbosa
75-106
Abstract: The study of minimal immersions of the $2$-sphere into the standard $n$-sphere of the euclidean space has been better accomplished by associating to each such immersion a certain holomorphic curve. This has been done in several ways in the literature. In the present paper we explore this technique applying some knowledge about the topological and analytical invariants of the particular set of holomorphic curves used to obtain further results. Some new examples are provided, a beginning of a general description of such immersions is given and a rigidity theorem is proved.
Topological dynamics and $C\sp{\ast} $-algebras
William L.
Green
107-121
Abstract: If $G$ is a group of automorphisms of a ${C^ \ast }$-algebra $A$ with identity, then $G$ acts in a natural way as a transformation group on the state space $S(A)$ of $A$. Moreover, this action is uniformly almost periodic if and only if $G$ has compact pointwise closure in the space of all maps of $A$ into $A$. Consideration of the enveloping semigroup of $ (S(A),G)$ shows that, in this case, this pointwise closure $\bar G$ is a compact topological group consisting of automorphisms of $A$. The Haar measure on $\bar G$ is used to define an analogue of the canonical center-valued trace on a finite von Neumann algebra. If $A$ possesses a sufficiently large group $ {G_0}$ of inner automorphisms such that $ (S(A),{G_0})$ is uniformly almost periodic, then $A$ is a central $ {C^ \ast }$-algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional $ {C^ \ast }$-algebra possess exactly one finite trace.
On automorphism groups and endomorphism rings of abelian $p$-groups
Jutta
Hausen
123-128
Abstract: Let $A$ be a noncyclic abelian $ p$-group where $p \geqslant 5$, and let $ {p^\infty }A$ be the maximal divisible subgroup of $A$. It is shown that $ A/{p^\infty }A$ is bounded and nonzero if and only if the automorphism group of $ A$ contains a minimal noncentral normal subgroup. This leads to the following connection between the ideal structure of certain rings and the normal structure of their groups of units: if the noncommutative ring $R$ is isomorphic to the full ring of endomorphisms of an abelian $p$-group, $ p \geqslant 5$, then $ R$ contains minimal twosided ideals if and only if the group of units of $ R$ contains minimal noncentral normal subgroups.
The similarity orbit of a normal operator
L. A.
Fialkow
129-137
Abstract: If $N$ is a bounded normal operator on a separable Hilbert space $ \mathcal{H}$, let $\mathcal{S}(N)$ denote the similarity orbit of $ N$ in $L(\mathcal{H})$ and let ${\mathcal{S}_k}(N)$ denote the set of all compact perturbations of elements of $\mathcal{S}(N)$. It is proved that $ \mathcal{S}(N)({\mathcal{S}_K}(N))$ is norm closed in $L(\mathcal{H})$ if and only if the spectrum (essential spectrum) of $N$ is finite. If the essential spectrum of $ N$ is infinite and $ M$ is a normal operator whose spectrum is connected and contains that of $ N$, then $M$ is in the closure of $\mathcal{S}(N)$. If the spectrum of $N$ is connected, this result characterizes the normal elements of the closure of $\mathcal{S}(N)$. A normal operator is similar to a nonquasidiagonal operator if and only if its essential spectrum contains more than two points.
The zeroes of nonnegative holomorphic curvature operators
A. M.
Naveira;
C.
Fuertes
139-147
Abstract: Here, we study the structure of points in a holomorphic Grassmann's submanifold where the holomorphic sectional curvature assumes its minimum and maximum. For spaces of nonnegative holomorphic sectional curvature we study the set of points on which it assumes the value zero. We show that the minimum and maximum sets of holomorphic sectional curvature are the intersections of a holomorphic Grassmann's submanifold with linear complex holomorphic subspaces of type (1, 1).
A classification theorem for abelian $p$-groups
R. B.
Warfield
149-168
Abstract: A new class of Abelian $p$-groups, called $S$-groups, is studied, and the groups in this class are classified in terms of cardinal invariants. The class of $S$-groups includes Nunke's totally projective $ p$-groups. The invariants consist of the Ulm invariants (which Hill has shown can be used to classify the totally projective groups) together with a new sequence of invariants indexed by limit ordinals which are not cofinal with $ \omega$. The paper includes a fairly complete discussion of dense isotype subgroups of totally projective $p$-groups, including necessary and sufficient conditions for two of them to be congruent under the action of an automorphism of the group. It also includes an extension of Ulm's theorem to a class of mixed modules over a discrete valuation ring.
Concerning first countable spaces. III
G. M.
Reed
169-177
Abstract: The primary purpose of this paper is (1) to provide a ``real'' example of a regular first countable ${T_1}$-space which has no dense developable subspace and (2) to provide a new technique for producing Moore spaces which fail to have dense metrizable subspaces. Related results are established which produce new examples of noncompletable Moore spaces and which show that each regular hereditary $M$-space with a $ {G_\delta }$-diagonal has a dense metrizable subspace.
