Listing 1 - 5 of 5 |
Sort by
|
Choose an application
The book provides the advanced reader with a deep insight into the exciting line of research, namely, proof that a solution exists has enabled an algorithm to find that solution itself with applications in many areas of computer science. It will inspire readers in deploying the techniques in their own further research.
Choose an application
Algebraic geometry --- Algebraic topology --- Semialgebraic sets. --- Semianalytic sets. --- Ensembles semi-algébriques. --- Ensembles semi-analytiques. --- Semialgebraic sets --- Semianalytic sets --- Semi-analytic sets --- Geometry, Algebraic --- Set theory
Choose an application
Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.
Polynomials. --- Semialgebraic sets. --- Topological fields. --- 512.62 --- Fields. Polynomials --- 512.62 Fields. Polynomials --- Polynomials --- Semialgebraic sets --- Topological fields --- Algebra --- Algebraic fields --- Geometry, Algebraic --- Set theory --- Algebra. --- Algebraic geometry. --- Functional analysis. --- Algebraic Geometry. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis
Choose an application
Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of -or the connected components of-a constructible set (Le. , a locally finite union of differ ences of real analytic sets) need not be constructible (e. g. , R - {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 -=I- O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic.
Differential geometry. Global analysis --- Semialgebraic sets --- Semianalytic sets --- Topology. --- Algebraic geometry. --- Algebraic topology. --- Mathematical logic. --- Geometry. --- Algebraic Geometry. --- Algebraic Topology. --- Mathematical Logic and Foundations. --- Mathematics --- Euclid's Elements --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Topology --- Algebraic geometry --- Geometry --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Algebras, Linear --- Semianalytic sets. --- Semialgebraic sets. --- Geometry, Algebraic --- Semi-analytic sets --- Géométrie algébrique --- Géométrie algébrique --- Espaces analytiques
Choose an application
Mathematical statistics --- Geometry, Algebraic --- Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.) --- Algebraic geometry -- Real algebraic and real analytic geometry -- Semialgebraic sets and related spaces. --- Algebraic geometry -- Special varieties -- Determinantal varieties. --- Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Biology and other natural sciences -- Genetics and population dynamics -- Problems related to evolution. --- Commutative algebra -- Computational aspects and applications -- GrÃjabner bases; other bases for ideals and modules (e.g., Janet and border bases) --- Convex and discrete geometry -- Polytopes and polyhedra -- Lattice polytopes (including relations with commutative algebra and algebraic geometry) --- Operations research, mathematical programming -- Mathematical programming -- Integer programming. --- Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces) --- Statistics -- Instructional exposition (textbooks, tutorial papers, etc.) --- Statistics -- Multivariate analysis -- Contingency tables. --- Statistics -- Parametric inference -- Hypothesis testing. --- Commutative algebra -- Computational aspects and applications -- Solving polynomial systems; resultants.
Listing 1 - 5 of 5 |
Sort by
|