arXiv subjects

Luis A. Cedeño-Pérez

Publications and source records attributed to Luis A. Cedeño-Pérez.

5 recordsLinked to original sources

arXiv2026

Invariant Measures in Hamiltonian Systems: The Analytical Foundations of Statistical Physics

We construct a measure in the hamiltonian function level sets that is invariant under the hamiltonian flow for short times and flow preserving for arbitrarily long times. This allows a probabilistic approach to the study of hamiltonian systems, in the space of states with fixed energy. We prove that this measure generates the microcanonical partition function employed in physics and show that it can be transformed into the canonical partition function in an asymptotic limit, hence reproducing classical Statistical Physics. We also argue that this gives an alternative solution to Simon's second problem.

math-ph↗

arXiv2026

Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure

We employ methods of geometry and generalized convergence to construct a geometric measure that serves as an alternative to the integer-dimension Hausdorff measure. This construction prioritizes integration, yields the Area Formula as a byproduct of the construction and the Coarea Formula follows indirectly from the smooth case. Furthermore, this is the smallest measure that satisfies the Area Formula. Though this construction is not as general as Hausdorff measure, it provides a much simpler introduction to the topic and is enough for certain applications to Geometric Analysis.

math.FA↗

arXiv2025

A General Theory of Operator-Valued Measures

We construct a new kind of measures, called projection families, which generalize the classical notion of vector and operator-valued measures. The maximal class of reasonable functions admits an integral with respect to a projection family, where the integral is defined as an element of the second dual instead of the original space. We show that projection families possess strong enough properties to satisfy the theorems of Monotone Convergence and Dominated convergence, but are much easier to come by than the more restrictive operator-valued measures.

math.FA↗

arXiv2025

Smooth Functional Calculus and Spectral Theorem in Banach Spaces

The notion of projection families generalizes the classical notions of vector- and operator-valued measures. We show that projection families are general enough to extend the Spectral Theorem to Banach algebras and operators between Banach spaces. To this end, we first develop a Smooth Functional Calculus in Banach algebras using the Cauchy-Pompeiu formula, which is further extended to a Continuous Functional Calculus. We also show that these theorems are proper generalizations of the usual result for operators between Hilbert spaces.

math.FA↗

arXiv2024

Functional measures associated to operators

We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to compute the integral of functions with respect to this measure and develop approximate formulas in terms of a perturbative expansion. We show that this method can be used to represent solutions of classical equations, such as the diffusion and Fokker-Plank equations, as Wiener and Martin-Siggia-Rose-Jansen-de Dominics integrals, and propose an extension to paths in infinite dimensional spaces.

math-ph↗