Covers in free lattices
Ralph
Freese;
J. B.
Nation
1-42
Abstract: In this paper we study the covering relation $ (u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $w \in {\text{FL}}(X)$, finds all the elements which cover or are covered by $w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in ${\text{FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.
Surgery in dimension four and noncompact $5$-manifolds
Daniel S.
Silver
43-50
Abstract: This paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact $5$-manifolds.
The determination of the Lie algebra associated to the lower central series of a group
John P.
Labute
51-57
Abstract: In this paper we determine the Lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions. Other central series, such as the lower $ p$-central series, are treated as well.
A reciprocity law for polynomials with Bernoulli coefficients
Willem
Fouché
59-67
Abstract: We study the zeros $ \pmod p$ of the polynomial ${\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1)$ for $p$ an odd prime, where ${B_k}$ denotes the $k$th Bernoulli number and the summation extends over $ 1 \leqslant k \leqslant p - 2$. We establish a reciprocity law which relates the congruence ${\beta _p}(r) \equiv 0\;\pmod p$ to a congruence $ {f_p}(n) \equiv 0\,\pmod r$ for $r$ a prime less than $p$ and $ n \in {\mathbf{Z}}$. The polynomial ${f_p}(x)$ is the irreducible polynomial over ${\mathbf{Q}}$ of the number $ \operatorname{Tr}_L^{{\mathbf{Q}}(\zeta )}\zeta$, where $\zeta$ is a primitive ${p^2}$ th root of unity and $ L \subset {\mathbf{Q}}(\zeta )$ is the extension of degree $p$ over $ {\mathbf{Q}}$. These congruences are closely related to the prime divisors of the indices $I(\alpha ) = (\mathcal{O}:{\mathbf{Z}}[\alpha ])$, where $ \mathcal{O}$ is the integral closure in $L$ and $ \alpha \in \mathcal{O}$ is of degree $p$ over $ {\mathbf{Q}}$. We establish congruences $\pmod p$ involving the numbers $I(\alpha )$ and show that their prime divisors $ r \ne p$ are closely related to the congruence ${r^{p - 1}} \equiv 1\,\pmod {p^2}$.
New inequalities for polynomials
C.
Frappier;
Q. I.
Rahman;
St.
Ruscheweyh
69-99
Abstract: Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Riesz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.
Coordinatization in superstable theories. I. Stationary types
Steven
Buechler
101-114
Abstract: Suppose $ T$ is superstable and $ P$ is a complete type over some finite set with $U(p) = \alpha + 1$ for some $\alpha$. We show how to associate with $ p$ an incidence geometry which measures the complexity of the family of extensions of $p$ of rank $\alpha$. When $p$ is stationary we give a characterization of the possible incidence geometries. As an application we prove Theorem. Suppose $M$ is superstable and has only one $ 1$-type $p \in S(\emptyset )$. Further suppose $ p$ is stationary with $U(p) = \alpha + 1$ for some $\alpha$. Then one of the following holds: (i) There is an equivalence relation $E \subset {M^2}$ with infinitely many infinite classes definable over $ \emptyset$. (ii) $ M$ is the algebraic closure of a set of Morley rank $1$. In particular, $ M$ is ${\aleph _0}$-stable of finite rank.
Stability of Godunov's method for a class of $2\times 2$ systems of conservation laws
Randall J.
LeVeque;
Blake
Temple
115-123
Abstract: We prove stability and convergence of the Godunov scheme for a special class of genuinely nonlinear $2 \times 2$ systems of conservation laws. The class of systems, which was identified and studied by Temple, is a subset of the class of systems for which the shock wave curves and rarefaction wave curves coincide. None of the equations of gas dynamics fall into this class, but equations of this type do arise, for example, in the study of multicomponent chromatography. To our knowledge this is the first time that a numerical method other than the random choice method of Glimm has been shown to be stable in the variation norm for a coupled system of nonlinear conservation laws. This implies that subsequences converge to weak solutions of the Cauchy problem, although convergence for $2 \times 2$ systems has been proved by DiPerna using the more abstract methods of compensated compactness.
