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We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral.…

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear…

We study possible factorizations of supersymmetric (SUSY) transformations in the one-dimensional quantum mechanics into chains of elementary Darboux transformations with nonsingular coefficients. A classification of irreducible (almost)…

Tensors are ubiquitous in science and engineering and tensor factorization approaches have become important tools for the characterization of higher order structure. Factorizations includes the outer-product rank Canonical Polyadic…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

We obtain some structural properties of a factorised group $G = AB$, given that the conjugacy class sizes of certain elements in $A\cup B$ are not divisible by $p^2$, for some prime $p$. The case when $G = AB$ is a mutually permutable…

Let $q$ be a prime power. This paper provides a new class of linear codes that arises from the action of the alternating group on $\mathbb F_q[x_1,\dots,x_m]$ combined with the ideas in (M. Datta and T. Johnsen, 2022). Compared with…

We use density functional theory to describe the phase behaviors of rigid molecules. The construction of kernel function G(x, P, x, P) is discussed. Excluded-volume potential is calculated for two types of molecules with C_{2v} symmetry.…

We prove that particle models whose action is given by the integrated $n$-th curvature function over the world line possess $n+1$ gauge invariances. A geometrical characterization of these symmetries is obtained via Frenet equations by…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…

In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling…

Inclusive deep inelastic scattering factorization combines two features that are often treated separately: an asymptotic reconstruction of the current-current matrix element from hard and long-distance data, and an invariance under finite…

Recently a systematic investigation of monoids of sequences of plus-minus weighted zero-sum sequences had been started, which is among others motivated by applications to monoids of norms of algebraic integers. In the current paper these…

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the…

Orders in an algebraic number field form a class of rings which are of special historical interest to the field of factorization theory. One of the primary tools used to study factorization is elasticity - a measure of how badly unique…

Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of…

Let $(X,\bullet )$ be a groupoid (binary algebra) and $Bin(X\dot{)}$ denote the collection of all groupoids defined on $X$. We introduce two methods of factorization for this binary system under the binary groupoid product \textquotedblleft…

The twisted partition monoid $\mathcal{P}_n^\Phi$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main…

Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent…

We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On…