arXiv subjects
Publications and source records attributed to Anselmo Torresblanca-Badillo.
The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various varieties of algebras, such as Lie, Leibniz, and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalised to the variety of all non-associative algebras.
This article describes a class of pseudo-differential operators \begin{equation*} (\mathcal{A}^αφ)(x)=\mathcal{F}^{-1}_{ξ\rightarrow x}\left(\left[\max\{|\boldsymbolψ_{1}(||ξ||_{p})|,|\boldsymbolψ_{2}(||ξ||_{p})|\}\right]^{-α}\widehatφ(ξ)\right), \end{equation*} $φ\in \mathcal{D}(\mathbb{Q}_{p}^{n})$ and $α\in\mathbb{C}$; here $\left[\max\{|\boldsymbolψ_{1}(||ξ||_{p})|,|\boldsymbolψ_{2}(||ξ||_{p})|\}\right]^{-α}$ is the symbol of the operator $\mathcal{A}^α$. These operators can be seen as a generalization of the Bessel potentials in the $p$-adic context. We show that the family $\left(K_α\right)_{α>0}$ of convolution kernels attached to generalized Bessel potentials $\mathcal{A}^α$, $α>0$, determine a convolution semigroup on $\mathbb{Q}_{p}^{n}$. Imposing certain conditions we have that $K_α$, $α>0$, is a probability measure on $\mathbb{Q}_{p}^{n}$. Moreover, we will study certain properties corresponding to the Green function of the operator $\mathcal{A}^α$ and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.
In this article, we study a large class of radial probability density functions defined on the p-adic numbers from which it is possible to obtain certain non-archimedean pseudo-differential operators. These operators are associated with certain p-adic master equations of some models of complex systems (such as glasses, macromolecules, and proteins). We prove via the theory of distributions some properties corresponding to the heat Kernel associated with these pseudo-differential operators. Also, study some properties corresponding to the fundamental solution of these p-adic equations. Finally, we will study strong Markov processes, the first passage time problem and the survival probability (of the trajectories of these processes) corresponding to radial probability density functions connected with energy landscapes of the linear and logarithmic types.
In this article, we study a class of non-archimedean pseudo-differential operators associated via Fourier transform to the Bessel potentials. These operators (which we will denote as $J^{α},$ $α>n$) are of the form (J^{α})(x)=\mathcal{F}_{ξ\rightarrow x}^{-1}\left[ (\max\{1,||ξ||_{p}\})^{-α}\widehat{φ}(ξ)\right] ,\text{ } φ\in \mathcal{D}\mathbb{Q}_{p}^{n}),\text{ } x\in\mathbb{Q}_{p}^{n}. We show that the fundamental solution $Z(x,t)$ of the $p-$adic heat equation naturally associated to these operators satisfies $Z(x,t)<= 0,x\in\mathbb{Q} _{p}^{n},t>0. So this equation describes the cooling (or loss of heat) in a given region over time. Unlike the archimedean classical theory, although the operator symbol -J^{α} is not a function negative definite, we show that the operator -J^{α} satisfies the positive maximum principle on C_{0}(\mathbb{Q}_{p}^{n}). Moreover, we will show that the closure \overline{-J^{α}} of the operator -J^{α} is single-valued and generates a strongly continuous, positive, contraction semigroup {T(t)} on C_{0}(\mathbb{Q}_{p}^{n}). On the other hand, we will show that the operator -J^{α} is m-dissipative and is the infinitesimal generator of a C_{0}-semigroup of contractions T(t), t>= 0, on L^{2}(\mathbb{Q}_{p}^{n}). The latter will allow us to show that for f\in L^{1}([0,T):L^{2}(\mathbb{Q}_{p}^{n})), the function u(t)=T(t)u_{0}+\int\nolimits_{0}^{t}T(t-s)f(s)ds,\text{ \ \ }0<=t <=T, is the mild solution of the initial value problem \frac{\partial u}{\partial t}(x,t)=-J^{α}u(x,t)+f(t) & t>0\text{,\ } x\in \mathbb{Q}_{p}^{n} \\ u(x,0)=u_{0}\in L^{2}(\mathbb{Q}_{p}^{n})\text{.}
In this article we prove that the heat kernel attached to the non-archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous C_0 transition function of some strong Markov processes X with state space (Q_p)^n. We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a positive constant, a continuous homomorphism l : (Q_p)^n to R and a non-negative, continuous quadratic form q : (Q_p)^n to R.
In this article we study certain ultradiffusion equations connected with energy landscapes of exponential type. These equations are connected with the p-adic models of complex systems introduced by Avetisov et al. We show that the fundamental solutions of these equations are transition density functions of Lévy processes, we also study some aspects of these processes including the first passage time problem.
In this article we study a class of non-Archimedean pseudodifferential operators whose symbols are negative definite functions. We prove that these operators extend to generators of Feller semigroups. In order to study these operators, we introduce a new class of anisotropic Sobolev spaces, which are the natural domains for the operators considered here. We also study the Cauchy problem for certain pseudodifferential equations.