Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu\sb{tt} = -Au + \mathcal{F}(u)$
Howard A.
Levine
1-21
Abstract: For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain $D \subseteq H$, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let $\mathcal{F}:D \to H$ be a continuously differentiable gradient operator with associated potential function $\mathcal{G}$. Assume that $ (x,\mathcal{F}(x)) \geq 2(2\alpha + 1)\mathcal{G}(x)$ for all $x \in D$ and some $\alpha > 0$. Let $ E(0) = \tfrac{1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})]$ where ${u_0} = u(0),{v_0} = {u_t}(0)$ and $u:[0,T) \to D$ be a solution to the equation in the title. The following statements hold: If $ \mathcal{G}({u_0}) > E(0)$, then ${\lim _{t \to {T^ - }}}(u,Pu) = + \infty$ for some $T < \infty $. If $ ({u_0},P{v_0}) > 0,0 < E(0) - \mathcal{G}({u_0}) < \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0})$ and if u exists on $ [0,\infty )$, then (u,Pu) grows at least exponentially. If $({u_0},P{v_0}) > 0$ and $\alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal{G}({u_0}) < \tfrac{1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0})$ and if the solution exists on $[0,\infty )$, then (u,Pu) grows at least as fast as ${t^2}$. A number of examples are given.
A divergent weighted orthonormal series of broken line Franklin functions
Coke S.
Reed
23-28
Abstract: The purpose of this paper is to define a differentiable function F and an inner product on the space of continuous functions on [0,1] in such a way that the Fourier expansion of F obtained by orthonormalizing the broken line Franklin functions according to this inner product is divergent.
A theory of Stiefel harmonics
Stephen S.
Gelbart
29-50
Abstract: An explicit theory of special functions is developed for the homogeneous space $ SO(n)/SO(n - m)$ generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on $n \times m$ matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of $SO(n)$ and the polynomial representations of $GL(m,{\mathbf{C}})$.
Lower semicontinuity of integral functionals
Leonard D.
Berkovitz
51-57
Abstract: It is shown that the integral functional $I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu$ is lower semicontinuous on its domain with respect to the joint strong convergence of $ {y_k} \to y$ in $ {L_p}(G)$ and the weak convergence of ${z_k} \to z$ in ${L_p}(G)$, where $1 \leq p \leq \infty$ and $1 \leq q \leq \infty$, under the following conditions. The function $f:(t,x,w) \to f(t,x,w)$ is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).
Sequence of regular finitely additive set functions
J. D.
Stein
59-66
Abstract: The purpose of this paper is to deduce versions of Phillips' lemma and the Vitali-Hahn-Saks theorem, with weaker conditions placed on the set functions and convergence conditions than is usually required.
Monotone decompositions of continua not separated by any subcontinua
Eldon J.
Vought
67-78
Abstract: Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type
Symmetric jump processes
Martin L.
Silverstein
79-96
Abstract: We use the theory of Dirichlet spaces to construct symmetric Markov processes of pure jump type and to identify the Lévy measures for these processes. Particular attention is paid to lattice and hard sphere systems which interact through speed change and exclusion.
Interpolation in a classical Hilbert space of entire functions
Robert M.
Young
97-114
Abstract: Let H denote the Paley-Wiener space of entire functions of exponential type $\pi$ which belong to ${L^2}( - \infty ,\infty )$ on the real axis. A sequence $\{ {\lambda _n}\}$ of distinct complex numbers will be called an interpolating sequence for H if $ TH \supset {l^2}$, where T is the mapping defined by $Tf = \{ f({\lambda _n})\} $. If in addition $\{ {\lambda _n}\}$ is a set of uniqueness for H, then $ \{ {\lambda _n}\}$ is called a complete interpolating sequence. The following results are established. If $ \operatorname{Re} ({\lambda _{n + 1}}) - \operatorname{Re} ({\lambda _n}) \geq \gamma > 1$ and if the imaginary part of ${\lambda _n}$ is sufficiently small, then $\{ {\lambda _n}\}$ is an interpolating sequence. If $\vert\operatorname{Re} ({\lambda _n}) - n\vert \leq L \leq (\log 2)/\pi \;( - \infty < n < \infty )$ and if the imaginary part of ${\lambda _n}$ is uniformly bounded, then $\{ {\lambda _n}\}$ is a complete interpolating sequence and $\{ {e^{i{\lambda _n}t}}\} $ is a basis for ${L^2}( - \pi ,\pi )$. These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.
Hereditary $QI$-rings
Ann K.
Boyle
115-120
Abstract: We consider in this paper rings in which every quasi-injective right R-module is injective. These rings will be referred to as right QI-rings. For a hereditary ring, this is equivalent to the condition that R be noetherian and a right V-ring. We also consider rings in which proper cyclic right R-modules are injective. These are right QI-rings which are either semisimple or right hereditary, right Ore domains in which indecomposable injective right R-modules are either simple or isomorphic to the injective hull of $ {R_R}$.
