Related papers: On diffeologies for power sets and measures

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without…

We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous…

We consider a geometrization, i.e., we identify geometrical structures, for the space of density states of a quantum system. We also provide few comments on a possible application of this geometrization for composite systems.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then…

Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…

We show that any multiplicative bijection between the algebras of differentiable functions, defined on differentiable manifolds of positive dimension, is an algebra isomorphism, given by composition with a unique diffeomorphism.

We study the divided power structures over a product of operads with distributive law. We give a systematic method to characterise the divided power algebras over such a product from the structures of divided power algebra coming from each…

We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular…

The quantum theory of a massless spin two particle is strongly constrained by diffeomorphism invariance, which is in turn implied by unitarity. We explicitly exhibit the space-time diffeomorphism algebra of string theory, realizing it in…

The paper is a survey of some results about Weil algebras applicable in differential geometry, especially in some classification questions on bundles of generalized velocities and contact elements. Mainly, a number of claims concerning a…

In this paper, we systematically investigate the nilpotentizer and nilpotent graph for a Lie superalgebra over the field of characteristic not equal to 2. First, we establish some fundamental properties of the nilpotentizer. Next, we show…

We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent…

The dressing field method is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M)-invariant objects, which has a natural relational…

Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the…

We propose a framework for performing differentiable geometric modeling based on the Function Representation (FRep). The framework is built on top of modern libraries for performing automatic differentiation allowing us to obtain…

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic…

The structural analysis of shape boundaries leads to the characterization of objects as well as to the understanding of shape properties. The literature on graphs and networks have contributed to the structural characterization of shapes…

We provide a framework for the construction of diffeomorphism invariant sheaves of nonlinear generalized functions spaces. As an application, global algebras of generalized functions for distributions on manifolds and diffeomorphism…