Generalized halfspaces in dimension groups

In a dimension group, the projection of a finite intersection of generalized halfspaces is a finite intersection of generalized halfspaces. The dimension groups obeying a stronger version of this result, true in dense Archimedean ordered groups, are characterized algebraically and provided with a simple set of axioms.     

Gradient Projection Method for Optimization Problems with a Constraint in the Form of the Intersection of a Smooth Surface and a Convex Closed Set

Computational Mathematics and Mathematical Physics - The gradient projection method is generalized to the case of nonconvex sets of constraints representing the set-theoretic intersection of a...     

Generalized projection operators in Geometric Algebra

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas [2] equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism ‘grade involution’ and the anti-automorphism ‘inverse’ on the semigroup of invertible versors. This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators. Other generalized projection operators include projections ontoany invertible element, or a weighted projection ontoany element. This last projection operator even implies a possible projection operator for the zero element.     

A Product Theorem for Intersection Families

We will consider a certain kind of intersection family: a family of vectors over a field of finite characteristic q, such that the generalized inner products of every q or fewer vectors take on prescribed values. For certain values of the parameters, these values distinguish the generalized inner product of any q distinct vectors, from that of any q vectors in which some vector is repeated.We will represent such an intersection family by its incidence matrix, and show that the matrix product of two such matrices is itself an intersection family of the type under consideration. If the factor families distinguish between generalized inner products of q vectors with or without repetition, then so does the product family.     

Convergence of the gradient projection method and Newton’s method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set

The gradient projection method and Newton’s method are generalized to the case of nonconvex constraint sets representing the set-theoretic intersection of a spherical surface with a convex closed set. Necessary extremum conditions are examined, and the convergence of the methods is analyzed.     

On a linear problem of tracing a given motion

In the present paper, we consider the problem on the optimal tracing of a given vector function with the use of a generalized projection of the trajectory of a linear plant. The deviation of a given motion is measured in the metric C ^m [0, T] of continuous vector functions of the corresponding dimension m. We suggest an efficient method for the construction of an approximate solution of this optimization problem with given accuracy.     

Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body

The stability for all generic equilibria of the Lie–Poisson dynamics of the \mathfrakso(4)\mathfrak{so}(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of \mathfrakso(n)\mathfrak{so}(n) are equilibrium points for the rigid body dynamics. In the case of \mathfrakso(4)\mathfrak{so}(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in \mathfrakso(3)\mathfrak{so}(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.     

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball Bd⊂Rd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to Bd in Banach–Mazur distance converge to Bd in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of Bd. We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.     

On a Generalized Finite Intersection Property

We continue the study of the right finite intersection property under a weaker condition on annihilators, introducing the concept of generalized right finite intersection property (simply, generalized right FIP). We observe the structure of rings with the generalized right FIP and examine the generalized right FIP for various kinds of basic extensions of rings with the property. We show that the generalized right FIP does not go up to polynomial rings, and that the 2-by-2 full matrix ring over a domain has the generalized right FIP. In the process, we also obtain an equivalent condition for which a nonzero polynomial, over the ring of integers modulo n ≥ 2, is a non-zero-divisor.     

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.     

The Busemann theorem for complex p-convex bodies

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.     

Optimal Rates of Linear Convergence of Relaxed Alternating Projections and Generalized Douglas-Rachford Methods for Two Subspaces

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis     

Intersections of random hypersurfaces and visibility

Summary For a wide class of stationary random hypersurfaces in ^d the notion of the projection body is introduced. It turns out that this convex body, a very special case of which is Matheron's Steiner compact associated with a Poisson process of hyperplanes, contains most of the information concerning certain intersection properties of the random hypersurface, while its polar reciprocal set is closely connected with the behaviour of the random hypersurface in visibility problems. This enables one to give a unified treatment of several intersection and visibility problems for random hypersurfaces. A detailed investigation of the case where the random hypersurface is generated by a Poisson process is given separately.     

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

Involutions of Iwahori–Hecke Algebras and Representations of Fixed Subalgebras

We establish branching rules between some Iwahori–Hecke algebra of type B and their subalgebras which are defined as fixed subalgebras by involutions including Goldman involution. The Iwahori–Hecke algebra of type D is one of such fixed subalgebras. We also obtain branching rules between those fixed subalgebras and their intersection subalgebra. We determine basic sets of irreducible representations of those fixed subalgebras and their intersection by making use of generalized Clifford theory.     

Minimum L1-Distance Projection onto the Boundary of a Convex Set: Simple Characterization

We show that the minimum distance projection in the L ₁-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec (Ref. 1).     

The Petty Projection Inequality for Lp-Mixed Projection Bodies

Recently, Lutwak, Yang and Zhang posed the notion of Lp-projection body and established the Lp-analog of the Petty projection inequality. In this paper, the notion of Lp-mixed projection body is introduced--the Lp-projection body being a special case. The Petty projection inequality, as well as Lutwak＇s quermassintegrals （Lp-mixed quermassintegrals） extension of the Petty projection inequality, is established for Lp-mixed projection body.     

On intersection and mixed intersection bodies

Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies.      

二阶曲线间的射影映射

平面上的射影变换,将二阶曲线变为另一二阶曲线,这个射影变换也可以称为这两个二阶曲线间的射影映射.若两个二阶曲线相切,则存在以切点为射影中心的两个二阶曲线间的射影映射;若两个二阶曲线相离,则存在以两个二阶曲线公切线交点为射影中心的射影映射;若两个二阶曲线相交,则存在以其中一交点为射影中心的两个二阶曲线间的射影映射.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.