Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.     

Top Varieties of Generalized MV-Algebras and Unital Lattice-Ordered Groups

In spite of the well-know fact that the system of ?-groups with strong unit (unital ?-groups) does not form a variety, there is a categorical connection between the category of unital ?-groups and the variety of generalized MV-algebras which enables us to naturally export equational machinery and terminology like “variety” from the latter category to the former.

Using this categorical equivalence, we study varieties, or equationally defined classes, and top varieties, varieties above the normal valued variety, of both structures. We generalize Chang's Completeness Theorem for generalized MV-algebras, and formulate some open questions for both structures.     

A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.     

Central Sets Theorem along filters and some combinatorial consequences

The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions and extensions of this theorem. All of these theorems deal with central sets, and its origin from the algebra of Stone–?ech compactification of arbitrary semigroup, say $\beta S$. It can be proved that every closed subsemigroup of $\beta S$ is generated by a filter. We will show that, under some restrictions, one can derive the Central Sets Theorem for any closed subsemigroup of $\beta S$. We will derive this theorem using the corresponding filter and its algebra. Later we will also deal with how the notions of largeness along filters are preserved under some well behaved homomorphisms and give some consequences.     

The second variation of nonorientable minimal submanifolds

Suppose is a complete nonorientable minimal submanifold of a Riemannian manifold . We derive a second variation formula for the area of with respect to certain perturbations, giving a sufficient condition for the instability of . Some simple applications are given: we show that the totally geodesic is the only stable surface in , and we show the non-existence of stable nonorientable cones in . We reproduce and marginally extend some known results in the truly non-compact setting.