Related papers: Cartesian Difference Categories
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we…
Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category.…
We demonstrate the possibility of classifying causal systems into kinds that share a common structure without first constructing an explicit dynamical model or using prior knowledge of the system dynamics. The algorithmic ability to…
This paper explores differential bundles in tangent categories, characterizing them as functors from a structure category. This is analogous to the actegory perspective of Garner and Leung, which we also use to describe the tangent…
Categorical predictors are omnipresent in everyday regression practice: in fact, most regression data involve some categorical predictors, and this tendency is increasing in modern applications with more complex structures and larger data…
A causal-net is a finite acyclic directed graph. In this paper, we introduce a category, denoted by $\mathbf{Cau}$ and called causal-net category, whose objects are causal-nets and morphisms between two causal-nets are the functors between…
Though the topic of causal inference is typically considered in the context of classical statistical models, recent years have seen great interest in extending causal inference techniques to quantum and generalized theories. Causal…
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on…
We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
We establish a set of general results to study how the Galois action on modular tensor categories interacts with fusion subcategories. This includes a characterization of fusion subcategories of modular tensor categories which are closed…
Differentiable programming is the combination of classical neural networks modules with algorithmic ones in an end-to-end differentiable model. These new models, that use automatic differentiation to calculate gradients, have new learning…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
A probabilistic model describes a system in its observational state. In many situations, however, we are interested in the system's response under interventions. The class of structural causal models provides a language that allows us to…
Tangent categories offer a categorical context for differential geometry, by categorifying geometric notions like the tangent bundle functor, vector fields, Euclidean spaces, vector bundles, connections, etc. In the last decade, the theory…
We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. Specifically, we use a model category structure on differential graded categories over k (a field of characteristic 0)…