Using flows to construct Hilbert space factors of function spaces
James
Keesling
1-24
Abstract: Let X and Y be metric spaces. Let $G(X)$ be the group of homeomorphisms of X with the compact open topology. The main result of this paper is that if X admits a nontrivial flow, then $G(X)$ is homeomorphic to $G(X) \times {l_2}$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The techniques are applied to other function spaces with the same result. Two such spaces for which our techniques apply are the space of imbeddings of X into Y, $E(X,Y)$, and the space of light open mappings of X into (or onto) Y, LO (X, Y). Some applications of these results are given. The paper also uses flows to show that if X is the $\sin (1/x)$-curve, then $G(X)$ is homeomorphic to ${l_2} \times N$, where N is the integers.
Oscillation properties of two term linear differential equations
G. A.
Bogar
25-33
Abstract: The two term differential equations $ {L_n}[y] + py = 0$, where $ \rho _1^{ - 1}(t)p(t)$ on $[a,\infty )$ when ${L_n}[y]$ is disconjugate. By changing the integral conditions slightly we then prove that the equation has n linearly independent oscillatory solutions.
Higher dimensional knots in tubes
Yaichi
Shinohara
35-49
Abstract: Let K be an n-knot in the $(n + 2)$-sphere and V a tubular neighborhood of K. Let $L'$ be an n-knot contained in a tubular neighborhood $V'$ of a trivial n-knot and L the image of $L'$ under an orientation preserving diffeomorphism of $ V'$ onto V. The purpose of this paper is to show that the higher dimensional Alexander polynomial and the signature of the n-knot L are determined by those of K and $L'$.
Degree of symmetry of a homotopy real projective space
H. T.
Ku;
L. N.
Mann;
J. L.
Sicks;
J. C.
Su
51-61
Abstract: The degree of symmetry $N(M)$ of a compact connected differentiable manifold M is the maximum of the dimensions of the compact Lie groups which can act differentiably and effectively on it. It is well known that $N(M) \leqq \dim \; SO(m + 1)$, for an m-dimensional manifold, and that equality holds only for the standard m-sphere and the standard real projective m-space. W. Y. Hsiang has shown that for a high dimensional exotic m-sphere M, $ N(M) < {m^2}/8 + 1 < \left( {\frac{1}{4}} \right)\dim SO(m + 1)$, and that $ N(M) = {m^2}/8 + 7/8$ for some exotic m-spheres. It is shown here that the same results are true for exotic real projective spaces.
Well distributed sequences of integers
William A.
Veech
63-70
Abstract: Niven's notion of a uniformly distributed sequence of integers is generalized to well distribution, and two classes of integer sequences are studied in terms of this generalization.
Studies in the representation theory of finite semigroups
Yechezkel
Zalcstein
71-87
Abstract: This paper is a continuation of [14], developing the representation theory of finite semigroups further. The main result, Theorem 1.24, states that if the group of units U of a mapping semigroup (X, S) is multiply transitive with a sufficiently high degree of transitivity, then for certain irreducible characters $\chi$ of U, $\chi$ can be ``extended'' formally to an irreducible character of S. This yields a partial generalization of a well-known theorem of Frobenius on the characters of multiply-transitive groups and provides the first nontrivial explicit formula for an irreducible character of a finite semigroup. The paper also contains preliminary results on the ``spectrum'' (i.e., the set of ranks of the various elements) of a mapping semigroup.
Implicitly defined mappings in locally convex spaces
Terrence S.
McDermott
89-99
Abstract: Results on existence, uniqueness, continuity and differentiability of implicit functions in locally convex, linear topological spaces are obtained, and certain of these results are applied to obtain results on the existence and continuous dependence on parameters of global solutions for a nonlinear Volterra integral equation.
