Related papers: Computing the Conley Index: a Cautionary Tale
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…
We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the…
Let $R$ be a commutative ring and $g(t) \in R[t]$ a monic polynomial. The commutative ring of polynomials $f(C_g)$ in the companion matrix $C_g$ of $g(t)$, where $f(t)\in R[t]$, is called the Companion Ring of $g(t)$. Special instances…
In statistics cumulants are defined to be functions that measure the linear independence of random variables. In the non-communicative case the Boolean cumulants can be described as functions that measure deviation of a map between algebras…
Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}$ to $G$ factors through the projection to some finite product $\mathbb Z^n$. Various authors have proposed…
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to…
A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for…
We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_{\theta}/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic…
In this paper we investigate Deep Learning Models using topological dynamical systems, index theory, and computational homology. These mathematical machinery was invented initially by Henri Poincare around 1900 and developed over time to…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
The notion of differentiation index for DAE systems of arbitrary order with generic second members is discussed by means of the study of the behavior of the ranks of certain Jacobian associated sub-matrices. As a by-product, we obtain upper…
We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an…
This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with…
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the…
We establish nonparametric identification in a class of so-called index models using a novel approach that relies on general topological results. Our proof strategy requires substantially weaker conditions on the functions and distributions…
Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure.…
Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined…
Motivated by applications in reliable and secure communication, we address the problem of tiling (or partitioning) a finite constellation in $\mathbb{Z}_{2^L}^n$ by subsets, in the case that the constellation does not possess an abelian…
We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules…
We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of…