Related papers: Lee-Yang Problems and The Geometry of Multivariate…
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic…
We study, in the setting of algebraic varieties, finite-dimensional spaces of functions V that are invariant under a ring D^V of differential operators, and give conditions under which D^V acts irreducibly. We show how this problem,…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.
We consider the problem of finding the set of classical polylogarithmic functions $\text{Li}_n$ with branching locus determined by the solution of $p_1\cdot p_2\cdot \ldots \cdot p_n=0$, where $p_1,\ldots, p_n$ are irreducible polynomials…
Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate…
We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…
We introduce and study a notion of dually Lorentzian polynomials, and show that if $s$ is non-zero and dually Lorentzian then the operator \[s(\partial_{x_1},\ldots,\partial_{x_n}):\mathbb R[x_1,\ldots,x_n] \to \mathbb R[x_1,\ldots,x_n]\]…
Basic properties of Lie-orthogonal operators on a finite-dimensional Lie algebra are studied. In particular, the center, the radical and the components of the ascending central series prove to be invariant with respect to any Lie-orthogonal…
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore…
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a…
We obtain a general class of polynomials for which the Schrodinger operator has a discrete spectrum. This class includes all the scalar potentials in membrane, 5-brane, p-branes, multiple M2 branes, BLG and ABJM theories. We provide a proof…
We consider algebras acting on Schur and Q-Schur polynomials, corresponding to Kadomtsev-Petviashvili (KP) and BKP hierarchies. We present them in the spirit of affine Yangians, paying special attention to commutative subalgebras, box…
We study a class of noncommutative gauge theory models on 2-dimensional Moyal space from the viewpoint of matrix models and explore some related properties. Expanding the action around symmetric vacua generates non local matrix models with…
In this paper we introduce new various generalizations of the classical Kadomtsev-Petviashvili hierarchy in the case of operators in several variables. These generalizations are the candidates for systems that should play the role,…
We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from…
We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers but each of them…
We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential…
The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise…
By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate L\'evy processes. Many properties of these polynomials are stated straightforwardly thanks to…