Fredholm Properties of the Difference of Orthogonal Projections in a Hilbert Space

Buckholtz (Proc. Amer. Math. Soc. 128 (2000), 1415–1418) gave necessary and sufficient conditions for the invertibility of the difference of two orthogonal projections in a Hilbert space. We generalize this result by investigating when the difference of such projections is a Fredholm operator, and give an explicit formula for its Fredholm inverse.     

On a Class of Projections on *-Rings

ABSTRACT

In this article we describe the structure of projections acting on semiprime *-rings and satisfying a certain functional identity. The main result is applied to bicircular projections on C *-algebras.     

Linear convergence rates for extrapolated fixed point algorithms

ABSTRACT

We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.     

COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS

Abstract

We consider the class of all commutative reduced rings for which there exists a finite subset T ? A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.     

Cohomology of normal bundles of special rational varieties

Abstract

We give a new method for calculating the cohomology of the normal bundles over rational varieties which are smooth projections of Veronese embeddings. The method can be used also when the projections are not smooth, in this case it provides information about the critical locus of maps between projective spaces.     

Finding Optimal Shadows of Polytopes

This paper deals with the problem of projecting polytopes in finite-dimensional Euclidean spaces on subspaces of given dimension so as to maximize or minimize the volume of the projection. As to the computational complexity of the underlying decision problems we show that maximizing the volume of the orthogonal projection on hyperplanes is already NP-hard for simplices. For minimization, the problem is easy for simplices but NP-hard for bipyramids over parallelotopes. Similar results are given for projections on lower-dimensional subspaces. Several other related NP-hardness results are also proved including one for inradius computation of zonotopes and another for a location problem. On the positive side, we present various polynomial-time approximation algorithms. In particular, we give a randomized algorithm for maximizing orthogonal projections of CH-polytopes in R ⁿ on hyperplanes with an error bound of essentially . Received February 17, 1999.     

Extrapolated cyclic subgradient projection methods for the convex feasibility problems and their numerical behaviour

ABSTRACT

We present two versions of the extrapolated cyclic subgradient projections method for solving the convex feasibility problem. Moreover, we present the results of numerical tests, where we compare the methods with the classical cyclic subgradient projections method.     

Harmonic analysis on SL(2,ℂ) and projectively adapted pattern representation) and projectively adapted pattern representation

Among all image transforms, the classical (Euclidean) Fourier transform has had the widest range of applications in image processing. Here its projective analogue, given by the double cover groupSL(2, ℂ) of the projective groupPSL(2, ℂ) for patterns, is developed. First, a projectively invariant classification of patterns is constructed in terms of orbits of the groupPSL(2, ℂ) acting on the image plane (with complex coordinates) by linear-fractional transformations. Then,SL(2, ℂ)-harmonic analysis, in the noncompact picture of induced representations, is used to decompose patterns into the components invariant under irreducible representations of the principal series ofSL(2, ℂ). Usefulness in digital image processing problems is studied by providing a camera model in which the action ofSL(2, ℂ) on the complex image plane corresponds to, and exhausts, planar central projections as produced when aerial images of the same scene are taken from different vantage points. The projectively adapted properties of theSL(2, ℂ)-harmonic analysis, as applied to the problems, in image processing, are confirmed by computational tests. Therefore, it should be an important step in developing a system for automated perspective-independent object recognition.     

2001 Change Index

Abstract

The following is an index by author of the six issues of Change, Vol. 35, January-December 2003.     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

How Publishing Can be More Than Not Perishing: Writing Relationships

Abstract

From time to time, the editors of Change will be spotlighting interesting research in progress. The following report by Barbara Burn, coordinator of U.S. participation in the Study Abroad Evaluation Project, is one of this series.     

Multivariate symmetry via projection pursuit

Blough (1985,Ann. Inst. Statist. Math.,37, 545–555) developed a multivariate location region for a randomp-vectorX. The dimension of this region provides information on the degree of symmetry possessed by the distribution ofX. By considering all one-dimensional projections ofX, it is possible to ascertain the dimension of the location region. Projection pursuit techniques can therefore be used to study symmetry in multivariate data sets. An example from an Entomology investigation is presented illustrating these methods.     

Reconstruction of Convex Bodies from Brightness Functions

Abstract. Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of facets, while the others do not restrict the number of facets. Convergence of the polytopes to the body is proved under certain essential assumptions including origin symmetry of the body. Also described is an oracle-polynomial-time algorithm for reconstructing an approximation to an origin-symmetric rational convex polytope of fixed and full dimension that is only accessible via its brightness function. Some of the algorithms have been implemented, and sample reconstructions are provided.     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

Reconstruction of Convex Bodies from Brightness Functions

   Abstract. Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of facets, while the others do not restrict the number of facets. Convergence of the polytopes to the body is proved under certain essential assumptions including origin symmetry of the body. Also described is an oracle-polynomial-time algorithm for reconstructing an approximation to an origin-symmetric rational convex polytope of fixed and full dimension that is only accessible via its brightness function. Some of the algorithms have been implemented, and sample reconstructions are provided.     

GROEBNER BASES FOR CONSTRUCTIBLE SETS

ABSTRACT

A constructible set can be defined in terms of a unique sequence of varieties. Given a monomial ordering, the reduced Groebner bases of the ideals of these varieties comprise a complete invariant for the constructible set. Morphic images of varieties, such as orbits under algebraic group actions and projections of varieties, can therefore be assigned complete invariants.     

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R ⁿ obtained by eliminating from R ⁿ the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles. Dedicated to N. N. Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 7–48.     

Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

ABSTRACT

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.     

An interior proximal linearized method for DC programming based on Bregman distance or second-order homogeneous kernels

Abstract

We present an interior proximal method for solving constrained nonconvex optimization problems where the objective function is given by the difference of two convex function (DC function). To this end, we consider a linearized proximal method with a proximal distance as regularization. Convergence analysis of particular choices of the proximal distance as second-order homogeneous proximal distances and Bregman distances are considered. Finally, some academic numerical results are presented for a constrained DC problem and generalized Fermat–Weber location problems.     

XGobi: Interactive Dynamic Data Visualization in the X Window System

Abstract

XGobi is a data visualization system with state-of-the-art interactive and dynamic methods for the manipulation of views of data. It implements 2-D displays of projections of points and lines in high-dimensional spaces, as well as parallel coordinate displays and textual views thereof. Projection tools include dotplots of single variables, plots of pairs of variables, 3-D data rotations, various grand tours, and interactive projection pursuit. Views of the data can be reshaped. Points can be labeled and brushed with glyphs and colors. Lines can be edited and colored. Several XGobi processes can be run simultaneously and linked for labeling, brushing, and sharing of projections. Missing data are accommodated and their patterns can be examined; multiple imputations can be given to XGobi for rapid visual diagnostics. XGobi includes an extensive online help facility. XGobi can be integrated in other software systems, as has been done for the data analysis language S, the geographic information system (GIS) Arc View?, and the interactive multidimensional scaling program XGvis. XGobi is implemented in the X Window System? for portability as well as the ability to run across a network.