Polar coordinates induced by actions of compact Lie groups
Jiri
Dadok
125-137
Abstract: Let $G$ be a connected Lie subgroup of the real orthogonal group $O(n)$. For the action of $G$ on $ {{\mathbf{R}}^n}$, we construct linear subspaces $ \mathfrak{a}$ that intersect all orbits. We determine for which $G$ there exists such an $\mathfrak{a}$ meeting all the $G$-orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible $G$ arise as the isotropy subgroups of Riemannian symmetric spaces.
A reflexivity theorem for weakly closed subspaces of operators
Hari
Bercovici
139-146
Abstract: It was proved in [4] that the ultraweakly closed algebras generated by certain contractions on Hilbert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to $ {H^\infty }$, was used in [3] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with ${H^\infty }$, and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [3,2 and 9] as particular cases.
A bijective proof of Stanley's shuffling theorem
I. P.
Goulden
147-160
Abstract: For two permutations $\sigma$ and $\omega$ on disjoint sets of integers, consider forming a permutation on the combined sets by "shuffling" $ \sigma$ and $ \omega$ (i.e., $ \sigma$ and $ \omega$ appear as subsequences). Stanley [10], by considering $ P$-partitions and a $ q$-analogue of Saalschutz's $_3{F_2}$ summation, obtained the generating function for shuffles of $\sigma$ and $\omega$ with a given number of falls (an element larger than its successor) with respect to greater index (sum of positions of falls). It is a product of two $q$-binomial coefficients and depends only on remarkably simple parameters, namely the lengths, numbers of falls and greater indexes of $\sigma$ and $\omega$. A combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce $ \sigma$ and $ \omega$ to canonical permutations, for which a direct evaluation of the generating function is given.
Singular integral operators of Calder\'on type and related operators on the energy spaces
Takafumi
Murai
161-178
Abstract: We show the boundedness of some singular integral operators on the energy spaces.
Best approximation and quasitriangular algebras
Timothy G.
Feeman
179-187
Abstract: If $\mathcal{P}$ is a linearly ordered set of projections on a Hilbert space and $\mathcal{K}$ is the ideal of compact operators, then $\operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ is the quasitriangular algebra associated with $\mathcal{P}$. We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given $T$ and $ \mathcal{P}$, is there an $ A$ in $ \operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ such that $\left\Vert {T - A} \right\Vert = \inf \{ \left\Vert {T - S} \right\Vert:S \in \operatorname{Alg}\,\mathcal{P} + \mathcal{K}\}$? We prove that if $\mathcal{A}$ is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition $\Delta$, then every operator $T$ has a best approximant in $\mathcal{A} + \mathcal{K}$. We also show that if $ \mathcal{E}$ is an increasing sequence of finite rank projections converging strongly to the identity then $ \operatorname{Alg}\,\mathcal{E}$ satisfies the condition $\Delta$. Also, we show that if $ T$ is not in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ then the best approximants in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ to $T$ are never unique.
Compact conformally flat hypersurfaces
Manfredo
do Carmo;
Marcos
Dajczer;
Francesco
Mercuri
189-203
Abstract: Roughly speaking, a conformal space is a differentiable manifold $ {M^n}$ in which the notion of angle of tangent vectors at a point $p \in {M^n}$ makes sense and varies differentiably with $p$; two such spaces are (locally) equivalent if they are related by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space $ {R^n}$. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure; in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, assume the ambient space to be $ {R^{n + 1}}$. Then, if $n \geqslant 4$, a conformally flat hypersurface $ {M^n} \subset {R^{n + 1}}$ can be described as follows. Diffeomorphically, $ {M^n}$ is a sphere $ {S^n}$ with ${b_1}(M)$ handles attached, where $ {b_1}(M)$ is the first Betti number of $M$. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of ${R^{n + 1}}$ that are foliated by complete round $ (n - 1)$-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of ${R^{n + 1}}$: (a) an open piece of an $ n$-sphere or an $ n$-plane bounded by round $ (n - 1)$-sphere, (b) a round $(n - 1)$-sphere, (c) a point.
