Vector fields on polyhedra
Michael A.
Penna
1-31
Abstract: This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra $_ \ast$ and applications. Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms.
A law of the iterated logarithm for stable summands
R. P.
Pakshirajan;
R.
Vasudeva
33-42
Abstract: Let ${X_1},{X_2}, \ldots$ be a sequence of independent indentically distributed stable random variables with parameters $\alpha \;(0 < \alpha < 2)$ and $\beta (\vert\beta \vert \leqslant 1)$. Let ${S_n} = \sum\nolimits_{i = 1}^n {{X_i}}$. Suppose that $({S_{1,n}})$ and $({S_{2,n}})$ are independent copies of the sequence $({S_n})$. In this paper we obtain the set of all limit points in the plane of the sequence $\displaystyle \left\{ {\vert{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){\vert^{1/(\lo... ...},\vert{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){\vert^{1/(\log \log n)}}} \right\}$ where $({a_n})$ is zero if $ \alpha \ne 1$ and is $(2\beta n\log n)/\pi$ if $\alpha = 1$.
The lattice of patterns induced by a positive cone of functions
D. J.
Hartfiel;
C. J.
Maxson
43-59
Abstract: This paper contains a study of a special type of lattice which arises by considering the supports of functions in a positive cone of functions. It is shown that many known combinatorial structures provide examples of such a lattice. The basic problem addressed in the paper is that of determining the structure of this lattice.
On inner ideals and ad-nilpotent elements of Lie algebras
Georgia
Benkart
61-81
Abstract: An inner ideal of a Lie algebra L over a commutative ring k is a k-submodule B of L such that $ [B[BL]] \subseteq B$. This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let L be a simple Lie algebra in which $D_y^2 = 0$ implies $y = 0$, where ${D_y}$ denotes the adjoint mapping determined by y. If L satisfies the descending chain condition on inner ideals and has proper inner ideals, then L contains a subalgebra $S = \langle e,f,h\rangle $, isomorphic to the split 3-dimensional simple Lie algebra, such that $D_e^3 = D_f^3 = 0$. Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when L is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic $p > 5$ this decomposition yields: Theorem. L is classical if and only if there is an $x \ne 0$ in L such that $D_x^{p - 1} = 0$ and if $ D_y^2 = 0$ implies $ y = 0$. The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.
Disjoint circles: a classification
Gary L.
Ebert
83-109
Abstract: For q a prime-power, let $ {\text{IP}}(q)$ denote the miquelian inversive plane of order q. The classification of certain translation planes of order $ {q^2}$, called subregular, has been reduced to the classification of sets of disjoint circles in $ {\text{IP}}(q)$. While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in $ {\text{IP}}(q)$. In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in ${\text{IP}}(q)$ is asymptotic to ${q^{12}}/1536$. Hence a judicious approach to the classification problem is to study ``interesting'' quadruples. In general, let ${C_1}, \ldots ,{C_n}$ be a nonlinear set of n disjoint circles in $ {\text{IP}}(q)$. Let H be the subgroup of the collineation group of ${\text{IP}}(q)$ composed of collineations that permute the ${C_i}$'s among themselves, and let K be that subgroup composed of collineations fixing each of the ${C_i}'s$. An interesting set of n disjoint circles would be one for which $K = 1$. It is shown that $K = 1$ if and only if $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{\text{(i)}}\;{\text{there}}\;{... ...}}\;{\text{other}}\;n - 1\;{\text{circles}}{\text{.}}} \end{array} } \right.$ ($ *$) When $n = 4$ and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions $( \ast )$ and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in $ {\text{IP}}(q)$, for large q, do indeed satisfy the conditions $ ( \ast )$. In addition, the cases when $n = 5,6,$ or 7 are explored to a lesser degree.
Generalized Hankel conjugate transformations on rearrangement invariant spaces
R. A.
Kerman
111-130
Abstract: The boundedness properties of the generalized Hankel conjugate transformations ${H_\lambda }$ on certain weighted Lebesgue spaces are studied. These are used to establish a boundedness criterion for the $ {H_\lambda }$ on the more general class of rearrangement invariant spaces. The positive operators in terms of which the criterion is given are used to construct pairs of spaces between which the $ {H_\lambda }$ are continuous; in particular, a natural analogue of a well-known result of Zygmund concerning the classical conjugate function operator is obtained for the ${H_\lambda }$.
Closed convex invariant subsets of $L\sb{p}(G)$
Anthony To Ming
Lau
131-142
Abstract: Let G be a locally compact group. We characterize in this paper closed convex subsets K of ${L_p}(G),1 \leqslant p < \infty$, that are invariant under all left or all right translations. We prove, among other things, that $K = \{ 0\}$ is the only nonempty compact (weakly compact) convex invariant subset of ${L_p}(G)\;({L_1}(G))$. We also characterize affine continuous mappings from ${P_1}(G)$ into a bounded closed invariant subset of $ {L_p}(G)$ which commute with translations, where ${P_1}(G)$ denotes the set of nonnegative functions in $ {L_1}(G)$ of norm one. Our results have a number of applications to multipliers from ${L_q}(G)$ into ${L_p}(G)$.
