Related papers: Computing Hermitian determinantal representations …
We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the…
We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In…
We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of…
Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the…
Inspired by the work about solutions of a system of real polynomial equations done by Hermite, this paper introduces a Hermitian form, which encodes information about solutions of a system of complex polynomial equations with conjugate…
In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary…
Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use…
Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation…
It is demonstrated that when the bundle of 2-forms on a four-dimensional manifold M admits an almost-complex structure any choice of "real + imaginary" subspace decomposition of the bundle defines a conjugation map, as well as a Hermitian…
Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is…
Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations,…
The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…
Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite…