Monotone approximation by algebraic polynomials
G. G.
Lorentz;
K. L.
Zeller
1-18
Abstract: A given real continuous function f on [a, b] is approximated by polynomials ${P_n}$ of degree n that are subject to certain restrictions. Let $1 \leqq {k_1} < \cdots < {k_p} \leqq n$ be given integers, ${\varepsilon _i} = \pm 1$, given signs. It is assumed that $ P_n^{({k_i})}(x)$ has the sign of ${\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b$. Theorems are obtained which describe the polynomials of best approximation, and (for $p = 1$) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where $ \vert f(x) - {P_n}(x)\vert$ attains its maximum, and the sets ${B_i}$ with $P_n^{({k_i})}(x) = 0$. Conditions are discussed which these sets must satisfy for a polynomial $ {P_n}$ of best approximation for f. Numbers of the points of sets A, $ {B_i}$ are studied, the possibility of certain extreme situations established. For example, if $ p = 1,{k_1} = 1,n = 2q + 1$, it is possible that $ \vert A\vert = 3,\vert B\vert = n$.
Separability of metric spaces
Prabir
Roy
19-43
The pseudo-circle is unique
Lawrence
Fearnley
45-64
On the Mann iterative process
W. G.
Dotson
65-73
On the maximality of sums of nonlinear monotone operators
R. T.
Rockafellar
75-88
On Wiener process sample paths
G. J.
Foschini;
R. K.
Mueller
89-93
Abstract: Let $\{ {X_t}(\omega )\}$ represent a version of the Wiener process having almost surely continuous sample paths on $( - \infty ,\infty )$ that vanish at zero. We present a theorem concerning the local nature of the sample paths. Almost surely the local behavior at each t is of one of seven varieties thus inducing a partition of $( - \infty ,\infty )$ into seven disjoint Borel sets of the second class. The process $\{ {X_t}(\omega )\}$ can be modified so that almost surely the sample paths are everywhere locally recurrent.
Generic bifurcation of periodic points
K. R.
Meyer
95-107
Abstract: This paper discusses the bifurcation of periodic points of a generic symplectic diffeomorphism of a two manifold that depends on a parameter. A complete classification of the types of bifurcation that can occur in the generic case is given.
Inequalities satisfied by entire functions and their derivatives
Boo-sang
Lee;
S. M.
Shah
109-117
Abstract: For a class of entire functions with simple and positive zeros, it is shown that the maximum of the moduli of the first two Taylor coefficients at any point z, dominate all the remaining Taylor coefficients, provided $ \vert z\vert$ is sufficiently large. Further, there is a subclass for which this result holds at every point z.
Two point boundary problems for second order matrix differential systems
Garret J.
Etgen
119-132
Abstract: This paper is concerned with second order matrix differential systems involving a parameter together with boundary conditions specified at two points. The object of the paper is to establish sufficient conditions for the existence of eigenvalues for the system. Although such problems have been considered using the results of and techniques from the calculus of variations, the methods and results here are entirely in the context of ordinary differential equations. Use is made of the matrix generalization of the polar coordinate transformation introduced by J. H. Barrett and the unitary transformation suggested by F. V. Atkinson and V. A. Jakubovič. The sufficient conditions for the existence of eigenvalues obtained here represent certain extensions of W. M. Whyburn's work concerning linear and nonlinear boundary problems for second order differential systems.
A noncommutative Hilbert basis theorem and subrings of matrices
S. A.
Amitsur
133-142
Abstract: A finitely generated central extension $A[{u_1}, \ldots ,{u_k}]$ of a commutative noetherian ring A, satisfies the ascending chain condition for ideals P for which $A[{u_1}, \ldots ,{u_k}]/P$ can be embedded in matrix rings ${M_n}(K)$ over arbitrary commutative rings K and n bounded. The method of proof leads to an example of a ring R which satisfies the same identities of ${M_n}(K)$ but nevertheless cannot be embedded in any matrix ring over a commutative ring of arbitrary finite order.
Balanced rings and a problem of Thrall
Victor P.
Camillo
143-153
Abstract: Balanced ring is defined and related to Thrall's QF-1 rings. Several theorems are obtained which show that balanced rings enjoy strong homological and chain conditions. The structure of commutative balanced rings is determined. Also, the structure of commutative artinian QF-1 rings is gotten. This is a generalization of a theorem of Floyd.
Noncompact simplices
S.
Simons
155-161
Abstract: A bounded, but not necessarily closed, (Choquet) simplex in $ {R^n}$ with nonempty interior is the intersection of $n + 1$ half-spaces. There is no bounded simplex with nonempty interior in an infinite dimensional Hausdorff real linear topological space.
