On ordinary linear $p$-adic differential equations
B.
Dwork;
P.
Robba
1-46
Abstract: We study the solutions of ordinary linear differential equations whose coefficients are analytic elements. As one application we show nonexistence of index for certain linear differential operators with rational function coefficients.
Associated and skew-orthologic simplexes
Leon
Gerber
47-63
Abstract: A set of $n + 1$ lines in n-space is said to be associated if every $ (n - 2)$-flat which meets n of the lines also meets the remaining line. Two Simplexes are associated if the joins of their corresponding vertices are associated. Two Simplexes are (skew-)orthologic if the perpendiculars from the vertices of one on the faces of the other are concurrent (associated); it follows that the reciprocal relation holds. In an earlier paper, Associated and Perspective Simplexes, we gave an affine necessary and sufficient condition for two simplexes to be associated that was so easy to apply that extensions to n-dimensions of nearly all known theorems, and a few new ones, were proved in a few lines of calculations. In this sequel we take a closer look at some of the results of the earlier paper and prove some new results. Then we give simple Euclidean necessary and sufficient conditions for two simplexes to be orthologic or skew-orthologic which yield as corollaries known results on altitudes, the Monge point and orthocentric simplexes. We conclude by discussing some of the qualitative differences between the geometries of three and higher dimensions.
Variation of conformal spheres by simultaneous sewing along several arcs
T. L.
McCoy
65-82
Abstract: Let M be a closed Riemann surface of genus zero, $\Gamma$ a tree on M with branches ${\Gamma _j}$, and ${p_0}$ a point of $M - \Gamma$. A family of neighboring topological surfaces $ M(\varepsilon )$ is formed by regarding each $ {\Gamma _j}$ as a slit with edges $\Gamma _j^ -$ and $\Gamma _j^ +$, and re-identifying p on ${\Gamma ^{{ - _j}}}$ with $ p + \varepsilon {\chi _j}(p,\varepsilon )$ on $ \Gamma _j^ +$, with $ {\chi _j}$ vanishing at the endpoints of $ {\Gamma _j}$. We assume the ${\Gamma _j}$ and ${\chi _j}$ are such that, under a certain natural choice of uniformizers, the $M(\varepsilon )$ are closed Riemann surfaces of genus zero. Then there exists a unique function $f(p,\varepsilon ;{p_0})$ mapping $M(\varepsilon )$ conformally onto the complex number sphere, with normalization $ \varepsilon$. Further, we obtain smoothness results for f as a function of $\varepsilon $. The problem is connected with the study of the extremal schlicht functions; that is, the schlicht mappings of the unit disc corresponding to boundary points of the coefficient bodies.
Asymptotic behavior of solutions of nonlinear functional differential equations in Banach space
John R.
Haddock
83-92
Abstract: Let X be a Banach space and let $C = C([ - r,0],X)$ denote the space of continuous functions from $[ - r,0]$ to X. In this paper the problem of convergence in norm of solutions of the nonlinear functional differential equation $\dot x = F(t,{x_t})$ is considered where $F:[0,\infty ) \times C \to X$. As a special case of the main theorem, stability results are given for the equation $\dot x(t) = f(t,x(t)) + g(t,{x_t})$, where $- f(t, \cdot ) - \alpha (t)I$ satisfies certain accretive type conditions and $g(t, \cdot )$ is Lipschitzian with Lipschitz constant $\beta (t)$ closely related to $\alpha (t)$.
Inequalities for polynomials on the unit interval
Q. I.
Rahman;
G.
Schmeisser
93-100
Abstract: Let $ {p_n}(z) = \sum\nolimits_{k = 0}^n {{a_k}{z^k}}$ be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if $ {p_n}(z)$ has at most $n - 1$ distinct zeros in $( - 1,1)$, then
A class of infinitely connected domains and the corona
W. M.
Deeb
101-106
Abstract: Let D be a bounded domain in the complex plane. Let ${H^\infty }(D)$ be the Banach algebra of bounded analytic functions on D. The corona problem asks whether D is weak$^\ast$ dense in the space $\mathfrak{M}(D)$ of maximal ideals of ${H^\infty }(D)$. Carleson [3] proved that the open unit disc ${\Delta _0}$ is dense in $\mathfrak{M}({\Delta _0})$. Stout [9] extended Carleson's result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens' idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques.
$D$-domains and the corona
W. M.
Deeb;
D. R.
Wilken
107-115
Abstract: Let D be a bounded domain in the complex plane C. Let ${H^\infty }(D)$ denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is weak$^\ast$ dense in the space $\mathfrak{M}(D)$ of maximal ideals of ${H^\infty }(D)$. In [2] Carleson proved that the unit disk ${\Delta _0}$ is dense in $\mathfrak{M}({\Delta _0})$. In [7] Stout extended Carleson's result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains. In this paper we extract some of the crucial ingredients of Behrens' methods and extend his results to a broader class of infinitely connected domains.
Multiplier criteria of Marcinkiewicz type for Jacobi expansions
George
Gasper;
Walter
Trebels
117-132
Abstract: It is shown how an integral representation for the product of Jacobi polynomials can be used to derive a certain integral Lipschitz type condition for the Cesàro kernel for Jacobi expansions. This result is then used to give criteria of Marcinkiewicz type for a sequence to be multiplier of type (p, p), $1 < p < \infty$, for Jacobi expansions.
Subgroups of finitely presented solvable linear groups
Michael W.
Thomson
133-142
Abstract: Let G be a finitely generated solvable linear group. It is shown that there exists a finitely presented solvable linear group H with G embedded in H.
