Compensated compactness and general systems of conservation laws
Ronald J.
DiPerna
383-420
Abstract: We outline a general program and present some new results dealing with oscillations in weakly convergent solution sequences to systems of conservation laws. The analysis employs the Young measure and the Tartar-Murat theory of compensated compactness and deals with systems of hyperbolic and elliptic type.
Algorithmic solution of extremal digraph problems
W. G.
Brown;
P.
Erdős;
M.
Simonovits
421-449
Abstract: For a given family $\mathcal{L}$ of digraphs, we study the "extremal" digraphs on $n$ vertices containing no member of $\mathcal{L}$, and having the maximum number of arcs, $ \operatorname{ex} (n,\mathcal{L})$. We resolve conjectures concerning the set $\{ {\lim _{n \to \infty }}(\operatorname{ex} (n,\mathcal{L})/{n^2})\}$ as $ \mathcal{L}$ ranges over all possible families, and describe a "finite" algorithm that can determine, for any $\mathcal{L}$, all matrices $A$ for which a sequence $\{ A(n)\}$ of "matrix digraphs" is asymptotically extremal ($A(n)$ contains no member of $\mathcal{L}$ and has $ \operatorname{ex} (n,\mathcal{L}) + o({n^2})$ arcs as $ n \to \infty$.) Résumé. Pour une famille donnée, $\mathcal{L}$, de digraphes, on étudie les digraphes "extrémaux" à $n$ sommets qui ne contiennent aucun membre de $ \mathcal{L}$, et qui possèdent le nombre maximal d'arêtes, $ \operatorname{ex} (n,\mathcal{L})$. On résolue des conjectures qui concernent l'ensemble $\{ {\lim _{n \to \infty }}(\operatorname{ex} (n,\mathcal{L})/{n^2})\}$ où $\mathcal{L}$ soit une famille quelconque, et on présente un algorithme "fini" qui peut déterminer, pour chaque $ \mathcal{L}$, toute matrice $A$ pour laquelle une suite $\{ A(n)\}$ de "digraphes matriciels" est extrémale asymptotiquement ($ A(n)$ ne contient aucun membre de $ \mathcal{L}$ et possède $\operatorname{ex} (n,\mathcal{L}) + o({n^2})$ arêtes lorsque $ n \to \infty$.)
Les groupes $\omega$-stables de rang fini
Daniel
Lascar
451-462
Abstract: We prove that a group $G$ which is $\omega$-stable of finite Morley rank is nonmultidimensional. If moreover it is connected and does not have any infinite normal abelian definable subgroup, then it is isomorphic to $ \Pi {H_i}/K$, where the $ {H_i}$ are ${\omega _1}$-categorical groups and $K$ is a finite group.
Linearizing certain reductive group actions
H.
Bass;
W.
Haboush
463-482
Abstract: Is every algebraic action of a reductive algebraic group $ G$ on affine space ${{\mathbf{C}}^n}$ equivalent to a linear action? The "normal linearization theorem" proved below implies that, if each closed orbit of $G$ is a fixed point, then ${{\mathbf{C}}^n}$ is $G$-equivariantly isomorphic to $ {({{\mathbf{C}}^n})^G} \times {{\mathbf{C}}^m}$ for some linear action of $ G$ on ${{\mathbf{C}}^m}$.
Graphic flows and multiple disjointness
Joseph
Auslander;
Nelson
Markley
483-499
Abstract: A graphic flow is a totally minimal flow for which the only minimal subsets of the product flow are the graphs of the powers of the generating homeomorphism. The POD flows of Furstenberg, Keynes, and Shapiro [5] are examples of graphic flows. Graphic flows are in some ways analogous to ergodic systems with minimal self-joinings [11]. Various disjointness results concerning graphic flows and their powers are obtained, and their regular factors are determined.
Radial limits of $M$-subharmonic functions
David
Ullrich
501-518
Abstract: "$ M$-subharmonic" functions are defined in the unit ball of ${{\mathbf{C}}^n}$. Their basic properties are developed, leading to the following generalization of a theorem of Littlewood: An $M$-subharmonic function such that its restrictions to spheres centered at the origin are bounded in $ {L^1}$ must have radial limits almost everywhere on the unit sphere.