On infinite deficiency in ${\bf R}\sp \infty$-manifolds
Vo Thanh
Liem
205-226
Abstract: Using the notion of inductive proper $ q - 1 - {\text{LCC}}$ introduced in this note, we will prove the following theorems. Theorem 1. Let $M$ be an $ {R^\infty }$-manifold and let $ H:X \times I \to M$ be a homotopy such that ${H_0}$ and ${H_1}$ are $ {R^\infty }$-deficient embeddings. Then, there is a homeomorphism $ F$ of $M$ such that $F \circ {H_0} = {H_1}$. Moreover, if $ H$ is limited by an open cover $\alpha$ of $M$ and is stationary on a closed subset $ {X_0}$ of $ X$ and $ {W_0}$ is an open neighborhood of $\displaystyle H[(X - {X_0}) \times I] \quad {in\;M,}$ then we can choose $ F$ to also be $\operatorname{St}^4(\alpha )$-close to the identity and to be the identity on $\dot X_{0} \cup (M - {W_0})$. Theorem 2. Every closed, locally ${R^\infty }({Q^\infty })$-deficient subset of an $ {R^\infty }({Q^\infty })$-manifold $M$ is ${R^\infty }({Q^\infty })$-deficient in $ M$. Consequently, every closed, locally compact subset of $M$ is ${R^\infty }({Q^\infty })$-deficient in $ M$.
Sobolev space projections in strictly pseudoconvex domains
Harold P.
Boas
227-240
Abstract: The orthogonal projection from a Sobolev space ${W^s}(\Omega )$ onto the subspace of holomorphic functions is studied. This analogue of the Bergman projection is shown to satisfy regularity estimates in higher Sobolev norms when $\Omega$ is a smooth bounded strictly pseudoconvex domain in $ {{\mathbf{C}}^n}$.
Nilpotent automorphism groups of Riemann surfaces
Reza
Zomorrodian
241-255
Abstract: The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automorphisms of a surface of genus $g \geqslant 2$ cannot exceed $ 16(g - 1)$. Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0;2,4,8).
Invariant means on an ideal
Michel
Talagrand
257-272
Abstract: Let $G$ be a compact abelian group and $ Q$ an invariant ideal of ${L^\infty }(G)$. Let ${M_Q}$ be the set of invariant means $ \nu$ on ${L^\infty }(G)$ that are zero on $Q$, that is $\nu ({\chi _A}) = 1$ for ${\chi _A} \in Q$. We show that ${M_Q}$ is very large in the sense that a nonempty $ {G_\delta }$ subset of $ {M_Q}$ must contain a copy of $ \beta {\mathbf{N}}$. Let $ {E_Q}$ be the set of extreme points of ${M_Q}$. We show that its closure is very small in the sense that it contains no nonempty ${G_\delta }$ of ${M_Q}$. We also show that ${E_Q}$ is topologically very irregular in the sense that it contains no nonempty ${G_\delta }$ of its closure. The proofs are based on delicate constructions which rely on combinatorial type properties of abelian groups. Assume now that $ G$ is locally compact, noncompact, nondiscrete and countable at infinity. Let $ M$ be the set of invariant means on $ {L^\infty }(G)$ and $ {M_t}$, the set of topologically invariant means. We show that ${M_t}$ is very small in $M$. More precisely, each nonempty ${G_\delta }$ subset of $M$ contains a $\nu$ such that $ \nu (f) = 1$ for some $f \in C(G)$]> with <![CDATA[ $0 \leqslant f \leqslant 1$ and the support of $ f$ has a finite measure. Under continuum hypothesis, we also show that there exists points in ${M_t}$ which are extremal in $M$ (but, in general, ${M_t}$ is not a face of $M$, that is, not all the extreme points of ${M_t}$ are extremal in $M$).
Decompositions into codimension-two manifolds
R. J.
Daverman;
J. J.
Walsh
273-291
Abstract: Let $M$ denote an orientable $(n + 2)$-manifold and let $ G$ denote an upper semicontinuous decomposition of $M$ into continua having the shape of closed, orientable $n$-manifolds. The main result establishes that the decomposition space $M/G$ is a $2$-manifold.
Simply transitive groups and K\"ahler structures on homogeneous Siegel domains
Josef
Dorfmeister
293-305
Abstract: We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain $D$ as modifications of standard normal $j$-algebras. We show that the Lie algebra of all automorphisms of $D$ is a "complete isometry algebra in standard position". This implies that $D$ carries a riemannian metric $ \tilde g$ with nonpositive sectional curvature satisfying Lie $ \operatorname{Iso}(D,\tilde g) = \operatorname{Lie}\; \operatorname{Aut}\,$ D. We determine all Kähler metrics $ f$ on $D$ for which the group $\operatorname{Aut}(D,f)$ of holomorphic isometries acts transitively. We prove that in this case $ \operatorname{Aut}(D,f)$ contains a simply transitive split solvable subgroup. The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.
