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There are two ways to compute Poincar\'e-Dulac normal forms of systems of ODEs. Under the original approach used by Poincar\'e the normalizing transformation is explicitly computed. On each step, the normalizing procedure requires the…

We present canonical forms for all indecomposable pairs $(A,B)$ of commuting nilpotent matrices over an arbitrary field under simultaneous similarity, where $A$ is the direct sum of two Jordan blocks with distinct sizes. We also provide the…

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by SLOCC…

This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition…

Using some modification of the standard fermion technique we derive factorized formula for spin operator matrix elements (form-factors) between general eigenstates of the Hamiltonian of quantum Ising chain in a transverse field of finite…

Let $L \subset \mathbb{C}^r \otimes \mathbb{C}[x_1^\pm, \ldots, x_n^\pm]$ be a finite dimensional subspace of vector-valued Laurent polynomials invariant under the action of torus $(\mathbb{C}^*)^n$. We study subvarieties in the torus,…

Let $\mathbb{K}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{K})$ be the algebra of all $n\times n$ matrices over $\mathbb{K}$. We consider the corresponding special Jordan algebra $\mathcal{A}:=M_n(\mathbb{K})^+$…

We introduce vector bundle techniques for finding equations of secant varieties. A test is established that determines when a secant variety is an irreducible component of the zero set of the equations found. We also prove an induction…

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix…

In this paper subcentral (resp., central) idempotent series and composition subcentral (resp., central) idempotent series in an inverse semigroup are introduced and investigated. It is shown that if $S=EG$ is a factorizable inverse monoids…

We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance…

It is well known that the matrix exponential of a non-normal matrix can exhibit transient growth even when all eigenvalues of the matrix have negative real part, and similarly for the powers of the matrix when all eigenvalues have magnitude…

In [14], the second named author constructed the bracket invariant [.] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following…

We provide and discuss complex analytic methods for overcoming the formal character of formal deformation quantization. This is a necessity for returning to physically meaningful statements, and accounts for the fact that the formal…

This paper uses tools in group theory and symbolic computing to give a classification of the representations of finite groups with order lower than 9 that can be derived from the study of local reversible-equivariant vector fields in…

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of…

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered, the starting point being the classical Bjorck-Pereyra algorithms for…

Linear neural network layers that are either equivariant or invariant to permutations of their inputs form core building blocks of modern deep learning architectures. Examples include the layers of DeepSets, as well as linear layers…

Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used L\"owner ordering, which relies on real-valued eigenvalues, is limited…