Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

Identification of the glial cell types containing carnosine-related peptides in the rat brain

The cellular localization of carnosine-like immunoreactivity was investigated in the adult rat forebrain and in glial cell cultures obtained from newborn rat brain. Using double staining methods, we showed that in vivo carnosine-like immunoreactivity was occurring in a large number of both glial fibrillary acidic protein (GFAP)-positive astrocytes and 2'3'-cyclic nucleotide 3'-phosphodiesterase (CNP)-positive oligodendrocytes. In vitro, the carnosine-immunoreactive staining was restricted to a subpopulation of completely differentiated oligodendrocytes, whereas no reaction was detected in immature oligodendrocytes and in astrocytes. These observations could have profound physiopathological Implications considering the role suggested for carnosine and related peptides as endogenous antioxidants, free radical scavengers and anti-glycating agents of the central nervous system (CNS)