Study of the permanent conjecture and some of its generalizations. II
O. S.
Rothaus
143-154
Abstract: In this paper we investigate in a more systematic manner some of the topics initiated in part I of the paper with same title [5]. More specifically, we study in greater detail the properties of the function $E(y)$ defined in [5] attached to convex polytopes, whose properties in the special case of the space of doubly stochastic matrices are connected with the permanent conjecture. Some close links with Perron-Frobenius theory are developed, and we obtain as a by-product of our study what is, I believe, a new expression for the maximum eigenvalue of a nonnegative matrix, which leads to some new estimates of the same. A final section of the paper investigates some purely algebraic properties of $E(y)$, and we obtain some very interesting information connecting a doubly stochastic matrix and its transversals. In order to keep this paper as self-contained as possible, facts used here drawn from part I are stated with as much explicit detail as possible.
Local and global factorizations of matrix-valued functions
K. F.
Clancey;
I.
Gohberg
155-167
Abstract: Let C be a simple closed Liapounov contour in the complex plane and A an invertible $n \times n$ matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space ${L_p}(C)$. The notion of local factorization of A relative to ${L_p}$ at a point ${t_0}$ in C is introduced. It is shown that A admits a factorization relative to $ {L_p}(C)$ if and only if A admits a local factorization relative to $ {L_p}$ at each point $ {t_0}$ in C. Several problems connected with local factorizations relative to ${L_p}$ are raised.
A bound on the rank of purely simple systems
Frank
Okoh
169-186
Abstract: A pair of complex vector spaces (V, W) is called a system if and only if there is a C-bilinear map from ${{\mathbf{C}}^2} \times V$ to W. The category of systems contains subcategories equivalent to the category of modules over the ring of complex polynomials. Many concepts in the latter generalize to the category of systems. In this paper the pure projective systems are characterized and a bound on the rank of purely simple systems is obtained.
On everywhere-defined integrals
Lester E.
Dubins
187-194
Abstract: Hardly any finite integrals can be defined for all real-valued functions. In contrast, if infinity is admitted as a possible value for the integral, then every finite integral can be extended to all real-valued functions.
Decomposition spaces having arbitrarily small neighborhoods with $2$-sphere boundaries
Edythe P.
Woodruff
195-204
Abstract: Let G be an u.s.c. decomposition of ${S^3}$. Let H denote the set of nondegenerate elements and P be the natural projection of $ {S^3}$ onto ${S^3}/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss $P(H)$. We prove in this paper that this condition implies that ${S^3}/G$ is homeomorphic to ${S^3}$. This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of $P(H)$.
A note on limits of unitarily equivalent operators
Lawrence A.
Fialkow
205-220
Abstract: Let $\mathcal{U}(\mathcal{H})$ denote the set of all unitary operators on a separable complex Hilbert space $\mathcal{H}$. If T is a bounded linear operator on $\mathcal{H}$, let ${\pi _T}$ denote the mapping of $\mathcal{U}(\mathcal{H})$ onto $\mathcal{U}(T)$ given by conjugation. It is proved that if T is normal or isometric, then there exists a locally defined continuous cross-section for $ {\pi _T}$ if and only if the spectrum of T is finite. Examples of nonnormal operators with local cross-sections are given.
PL involutions of fibered $3$-manifolds
Paik Kee
Kim;
Jeffrey L.
Tollefson
221-237
Abstract: Let h be a PL involution of $F \times [0,1]$ such that $ h(F \times \{ 0,1\} ) = F \times \{ 0,1\}$, where F is a compact 2-manifold. It is shown that h is equivalent to an involution $h'$ of the form
The Huygens property for the heat equation
D. V.
Widder
239-244
Abstract: This note summarizes several criteria which guarantee that a solution of the heat equation should also have the semigroup property described in equation (1.2) below. In particular, it corrects a mistake in an earlier proof of one of these.
Products of sequentially compact spaces and the $V$-process
M.
Rajagopalan;
R. Grant
Woods
245-253
Abstract: In this paper we produce a family of sequentially compact, locally compact, ${T_2}$ first countable, scattered and separable spaces whose product is not countably compact and thus answer a problem of C. T. Scarborough and A. H. Stone [11] in the negative. We do this using the continuum hypothesis. We also produce a completely regular, $ {T_2}$, sequentially compact space K which is not p-compact for any $p \in \beta N - N$.
