Related papers: Differentiation in Bundles with a Hyperspace Base
Convolutional Neural Networks (CNNs) have become the state-of-the-art in supervised learning vision tasks. Their convolutional filters are of paramount importance for they allow to learn patterns while disregarding their locations in input…
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star,…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions is extremely important. One of the reasons is that its error…
The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such…
We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior…
Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow…
We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with…
Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…
Ultrafunctions are a particular class of functions defined on a hyperreal field $\mathbb{R}^{\ast}\supset\mathbb{R}$. They have been introduced and studied in some previous works. In this paper we introduce a particular space of…
In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus…
This work is the first in a series of papers that, among other things, extends the formalism of diolic differential calculus, wherein a new context for obtaining differential calculus in vector bundles was established. Here we provide a…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
As the development of measuring instruments and computers has accelerated the collection of massive amounts of data, functional data analysis (FDA) has experienced a surge of attention. The FDA methodology treats longitudinal data as a set…
Existing approaches to combine both additive and multiplicative neural units either use a fixed assignment of operations or require discrete optimization to determine what function a neuron should perform. However, this leads to an…
In this paper we study two operations, Pasting and Reversing, defined from a natural way to be applied over some rings such as the ring of polynomials and the ring of linear differential operators, which is a differential ring. We obtain…
Anomaly detection (AD) is the machine learning task of identifying highly discrepant abnormal samples by solely relying on the consistency of the normal training samples. Under the constraints of a distribution shift, the assumption that…
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas…