The geometry of basic, approximate, and minimum-norm solutions of linear equations

AbstractThe basic solutions of the linear equations Ax = b are the solutions of subsystems corresponding to maximal nonsingular submatrices of A. The convex hull of the basic solutions is denoted by C = C(A, b). Given 1 ≤ p ≤ ∞, the lp-approximate solutions of Ax = b, denoted x{p}, are minimizers of ∥Ax − b∥p. Given M ∈ Dm, the set of positive diagonal m × m matrices, the solutions of minx ∥M(Ax − b)∥p are called scaledlp-approximate solutions. For 1 ≤ p1, p2 ≤ ∞, the minimum-lp2-norm lp1-approximate solutions are denoted x{p1}{p2}. Main results: 1.(1) If A ∈ Rm × nm, then C contains all [some] minimum lp-norm solutions, for 1 ≤ p < ∞ [p = ∞].2.(2) For general A and any 1 ≤ p1, p2 < ∞ the set C contains all x{p1}{p2}.3.(3) The set of scaled lp-approximate solutions, with M ranging over Dm, is the same for all 1 < p < ∞.4.(4) The set of scaled least-squares solutions has the same closure as the set of solutions of minx f (|Ax − b|), where f:Rm+ → R ranges over all strictly isotone functions

A Generalized Weiszfeld method for the Multi-Facility Location Problem

a b s t r a c t An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities

The Soreq Applied Research Accelerator Facility (SARAF) - Overview, Research Programs and Future Plans

The Soreq Applied Research Accelerator Facility (SARAF) is under construction in the Soreq Nuclear Research Center at Yavne, Israel. When completed at the beginning of the next decade, SARAF will be a user facility for basic and applied nuclear physics, based on a 40 MeV, 5 mA CW proton/deuteron superconducting linear accelerator. Phase I of SARAF (SARAF-I, 4 MeV, 2 mA CW protons, 5 MeV 1 mA CW deuterons) is already in operation, generating scientific results in several fields of interest. The main ongoing program at SARAF-I is the production of 30 keV neutrons and measurement of Maxwellian Averaged Cross Sections (MACS), important for the astrophysical s-process. The world leading Maxwellian epithermal neutron yield at SARAF-I ($5\times 10^{10}$ epithermal neutrons/sec), generated by a novel Liquid-Lithium Target (LiLiT), enables improved precision of known MACSs, and new measurements of low-abundance and radioactive isotopes. Research plans for SARAF-II span several disciplines: Precision studies of beyond-Standard-Model effects by trapping light exotic radioisotopes, such as $^6$He, $^8$Li and $^{18,19,23}$Ne, in unprecedented amounts (including meaningful studies already at SARAF-I); extended nuclear astrophysics research with higher energy neutrons, including generation and studies of exotic neutron-rich isotopes relevant to the rapid (r-) process; nuclear structure of exotic isotopes; high energy neutron cross sections for basic nuclear physics and material science research, including neutron induced radiation damage; neutron based imaging and therapy; and novel radiopharmaceuticals development and production. In this paper we present a technical overview of SARAF-I and II, including a description of the accelerator and its irradiation targets; a survey of existing research programs at SARAF-I; and the research potential at the completed facility (SARAF-II).Comment: 32 pages, 31 figures, 10 tables, submitted as an invited review to European Physics Journal

Ordered Incidence geometry and the geometric foundations of convexity theory

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42995/1/22_2005_Article_BF01227810.pd

On linear optimal control problems with multiple quadratic criteria / BEBR No. 65

Includes bibliographical references (leaf 7)