Related papers: A formula for the Jack super nabla operator
We introduce a formal $\hbar$-differential operator $\Delta$ that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $P_{\infty}$-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in…
In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+\mu(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla…
If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go…
The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$, which is obtained by replacing the Cartesian components of $\bm{r}$ by the Cartesian components of $\nabla$ in the regular solid harmonic ${\mathcal{Y}}_{\ell}^{m}…
It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the…
In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the…
In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability…
The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the…
We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…
We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free…
We discuss Sekiguchi-type differential operators, their eigenvalues, and a generalization of Andrews-Goulden-Jackson formula. These will be applied to extract explicit formulae involving shifted partitions and hook lengths.
We obtain the bi-Hamiltonian structure of the super KP hierarchy based on the even super KP operator $\Lambda = \theta^{2} + \sum^{\infty}_{i=-2}U_{i} \theta^{-i-1}$, as a supersymmetric extension of the ordinary KP bi-Hamiltonian…
We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables $x_1,x_2,...$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised…
It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik-Schouten bracket. This is a particular case of…
We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla…
We prove that Euler supercharacters for orthosymplectic Lie superalgebras can be obtained as a certain specialization of super Jacobi polynomials. A new version of Weyl type formula for super Schur functions and specialized super Jacobi…
For permutations $v,w \in \mathfrak S_n$, Macdonald defines the skew divided difference operators $\partial_{w/v}$ as the unique linear operators satisfying $\partial_w(PQ) = \sum_v v(\partial_{w/v}P) \cdot \partial_vQ$ for all polynomials…
The first part of the paper is devoted to the $G$-type spaces i.e. the spaces $G^\alpha_\alpha (\mathbb R^d_+)$, $\alpha\geq 1$ and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with…
We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras…
We first introduce a new presentation for the mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$ over an arbitrary commutative ring $R$ and derive a new basis. Based on this presentation, specializing to the case of $\mathscr{H}_n(q)$ over the…