On the Wedderburn principal theorem for nearly $(1,\,1)$ algebras
T. J.
Miles
101-110
Abstract: A nearly $ (1,1)$ algebra is a finite dimensional strictly power-associative algebra satisfying the identity $(x,x,y) = (x,y,x)$ where the associator $(x,y,z) = (xy)z - x(yz)$. An algebra A has a Wedderburn decomposition in case A has a subalgebra $S \cong A - N$ with $A = S + N$ (vector space direct sum) where N denotes the radical (maximal nil ideal) of A. D. J. Rodabaugh has shown that certain classes of nearly $(1,1)$ algebras have Wedderburn decompositions. The object of this paper is to expand these classes. The main result is that a nearly $(1,1)$ algebra A containing 1 over a splitting field of characteristic not 2 or 3 such that A has no nodal subalgebras has a Wedderburn decomposition.
On the differentiability of arbitrary real-valued set functions. II.
Harvel
Wright;
W. S.
Snyder
111-122
Abstract: Let f be a real-valued function defined and finite on sets from a family $\mathcal{F}$ of bounded measurable subsets of Euclidean n-space such that if $T \in \mathcal{F}$, the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.
Repairing embeddings of $3$-cells with monotone maps of $E\sp{3}$
William S.
Boyd
123-144
Abstract: If ${S_1}$ is a 2-sphere topologically embedded in Euclidean 3-space ${E^3}$ and ${S_2}$ is the unit sphere about the origin, then there may not be a homeomorphism of ${E^3}$ onto itself carrying ${S_1}$ onto ${S_2}$. We show here how to construct a map f of ${E^3}$ onto itself such that $f\vert{S_1}$ is a homeomorphism of $ {S_1}$ onto ${S_2}$, $f({E^3} - {S_1}) = {E^3} - {S_2}$ and ${f^{ - 1}}(x)$ is a compact continuum for each point x in ${E^3}$. Similar theorems are obtained for 3-cells and disks topologically embedded in ${E^3}$.
Borel measurable mappings for nonseparable metric spaces
R. W.
Hansell
145-169
Abstract: The main object of this paper is the extension of part of the basic theory of Borel measurable mappings, from the ``classical'' separable metric case, to general metric spaces. Although certain results of the standard theory are known to fail in the absence of separability, we show that they continue to hold for the class of ``$\sigma $-discrete'' mappings. This class is shown to be quite extensive, containing the continuous mappings, all mappings with a separable range, and any Borel measurable mappings whose domain is a Borel subset of a complete metric space. The last result is a consequence of our Basic Theorem which gives a topological characterization of those collections which are the inverse image of an open discrete collection under a Borel measurable mapping. Such collections are shown to possess a strong type of $ \sigma$-discrete refinement. The properties of $\sigma$-discrete mappings together with the known properties of ``locally Borel'' sets allow us to extend, to general metric spaces, well-known techniques used for separable spaces. The basic properties of ``complex'' and ``product'' mappings, well known for separable spaces, are proved for general metric spaces for the class of $\sigma$-discrete mappings. A consequence of these is a strengthening of the basic theorem of the structure theory of nonseparable Borel sets due to A. H. Stone. Finally, the classical continuity properties of Borel measurable mappings are extended, and, in particular, a generalization of the famous theorem of Baire on the points of discontinuity of a mapping of class 1 is obtained.
A characterization of the equicontinuous structure relation
Robert
Ellis;
Harvey
Keynes
171-183
Abstract: The main result in the paper is to show that in a large class of minimal transformation groups (including those with abelian phase groups, and point-distal transformation groups), the equicontinuous structure relation is precisely the regionally proximal relation. The techniques involved enable one to recover and extend the previously known characterizations. Several corollaries are indicated, among which the most important is a new criterion (which is easily applicable) for the existence of a nontrivial equicontinuous image of a given transformation group.
The space of all self-homeomorphisms of a two-cell which fix the cell's boundary is an absolute retract
W. K.
Mason
185-205
Abstract: The theorem mentioned in the title is proved. A corollary of the title theorem is: any homeomorphism between two compact subsets of the function space mentioned in the title can be extended to a homeomorphism of the function space onto itself.
