On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

The problem of two fixed centers: bifurcation diagram for positive energies

We give a comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy. These systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy-momentum mapping from the phase space to the space of the integration constants. In this setting, we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram, i.e., the set of critical value of the energy-momentum map.

Estrogens, brain and behavior: studies in fundamental neurobiology and observations related to women's health

Mechanisms and consequences of the effects of estrogen on the brain have been studied both at the fundamental level and with therapeutic applications in mind. Estrogenic hormones binding in particular neurons in a limbic-hypothalamic system and their effects on the electrophysiology and molecular biology of medial hypothalamic neurons were central in establishing the first circuit for a mammalian behavior, the female-typical mating behavior, lordosis. Notably, the ability of estradiol to facilitate transcription from six genes whose products are important for lordosis behavior proved that hormones can turn on genes in specific neurons at specific times, with sensible behavioral consequences. The use of a gene knockout for estrogen receptor alpha (ERalpha) revealed that homozygous mutant females simply would not do lordosis behavior and instead were extremely aggressive, thus identifying a specific gene as essential for a mammalian social behavior. In dramatic contrast, ERbeta knockout females can exhibit normal lordosis behavior. With the understanding, in considerable mechanistic detail, of how the behavior is produced, now we are also studying brain mechanisms for the biologically adaptive influences which constrain reproductive behavior. With respect to cold temperatures and other environmental or metabolic circumstances which are not consistent with successful reproduction, we are interested in thyroid hormone effects in the brain. Competitive relations between two types of transcription factors - thyroid hormone receptors and estrogen receptors have the potential of subserving the blocking effects of inappropriate environmental circumstances on female reproductive behaviors. TRs can compete with ERalpha both for DNA binding to consensus and physiological EREs and for nuclear coactivators. In the presence of both TRs and ERs, in transfection studies, thyroid hormone coadministration can reduce estrogen-stimulated transcription. These competitive relations apparently have behavioral consequences, as thyroid hormones will reduce lordosis, and a TRbeta gene knockout will increase it. In sum, we not only know several genes that participate in the selective control of this sex behavior, but also, for two genes, we know the causal routes. Estrogenic hormones are also the foci of widespread attention for their potential therapeutic effects improving, for example, certain aspects of mood and cognition. The former has an efficient animal analog, demonstrated by the positive effects of estrogen in the Porsolt forced swim test. The latter almost certainly depends upon trophic actions of estrogen on several fundamental features of nerve cell survival and growth. The hypothesis is raised that the synaptic effects of estrogens are secondary to the trophic actions of this type of hormone in the nucleus and nerve cell body.