Related papers: Cayley's hyperdeterminant: a combinatorial approac…
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete…
We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated…
The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a…
Type B 3-fold supersymmetry is a necessary and sufficient condition for a quantum Hamiltonian to admit three linearly independent local solutions in closed form. We show that any such a system is invariant under GL(3,C) homogeneous linear…
We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence…
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no…
We study holomorphic 2-forms on projective (or compact Kaehler) threefolds not of general type and prove that in almost all cases the 2-form is created by some standard process. This means roughly that every 2-form is induced by a…
For a Lie algebra ${\mathcal L}$ with basis $\{x_1,x_2,\cdots,x_n\}$, its associated characteristic polynomial $Q_{{\mathcal L}}(z)$ is the determinant of the linear pencil $z_0I+z_1\text{ad} x_1+\cdots +z_n\text{ad} x_n.$ This paper shows…
We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that…
We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite…
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer $d$. It outputs the ideal of that family intersected with…
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we…
In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights…
We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative…
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for…
This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding…
We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove…
A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic…
We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…
Let g be a basic classical Lie superalgebra over C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of…