Inseparable Galois theory of exponent one
Shuen
Yuan
163-170
Abstract: An exponent one inseparable Galois theory for commutative ring extensions of prime characteristic $p \ne 0$ is given in this paper.
Markuschevich bases and duality theory
William B.
Johnson
171-177
Abstract: Several duality theorems concerning Schauder bases in locally convex spaces have analogues in the theory of Markuschevich bases. For example, a locally convex space with a Markuschevich basis is semireflexive iff the basis is shrinking and boundedly complete. The strong existence Theorem III.1 for Markuschevich bases allows us to show that a separable Banach space is isomorphic to a conjugate space iff it admits a boundedly complete Markuschevich basis, and that a separable Banach space has the metric approximation property iff it admits a Markuschevich basis which is a generalized summation basis in the sense of Kadec.
Identities involving the coefficients of a class of Dirichlet series. IV
Bruce C.
Berndt
179-185
Abstract: We consider a class of Dirichlet series satisfying a functional equation with gamma factors. We define a generalized Dirichlet series that is analogous to the generalized zeta-function of Riemann. An analytic continuation for these generalized series is derived, and a few simple properties are established. Secondly, we prove a theorem on the Abel summation of Dirichlet series that satisfy Hecke's functional equation.
The continuity of functions on Cartesian products
N.
Noble
187-198
Zero-one laws for Gaussian processes
G.
Kallianpur
199-211
Abstract: Some zero-one laws are proved for Gaussian processes defined on linear spaces of functions. They are generalizations of a result for Wiener measure due to R. H. Cameron and R. E. Graves. The proofs exploit an interesting relationship between a Gaussian process and its reproducing kernel Hilbert space. Applications are discussed.
Harmonic analysis on certain vector spaces
J.
Kuelbs;
V.
Mandrekar
213-231
The asymptotic manifolds of a perturbed linear system of differential equations
T. G.
Hallam;
J. W.
Heidel
233-241
Quadratic variation of potentials and harmonic functions
Gunnar A.
Brosamler
243-257
Abstract: We prove the existence of a finite quadratic variation for stochastic processes $u(Y)$, where Y is Brownian motion on a Green domain of ${R^n}$, stopped upon reaching the Martin boundary, and u is a positive superharmonic function on the domain. As by-products we have results which are also of interest from a non-probabilistic point of view.
An elementary theory of the category of topological spaces
Dana I.
Schlomiuk
259-278
Abstract: An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category of topological spaces and continuous mappings and to prove that any model of these axioms is equivalent to ``the category of topological spaces'' constructed over some model of Lawvere's axioms. Furthermore, we prove that any complete category, model of the given axioms is equivalent to the category of topological spaces.
State spaces for Markov chains
J. L.
Doob
279-305
Abstract: If $p(t,i,j)$ is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with $p(0 + ,i,i) = 1$, entrance and exit spaces for p are defined. If $L[{L^ \ast }]$ is an entrance [exit] space, the function $p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )]$ has a continuous extension to $ (0,\infty ) \times L[(0,\infty ) \times {L^ \ast }$, for a certain norming function h on $ {L^ \ast }$]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b.
Extension methods in cardinal arithmetic
Erik
Ellentuck
307-325
Abstract: Functions (relations) defined on the nonnegative integers are extended to the cardinal numbers by the method of Myhill (Nerode) respectively. We obtain various results relating these extensions and conclude with an analysis of AE Horn sentences interpreted in the cardinal numbers. Let $ \mathfrak{A}$ be the sentence $(\forall {x_1}) \cdots (\forall {x_n})(\exists !y)\mathfrak{b}$ where quantifiers are restricted to the Dedekind cardinals and $ \mathfrak{b}$ is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that $ \mathfrak{A}$ is a theorem of set theory (with the axiom of choice replaced by the axiom of choice for sets of finite sets) if and only if we can prove that the uniquely determined Skolem function for $ \mathfrak{A}$ extends an almost combinatorial function.
Representations of certain compact semigroups by ${\rm HL}$-semigroups
J. H.
Carruth;
C. E.
Clark
327-337
Abstract: An HL-semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each $ \mathcal{H}$-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S.
Two-sided semisimple maximal quotient rings
Vasily C.
Cateforis
339-349
Abstract: Let R be an associative ring with singular right ideal zero and finite right Goldie dimension; F. L. Sandomierski has shown that the (R. E. Johnson) maximal right quotient ring Q of R is then semisimple (artinian). In this paper necessary and sufficient conditions are sought that Q be also a left (necessarily the maximal) quotient ring of R. Flatness of Q as a right R-module is shown to be such a condition. The condition that R have singular left ideal zero and finite left Goldie dimension, though necessary, is shown to be not sufficient in general. Conditions of two-sidedness of Q are also obtained in terms of the homogeneous components (simple subrings) of Q and the subrings of R, they induce.