A singular function and its relation with the number systems involved in its definition

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.

On the concept of optimality interval

The approximants to regular continued fractions constitute `best approximations' to the numbers they converge to in two ways known as of the first and the second kind.This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a baseball player is 0.334, what is the minimum number of times he has been at bat? In this paper, we tackle somehow the inverse question: given a rational number P/Q, what is the set of all numbers for which P/Q is a `best approximation' of one or the other kind? We prove that inboth cases these `Optimality Sets' are intervals and we give aprecise description of their endpoints.

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results.

Most of the Abundant Protein Fractions of Embryonic Cerebrospinal Fluid are Produced Out of the Brain Anlagen

The microenvironment of the central nervous system is important for neuronal function and development. During the early stages of embryo development the cephalic vesicles are filled by embryonic cerebrospinal fluid, a complex fluid containing different protein fractions, which contributes to the regulation of the survival, proliferation and neurogenesis of neuroectodermal stem cells. The protein content of embryonic cerebrospinal fluid from chick and rat embryos at the start of neurogenesis has already been determined. Most of the identified gene products are thought to be involved in the regulation of developmental processes during embryogenesis. However, due to the crucial roles played by embryonic cerebrospinal fluid during brain development, the embryological origin of the gene products it contains remains an intriguing question. According to the literature most of these products are synthesised in embryonic tissues other than the neuroepithelium. In this study we examined the embryological origin of the most abundant embryonic cerebrospinal fluid protein fractions by means of slot-blot analysis and by using several different embryonic and extraembryonic protein extracts, immunodetected with polyclonal antibodies. This first attempt to elucidate their origin is not based on the proteins identified by proteomic methods, but rather on crude protein fractions detected by SDS-PAGE analysis and to which polyclonal antibodies were specifically generated. Despite some of the limitations of this study, i.e. that one protein fraction may contain more than one gene product, and that a specific gene product may be contained in different protein fractions depending on post-translational modifications, our results show that most of the analysed protein fractions are not produced by the cephalic neuroectoderm but are rather stored in the egg reservoir; furthermore, few are produced by embryo tissues, thus indicating that they must be transported from their production or storage sites to the cephalic cavities, most probably via embryonic serum. These results raise the question as to whether the transfer of proteins from these two embryo compartments is regulated at this early developmental stage.