The bifurcation of solutions in Banach spaces
William S.
Hall
207-218
Abstract: Let $L:D \subset X \to D \subset {X^ \ast }$ be a densely defined linear map of a reflexive Banach space X to its conjugate ${X^\ast}$. Define M and $ {M^\ast}$ to be the respective null spaces of L and its formal adjoint $ {L^\ast}$. Let $f:X \to {X^\ast}$ be continuous. Under certain conditions on ${L^\ast}$ and f there exist weak solutions to $Lu = f(u)$ provided for each $w \in X,v(w) \in M$ can be found such that $f(v(w) + w)$ annihilates $ {M^ \ast }$. Neither M and ${M^\ast}$ nor their annihilators need be the ranges of continuous linear projections. The results have applications to periodic solutions of partial differential equations.
Prime entire functions
Fred
Gross
219-233
Abstract: Factorizations of various functions are discussed. Complete factorizations of certain classes of functions are given. In particular it is shown that there exist primes of arbitrary growth.
The powers of a maximal ideal in a Banach algebra and analytic structure
T. T.
Read
235-248
Abstract: Sufficient conditions are given for the existence of an analytic variety at an element $\phi$ of the spectrum of a commutative Banach algebra with identity. An associated graded algebra first considered by S. J. Sidney is used to determine the dimension of the analytic variety in terms of the closed powers of the maximal ideal which is the kernel of $\phi$.
Weighted norm inequalities for singular and fractional integrals
Benjamin
Muckenhoupt;
Richard L.
Wheeden
249-258
Abstract: Inequalities of the form ${\left\Vert {{{\left\vert x \right\vert}^\alpha }Tf} \right\Vert _q} \leqq C{\left\Vert {{{\left\vert x \right\vert}^\alpha }f} \right\Vert _p}$ are proved for certain well-known integral transforms, T, in ${E^n}$. The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.
The two-piece property and tight $n$-manifolds-with-boundary in $E\sp{n}$
Thomas F.
Banchoff
259-267
Abstract: The two-piece property for a set A is a generalization of convexity which reduces to the condition of minimal total absolute curvature if A is a compact 2-manifold. We show that a connected compact 2-manifold-with-boundary in ${E^2}$ has the TPP if and only if each component of the boundary has the TPP. The analogue of this result is not true in higher dimensions without additional conditions, and we introduce a stronger notion called k-tightness and show that an $(n + 1)$-manifold-with-boundary ${M^{n + 1}}$ embedded in $ {E^{n + 1}}$ is 0- and $ (n - 1)$-tight if and only if its boundary is 0- and $(n - 1)$-tight.
Local theory of complex functional differential equations
Robert J.
Oberg
269-281
Abstract: We consider the equation $g({z_0}) = {z_0}$. We classify fixed points $ {z_0}$ of g as attractive if $ f({z_0}) = {w_0}$. This solution depends continuously on ${w_0}$ and on the functions F and g. For ``most'' indifferent fixed points the initial-value problem has a unique solution. Around a repulsive fixed point a solution in general does not exist, though in exceptional cases there may exist a singular solution which disappears if the equation is subjected to a suitable small perturbation.
Almost-arithmetic progressions and uniform distribution
H.
Niederreiter
283-292
Abstract: In a recent paper, P. E. O'Neil gave a new criterion for uniform distribution modulo one in terms of almost-arithmetic progressions. We investigate the relation between almost-arithmetic progressions and uniformly distributed sequences from a quantitative point of view. An upper bound for the discrepancy of almost-arithmetic progressions is given which is shown to be best possible. Estimates for more general sequences are also obtained. As an application, we prove a quantitative form of Fejér's theorem on the uniform distributivity of slowly increasing sequences.