A property of finite $p$-groups with trivial multiplicator
Michael R.
Jones
179-183
Abstract: A sufficient condition for a finite $2$-generator $p$-group to have nontrivial multiplicator is given. To show that this result is best possible, a finite $2$-group with trivial multiplicator is constructed.
On the variety of manifolds without conjugate points
Robert
Gulliver
185-201
Abstract: The longest geodesic segment in a convex ball of a riemannian manifold, where the convexity is ensured by an upper bound on sectional curvatures, is the diameter. This and related results are demonstrated and applied to show that there exist manifolds with sectional curvatures of both signs but with-out conjugate points.
On the construction of the Bockstein spectral sequence
Jerrold
Siegel
203-224
Abstract: The Bockstein spectral sequence is developed from a direct limit construction. This is shown to clarify its relation to certain associated structures, in particular the divided power operations. Finally, the direct limit construction is used to study the problem of enumerating the Bockstein spectral sequences over a given simple $R$-module.
Two weight function norm inequalities for the Poisson integral
Benjamin
Muckenhoupt
225-231
Abstract: Let $f(x)$ denote a complex valued function with period $2\pi$, let $\displaystyle {P_r}(f,x) = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\frac{{(1 - {r^2})f(y)dy}}{{1 - 2r\cos (x - y) + {r^2}}}}$ be the Poisson integral of $f(x)$ and let $\vert I\vert$ denote the length of an interval $ I$. For $1 \leqslant p < \infty $ and nonnegative $ U(x)$ and $V(x)$ with period $2\pi$ it is shown that there is a $ C$, independent of $ f$, such that $\displaystyle \mathop {\sup }\limits_{0 \leqslant r < 1} \int_{ - \pi }^\pi {\v... ...,x){\vert^p}U(x)dx \leqslant C\int_{ - \pi }^\pi {\vert f(x){\vert^p}V(x)dx} }$ if and only if there is a $ B$ such that for all intervals $I$ $\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {U(x)dx} } \right]{\left[ ... ...t I\vert}}\int_I {{{[V(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1.}} \leqslant B.$ Similar results are obtained for the nonperiodic case and in the case where $U(x)dx$ and $V(x)dx$ are replaced by measures.
Polynomials related to the Bessel functions
F. T.
Howard
233-248
Abstract: In this paper we examine the polynomials ${W_n}(a)$ defined by means of $\displaystyle - 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum\limits_{n = 0}^\infty {{W_n}(a){x^n}/n!} .$ These polynomials are closely related to the zeros of the Bessel function of the first kind of index --3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of ${W_n}(a)$ are investigated. We show that if $n$ is even then $ {W_n}(a) > 0$ for all $ a$, and if $n$ is odd then ${W_n}(a)$ has only the one real root $ a = 1/2$. Also we find upper and lower bounds for all $b$ such that $ {W_n}(a + bi) = 0$. The problem of multiple roots is discussed and we show that if $n \equiv 0,1,5,8$ or 9 $(\bmod\; 12)$, then ${W_n}(a)$ has no multiple roots. Finally, if $ n \equiv 0,1,2,5,6$ or 8 $(\bmod \; 12)$, then ${W_n}(a)$ has no factor of the form ${a^2} + ca + d$ where $c$ and $ (\bmod\; 12)$ are integers.
The center of an order with finite global dimension
Mark
Ramras
249-257
Abstract: Let $\Lambda$ be a quasi-local ring of global dimension $n < \infty $. Assume that its center $ R$ is a noetherian domain, that $\Lambda$ is finitely generated torsion-free as an $ R$-module, and that $ R$ is an $R$-direct summand of $\Lambda$. Then $R$ is integrally closed in its quotient field $K$ and Macauley of dimension $n$. Furthermore, when $n = 2,\Lambda$ is a maximal $R$-order in the central simple $ K$-algebra $\Lambda { \otimes _R}K$. This extends an earlier result of the author, in which $R$ was assumed to have global dimension 2. Examples are given to show that in the above situation $ R$ can have infinite global dimension.
Newman's theorem in the Riemannian category
L. N.
Mann;
J. L.
Sicks
259-266
Abstract: In 1931 M. H. A. Newman proved that if $M$ is a connected topological manifold with metric $d$, there exists a number $\varepsilon > 0$, depending only upon $ M$ and $d$, such that every compact Lie group $ G$ acting effectively on $ M$ has at least one orbit of diameter at least $ \varepsilon$. Aside from isolated results nothing appears to be known about $\varepsilon$. In order to learn more about the invariant $ \varepsilon$, attention is restricted here to groups of isometries on a Riemannian manifold. It is found that the invariant $\varepsilon$ of $M$ is connected with the notion of convexity introduced by J. H. C. Whitehead in 1932.
