Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems
G.
Keller
433-497
Abstract: Transfer operators and zeta functions of piecewise monotonic and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.
The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories
D. A.
Bayer;
J. C.
Lagarias
499-526
Abstract: This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call $P$-trajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called $A$-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of $P$-trajectories and $A$-trajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for $P$-trajectories and $A$-trajectories. It shows that projective transformations map $P$-trajectories into $P$-trajectories. It presents Karmarkar's interpretation of $A$-trajectories as steepest descent paths of the objective function $ \left\langle {{\mathbf{c}},{\mathbf{x}}} \right\rangle $ with respect to the Riemannian geometry $d{s^2} = \sum\nolimits_{i = 1}^n {d{x_i}d{x_i}/x_i^2}$ restricted to the relative interior of the polytope of feasible solutions. $ P$-trajectories of a canonical form linear program are radial projections of $ A$-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that $P$-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The $A$-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry.
The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories
D. A.
Bayer;
J. C.
Lagarias
527-581
Abstract: Karmarkar's projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called $P$-trajectories, as well as a related set of trajectories, called $A$-trajectories. The $ A$-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The $ A$-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesics of a Riemannian geometry isometric to Euclidean geometry. Each $A$-trajectory is part of a real algebraic curve. Each $P$-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The $P$-trajectory through $ {\mathbf{0}}$ in Legendre transform coordinates, called the central $ P$-trajectory, is part of a straight line, and is contained in the $ A$-trajectory through ${\mathbf{0}}$, called the central $ A$-trajectory. Each $ P$-trajectory is part of a real algebraic curve. The central $A$-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form $\langle {\mathbf{c}},{\mathbf{x}}\rangle = \mu$. It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central $ A$-trajectory, which is also the central $P$-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm. $ A$-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point ${{\mathbf{x}}_\infty }$ of a central $A$-trajectory on the boundary of the feasible solution polytope $P$ is the center of the unique face of $ P$ containing ${{\mathbf{x}}_\infty }$ in its relative interior. The central trajectory of a combined primal-dual linear program has a simple set of polynomial relations determining it as an algebraic curve. These relations are a relaxed form of the complementary slackness conditions. This central trajectory algebraically projects onto the central trajectories of both the primal and dual linear programs, and this gives an algebraic correspondence between points on the positive parts of the central trajectories of the primal and dual linear programs. Two Lagrangian dynamical systems with simple Lagrangians are shown to have $ A$-trajectories as ${\mathbf{\dot q}}$-trajectories. The Hamiltonian dynamical systems associated to these Lagrangian systems are completely integrable.
Lorentz spaces that are isomorphic to subspaces of $L\sp 1$
Carsten
Schütt
583-595
Abstract: We show which Lorentz spaces are isomorphic to subspaces of ${L^1}$ and which are not.
A Picard theorem with an application to minimal surfaces
Peter
Hall
597-603
Abstract: We prove a Picard theorem for holomorphic maps from ${\mathbf{C}}$ to a quadric hypersurface. This implies a theorem on the number of directions in general position omitted by the normals to a minimal surface of the conformal type of $ {\mathbf{C}}$.
Some properties of the curve straightening flow in the plane
Anders
Linnér
605-618
Abstract: We will explicitly compute the gradient of the total squared curvature functional on a space of closed curves. An example shows that the flow along the gradient trajectory may cause curves to develop self-intersections. We prove the existence of strictly convex curves that momentarily turn nonconvex. In conclusion we use computer graphics to illustrate how self-intersections come about.
Stability in the isoperimetric problem for convex or nearly spherical domains in ${\bf R}\sp n$
Bent
Fuglede
619-638
Abstract: For convex bodies $ D$ in ${{\mathbf{R}}^n}$ the deviation $d$ from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency $\Delta$ of $D$ as follows: $d \leq f(\Delta)$ (for $ \Delta$ sufficiently small). Here $f$ is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of $f$. The dimensions $n = 2$ and $3$ present anomalies as to the form of $ f$. In the planar case $ n = 2$ the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies $D$ in $ {{\mathbf{R}}^n}$ in the sense that, as $D$ varies, $d \to 0$ for $ \Delta \to 0$. The proof of the estimate $ d \leq f(\Delta)$ is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.
A uniform algebra of analytic functions on a Banach space
T. K.
Carne;
B.
Cole;
T. W.
Gamelin
639-659
Abstract: Let $A(B)$ be the uniform algebra on the unit ball of a dual Banach space $\mathcal{Z} = {\mathcal{Y}^\ast}$ generated by the weak-star continuous linear functionals. We focus on three related problems: (i) to determine when $A(B)$ is a tight uniform algebra; (ii) to describe which functions in $ {H^\infty }(B)$ are approximable pointwise on $B$ by bounded nets in $A(B)$; and (iii) to describe the weak topology of $B$ regarded as a subset of the dual of $ A(B)$. With respect to the second problem, we show that any polynomial in elements of $ {\mathcal{Y}^{\ast\ast}}$ can be approximated pointwise on $B$ by functions in $A(B)$ of the same norm. This can be viewed as a generalization of Goldstine's theorem. In connection with the third problem, we introduce a class of Banach spaces, called $\Lambda$-spaces, with the property that if $\{ {x_j}\}$ is a bounded sequence in $\mathcal{X}$ such that $P({x_j}) \to 0$ for any $m$-homogeneous analytic function $P$ on $\mathcal{X}, m \geq 1$, then ${x_j} \to 0$ in norm. We show for instance that a Banach space has the Schur property if and only if it is a $\Lambda$-space with the Dunford-Pettis property.
