arXiv subjects

Petr Vlachopulos

Publications and source records attributed to Petr Vlachopulos.

4 recordsLinked to original sources

arXiv2026

The Cheeger constant of curved tubes in real space forms

Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.

math.DG↗

arXiv2026

On the generalized inverse tangent integral and Catalan's constant

In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The core idea is based on the introduction of an auxiliary integral depending on two parameters and analyzing it via a generating-function perspective. This converts the integral into an explicit formula, yielding a compact representation in terms of the real part of a dilogarithmic expression plus a companion dilogarithm contribution. In parallel, the inverse tangent integral is rewritten through standard integral transformations into a form governed by the imaginary part of a dilogarithm evaluated at a complex argument, producing a clean polylogarithmic description. Overall, we establish a coherent bridge between arctangent-type integrals, identities connected to Catalan's constant, and the systematic generation of new integral representations and decompositions.

math.NT↗

arXiv2026

Entropy and Holography through Adjunctions: A Bicategorical Perspective on Landauer's Principle

We develop a bicategorical formulation of entropy and Landauer's principle in which deterministic monotone maps between logical and thermodynamic state spaces are embedded into a broader theory of open, many-to-many feasibility interfaces. Entropy-ordered state spaces form the objects of a locally posetal bicategory, Boolean profunctors encode feasible implementations, and refinements define 2-morphisms. Although this bicategory admits genuinely non-representable interfaces, every exact Landauer adjunction lies in its representable sector and is generated by a unique monotone implementation map. Such an adjunction also induces an idempotent boundary monad, together with a dual idempotent bulk comonad. Our main result identifies the Eilenberg-Moore object of the boundary monad with a universal boundary-stable quotient determined by the bulk-mediated round-trip relation. This provides a canonical reconstruction principle: the quotient captures exactly those boundary distinctions that remain observable after implementation in the bulk and subsequent readout. In the representable case, two boundary states are identified precisely when they have the same image under the implementation map. Therefore, the result turns irreversible information loss into an explicit universal object and provides a compositional framework for studying reconstruction, coarse-graining, and bulk-boundary visibility in open thermodynamic systems. Finally, we outline a potential cost-enriched extension in which interfaces carry dissipation costs and composition minimizes accumulated dissipation over admissible intermediate realizations, opening a route from Boolean feasibility to quantitative thermodynamic implementation.

physics.gen-ph↗

arXiv2018

The Cheeger constant of curved tubes

We compute the Cheeger constant of spherical shells and tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space.

math.OC↗