A Dowker product
Amer
Bešlagić
519-530
Abstract: $\diamondsuit$ implies that there is a (normal) countably paracompact space $X$ such that ${X^2}$ is normal and not countably paracompact.
Stationary solutions of chemotaxis systems
Renate
Schaaf
531-556
Abstract: The Keller-Segel Model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance (Chemotaxis). This paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns. Examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response.
Representation of set valued operators
Nikolaos S.
Papageorgiou
557-572
Abstract: In this paper we prove representation theorems for set valued additive operators acting on the spaces $L_X^1(X = {\text{separable Banach space)}}$, ${L^1}$ and $ {L^\infty }$. Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be $ w$-compact in $ L_X^1$. This result is a new very general result about $w$-compactness in the Lebesgue-Bochner space $ L_X^1$.
Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates
Mark Alan
Mostow;
Steven
Shnider
573-583
Abstract: In this paper we study the question of the existence of a continuous inverse to the multiplication mapping $(f,g) \to (fg,g)$ defined on pairs of ${C^\infty }$ functions on a manifold $M$. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of $(f,G) \to (f(p \circ G),G)$, where $ G$ is a ${C^\infty }$ mapping of the manifold $M$ into an analytic manifold $ N$ and $p$ is a fixed analytic function on $ N$. We prove a theorem adequate for application to the study of inverting the mapping $(A,X) \to (A,AX)$, where $ X$ is a vector valued ${C^\infty }$ function and $A$ is a square matrix valued ${C^\infty }$ function on $M$ whose determinant may vanish on a nowhere dense set.
Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold
Mark Alan
Mostow
585-594
Abstract: Let $S$ be a closed set in ${{\mathbf{R}}^m}$, and let a ${C^1}$ Whitney stratification of $ S$ be given. (Actually, only Whitney (a)-regularity is needed.) Let $ f:{{\mathbf{R}}^n} \to {{\mathbf{R}}^m}$ be a ${C^1}$ map transversal to all the strata. Assume that the image of $f$ intersects $S$. Then for each compact set $K$ in $ {{\mathbf{R}}^n}$, the Euclidean distances $\rho (x,{f^{ - 1}}(S))$ and $\rho (f(x),S)$, for $x$ in $K$, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps $g$ which are close enough to $f$ in a ${C^1}$ sense on a neighborhood of $ K$. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS].
Polyhedral resolutions of algebraic varieties
James A.
Carlson
595-612
Abstract: We give a method for constructing relatively small smooth simplicial resolutions of singular projective algebraic varieties. For varieties of dimension $n$, at most $n$ applications of the basic process yields a resolution of combinatorial dimension at most $ n$. The object so obtained may be used to compute the mixed Hodge stucture of the underlying variety.
Embeddings of locally connected compacta
Gerard A.
Venema
613-627
Abstract: Let $X$ be a $k$-dimensional compactum and $f:X \to {M^n}$ a map into a piecewise linear $ n$-manifold, $n \geqslant k + 3$. The main result of this paper asserts that if $X$ is locally $(2k - n)$-connected and $f$ is $(2k - n + 1)$-connected, then $ f$ is homotopic to a CE equivalence. In particular, every $ k$-dimensional, $ r$-connected, locally $ r$-connected compactum is CE equivalent to a compact subset of ${{\mathbf{R}}^{2k - r}}$ as long as $r \leqslant k - 3$.
Nonmonotoneity of Picard principle
Mitsuru
Nakai;
Toshimasa
Tada
629-644
Abstract: Two nonnegative ${C^\infty }$ functions $P(z)$ and $Q(z)$ on the punctured unit disk $0 < \vert z\vert \leqslant 1$ are constructed such that $ Q(z) \leqslant P(z)$ and there exists only one Martin minimal boundary point for the equation $ \Delta u = Pu$ over $ z = 0$ and, nevertheless, there exist exactly two Martin minimal boundary points for the equation $ \Delta u = Qu$ over $ z = 0$.
An irreducible representation of a symmetric star algebra is bounded
Subhash J.
Bhatt
645-652
Abstract: A $ {\ast}$-algebra $ A$ is called symmetric if $ (1 + {x^{\ast}}x)$ is invertible in $A$ for each $x$ in $A$. An irreducible hermitian representation of a symmetric ${\ast}$-algebra $A$ maps $A$ onto an algebra of bounded operators.
