arXiv subjects

Frank Sottile

Publications and source records attributed to Frank Sottile.

At least 73 records · Page 4Linked to original sources

arXiv2013

Complex static skew-symmetric output feedback control

We study the problem of feedback control for skew-symmetric and skew-Hamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke, et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, give necessary and sufficient conditions for pole placement by static skew-symmetric complex feedback, and use Schubert calculus for the orthogonal Grassmannian to count the number of complex feedback laws when there are finitely many of them. Finally, we also construct a real skew-symmetric linear system with only real feedback for any set of real poles.

math.OC↗

arXiv2012

Certifiable Numerical Computations in Schubert Calculus

Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and (in local coordinates) typically of degree exceeding two. We present an alternative primal-dual formulation using parametrizations of Schubert cells in the dual Grassmannians in which intersections of Schubert varieties become complete intersections of bilinear equations. This formulation enables the numerical certification of problems in the Schubert calculus.

math.AG↗

arXiv2012

Newton polytopes and witness sets

We present two algorithms that compute the Newton polytope of a polynomial defining a hypersurface H in C^n using numerical computation. The first algorithm assumes that we may only compute values of f - this may occur if f is given as a straight-line program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes the Newton polytope of H using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when H is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face.

math.AG↗

arXiv2012

Discriminant coamoebas through homology

Understanding the complement of the coamoeba of a (reduced) A-discriminant is one approach to studying the monodromy of solutions to the corresponding system of A-hypergeometric differential equations. Nilsson and Passare described the structure of the coamoeba and its complement (a zonotope) when the reduced A-discriminant is a function of two variables. Their main result was that the coamoeba and zonotope form a cycle which is equal to the fundamental cycle of the torus, multiplied by the normalized volume of the set A of integer vectors. That proof only worked in dimension two. Here, we use simple ideas from topology to give a new proof of this result in dimension two, one which can be generalized to all dimensions.

math.AG↗

arXiv2012

An inequality of Kostka numbers and Galois groups of Schubert problems

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of sl_2(C)-modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.

math.CO↗

arXiv2012

The Phase Limit Set of a Variety

A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which we relate to its logarithmic limit set. Detailed examples of lines in three-dimensional space illustrate and motivate these results.

math.AG↗

arXiv2012

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.

math.AG↗

arXiv2011

Non-Archimedean coamoebae

A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. Similarly, a non-archimedean coamoeba is the image of a subvariety of a torus over a non-archimedean field with complex residue field under an argument map. The phase tropical variety is the closure of the image under the pair of maps, tropicalization and argument. We describe the structure of non-archimedean coamoebae and phaseropical varieties in terms of complex coamoebae and their phase limit sets. The argument map depends upon a section of the valuation map, and we explain how this choice (mildly) affects the non-archimedean coamoeba. We also identify a class of varieties whose non-archimedean coamoebae and phase tropical varieties are objects from polyhedral combinatorics.

math.AG↗

arXiv2011

alphaCertified: certifying solutions to polynomial systems

Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on alpha-theory to certify solutions to polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements an algorithm to certify whether a given point corresponds to a real solution to a real polynomial system, as well as algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

math.NA↗

arXiv2011

Injectivity of 2D Toric Bézier Patches

Rational Bézier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric Bézier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric Bézier patch with those control points for all possible choices of weights. This condition refines that of Craciun, et al., which only implied injectivity on the interior of a patch.

cs.GR↗

arXiv2011

Orbitopes

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic geometry. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n). These include Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese orbitopes and Grassmann orbitopes. We study their face lattices, their algebraic boundary hypersurfaces, and representations as spectrahedra or projected spectrahedra.

math.AG↗

arXiv2010

Toric degenerations of Bezier patches

The control polygon of a Bezier curve is well-defined and has geometric significance---there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. We propose a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary.

cs.GR↗

arXiv2010

Cofree compositions of coalgebras (extended abstract)

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes.

math.CO↗

arXiv2010

Cofree compositions of coalgebras

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient conditions that ensure the composition is a one-sided Hopf algebra. We show these conditions are satisfied when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.

math.CO↗

arXiv2010

Dense Fewnomials

We derive new bounds of fewnomial type for the number of real solutions to systems of polynomials that have structure intermediate between fewnomials and generic (dense) polynomials. This uses a modified version of Gale duality for polynomial systems. We also use stratified Morse theory to bound the total Betti number of a hypersurface defined by such a dense fewnomial. These bounds contain and generalize previous bounds for ordinary fewnomials obtained by Bates, Bertrand, Bihan, and Sottile.

math.AG↗

arXiv2010

Real solutions to equations from geometry

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure. Particularly fruitful--both for information on real solutions and for applicability--are systems whose additional structure comes from geometry. Such equations from geometry for which we have information about their real solutions are the subject of these lecture notes, which focuses on bounds, both upper and lower, as well as situations in which all solutions can be real.

math.AG↗

arXiv2010

Solving Schubert Problems with Littlewood-Richardson Homotopies

We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.

math.NA↗

arXiv2009

Experimentation at the Frontiers of Reality in Schubert Calculus

We describe a general framework for large-scale computational experiments in mathematics using computer resources that are available in most mathematics departments. This framework was developed for an experiment that is helping to formulate and test conjectures in the real Schubert calculus. Largely using machines in instructional computer labs during off-hours and University breaks, it consumed in excess of 350 GigaHertz-years of computing in its first six months of operation, solving over 1.1 billion polynomial systems.

math.AG↗