Rings and Algebras Papers of the Week

To this end, we prove that, for every topological MV-algebra, the associated Mundici group can be endowed with a topology that extends the topology of the MV-algebra and makes it into an abelian topological $\ell$-group.

This paper introduces the ${\rm P}$-generalized determinant, a more general class of quaternion determinants.

For any $m$ pairwise commuting matrices $\{A_i\}_{i=1}^{m}\subset \M_n(\F)$, we prove $\dim_\F \F[A_1,\cdots, A_m]\leq (2^m-1)(n-1)+1$, answering Bergman's question on the existence of a linear bound for a fixed number $m$.

We determine the least $m$ for which every point of each arc occurs as an eigenvalue in our family.

We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension.

We show that the Lie algebra of algebraic vector fields for any connected, semi-simple matrix group over $\mathbb{F}$, for $\mathbb{F}=\mathbb{R},\mathbb{C}$, can be generated by three $\mathbb{F}$-complete algebraic vector fields.

However, even if the sampled matrices equal the original matrix in expectation, the expected squared norm of the random iterates may grow while the original iteration converges. Under a fixed rank bound, we study how to choose the distribution of these random matrices to minimize this growth when sampling independently at each step.

In this paper, we study quasi-projective dimensions of modules over rings connected by homological ring epimorphisms.

We prove that every composition factor algebra is representation-finite.

A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block.

We determine the high-degree restricted first cohomology of the positive subalgebra of the rank-one contact algebra in characteristics five and seven, with parabolically induced coefficients.

At height two in characteristic seven, we present the Yoneda algebra as an eight-dimensional symmetric algebra tensored with a polynomial algebra on three central degree-two classes.

We study the representability of actions in varieties of algebras with $n \geq 1$ not necessarily associative bilinear multiplications.

An anti-quadratic nearly associative algebra $(A, \bullet, \omega)$ is a nearly associative algebra equipped with a non-degenerate skew-symmetric bilinear form $\omega$ satisfying the anti-associativity condition $\omega(u \bullet v, w) = -\omega(u, v \bullet w)$ for all $u, v, w \in A.$ In this paper, we investigate the structural properties and extension theory of these algebras.

We find bases for polynomial identities and calculate their codimensions for the Sweedler algebra, treating it first as an ordinary associative algebra over a field and second as a Hopf algebra.

We study four-dimensional binomial Artin-Schelter regular algebras and their connection with twisted monoid algebras.

We classify linear maps $\phi$ between possibly different rectangular matrix spaces over a field $\mathbb F$ with $|\mathbb F|\ge5$ such that $\operatorname{rank} \phi(A)\le2$ whenever $\operatorname{rank} A\le2$.

We prove that a finite-dimensional connected monomial algebra over an algebraically closed field is derived tame if and only if every finite connected convex subcategory of its universal cover has non-negative Euler form.

Although non-dimensionalisation is a fundamental step in modelling, it is rarely obvious, especially for large dynamical systems, how to systematically construct optimal non-dimensionalisations. In this work, we develop a general mathematical framework for non-dimensionalising polynomial ODEs by interpreting them as chemical reaction networks (CRNs) and characterise the space of admissible scalings.