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We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that...
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics ha...
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real ...
Geometry of the tetrahedron space     Mark  smooth space       2014/12/25
Let X be the space of all labeled tetrahedra in . In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285–312] we constructed a smooth symmetric compactifica...
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry a...
Likelihood Geometry     Likelihood  Geometry       2013/6/17
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded...
This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. I...
Many algorithms for inferring causality rely heavily on the faithfulness assumption.The main justi cation for imposing this assumption is that the set of unfaithful distribu-tions has Lebesgue measure...
We determine an explicit Gr¨obner basis, consisting of linear forms and determi-nantal quadrics, for the prime ideal of Raftery’s mixture transition distribution model for Markov chains. When the stat...
Riemannian statistics geometry is proposed in this work as a counterpart approach of inference geometry.
We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and T...
We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and T...
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observation...
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimensi...
Consider an i.i.d. sequence of random variables whose distribution f lies in one of a nested family of models (Mq)q2N,Mq  Mq+1. The smallest index q such that Mq contains f is called the model orde...

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