Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

Truncated Local Uniformization of Formal Integrable Differential Forms

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

Crepant semi-divisorial log terminal model

We prove the existence of a crepant sdlt model for slc pairs whose irreducible components are normal in codimension one.

Higher connectivity of tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is $$(d-\ell )$$ ( d - ℓ ) -connected through codimension one, where $$\ell $$ ℓ is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.

Cooperation-Based Modeling of Sustainable Development: An Approach from Filippov’s Systems

The concept of Sustainable Development has given rise to multiple interpretations. In this article, it is proposed that Sustainable Development should be interpreted as the capacity of territory, community, or landscape to conserve the notion of well-being that its population has agreed upon. To see the implications of this interpretation, a Brander and Taylor model, to evaluate the implications that extractivist policies have over an isolated community and cooperating communities, is proposed. For an isolated community and through a bifurcation analysis in which the Hopf bifurcation and the heteroclinic cycle bifurcation are detected, 4 prospective scenarios are found, but only one is sustainable under different extraction policies. In the case of cooperation, the exchange between communities is considered by coupling two models such as the one defined for the isolated community, with the condition that their transfers of renewable resources involve conservation policies. Since human decisions do not occur in a continuum, but rather through jumps, the mathematical model of cooperation used is a Filippov System, in which the dynamics could involve two switching manifolds of codimension one and one switching manifold of codimension two. The exchange in the cooperation model, for specific parameter arrangements, exhibits n -periodic orbits and chaos. It is notable that, in the cases in which the system shows sliding, it could be interpreted as a recovery delay related to the time needed by the deficit community to recover, until its dependence on the other community stops. It is concluded (1) that a sustainability analysis depends on the way well-being is defined because every definition of well-being is not necessarily sustainable, (2) that sustainability can be visualized as invariant sets in the nonzero region of the space of states (equilibrium points, n -periodic orbits, and strange attractors), and (3) that exchange is key to the prevalence of the human being in time. The results question us on whether Sustainable Development is only to keep us alive or if it also implies doing it with dignity.

Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.

Hexagon bootstrap in the double scaling limit

We study the six-particle amplitude in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory in the double scaling (DS) limit, the only nontrivial codimension-one boundary of its positive kinematic region. We construct the relevant function space, which is significantly constrained due to the extended Steinmann relations, up to weight 13 in coproduct form, and up to weight 12 as an explicit polylogarithmic representation. Expanding the latter in the collinear boundary of the DS limit, and using the Pentagon Operator Product Expansion, we compute the non-divergent coefficient of a certain component of the Next-to-Maximally-Helicity-Violating amplitude through weight 12 and eight loops. We also specialize our results to the overlapping origin limit, observing a general pattern for its leading divergences.