Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

Lower Bounds in Real Algebraic Geometry and Orientability of Real Toric Varieties

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishimura, who characterized the orientability of smooth real toric varieties.     

Grid Methods in Computational Real Algebraic (and Semialgebraic) Geometry

In recent years, a family of numerical algorithms to solve problems in real algebraic and semialgebraic geometry has been slowly growing. Unlike their counterparts in symbolic computation they are numerically stable. But their complexity analysis, based on the condition of the data, is radically different from the usual complexity analysis in symbolic computation as these numerical algorithms may run forever on a thin set of ill-posed inputs.     

Real-World O.R./M.S. Applications in Journals

In this paper, we present the results of a study of real-world applications of O.R./M.S. as seen in journals. Five leading journals in the field are surveyed, and real-world application articles are classified using a two dimensional framework consisting of orientation and decision. The orientation dimension separates strategically oriented applications with long-term implications and tactically oriented applications with medium- and short-term implications. The decision dimension refers to the type of decision in the application-largely structured or largely unstructured. The major O.R./M.S. topics are placed in the resulting four quadrants, and articles published in the four most recent volumes of the five journals are classified. Based on the results of this survey, the thrust and shortcomings of implementation research are discussed. Some measures for enhancing publication of field-based research are also proposed.     

Algorithms in Algebraic Topology and Homological Algebra: Problem of Complexity

This review tackles the problem of the high computational complexity lying in most algorithms in algebraic topology and homological algebra. Three particular algorithms are considered: the computation of the homology of commutative differential graded algebras, the homology of principal twisted Cartesian products of EilenbergMacLane spaces, and a combinatorial method of computing Steenrod squares. Bibliography: 54 titles.     

N.M.R. and I.R. studies of 1, 1, 1 trifluoroacetone in proton donor solvents

Interactions of 1, 1, 1 trifluoroacetone with proton donor solvents of the type RH [where R=HO, CH₃S, CH₃CH₂CH₂CH₂S, CH₃O, (CH₃)₂N, (C₂H₅)₂N] have been investigated using NMR and infra-red techniques. Evidence of the formation of addition products of the type CF₃.C(OH)R.CH₃ where OH group is hydrogen-bonded with the fluorine of the CF₃ group has been obtained.