The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a “weak ε-Kronecker set” if for every φ:E → there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not “too many”...

Let p,q,n be natural numbers such that p+q = n. Let be either ℂ, the complex numbers field, or ℍ, the quaternionic division algebra. We consider the Heisenberg group N(p,q,) defined ⁿ × ℑ , with group law given by (v,ζ)(v’,ζ’) = (v + v’, ζ + ζ’- 1/2 ℑ B(v,v’)), where $B(v,w)={\sum }_{j=1}^p{v}_j\overline{w_j}-{\sum }_{j=p+1}^n{v}_j\overline{w_j}$. Let U(p,q,) be the group of n × n matrices with coefficients in that leave the form B invariant. We compute explicit fundamental solutions of some second order differential operators on N(p,q,) which are canonically associated to...

Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $\left(T\right)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma$ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r\subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma$ is described explicitly....

On munit la classe des algèbres de Kac d’une nouvelle classe de morphismes, stable par dualité. Cela permet de rendre compte, dans les cas abélien ou symétrique, de la catégorie des groupes localement compacts munis des morphismes continus de groupe. Le lien avec les morphismes précédemment définis et beaucoup plus restrictifs est établi.

From the fact that the two-dimensional moment problem is not always solvable, we can deduce that there must be extreme ray generators of the cone of positive definite double sequences which are nor moment sequences. Such an argument does not lead to specific examples. In this paper it is shown how specific examples can be constructed if one is given an example of an N-extremal indeterminate measure in the one-dimensional moment problem (such examples exist in the literature). Konrad Schmüdgen had...