Weighted norm inequalities for the continuous square function
J. Michael
Wilson
661-692
Abstract: We prove new weighted norm inequalities for real-variable analogues of the Lusin area function. We apply our results to obtain new: (i) weighted norm inequalities for singular integral operators; (ii) weighted Sobolev inequalities; (iii) eigenvalue estimates for degenerate Schrödinger operators.
On two-cardinal properties of ideals
Jacek
Cichoń
693-708
Abstract: We investigate two-cardinal properties of ideals. These properties involve notions such as Luzin sets, special coverings, etc. We apply our results to the ideals of meagre sets and of negligible sets in the real line. In case of the negligible sets, we relate these properties to caliber and precalibers of the measure algebra.
Boundary layers in approximate solutions
K. T.
Joseph
709-726
Abstract: In this paper we study the development of boundary layers in parabolic approximate solutions of the initial boundary value problem for linear strictly hyperbolic systems of equations in one space variable. We also analyse the boundary layer behaviour.
Sobolev inequalities for products of powers
A. Eduardo
Gatto;
Richard L.
Wheeden
727-743
Abstract: We derive weighted Sobolev inequalities of the form ${\left\Vert f \right\Vert _{L_u^q}} \leq C{\left\Vert {\nabla f} \right\Vert _{L_v^p}}$, $f \in C_0^\infty ({{\mathbf{R}}^n})$, $1 < p \leq q < \infty$, for classes of weight functions $u$, $v$ which include $v$'s that are a finite product of certain power weights times an ${A_p}$ function.
The graded Witt ring and Galois cohomology. II
Jón Kr.
Arason;
Richard
Elman;
Bill
Jacob
745-780
Abstract: A primary problem in the theory of quadratic forms over a field $ F$ of characteristic different from two is to prove that the rings $H_q^\ast F$ and $GWF$ are isomorphic. Here $H_q^\ast F = {H^\ast }(\operatorname{Gal}({F_q}/F),{\mathbf{Z}}/2{\mathbf{Z}}))$, where ${F_q}$ is the quadratic closure of $ F$, and $GWF$ is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring $WF$ of $F$. In this paper, we assume we are given a field extension $K$ of $F$ such that $WK$ is 'close' to $WF$ or $ H_q^{\ast} K$ is 'close' to $H_q^\ast F$. A method is developed to obtain information about these graded rings over $F$ and its $2$-extensions from information about the corresponding graded ring of $K$. This relative theory extends and includes the previously developed absolute case where $K = {F_q}$. Applications are also given to show that $ H_q^\ast F$ and $ GWF$ are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.
On certain integrals associated to CR-functions
Telemachos
Hatziafratis
781-802
Abstract: We construct explicit $\bar \partial $-primitives of Cauchy-Fantappiè type kernels on analytic subvarieties of domains in $ {{\mathbf{C}}^n}$, outside the zero set of a holomorphic map defined on the variety. Then we use these primitives to derive, by means of a process of passing to a residue, integral formulas for $ {\text{CR}}$-functions defined on subsets of the boundary of a variety. Certain geometric restrictions on the varieties are imposed. These primitives apply in the particular case of the Bochner-Martinelli kernel in domains in ${{\mathbf{C}}^n}$ and we use these primitives, in this case, to prove a criterion for holomorphic extendability of $ {\text{CR}}$-functions defined on certain subsets of the boundary of a domain in $ {{\mathbf{C}}^n}$.
Topologically principal part of analytic functions
Etsuo
Yoshinaga
803-814
Abstract: The problem of $ {C^0}$-sufficiency of jets is one of the most interesting problems in the theory of functions or singularities. Roughly speaking, it is the problem of determining a topologically principal part of the Taylor expansion of a given function $f(x)$ at the origin of Euclidean space. Here, the topologically principal part should satisfy the properties that it is as small as possible a part of the Taylor expansion of $f(x)$ and that the local topological type of $ f(x)$ at the origin is determined by it. If a function $f(x)$ is an isolated singularity at the origin or has a nondegenerate Newton principal part (see (1.2)), then we know some answers to this problem (see (1.1), (1.3)). The purpose of this paper is to give some results for this problem for any analytic function. The main results are formulated in (1.5), (1.6), and (1.7).
Isospectral potentials on a discrete lattice. III
Thomas
Kappeler
815-824
Abstract: Continuing prior work of the author, spectral problems for a discretized version of the Schrödinger equation 'Laplacian plus potential' on the torus are considered. Spectral invariants are constructed and with their help isospectral sets of certain nongeneric potentials are determined.
Projective $n$-folds of log-general type. I
M.
Beltrametti;
A.
Biancofiore;
A. J.
Sommese
825-849
Abstract: Let $X$ be a normal $n$-dimensional Gorenstein irreducible projective subvariety of a complex projective space with codimension $3$ singularities. This article introduces new projective invariants of $X$ and shows that they satisfy certain inequalities and relations making them useful tools in the projective classification of varieties. The main tools used are the recent precise adjunction theoretic results on classification of projective varieties.
The group of automorphisms of $L\sp 1(0,1)$ is connected
F.
Ghahramani
851-859
Abstract: It is shown that the group of the automorphisms of the radical convolution algebra $ {L^1}(0,1)$ is connected in the operator norm topology, and thus every automorphism is of the form ${e^{\lambda d}}{e^q}$, where $\lambda$ is a complex number, $ d$ is the derivation $df(x) = xf(x)$ and $q$ is a quasinilpotent derivation.