Monofunctors as reflectors
Claus Michael
Ringel
293-306
Abstract: In a well-powered and co-well-powered complete category $\mathcal{K}$ with weak amalgamations, the class M of all reflective subcategories with a monofunctor as reflector forms a complete lattice; the limit-closure of the union of any class of elements of M belongs to M. If $\mathcal{K}$ has injective envelopes, then the set-theoretical intersection of any class of elements of M belongs to M.
Singular integrals and fractional powers of operators
Michael J.
Fisher
307-326
Abstract: Recently R. Wheeden studied a class of singular integral operators, the hypersingular integrals, as operators from $L_p^\alpha (H)$ to ${L_p}(H);L_p^\alpha (H)$ is the range of the $ \alpha$th order Bessel potential operator acting on ${L_p}(H)$ with the inherited norm. The purposes of the present paper are to extend the known results on hypersingular integrals to complex indices, to extend these results to operators defined over a real separable Hilbert space, and to use Komatsu's theory of fractional powers of operators to show that the hypersingular integral operator $ {G^\alpha }$ is ${\smallint _H}{( - {A_y})^\alpha }f\,d\mu (y)$ when ${\mathop{\rm Im}\nolimits} (\alpha ) \ne 0$ or when ${\mathop{\Re}\nolimits} (\alpha )$ is not a positive integer where ${A_y}g$ is the derivative of g in the direction y. The case where ${\mathop{\rm Im}\nolimits} (\alpha ) = 0$ and $ {\mathop{\Re}\nolimits} (\alpha )$ is a positive integer is treated in a sequel to the present paper.
Functions of finite $\lambda $-type in several complex variables
Robert O.
Kujala
327-358
Abstract: If $ \lambda :{{\bf {R}}^ + } \to {{\bf {R}}^ + }$ is continuous and increasing then a meromorphic function f on ${C^k}$ is said to be of finite $ \lambda$-type if there are positive constants s, A, B and R such that $ {T_f}(r,s) \leqq A\lambda (Br)$ for all $r > R$ where $ {T_f}(r,s)$ is the characteristic of f. It is shown that if $\lambda (Br)/\lambda (r)$ is bounded for r sufficiently large and $B > 1$, then every meromorphic function of finite $\lambda$-type is the quotient of two entire functions of finite $\lambda$-type. This theorem is the result of a careful and detailed analysis of the relation between the growth of a function and the growth of its divisors. The central fact developed in this connection is: A nonnegative divisor $\nu$ on ${C^k}$ with $ \nu ({\bf {0}}) = 0$ is the divisor of an entire function of finite $ \lambda$-type if and only if there are positive constants A, B and R such that \begin{displaymath}\begin{array}{*{20}{c}} {{N_{\nu \vert\xi }}(r) \leqq A\lambd... ...ambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \end{array}\end{displaymath} for all $r \geqq s > R$, all unit vectors $\xi$ in ${C^k}$, and all natural numbers p. Here $ \nu \vert\xi$ represents the lifting of the divisor $\nu$ to the plane via the map $z \mapsto z\xi$ and ${N_{\nu \vert\xi }}$ is the valence function of that divisor. Analogous facts for functions of zero $ \lambda$-type are also presented.
The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains
Michael Frederick
Behrens
359-379
Closure theorems with applications to entire functions with gaps
J. M.
Anderson;
K. G.
Binmore
381-400
Abstract: In this paper we consider questions of completeness for spaces of continuous functions on a half line which satisfy appropriate growth conditions. The results obtained have consequences in the theory of entire functions with gap power series. In particular we show that, under an appropriate gap hypothesis, the rate of growth of an entire function in the whole plane is determined by its rate of growth along any given ray.
