Infinitesimally rigid polyhedra. I. Statics of frameworks
Walter
Whiteley
431-465
Abstract: From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in $3$-space, detailed analogues of Alexandrov's theorem are presented for convex $4$-polytopes built as bar and joint frameworks in $4$-space.
On the realization of invariant subgroups of $\pi \sb\ast (X)$
A.
Zabrodsky
467-496
Abstract: Let $p$ be a prime and $T:X \to X$ a self map. Let $A$ be a multiplicatively closed subset of the algebraic closure of ${F_p}$. Denote by ${V_{T,A}}$ the set of characteristic values of $ {\pi_{\ast} }(T) \otimes {F_p}$ lying in $A$. It is proved that under certain conditions $ {V_{T,A}}$ is realizable by a pair $ \tilde X,\tilde T$: There exist a space $\tilde X$, maps $\tilde T:\tilde X \to \tilde X$ and $f:\tilde X\: \to \:X$ so that $ f\,\circ \,\tilde T\sim T\,\circ \,f,{\pi _ * }(F)$ is $\bmod\, p$ injective and $ {\rm {im}}({\pi_{\ast} }(f) \otimes {F_p}) = {V_{T,A}}$. This theorem yields, among others, examples of spaces whose $ \bmod\, p$ cohomology rings are polynomial algebras.
On Kummer's twenty-four solutions of the hypergeometric differential equation
B.
Dwork
497-521
Abstract: The $p$-adic analyticity of the Boyarsky matrix associated with the hypergeometric function $F(a,b;c;x)$ has been investigated in an earlier article. The transformation of this matrix under translation of $(a, b, c)$ by $ {{\bf {Z}}^3}$ was determined at that time. This article gives the transformation of this matrix under the extended Kummer group. The $p$-adic implications of quadratic and higher-degree transformations remain open.
Normal structure and weakly normal structure of Orlicz sequence spaces
Thomas
Landes
523-534
Abstract: For a convex Orlicz function $\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}$ and the associated Orlicz sequence space $ {l_\varphi }$, we consider the following five properties: (1) ${l_\varphi }$ has a subspace isometric to $ {l_1}$. (2) ${l_\varphi }$ is Schur. (3) ${l_\varphi }$ has normal structure. (4) Every weakly compact subset of $ {l_\varphi }$ has normal structure. (5) Every bounded sequence in ${l_\varphi }$ has a subsequence $({x_n})$ which is pointwise and almost convergent to $ x \in {l_\varphi }$, i.e., $ \lim \,{\sup_{n \to \infty }}\parallel {x_n} - x{\parallel_{\varphi }} < \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel_\varphi }$ for all $y \ne x$. Our results are: (1) $ \Leftrightarrow \;\varphi$ is either linear at $0\;(\varphi (s)/s = c > 0,0 < s \leqslant t)$ or does not satisfy the $ {\Delta_2}$-condition at 0. (2) $\Leftrightarrow \;{l_\varphi }$ is isomorphic to ${l_1}\; \Leftrightarrow \;\varphi^{\prime}(0) = {\lim_{t \to 0}}\,\varphi \,(t)/t > 0$. (3) $\Leftrightarrow \varphi$ satisfies the ${\Delta_2}$-condition at $ 0, \varphi$ is not linear at 0 and $ C(\varphi ) = \sup \,\{ \varphi \,(t) < 1\} > \frac{1}{2}$. (4) $ \Leftrightarrow \,\varphi$ satisfies the $ {\Delta_2}$-condition at 0 and $C\,(\varphi ) > \frac{1}{2}\;{\rm {or}}\;\varphi^{\prime}(0) > 0$. (5) $\Leftrightarrow \;\varphi $ satisfies the ${\Delta_2}$-condition at 0 and $C(\varphi ) = 1$. The last equivalence contains a result of Lami-Dozo [10].
On the representation of order continuous operators by random measures
L.