Sobolev-Galpern equations of order $n+2$ in $R\sp{m}\times R$, $m\geq 2$
V. R. Gopala
Rao
267-278
Abstract: Equations with mixed time and space derivatives play an important role in several branches of physics. Here we establish existence and uniqueness results for such equations. In addition, we also prove a regularity result which employs a regularity result for nonhomogeneous elliptic equations whose proof is also included.
Localization and sheaf reflectors
J.
Lambek;
B. A.
Rattray
279-293
Abstract: Given a triple $(S,\eta ,\mu )$ on a category $\mathcal{A}$ with equalizers, one can form a new triple whose functor $Q$ is the equalizer of $\eta S$ and $S\eta$. Fakir has studied conditions for $ Q$ to be idempotent, that is, to determine a reflective subcategory of $\mathcal{A}$. Here we regard $S$ as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos $ \mathcal{A}$ from an injective object $I$, taking $ S = {I^{{I^{( - )}}}}$. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in $\mathcal{A}$ can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form $S = {I^{( - ,I)}},I$ injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.
On bilateral derivates and the derivative
K. M.
Garg
295-329
Abstract: In this paper we prove a new result on the monotonicity of a function in terms of its bilateral derivates, and obtain from it extensions of several existing results on such derivates and the derivative of a function. Let $ f:R \to R$, where $ R$ denotes the set of real numbers. If its lower derivate $\underline{D} f > 0$ at a nonmeager set of points, we prove $f$ to be ``adequately'' increasing in some interval, viz. even the function $ f(x) - \alpha x$ is increasing for some $ \alpha > 0$. When $ f$ is nowhere adequately monotone, it follows that there exists a residual set of points where $f$ has a zero ``median'' derivate, i.e. either $ D\_f \leqslant 0 \leqslant {D^ - }f$ or ${D_ + }f \leqslant 0 \leqslant {D^ + }f$. These results remain valid for functions defined on an arbitrary set $ X \subset R$ under a mild continuity hypothesis, e.g. the absence of ordinary discontinuity at the unilateral limit points of $ X$. The last result leads to a new version of A. P. Morse's theorem for median derivates, and this in turn yields an improved version of the Goldowski-Tonelli theorem. We also obtain some necessary and sufficient conditions for a function to be nondecreasing, and extensions of the mean-value theorem and the Denjoy and other properties of a derivative. If $f:X \to R$, where $ X \subset R$, and both the derivates of $f$ are finite at a set of points that is not meager in $X$, then $f$ is further proved to satisfy the Lipschitz condition on some portion of $X$. When $f$ has a finite derivate almost everywhere and $ X$ has a finite measure, it is shown that $f$ can be made Lipschitz by altering its values on a set with arbitrarily small measure. Some results on singular functions are also strengthened. The results and the methods of this paper further provide extensions of some results of Young, Tolstoff, Kronrod, Zahorski, Brudno, Fort, Hájek, Filipczak, Neugebauer and Lipiński on derivates and the derivability of a function.
On the generation of one-relator groups
Stephen J.
Pride
331-364
Abstract: This paper is concerned with obtaining information about the Nielsen equivalence classes and $T$-systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if $G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r})$ where the $ {\alpha _j}$ are positive, the ${\beta _i}$ are nonzero, ${E_r}$ has the form $[{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]]$ with $ \vert{ \in _1}\vert = \vert{ \in _2}\vert = \cdots = \vert{ \in _r}\vert = 1$, then in a large number of cases $G$ has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen's reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one $ T$-system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning $T$-systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.
Topological extension properties
R. Grant
Woods
365-385
Abstract: It is known that if a topological property $ \mathcal{P}$ of Tychonoff spaces is closed-hereditary, productive, and possessed by all compact $ \mathcal{P}$-regular spaces, then each $ \mathcal{P}$-regular space $ X$ is a dense subspace of a space $ {\gamma _\mathcal{P}}X$ with $\mathcal{P}$ such that if $Y$ has $ \mathcal{P}$ and $ f:X \to Y$ is continuous, then $f$ extends continuously to ${f^\gamma }:{\gamma _\mathcal{P}}X \to Y$. Such topological properties are called extension properties; $ {\gamma _\mathcal{P}}X$ is called the maximal $ \mathcal{P}$-extension of $ X$. In this paper we study the relationships between pairs of extension properties and their maximal extensions. A basic tool is the concept of $ \mathcal{P}$-pseudocompactness, which is studied in detail (a $\mathcal{P}$-regular space $X$ is $ \mathcal{P}$-pseudocompact if $ {\gamma _\mathcal{P}}X$ is compact). A classification of extension properties is attempted, and several means of constructing extension properties are studied. A number of examples are considered in detail.
Nonlinear techniques for linear oscillation problems
Zeev
Nehari
387-406
Abstract: It is shown that for differential equations of the form ${y^{(n)}} + py = 0$ there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case $n = 2$. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation. If $ p$ is of constant sign, the coefficients of the ``Riccati systems'' are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.