Anisotropic $H\sp{p}$ real interpolation, and fractional Riesz potentials
W. R.
Madych
255-263
Abstract: We observe that the anisotropic variants of ${H^p}$ interpolate by the real method in the usual manner. Using this fact we show that the corresponding fractional Riesz potentials and related operators perform an embedding in $ {H^p},p > 0$, analogous to the one for $ {L^p},p > 1$. We also state a theorem concerning the mapping properties of $f \to h \ast f$, where h is in $B_\alpha ^{1,\infty }$, which hold only for a restricted range of p.
Symplectic Stiefel harmonics and holomorphic representations of symplectic groups
Tuong
Ton-That
265-277
Abstract: Let ${I_k}$ denote the identity matrix of order k and set $\displaystyle {s_k} = \left[ {\begin{array}{*{20}{c}} 0 & { - {I_k}} {{I_k}} & 0 \end{array} } \right].$ Let $ {\text{Sp}}(k,{\mathbf{C}})$ denote the group of all complex $2k \times k$ matrices which satisfy the equation $g{s_k}{g^t} = {s_k}$. Let E be the linear space of all $n \times 2k$ complex matrices with $k \geqslant n$, and let $S({E^\ast})$ denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of $ {\text{Sp}}(k,{\mathbf{C}})$, which is obtained by right translation on $S({E^\ast})$, leads to a concrete and simple realization of all irreducible holomorphic representations of $ {\text{Sp}}(k,{\mathbf{C}})$. In connection with this realization, a theory of symplectic Stiefel harmonics is also established. This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold.
Two-descent for elliptic curves in characteristic two
Kenneth
Kramer
279-295
Abstract: This paper is a study of two-descent to find an upper bound for the rank of the Mordell-Weil group $A(F)$ of an elliptic curve A defined over a field F of characteristic two. It includes local and global duality theorems which are the analogs of known results for descent by an isogeny whose degree is relatively prime to the characteristic of the field of definition.
Strange billiard tables
Benjamin
Halpern
297-305
Abstract: A billiard table is any compact convex body T in the plane bounded by a continuously differentiable curve $\partial T$. An idealized billiard ball is a point which moves at unit speed in a straight line except when it hits the boundary $ \partial T$ where it rebounds making the angle of incidence equal to the angle of reflection. A rather surprising phenomenon can happen on such a table.
The level structure of a residual set of continuous functions
A. M.
Bruckner;
K. M.
Garg
307-321
Abstract: Let C denote the Banach space of continuous real-valued functions on $[0,1]$ with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in C intersect with different straight lines. It is proved that there exists a residual set A in C such that, for every function $f \in A$, the top and the bottom (horizontal) levels of f are singletons, in between these two levels there are countably many levels of f that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of f are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the minimum and the maximum values assumed by the function. As for the levels in different directions, there exists a residual set B in C such that, for every function $f \in B$, the structure of the levels of f is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of f is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function $f \in C$ a theorem is proved establishing the existence of singleton levels of f, and of the levels of f that contain isolated points.
Holomorphic continuation of smooth functions over Levi-flat hypersurfaces
Eric
Bedford
323-341
Abstract: Here we consider the singularities of a Levi-flat real hypersurface S in C that lie in an analytic variety of codimension 2. It is shown that, from the geometric point of view, there are two kinds of singularities, and the type of singularity determines whether S bounds a domain of holomorphy of type ${A^\infty }$.
The structure of generalized Morse minimal sets on $n$ symbols
John C.
Martin
343-355
Abstract: A class of bisequences on n symbols is constructed which includes the generalized Morse sequences introduced by Keane. Those which give rise to strictly ergodic sets are characterized, and the spectrum of the shift operator on these systems is investigated. It is shown that in certain cases the shift operator has partly discrete and partly continuous spectrum. The theorems generalize results of Keane on generalized Morse sequences and a theorem of Kakutani regarding a particular strictly transitive sequence on four symbols. Another special case yields information on the spectrum of certain substitution minimal sets.
The Kobayashi pseudometric on algebraic manifolds of general type and in deformations of complex manifolds
Marcus
Wright
357-370
Abstract: This paper deals with regularity properties of the infinitesimal form of the Kobayashi pseudo-distance. This form is shown to be upper semicontinuous in the parameters of a deformation of a complex manifold. The method of proof involves the use of a parametrized version of the Newlander-Nirenberg Theorem together with a theorem of Royden on extending regular mappings from polydiscs into complex manifolds. Various consequences and improvements of this result are discussed; for example, if the manifold is compact hyperbolic the infinitesimal Kobayashi metric is continuous on the union of the holomorphic tangent bundles of the fibers of the deformation. This result leads to the fact that the coarse moduli space of a compact hyperbolic manifold is Hausdorff. Finally, the infinitesimal form is studied for a class of algebraic manifolds which contains algebraic manifolds of general type. It is shown that the form is continuous on the tangent bundle of a manifold in this class. Many members of this class are not hyperbolic.