Weis
535-563
Abstract: Using the representation $ Tf(y) = \smallint f\;d{v_y}$, where $({v_y})$ is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator $T$ is (lattice-)orthogonal to all integral operators (i.e. all ${v_y}$ are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all ${v_y}$ are atomic) if and only if $ T$ satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all $ {v_y}$ are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.
Embeddings of Harish-Chandra modules, ${\germ n}$-homology and the composition series problem: the case of real rank one
David H.
Collingwood
565-579
Abstract: Let $G$ be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup $P = MAN$ and form the (normalized) principal series representations $I_P^G(U)$. In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands' classification) all irreducible submodules and quotients of $ I_P^G(U)$. As a corollary, all embeddings of an irreducible Harish-Chandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero $ {\mathfrak{n}}$-homology.
Asymptotic expansions of traces for certain convolution operators
Raymond
Roccaforte
581-602
Abstract: A version of Szegö's theorem in Euclidean space gives the first two terms of the asymptotics as $\alpha \to \infty$ of the determinant of convolution operators on ${L_2}(\alpha \,\Omega )$, where $\Omega$ is a bounded subset of ${{\mathbf{R}}^n}$ with smooth boundary. In this paper the more general problem of the asymptotics of traces of certain analytic functions of the operators is considered and the next term in the expansion is obtained.
Structure sets vanish for certain bundles over Seifert manifolds
Christopher W.
Stark
603-615
Abstract: Let ${M^{n + 3}}$ be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold ${K^3}$ with an effective circle action of hyperbolic type. Assume that the fiber ${N^n}$ in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set $ {S_{\text{TOP}}}\,({N^n} \times {D^k},\partial )$ of topological surgery vanishes for sufficiently large $k$. Then the projective class and Whitehead groups of $M$ vanish and $ {S_{\text{TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0$ if $n + k \geqslant 3$ or if ${K^3}$ is closed and $n = 2$. The UNil groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that ${\text{UNil}}_j$ vanishes in spite of the failure of Cappell's sufficient condition.
$\omega $-morasses, and a weak form of Martin's axiom provable in ${\rm ZFC}$
Dan
Velleman
617-627
Abstract: We prove, in ZFC, that simplified gap-$ 1$ morasses of height $ \omega$ exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martin's axiom-type forcing axiom is provable in ZFC. We show that this forcing axiom can be thought of as a weak form of ${\text{MA}}_{\omega_1}$ and give some applications.
Boundedness of fractional operators on $L\sp{p}$ spaces with different weights
Eleonor
Harboure;
Roberto A.
Macías;
Carlos
Segovia
629-647
Abstract: Let ${T_\alpha }$ be either the fractional integral operator $\smallint f(y)\vert x - y{\vert^{\alpha - n}}\; dy$, or the fractional maximal operator $\sup \left\{ {{r^{\alpha - n}}{\smallint_{\vert x - y\vert < r}}\vert f(y)\vert dy:\,r > 0} \right\}$. Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $ \upsilon$ (resp. $ w$) such that ${(\smallint \vert{T_\alpha }f{\vert^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint\vert f{\vert^p}w\;dx)^{1/p}}$ holds. Weak type substitutes in limiting cases are considered.
The Dieudonn\'e property on $C(K,\,E)$
Fernando
Bombal;
Pilar
Cembranos
649-656
Abstract: In this paper we prove that if $E$ is a Banach space with separable dual, then the space $C(K,E)$ of all continuous $E$-valued functions on a compact Hausdorff topological space $K$ has the Dieudonné property.
Infinite-to-one codes and Markov measures
Mike
Boyle;
Selim
Tuncel
657-684
Abstract: We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.
Surgery on Poincar\'e complexes
J. P. E.
Hodgson
685-701
Abstract: A geometric approach to surgery on Poincaré complexes is described. The procedure mimics the original techniques for manifolds. It is shown that the obstructions to surgering a degree-one normal map of Poincaré complexes to a homotopy equivalence lie in the Wall groups, and that all elements in these groups can arise as obstructions.