The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences
A.
Bellow;
V.
Losert
307-345
Abstract: In this paper on the weighted pointwise ergodic theorem we bring together a substantial amount of previous work by a number of mathematicians and integrate it into a systematic consistent scheme; we also bring some original contributions to the subject which extend its boundaries and suggest further avenues of research. The paper is divided into six sections. The most significant new results are contained in $\S\S5$ and $6$.
Multipliers on weighted $L\sb p$-spaces over certain totally disconnected groups
C. W.
Onneweer
347-362
Abstract: Let $G$ be a locally compact totally disconnected group with a suitable sequence of open compact subgroups. We prove a multiplier theorem for certain weighted ${L_p}$-spaces over $G$, which is a generalization of a Hörmander-type multiplier theorem for ${L_p}$-spaces over a local field, due to Taibleson.
Regularization of $L\sp 2$ norms of Lagrangian distributions
Steven
Izen
363-380
Abstract: Let $X$ be a compact smooth manifold, $\dim X = n$. Let $\Lambda$ be a fixed Lagrangian submanifold of $ {T^\ast}X$. The space of Lagrangian distributions $ {I^k}(X,\Lambda )$ is contained in ${L^2}(X)$ if $k < - n/4$. When $k = n/4$, $ {I^{ - n/4}}(X,\Lambda )$ just misses ${L^2}(X)$. A new inner product ${\langle u,v\rangle _R}$ is defined on $ {I^{ - n/4}}(X,\Lambda )/{I^{ - n/4 - 1}}(X,\Lambda )$ in terms of symbols. This inner product contains "${L^2}$ information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in ${L^2}(X)$. The asymptotic behavior of the $ {L^2}$ inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case, ${\langle u,v\rangle _R}$ is found to regulate the growth of the ordinary ${L^2}$ inner product.
Cubes of conjugacy classes covering the infinite symmetric group
Manfred
Droste
381-393
Abstract: Using combinatorial methods, we prove the following theorem on the group $S$ of all permutations of a countably-infinte set: Whenever $p \in S$ has infinite support without being a fixed-point-free involution, then any $s \in S$ is a product of three conjugates of $p$. Furthermore, we present uncountably many new conjugacy classes $C$ of $S$ satisfying that any $s \in S$ is a product of two elements of $ C$. Similar results are shown for permutations of uncountable sets.
Functions of uniformly bounded characteristic on Riemann surfaces
Shinji
Yamashita
395-412
Abstract: A characteristic function $T(D,w,f)$ of Shimizu and Ahlfors type for a function $ f$ meromorphic in a Riemann surface $R$ is defined, where $D$ is a regular subdomain of $R$ containing a reference point $ w \in R$. Next we suppose that $R$ has the Green functions. Letting $ T(w,f) = {\lim _{D \uparrow R}}T(D,w,f)$, we define $f$ to be of uniformly bounded characteristic in $ R$, $f \in {\text{UBC}}(R)$ in notation, if ${\sup _{w \in R}}T(w,f) < \infty$. We shall propose, among other results, some criteria for $ f$ to be in ${\text{UBC}}(R)$ in various terms, namely, Green's potentials, harmonic majorants, and counting functions. They reveal that ${\text{UBC}}(\Delta )$ for the unit disk $ \Delta$ coincides precisely with that introduced in our former work. Many known facts on $ {\text{UBC}}(\Delta )$ are extended to $ {\text{UBC}}(R)$ by various methods. New proofs even for $R = \Delta$ are found. Some new facts, even for $ \Delta$, are added.
Lowest order invariants for real-analytic surfaces in ${\bf C}\sp 2$
Gary A.
Harris
413-422
Abstract: Suppose $ M$ is a general real-analytic surface in complex euclidean two-space with complex tangent space at a point $p$. Further suppose $M$ is tangent to order $k$ at $p$. This paper determines a complete set of $ k$th order local holomorphic invariants for $M$ at $p$.
An upper bound for the waiting time for nonlinear degenerate parabolic equations
Michel
Chipot;
Thomas
Sideris
423-427
Abstract: An upper bound is obtained for the time when the support of the solution of some nonlinear, degenerate parabolic equations begins to spread.