Continuous dependence of solutions of operator equations. I
Zvi
Artstein
143-166
Abstract: Continuous dependence of the solutions of the operator equation $x = Tx + z$ in a topological vector space is the main subject of the paper. We find sufficient and necessary conditions for the continuous dependence on the data (T, z) or on a parameter. We do it for the space of all closed operators. Equivalent conditions for particular subfamilies are discussed. Among other families we deal with compact operators, compact perturbations of the identity, condensing operators and demicompact operators.
Conjugate points of vector-matrix differential equations
Roger T.
Lewis
167-178
Abstract: The system of equations $\displaystyle \sum\limits_{k = 0}^n {{{( - 1)}^{n - k}}{{\left( {{P_k}(x){y^{(n - k)}}(x)} \right)}^{(n - k)}}} = 0\quad (0 \leqslant x < \infty )$ is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and ${P_0}(x)$ is positive definite. It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.
On the transformation group of a real hypersurface
S. M.
Webster
179-190
Abstract: The group of biholomorphic transformations leaving fixed a strongly pseudoconvex real hypersurface in a complex manifold is a Lie group. In this paper it is shown that the Chern-Moser invariants must vanish if this group is noncompact and the hypersurface is compact. Also considered are transformation groups of flat hypersurfaces and intransitive groups.
Minimal invariant functions of the space-time Wiener process
Kai Yuen
Woo
191-200
Abstract: Minimal invariant functions of the space-time Wiener process are obtained.
An example where topological entropy is continuous
Louis
Block
201-213
Abstract: Let ent denote topological entropy, and let ${C^r}({S^1},{S^1})$ denote the space of continuous functions of the circle to itself having r continuous derivatives with the ${C^r}$ (uniform) topology. Let $ {f_0}$ denote a particular $ {C^2}$ map of the circle ($ {f_0}$ is the first bifurcation point one comes to in a bifurcation from a full three shift to a map with finite nonwandering set). The main results of this paper are the following: Theorem A. The map ent: $ {C^0}({S^1},{S^1}) \to R \cup \{ \infty \}$ is lower-semicontinuous at $ {f_0}$. Theorem B. The map ent: ${C^2}({S^1},{S^1}) \to R$ is continuous at $ {f_0}$. In proving these two theorems several general results on entropy of mappings of the circle are proved.
Functional calculus and positive-definite functions
Colin C.
Graham
215-231
Abstract: For a LCA group G with dual group Ĝ, let $D(G) = D(\hat G)$ denote the convex (not closed) hull of $\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}$. The set $D(G)$ is the natural domain for functions that operate by composition from the class, $P{D_1}(\hat G)$, of Fourier-Stieltjes transforms of probability measures on G to $B(\hat G)$, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of $ D(G)$. In §1, we show (1) if F operators from $P{D_1}(G)$ to $B(G)$ and G is compact, then $ K(z) = {\lim _{t \to {1^ - }}}F(tz)$ exists for all $ z \in D(G)$ and K operates from $ P{D_1}(\hat G)$ to $ B(\hat G)$; (2) if F operates from $ P{D_1}(\hat G)$ to $ PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)$ and G is compact, then K operates from $ P{D_1}(\hat G)$ to $PD(\hat G)$, and so also does $F - K$; (3) if $G = {{\mathbf{D}}_q},q \geqslant 2$, and F operates from $ P{D_1}(\hat G)$ to $ B(\hat G)$, then $ F = K$ on $D(G) \cap \{ z:\vert z\vert < 1\}$. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from $P{D_1}(\hat G)$ to $B(\hat G)$. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.
Almost sure behavior of linear functionals of supercritical branching processes
Søren
Asmussen
233-248
Abstract: The exact a.s. behavior of any linear functional ${Z_n} \cdot a$ of a supercritical positively regular p-type $ (1 < p < \infty )$ Galton-Watson process $ \{ {Z_n}\}$ is found under a second moment hypothesis. The main new results are of iterated logarithm type, with normalizing constants depending on the decomposition of a according to the Jordan canonical form of the offspring mean matrix.
Of regulated and steplike functions
Gadi
Moran
249-257
Abstract: Let C denote the class of regulated real-valued functions on the unit interval vanishing at the origin, whose positive and negative jumps sum to infinity in every nontrivial subinterval of I. Goffman [2] showed that every f in C is (essentially) a sum $g + s$ where g is continuous and s is steplike. In this sense, a function in C is like a function of bounded variation, that has a unique such g and s. The import of this paper is that for f in C the representation $f = g + s$ is not only not unique, but by far the opposite holds: g can be chosen to be any continuous function on I vanishing at 0, at the expense of a rearrangement of s.
Pointwise and norm convergence of a class of biorthogonal expansions
Harold E.
Benzinger
259-271
Abstract: Let $\{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1$, be sequences of functions in $ {L^\infty }(0,1)$, such that $ ({u_k},{v_j}) = {\delta _{kj}}$. Let ${\phi _k}(x) = \exp \;2k\pi ix$. It is shown that if for a given p, $1 < p < \infty$, the sequence $\{ {u_k}\}$ is complete in ${L^p}(0,1)$, and $ \{ {v_k}\}$ is complete in ${L^q}(0,1),pq = p + q$, and if the $ {u_k}$'s, ${v_j}$'s are asymptotically related to the ${\phi _k}$'s, in a sense to be made precise, then $\{ {u_k}\}$ is a basis for ${L^p}(0,1)$, equivalent to the basis $\{ {\phi _k}\}$, and for every f in ${L^p}(0,1)$ a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.
Remarks on ``Embedding theorems and generalized discrete ordered abelian groups'' (Trans. Amer. Math. Soc. {\bf 175} (1973), 283--297) by P. Hill and J. L. Mott
Isidore
Fleischer
273-274