相关论文: The quaternionic determinat
We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…
We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime…
This is a complement to our paper arXiv:0802.1461. We study irreducibility of spectral determinants of some one-parametric eigenvalue problems in dimension one with polynomial potentials.
As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts. In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly…
Recent innovations in the differential calculus for functions of non-commuting variables, beginning with a quaternionic variable, are now extended to consider some integration.
We develop a method of reducing the size of quantum minors in the algebra of n x n quantum matrices. The method is used to show that quantum determinantal factor rings of n x n quantum matrices over the complex numbers are maximal orders,…
We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…
We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd…
We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…
We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d…
We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with…
A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several…
Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a…
In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A…
Hadamard's maximal determinant problem consists in finding the maximal value of the determinant of a square $n\times n$ matrix whose entries are plus or minus ones. This is a difficult mathematical problem which is not yet solved. In the…
The literature presents the characteristic function of the Wishart distribution on m times m matrices as an inverse power of the determinant of the Fourier variable, the exponent being the positive, real shape parameter. I demonstrate that…
The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates,…
Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete…
The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation,…
A formula expressing the fermionic determinant (a large order polynomial) as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and…