Approximating groups of bundle automorphisms by loop spaces
Roberto
Bencivenga
703-715
Abstract: D. H. Gottlieb proved in 1972 that the group of automorphisms of a numerable $ G$-bundle $p:X \to B$ is weakly homotopy equivalent to $\Omega \;\operatorname{Map}(B,{B_G};k)$, where $ k:B \to {B_G}$ is a classifying map for $p$. We refine this classical result by constructing a genuine homotopy equivalence between these two spaces which is natural with respect to numerable bundle morphisms, can be generalized to fibre bundles, and can be interpreted as a natural isomorphism between two suitably defined functors.
The interfaces of one-dimensional flows in porous media
Juan L.
Vázquez
717-737
Abstract: The solutions of the equation $ {u_t} = {({u^m})_{x\,x}}$ for $x \in {\mathbf{R}},0 < t < T,m > 1$, where $ u(x,0)$ is a nonnegative Borel measure that vanishes for $x > 0$ (and satisfies a growth condition at $- \infty$), exhibit a finite, monotone, continuous interface $ x = \zeta (t)$ that bounds to the right the region where $u > 0$. We perform a detailed study of $ \zeta$: initial behaviour, waiting time, behaviour as $ t \to \infty$. For certain initial data the solutions blow up in a finite time ${T^{\ast}}$: we calculate ${T^{\ast}}$ in terms of $u(x,0)$ and describe the behaviour of $\zeta$ as $t\, \uparrow \,{T^{\ast}}$.
On Fourier multiplier transformations of Banach-valued functions
Terry R.
McConnell
739-757
Abstract: We obtain analogues of the Mihlin multiplier theorem and Littlewood-Paley inequalities for functions with values in a suitable Banach space $B$. The requirement on $B$ is that it have the unconditionality property for martingale difference sequences.
Real zeros of derivatives of meromorphic functions and solutions of second order differential equations
Simon
Hellerstein;
Li-Chien
Shen;
Jack
Williamson
759-776
Abstract: We classify all functions $F$ meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that $F^{\prime}$ has no zeros and $ F''$ only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation $w'' + H(z)w = 0,H$ entire.
Central limit theorem for products of random matrices
Marc A.
Berger
777-803
Abstract: Using the semigroup product formula of P. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.
A quasilinear hyperbolic integro-differential equation related to a nonlinear string
Melvin L.
Heard
805-823
Abstract: We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation \begin{displaymath}\begin{array}{*{20}{c}} {{u_{t\,t}}\,(t,x) + c\,(t)\,{u_t}(t,... ..._t}(0,x) = {u_1}(x), \qquad x \in {\mathbf{R}}.} \end{array}\end{displaymath} This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data ${u_0}\,(x),{u_1}\,(x)$ is small in a suitable sense, and if the damping coefficient $c\,(t)$ grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms $ h\,(t,x,u)$ which are sublinear in $u$. In the nonlinearity we require that $M \in {C^1}\,[0,\infty )$ and, in addition, satisfies $ M( \lambda ) \geq {m_0} > 0$ for all $ \lambda \geq 0$.
Generic reducing fields of Jordan pairs
Holger P.
Petersson
825-843
Abstract: Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.
Generic representations are induced from square-integrable representations
Ronald L.
Lipsman
845-854
Abstract: It is proven for arbitrary real algebraic groups that the generic irreducible unitary representation is induced from a square-integrable representation (modulo the projective kernel). This generalizes the well-known result for reductive groups that the generic representations are either discrete series, or induced from discrete series (modulo the nilradical) representations of cuspidal parabolic subgroups.
The behavior under projection of dilating sets in a covering space
Burton
Randol
855-859
Abstract: Let $M$ be a compact Riemannian manifold with covering space $S$, and suppose $d{\mu_r}\;(0 < r < \infty )$ is a family of Borel probability measures on $S$, all of which arise from some fixed measure by $ r$-homotheties of $ S$ about some point, followed by renormalization of the resulting measure. In this paper we study the ergodic properties, as a function of $r$, of the corresponding family of projected measures on $M$ in the Euclidean and hyperbolic cases. A typical example arises by considering the behavior of a dilating family of spheres under projection.