arXiv subjects
Publications and source records attributed to Ahmed Sani.
New copulas, based on perturbation theory, are introduced to clarify a \emph{symmetrization} procedure for asymmetric copulas. We give also some properties of the \emph{symmetrized} copula. Finally, we examine families of copulas with a prescribed symmetrized one. By the way, we study topologically, the set of all symmetric copulas and give some of its classical and new properties.
We give a new sufficient condition which allows to test primality of Fermat's numbers. This characterization uses uniquely values at most equal to tested Fermat number. The robustness of this result is due to a strict use of elementary arithmetic technical tools and it will be susceptible to open gates for revolutionary statement that all Fermat's numbers are all decomposable.
Preserver problems concern the characterization of operators on general spaces that leave invariant some categories of subsets or ratios. The most known in the mathematical literature are those of linear preserver problems (LPP) which date back to the ninth century. Here, we treat the preserver problem in the recent and emerging field of copulas. Precisely, we prove that \emph{copula property} is preserved uniquely under increasing transformations.
The concept of asymmetric copulas is revisited and is made more precise. We give a rigorous topological argument for opportunity to define asymmetry measures defined recently by K.F Siburg [6] through exhibiting at least three ordered classes of copulas according to a suitable equivalence relation. We define a process of ordering subcopulas which makes clearer the degree of asymmetry. As illustration, we treat the asymmetric Cobb-Douglas utility model.
The concept of boundary values of holomorphic semigroups in a general Banach space is studied. As an application, we consider the Riemann-Liouville semigroup of integration operator in the little Hölder spaces $\rm{lip}_0^α[0,\, 1] , \, 0<α<1$ and prove that it admits a strongly continuous boundary group, which is the group of fractional integration of purely imaginary order. The corresponding result for the $L^p$-spaces ($1<p<\infty$) has been known for some time, the case $p=2$ dating back to the monograph by Hille and Phillips. In the context of $L^p$ spaces, we establish the existence of the boundary group of the Hadamard fractional integration operators using semigroup methods. In the general framework, using a suitable spectral decomposition,we give a partial treatment of the inverse problem, namely: Which $C_0$-groups are boundary values of some holomorphic semigroup of angle $π/2$?
We prove $L^2$-maximal regularity of linear non-autonomous evolutionary Cauchy problem \begin{equation}\label{eq00}\nonumber \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time dependent sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed in \cite{ELKELA11}, \cite{ELLA13} and \cite{LH}. As a consequence, we obtain an invariance criterion for convex and closed sets of $H.$