Countable box products of ordinals
Mary Ellen
Rudin
121-128
Abstract: The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many $ \sigma$-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product.
Representing measures and topological type of finite bordered Riemann surfaces
David
Nash
129-138
Abstract: A finite bordered Riemann surface $ \mathcal{R}$ with s boundary components and interior genus g has first Betti number $r = 2g + s - 1$. Let a be any interior point of $ \mathcal{R}$ and $ {e_a}$ denote evaluation at a on the usual hypo-Dirichlet algebra associated with $ \mathcal{R}$. We establish some connections between the topological and, more strongly, the conformal type of $\mathcal{R}$ and the geometry of ${\mathfrak{M}_a}$ the set of representing measures for ${e_a}$. For example, we show that if ${\mathfrak{M}_a}$ has an isolated extreme point, then $\mathcal{R}$ must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case $r = 2$.
Linear operators and vector measures
J. K.
Brooks;
P. W.
Lewis
139-162
Abstract: Compact and weakly compact operators on function spaces are studied. Those operators are characterized by properties of finitely additive set functions whose existence is guaranteed by Riesz representation theorems.
Global residues and intersections on a complex manifold
James R.
King
163-199
Abstract: This paper is the study of a class of forms $\eta$ on a complex manifold V which are smooth on $V - W$ and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form $\varphi$ on W of filtration s (no (p, q) components with $p < s$) is the residue of a form $\eta$ of filtration $s + d$ such that $d\eta$ is smooth on V. This result is used to construct global kernels for $\bar \partial$ which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of ${f^\ast}\eta$ where f is a rather general holomorphic map.
K\"ahler differentials and differential algebra in arbitrary characteristic
Joseph
Johnson
201-208
Abstract: Let L and K be differential fields with L an extension of K. It is shown how the module of Kähler differentials $ \Omega _{L/K}^1$ can be used to ``linearize'' properties of a differential field extension $L/K$. This is done without restriction on the characteristic p and yields a theory which for $p \ne 0$ is no harder than the case $p = 0$. As an application a new proof of the Ritt basis theorem is given.
The geometry of flat Banach spaces
R. E.
Harrell;
L. A.
Karlovitz
209-218
Abstract: A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.
Analytic equivalence in the disk algebra
Hugh E.
Warren
219-226
Abstract: The notion of analytically equivalent domains can be extended from the complex plane to commutative Banach algebras with identity. In $C(X)$ a domain equivalent to the unit ball must have a boundary that is in a certain sense continuous. This paper shows that in the disk algebra ``continuous'' must be replaced with ``analytic.'' These results set limits in the classical Riemann mapping theorem on how smoothly the mapping can respond to changes in the domain being mapped.
Groups, semilattices and inverse semigroups. I, II
D. B.
McAlister
227-244
Abstract: An inverse semigroup S is called proper if the equations $ea = e = {e^2}$ together imply ${a^2} = a$ for each a, $a,e \in S$. In this paper a construction is given for a large class of proper inverse semigroups in terms of groups and partially ordered sets; the semigroups in this class are called P-semigroups. It is shown that every inverse semigroup divides a P-semigroup in the sense that it is the image, under an idempotent separating homomorphism, of a full subsemigroup of a P-semigroup. Explicit divisions of this type are given for $\omega $-bisimple semigroups, proper bisimple inverse semigroups, semilattices of groups and Brandt semigroups.
A probabilistic approach to $H\sp{p}(R\sp{d})$
D.
Stroock;
S. R. S.
Varadhan
245-260
Abstract: The relationship between ${H^p}({R^d}),1 \leq p < \infty$, and the integrability of certain functionals of Brownian motion is established using the connection between probabilistic and analytic notions of functions with bounded mean oscillation. An application of this relationship is given in the derivation of an interpolation theorem for operators taking $ {H^1}({R^d})$ to ${L^1}({R^d})$.
Weighted norm inequalities for fractional integrals
Benjamin
Muckenhoupt;
Richard
Wheeden
261-274
Abstract: The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left\Vert{T_\gamma }f(x)V(x)\right\Vert _q \leq C\left\Vert f(x)V(x)\right\Vert _p$ where the functions are defined on ${R^n},0 < \gamma < n,1 < p < n/\gamma ,1/q = 1/p - \gamma /n$, C is a constant independent of f and ${T_\gamma }f(x) = \smallint f(x - y)\vert y{\vert^{\gamma - n}}dy$. The main result is that $ V(x)$ is such a function if and only if $\displaystyle {\left( {\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^q}dx} } \right... ...{\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^{ - p'}}dx} } \right)^{1/p'}} \leq K$ where Q is any n dimensional cube, $\vert Q\vert$ denotes the measure of Q, $q = \infty$ and a weighted version of the Sobolev imbedding theorem are also proved.
