Related papers: Integration on valuation fields over local fields
These notes follow my articles [1, 6], and give some new important details. We propose here a new combinatorial method of encoding of measure spaces with measure preserving transformations, (or groups of transformations) in order to give…
We introduce a notion of integration on the category of proper birational maps to a given variety $X$, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers…
A construction of integration, function calculus, and exterior calculus is made, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann…
A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that…
The development of the theory of three-dimensional harmonic mappings is considered. The new classes of mappings that generate three-dimensional harmonic functions are introduced. The physical interpretation of these mappings is applied to…
The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This version deals with a more general situation that is a priori much more difficult. In this paper, we…
We discuss scalar quantum field theories in a Lorentz-invariant three-dimensional noncommutative space-time. We first analyze the one-loop diagrams of the two-point functions, and show that the non-planar diagrams are finite and have…
Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant…
Field theories with global continuous symmetries may admit configurations in which time translation invariance is broken by the movement of homogeneous background fields evolving along the flat directions implied by the symmetries. In this…
A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance…
All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complex-valued valuations on spherical Lebesgue measurable functions are also…
A generalized translational invariant noncommutative field theory is analyzed in detail, and a complete description of translational invariant noncommutative structures is worked out. The relevant gauge theory is described, and the planar…
A translation-invariant gapped local Hamiltonian is in the trivial phase if it can be connected to a completely decoupled Hamiltonian with a smooth path of translation-invariant gapped local Hamiltonians. For the ground state of such a…
Based on the notion of a $\Delta$-group(oid), ring-valued invariants of pairs of topological spaces can be defined in intrinsic topological terms.
Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit…
The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto &…
We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields.…
This survey paper explains how one can attach geometric invariants to semialgebraic sets defined over non-archimedean fields, using the theory of motivic integration of Hrushovski and Kazhdan. It also discusses tropical methods to compute…
The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of…
Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic…