Related papers: A descriptive approach to higher derived limits
We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris Lambie-Hanson which…
This paper significantly strengthens directed low-diameter decompositions in several ways. We define and give the first results for separated low-diameter decompositions in directed graphs, tighten and generalize probabilistic guarantees,…
We consider the question of the additivity of strong homology. This entails isolating the set-theoretic content of the higher derived limits of an inverse system indexed by the functions from $\mathbb{N}$ to $\mathbb{N}$. We show that this…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to…
We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…
In this paper we examine the concept of complexity as it applies to generative art and design. Complexity has many different, discipline specific definitions, such as complexity in physical systems (entropy), algorithmic measures of…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and…
We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\mathcal{F}(z)=\prod_{\ell=1}^\infty…
Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain…
We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
The ability to differentiate through optimization problems has unlocked numerous applications, from optimization-based layers in machine learning models to complex design problems formulated as bilevel programs. It has been shown that…
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
We investigate higher derivative estimates for the Lam\'e system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. As the distance $\varepsilon$ between two closely spaced hard inclusions approaches zero, the stress in…
\v{C}ech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_\beta$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets…