20,037 research outputs found
Algebraic statistical models
Many statistical models are algebraic in that they are defined in terms of
polynomial constraints, or in terms of polynomial or rational parametrizations.
The parameter spaces of such models are typically semi-algebraic subsets of the
parameter space of a reference model with nice properties, such as for example
a regular exponential family. This observation leads to the definition of an
`algebraic exponential family'. This new definition provides a unified
framework for the study of statistical models with algebraic structure. In this
paper we review the ingredients to this definition and illustrate in examples
how computational algebraic geometry can be used to solve problems arising in
statistical inference in algebraic models
The Secant Conjecture in the real Schubert calculus
We formulate the Secant Conjecture, which is a generalization of the Shapiro
Conjecture for Grassmannians. It asserts that an intersection of Schubert
varieties in a Grassmannian is transverse with all points real, if the flags
defining the Schubert varieties are secant along disjoint intervals of a
rational normal curve. We present theoretical evidence for it as well as
computational evidence obtained in over one terahertz-year of computing, and we
discuss some phenomena we observed in our data.Comment: 19 page
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
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