$H\sp{r,}\,\sp{\infty }(R)$- and $W\sp{r,\infty }(R)$-splines
Philip W.
Smith
275-284
Abstract: Let E be a subset of R the real line and $f:E \to R$. Necessary and sufficient conditions are derived for $\inf (\left\Vert{D^r}x\right\Vert _{{L^\infty }}:x{\vert _E} = f)$ to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in ${L^\infty }$. For finite partitions E it is shown that a solution to the $ {L^\infty }$ infimum problem can be obtained by solving $\inf (\left\Vert{D^r}x\right\Vert _{{L^p}}:x{\vert _E} = f)$ and letting p go to infinity. In this way it was discovered that solutions to the $ {L^\infty }$ problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the ${L^p}$ problem are called $ {H^{r,p}}$-splines and were studied extensively by Golomb in [3].
Convex hulls and extreme points of some families of univalent functions
D. J.
Hallenbeck
285-292
Abstract: The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on $( - 1,1)$. We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on $( - 1,1)$. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functions $f(z)$ analytic in the unit disk, normalized and satisfying $\alpha < 1$. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfying
The norm of the $L\sp{p}$-Fourier transform on unimodular groups
Bernard
Russo
293-305
Abstract: We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real line as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest. Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian.
A summation formula involving $\sigma (N)$
C.
Nasim
307-317
Abstract: The ${L^2}$ theories are known of the summation formula involving ${\sigma _k}(n)$, the sum of the kth power of divisors of n, as coefficients, for all k except $k = 1$. In this paper, techniques are used to overcome the extra convergence difficulty of the case $k = 1$, to establish a symmetric formula connecting the sums of the form $ \sum {\sigma _1}(n){n^{ - 1/2}}f(n)$ and $\sum {{\sigma _1}} (n){n^{ - 1/2}}g(n)$, where $ f(x)$ and $g(x)$ are Hankel transforms of each other.
The concordance diffeomorphism group of real projective space
Robert
Wells
319-337
Abstract: Let ${P_r}$ be r-dimensional real projective space with r odd, and let ${\pi _0}{\text{Diff}^ + }:{P_r}$ be the group of orientation preserving diffeomorphisms $ {P_r} \to {P_r}$ factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for $r \equiv 11 \bmod 16$ the group ${\pi _0}{\text{Diff}^ + }:{P_r}$ is isomorphic to the homotopy group ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$, where $ k = d{2^L} - r - 1$ with $L \geq \varphi ((r + 1)/2)$ and $d{2^L} \geq r + 1$. The function $\varphi$ is denned by $\varphi (l) = \{ i\vert < i \leq l,i \equiv 0,1,2,4 \bmod (8)\}$. The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism $t:{\pi _0}{\text{Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v)$ for $ r \equiv 11 \bmod 16$ and ${\Omega _{r + 1}}(v)$ a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup ${\Omega _{r + 1}}(v)$, and that this subgroup is isomorphic to ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$ as above. Then one reads off from Mahowald's tables of ${\pi _{n + m}}({P_\infty }/{P_{m - 1}})$ that $ {\pi _0}{\text{Diff}^ + }:{P_{11}} = {Z_2}$ and ${\pi _0}{\text{Diff}^ + }:{P_{27}} = 6{Z_2}$.
Fields of fractions for group algebras of free groups
Jacques
Lewin
339-346
Abstract: Let KF be the group algebra over the commutative field K of the free group F. It is proved that the field generated by KF in any Mal'cev-Neumann embedding for KF is the universal field of fractions $ U(KF)$ of KF. Some consequences are noted. An example is constructed of an embedding $ KF \subset D$ into a field D with $ D\;\not\simeq\;U(KF)$. It is also proved that the generalized free product of two free groups can be embedded in a field.
Further results on prime entire functions
Fred
Gross;
Chung Chun
Yang
347-355
Abstract: Let H denote the set of all the entire functions $f(z)$ of the form: $f(z) \equiv h(z){e^{p(z)}} + k(z)$ where $ p(z)$ is a nonconstant polynomial of degree m, and $h(\nequiv\;0)$, $k(\nequiv$ constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna's theory of meromorphic functions and some elementary facts about algebraic functions.
van Kampen's theorem for $n$-stage covers
J. C.
Chipman
357-370
Abstract: A version of van Kampen's theorem is obtained for covers whose members do not share a common point and whose pairwise intersection need not be connected.
Expansion of entire functions of several complex variables having finite growth
P. K.
Kamthan;
Manjul
Gupta
371-382
Abstract: We consider the space $\chi$ of entire functions of two complex variables having a finite nonzero order point and type, equip it with the natural locally convex topology, such that $\chi$ becomes a Fréchet space. Apart from finding the characterization of continuous linear functionals, linear transformations on $ \chi$, we have obtained the necessary and sufficient conditions for a double sequence in $\chi$ to be a proper base.