Related papers: Non-standard Analysis in Dynamic Geometry
Complex geometries as common in industrial applications consist of multiple patches, if spline based parametrizations are used. The requirements for the generation of analysis-suitable models are increasing dramatically since isogeometric…
This article introduces a novel nonparametric methodology for Generalized Linear Models which combines the strengths of the binary regression and latent variable formulations for categorical data, while overcoming their disadvantages.…
We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental…
We consider reusing established non-probabilistic output analyses (either forward or backwards) that yield over-approximations of a program's pre-image or image relation, e.g., interval analyses. We assume a probability measure over the…
It is time to renew old ways of thinking about dimensional analysis. Specifically, more than $n-r$ invariants and more than one functional relation between invariants need to be considered simultaneously. Thus generalized, dimensional…
The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a…
The asymptotic expansion method is generalized from the periodic setting to stationary ergodic stochastic geometries. This will demonstrate that results from periodic asymptotic expansion also apply to non-periodic structures of a certain…
Thanks to the nonstandard formalization of fast oscillating functions, due to P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for new non-asymptotic estimation techniques, which do not necessitate any statistical…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
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…
Distributional shift between domains poses great challenges to modern machine learning algorithms. The domain generalization (DG) signifies a popular line targeting this issue, where these methods intend to uncover universal patterns across…
The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are…
Classical dimensional analysis is one of the cornerstones of qualitative physics and is also used in the analysis of engineering systems, for example in engineering design. The basic power product relationship in dimensional analysis is…
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the…
We apply the method of differential renormalization to two and three dimensional abelian gauge theories. The method is especially well suited for these theories as the problems of defining the antisymmetric tensor are avoided and the…
Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of…
Nested space-filling designs are nested designs with attractive low-dimensional stratification. Such designs are gaining popularity in statistics, applied mathematics and engineering. Their applications include multi-fidelity computer…
Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…