Lipschitz symmetric functions on Banach spaces with symmetric bases

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

Application of symmetric analytic functions to spectra of linear operators

The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Linear Atomic Cluster Expansion Force Fields for Organic Molecules: beyond RMSE

We demonstrate that fast and accurate linear force fields can be built for molecules using the Atomic Cluster Expansion (ACE) framework. The ACE models parametrize the Potential Energy Surface in terms of body ordered symmetric polynomials making the functional form reminiscent of traditional molecular mechanics force fields. We show that the 4 or 5-body ACE force fields improve on the accuracy of the empirical force fields by up to a factor of 10, reaching the accuracy typical of recently proposed machine learning based approaches. We not only show state of the art accuracy and speed on the widely used MD17 and ISO17 benchmark datasets, but also go beyond RMSE by comparing a number of ML and empirical force fields to ACE on more important tasks such as normal mode prediction, high temperature molecular dynamics, dihedral torsional profile prediction and even bond breaking. We also demonstrate the smoothness, transferability and extrapolation capabilities of ACE on a new challenging benchmark dataset comprising a potential energy surface of a flexible drug-like molecule.

Symmetric polynomials in the variety generated by Grassmann algebras

Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].

Quadrature domains and the real quadratic family

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of P S L ( 2 , Z ) \mathrm {PSL}(2,\mathbb {Z}) , and that this conformal mating is the Schwarz function of a simply connected quadrature domain.