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Abstract. We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann
variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described. 相似文献
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We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set \(\Sigma \subset {\mathbb R}^n\) is bounded from below by where \(\mathrm{b}_m(\Sigma )\) is the mth Betti number of \(\Sigma \) with respect to “ordinary” (singular) homology and \(c_1,\ c_2\) are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36–43, 1997) for locally closed semialgebraic sets in terms of the total Borel–Moore Betti number. We also prove that if \(\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}\) is the projection map, then the height of any tree deciding membership in \(\Sigma \) is bounded from below by for some positive constants \(c_1,\ c_2\). We illustrate these general results by examples of lower complexity bounds for some specific computational problems.
相似文献
$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$
$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$
3.
An effective algorithm for a smooth (weak) stratification of a real semi-Pfaffian set is suggested, provided an oracle deciding
consistency of a system of Pfaffian equations and inequalities is given. An explicit estimate of the complexity of the algorithm
and of the resulting stratification is given, in terms of the parameters of the Pfaffian functions defining the original semi-Pfaffian
set. The algorithm is applied to sets defined by sparse polynomials and exponential polynomials.
This work was supported in part by the United States Army Research Office through the Army Center of Excellence for Symbolic
Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, under Grant DAAL03-91-C-0027. 相似文献
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Betti Numbers of Semialgebraic and Sub-Pfaffian Sets 总被引:1,自引:0,他引:1
Let X be a subset in [1,1]n0Rn0 defined by the formula X={x0|Q1x1Q2x2...Qx ((x0,x1,...x)X)}, where Qi{ }, Qi Qi+1, xi [1, 1]ni, and X may be eitheran open or a closed set in [1,1]n0+...+n, being the differencebetween a finite CW-complex and its subcomplex. An upper boundon each Betti number of X is expressed via a sum of Betti numbersof some sets defined by quantifier-free formulae involving X. In important particular cases of semialgebraic and semi-Pfaffiansets defined by quantifier-free formulae with polynomials andPfaffian functions respectively, upper bounds on Betti numbersof X are well known. The results allow to extend the boundsto sets defined with quantifiers, in particular to sub-Pfaffiansets. 相似文献
8.
Andrei Gabrielov 《Inventiones Mathematicae》1996,125(1):1-12
We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation
is a subanalytic set defined by the functions from the same subalgebra. This result has an equivalent formulation in logic:
Consider an expression built from functions as above using equalities and inequalities as well as existential and universal
quantifiers. Such an expression is equivalent to an existential expression involving functions from the same class, provided
that the variables approach neither infinity nor the boundary of the domain.
Oblatum 17-VIII-1995 相似文献
9.
We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane. 相似文献
10.
Clustering analysis of seismicity and aftershock identification 总被引:1,自引:0,他引:1
We introduce a statistical methodology for clustering analysis of seismicity in the time-space-energy domain and use it to establish the existence of two statistically distinct populations of earthquakes: clustered and nonclustered. This result can be used, in particular, for nonparametric aftershock identification. The proposed approach expands the analysis of Baiesi and Paczuski [Phys. Rev. E 69, 066106 (2004)10.1103/PhysRevE.69.066106] based on the space-time-magnitude nearest-neighbor distance eta between earthquakes. We show that for a homogeneous Poisson marked point field with exponential marks, the distance eta has the Weibull distribution, which bridges our results with classical correlation analysis for point fields. The joint 2D distribution of spatial and temporal components of eta is used to identify the clustered part of a point field. The proposed technique is applied to several seismicity models and to the observed seismicity of southern California. 相似文献