scholarly journals Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

2016 ◽  
Vol 18 (4) ◽  
pp. 733-772 ◽  
Author(s):  
Damien Gayet ◽  
Jean-Yves Welschinger
Keyword(s):  
Betti Numbers ◽  
Real Hypersurfaces ◽  
Symmetric Matrices ◽  
Random Real

scholarly journals EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

2014 ◽  
Vol 14 (4) ◽  
pp. 673-702 ◽  
Author(s):  
Damien Gayet ◽  
Jean-Yves Welschinger
Keyword(s):  
Holomorphic Vector Bundle ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Connected Components ◽  
Projective Manifold ◽  
Positive Probability ◽  
Random Real ◽  
Smooth Complex ◽  
Real Locus ◽  
Complex Projective

Let$X$be a smooth complex projective manifold of dimension$n$equipped with an ample line bundle$L$and a rank$k$holomorphic vector bundle$E$. We assume that$1\leqslant k\leqslant n$, that$X$,$E$and$L$are defined over the reals and denote by$\mathbb{R}X$the real locus of$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in$\mathbb{R}X$of holomorphic real sections of$E\otimes L^{d}$, where$d$is a large enough integer. Moreover, given any closed connected codimension$k$submanifold${\it\Sigma}$of$\mathbb{R}^{n}$with trivial normal bundle, we prove that a real section of$E\otimes L^{d}$has a positive probability, independent of$d$, of containing around$\sqrt{d}^{n}$connected components diffeomorphic to${\it\Sigma}$in its vanishing locus.

scholarly journals Topology of Random Real Hypersurfaces

2015 ◽  
Vol 49 (1) ◽  
pp. 139-160
Author(s):  
Jean-Yves Welschinger
Keyword(s):  
Real Hypersurfaces ◽  
Random Real

Las siguientes son las notas de un mini curso que dí durante la escuela de verano CIMPA en Villa de Leyva, Colombia, en julio de 2014. El tema fue el trabajo que en conjunto se desarrolló con Damien Gayet sobre la topología de las hipersuperficies reales aleatorias, restringiéndonos al caso de los espacios proyectivos y enfocándonos en nuestras estimaciones inferiores. Particularmente, estimamos (por arriba y) por abajo la esperanza matemática de todos los números de Betti de las hipersuperficies reales proyectivas aleato- rias de grado d. De hecho, para cualquier hipersuperficie cerrada y conexa ∑ de Rn, estimamos por abajo la esperanza del número de componentes conexas de éstas hipersuperficies reales proyectivas aleatorias de grado d, las cuales son difeomorfas a ∑.

Extremal Betti numbers of graded ideals

2003 ◽  
Vol 43 (3-4) ◽  
pp. 235-244 ◽  
Author(s):  
Marilena Crupi ◽  
Rosanna Utano
Keyword(s):  
Betti Numbers

REMARKS ON THE KIELANOWSKI PARAMETRIZATION OF THE KOBAYASHI-MASKAWA MATRIX

1996 ◽  
Vol 11 (31) ◽  
pp. 2531-2537 ◽  
Author(s):  
TATSUO KOBAYASHI ◽  
ZHI-ZHONG XING
Keyword(s):  
Symmetric Matrices

We study the Kielanowski parametrization of the Kobayashi-Maskawa (KM) matrix V. A new two-angle parametrization is investigated explicitly and compared with the Kielanowski ansatz. Both of them are symmetric matrices and lead to |V13/V23|=0.129. Necessary corrections to the off-diagonal symmetry of V are also discussed.

Converting immanants on skew-symmetric matrices

2021 ◽  
Vol 618 ◽  
pp. 76-96
Author(s):  
M.A. Duffner ◽  
A.E. Guterman ◽  
I.A. Spiridonov
Keyword(s):  
Symmetric Matrices

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

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