Degenerate evolution equations in Hilbert space
Avner
Friedman;
Zeev
Schuss
401-427
Abstract: We consider the degenerate evolution equation ${c_1}(t)du/dt + {c_2}(t)A(t)u = f(t)$ in Hilbert space, where ${c_1} \geqq 0,{c_2} \geqq 0,{c_1} + {c_2} > 0;A(t)$ is an unbounded linear operator satisfying the usual conditions which ensure that there is a unique solution for the Cauchy problem $du/dt + A(t)u = f(t){\rm {in}}(0,T],u(0) = {u_0}$. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.
$\sp{\ast} $-taming sets for crumpled cubes. I. Basic properties
James W.
Cannon
429-440
Abstract: Is a surface in a 3-manifold tame if it is tame modulo a tame set? This question was answered by the author through the introduction and characterization of taming sets. The purpose of this paper is to introduce and establish the basic properties of the more general and more flexible, but closely related, $^ \ast$-taming set.
$\sp{\ast} $-taming sets for crumpled cubes. II. Horizontal sections in closed sets
James W.
Cannon
441-446
Abstract: We prove that a closed subset X of ${E^3}$ is a $^ \ast$-taming set if no horizontal section of X has a degenerate component. This implies, for example, that a 2-sphere S in ${E^3}$ is tame if no horizontal section of S has a degenerate component. It also implies (less obviously) that a 2-sphere S in $ {E^3}$ is tame if it can be touched at each point from each side of S by a pencil.
$\sp{\ast} $-taming sets for crumpled cubes. III. Horizontal sections in $2$-spheres
James W.
Cannon
447-456
Abstract: We prove that a 2-sphere S in ${E^3}$ is tame if each horizontal section of S has at most four components. Since there are wild spheres in ${E^3}$ whose horizontal sections have at most five components, this result is, in a sense, best possible. Much can nevertheless be said, however, even if certain sections have more than five components; and we show that the wildness of a 2-sphere S in $ {E^3}$ is severely restricted by the requirement that each of the horizontal sections of S have at most finitely many components that separate S.
Endomorphism rings of torsionless modules
Arun Vinayak
Jategaonkar
457-466
Abstract: Let A be a right order in a semisimple ring $\Sigma ,{M_A}$ be a finite-dimensional torsionless right A-module and $ {\hat M_A}$ be the injective hull of M. J. M. Zelmanowitz has shown that $Q = {\rm {End}}\;{\hat M_A}$ is a semisimple ring and $S = {\rm {End}}\;{M_A}$ is a right order in Q. Further, if A is a two-sided order in $ \Sigma$ then S is a two-sided order in Q. We give a conceptual proof of this result. Moreover, we show that if A is a bounded order then so is S. The underlying idea of our proofs is very simple. Rather than attacking $S = {\rm {End}}\;{M_A}$ directly, we prove the results for $B = {\rm {End}}\;({M_A} \oplus {A_A})$. If $e:{M_A} \oplus {A_A} \to {M_A} \oplus {A_A}$ is the projection on M along ${A_A}$ then, of course, $S \cong eBe$ and it is easy to transfer the required information from B to S. The reason why it is any easier to look at B rather than S is that $ {M_A} \oplus {A_A}$ is a generator in $ \bmod$-$A$ and a Morita type transfer of properties from A to B is available. We construct an Artinian ring resp. Noetherian prime ring containing a right ideal whose endomorphism ring fails to be Artinian resp. Noetherian from either side.
A generalization of the strict topology
Robin
Giles
467-474
Abstract: The strict topology $ \beta$ on the space $ C(X)$ of bounded real-valued continuous functions on a topological space X was defined, for locally compact X, by Buck (Michigan Math. J. 5 (1958), 95-104). Among other things he showed that (a) $C(X)$ is $\beta$-complete, (b) the dual of $C(X)$ under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for $\beta$-closed subalgebras of $ C(X)$. In this paper the definition of the strict topology is generalized to cover the case of an arbitrary topological space and these results are established under the following conditions on X: for (a) X is a k-space; for (b) X is completely regular; for (c) X is unrestricted.