Let $$M_{n}$$ denote a random symmetric $$n\times n$$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher variables (taking values $$\pm 1$$ with probability 1/2 each). Resolving conjecture of Vu, we prove that permanent has magnitude $$n^{n/2+o(n)}$$ $$1-o(1)$$ . Our result can also be extended to more general models matrices. In our proof, build extend some techniques introd...

Matrix Schubert varieties are the closures of orbits B×B acting on all n×n matrices, where B is group invertible lower triangular matrices. Extending work Fulton, Knutson and Miller identified a Gröbner basis for prime ideals these varieties. They also showed that corresponding initial Stanley-Reisner shellable simplicial complexes, derived related primary decomposition in terms reduced pipe dr...

We construct interpolation operators for functions taking values in a symmetric space—a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition—a generalization of the well-known factorizat...

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positivesemidefinite matrix and a symmetric indefinite matrix. ...

The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.This comput...