On the spectrum of a self-adjoint polynomial pencil

1997 ◽  
Vol 31 (3) ◽  
pp. 204-207
Author(s):  
K. Kh. Boimatov
Keyword(s):  
Adjoint Polynomial

scholarly journals On the minimum real roots of the adjoint polynomial of a graph

2009 ◽  
Vol 309 (13) ◽  
pp. 4635-4641 ◽  
Author(s):  
Haixing Zhao ◽  
Ruying Liu
Keyword(s):  
Real Roots ◽  
Adjoint Polynomial

scholarly journals Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhigang Jia ◽  
Meixiang Zhao ◽  
Minghui Wang ◽  
Sitao Ling
Keyword(s):  
Iterative Algorithm ◽  
Perturbation Analysis ◽  
Matrix Equation ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Polynomial Matrix ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Solvability Theory ◽  
Adjoint Polynomial

The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by numerical experiments.

scholarly journals Two remarks on the adjoint polynomial

2012 ◽  
Vol 33 (4) ◽  
pp. 583-591 ◽  
Author(s):  
Péter Csikvári
Keyword(s):  
Adjoint Polynomial

scholarly journals Projective Geometry of Wachspress Coordinates

2019 ◽  
Vol 20 (5) ◽  
pp. 1135-1173 ◽  
Author(s):  
Kathlén Kohn ◽  
Kristian Ranestad
Keyword(s):  
Hyperplane Arrangement ◽  
Probability Distributions ◽  
Linear Span ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Intersection Theory ◽  
Barycentric Coordinates ◽  
Rational Map ◽  
Adjoint Polynomial ◽  
Wachspress Coordinates

Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

Best restriction L1-approximation of periodic convolution class by generalized splines

2014 ◽  
Vol 12 (05) ◽  
pp. 1461006
Author(s):  
Rong Huang ◽  
Yongping Liu
Keyword(s):  
Differential Operator ◽  
L1 Approximation ◽  
Exact Constants ◽  
Adjoint Polynomial

We consider the estimate of the best L1-approximation of convolution classes [Formula: see text], defined by a self-adjoint polynomial differential operator Pr(D), in the term of some kinds of generalized splines and get some exact constants.

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