Isometric homotopy in codimension two
John Douglas
Moore
653-663
Abstract: This article investigates the structure of the space of isometric immersions from a simply connected $n$-dimensional Riemannian manifold with positive sectional curvatures into $(n + 2)$-dimensional Euclidean space ${E^{n + 2}}$. It is proven that if $n \geqslant 4$ and ${M^n}$ is such a manifold which admits a ${C^\infty }$ isometric immersion as a hypersurface in ${E^{n + 1}}$, then any ${C^\infty }$ isometric immersion from $ {M^n}$ into ${E^{n + 2}}$ is $ {C^{2n - 4}}$ homotopic through isometric immersions to an immersion whose image lies in some hyperplane.
Local and global envelopes of holomorphy of domains in ${\bf C}\sp 2$
Eric
Bedford
665-674
Abstract: A criterion is given for a smoothly bounded domain $D \subset {{\mathbf{C}}^2}$ to be locally extendible to a neighborhood of a point ${z_0} \in \partial D$. (This result may also be formulated in terms of extension of CR functions on $ \partial D$.) This is related to the envelope of holomorphy of the semitubular domain $\displaystyle \Omega (\Phi ) = \{ (z,w) \in {{\mathbf{C}}^2}:\operatorname{Re} w + {r^k}\Phi (\theta ) < 0\} ,$ where $r = \vert z\vert$, $ \theta = \arg (z)$. Necessary and sufficient conditions are given for the envelope of holomorphy of $\Omega (\Phi )$ to be ${{\mathbf{C}}^2}$. These conditions are equivalent to the existence of a subharmonic minorant for ${r^k}\Phi (\theta )$.
Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines
Wolfgang
Maass
675-693
Abstract: We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic $ 1$-tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of $2$-tape Turing machines by $1$-tape Turing machines an optimal quadratic lower bound in the deterministic case and a nearly optimal lower bound in the nondeterministic case. This answers the rather old question whether the computing power of the considered types of Turing machines is significantly increased when more than one tape is used (problem Nos. 1 and 7 in the list of Duris, Galil, Paul, Reischuk [3]). Further, we demonstrate a substantial superiority of nondeterminism over determinism and of co-nondeterminism over nondeterminism for $ 1$-tape Turing machines.
A dynamical meaning of fractal dimension
Steve
Pelikan
695-703
Abstract: When two attractors of a dynamical system have a common basin boundary $B$, small changes in initial conditions which lie near $B$ can result in radically different long-term behavior of the trajectory. A quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary $B$.
Noncommutative algebras of dimension three over integral schemes
Rick
Miranda;
Mina
Teicher
705-712
Abstract: In this article we describe the algebraic data which is equivalent to giving an associative, noncommutative algebra ${\mathcal{O}_X}$ over an integral $k$-scheme $Y$ (where $k$ is an algebraically closed field of characteristic $\ne 3$), which is locally free of rank $3$. The description allows us to conclude that, essentially, all such are locally upper triangular $2 \times 2$ matrices, with degenerations of a restricted form allowed.
A canonical partition of the periodic orbits of chaotic maps
Kathleen T.
Alligood
713-719
Abstract: We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period $k,\;2k,\;4k,\;8k \ldots $, for some positive integer $k$. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period ${2^m}k$, for each nonnegative integer $ m$.
Rational homotopy of spaces of maps into spheres and complex projective spaces
Jesper Michael
Møller;
Martin
Raussen
721-732
Abstract: We investigate the rational homotopy classification problem for the components of some function spaces with $ {S^n}$ or ${\mathbf{C}}{P^n}$ as target space.
Boundary behavior of ${\rm BMO}(B\sb n)$
Paula A.
Russo
733-740
Abstract: If $f$ is a holomorphic function of bounded mean oscillation in the unit ball of ${{\mathbf{C}}^n}$, then it has radial limits at almost every point of the boundary of the ball. The question remains as to how nicely one can expect this function to behave on subsets of the boundary of zero measure. For example, there is a holomorphic BMO function in the ball that has a finite radial limit at no point of the $n$-torus. We show here that this is not an isolated phenomenon; there exists at least one other $ n$-dimensional submanifold of the boundary of the ball with this same behavior.