Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Vector functions and curves

Exercises and solutions about vector functions and curves.

Hematopoietic stem cells, overview and pathways implied on their self-renewal mechanisms

Blood tissue is composed approximately in 45% by cells and its derivatives, with a life span of around 120 days for erythrocytes and 3 years for certain type of lymphocytes. This lost is compensated with the hematopoietic system activity and the presence of an immature primitive cell population known as Hematopoietic Stem Cells (HSCs) which perform the hematopoiesis, a process that is active from the beginning of the fetal life and produces near to 2 x 1011 eritrocytes and 1010 white blood cells per day (1). Hematopoietic Stem Cells are capable of both self-renewal and differentiation into multiple lineages, are located in a particular niche and are identified by their own cell surface markers, as the CD34 antigen. Recently it has been possible to advance in the understanding of self-renewal, differentiation and proliferation processes and in the involvement of the signaling pathways Hedgehog, Notch and Wnt. Studying the influence of these mechanisms on in vivo and in vitro behavior and the basic biology of HSCs, has given valuable tools for the generation of alternative therapies for hematologic disorders as leukemias.

Special needs and early years : a practitioner's guide.

Recurso para profesionales, y para todos aquellos que trabajan con niños pequeños con necesidades especiales y sus familias. Examina el verdadero problema del desarrollo afectivo, cómo esto puede influir en el progreso del niño, y reconoce la importancia de la colaboración con los padres. La autora combina la teoría y la práctica para proporcionar un análisis detallado de la prestación para los niños pequeños con necesidades adicionales. Ofrece los objetivos de aprendizaje; puntos de reflexión para facilitar el estudio independiente y el pensamiento crítico; estudio de casos, que cubren desde el nacimiento hasta los ocho años; el papel de los padres, la expresión del alumno, escuchar a los niños, toma de una visión integral del niño, y temas de actualidad en torno a la diversidad y la inserción.

Los equipos de orientación del sector : funciones y modelos de intervención = External guidance services : functions and intervention models.

Resumen basado en el del autor

A comparison of diatom phosphorus transfer functions and export coefficient models as tools for reconstructing lake nutrient histories

1. We compared the baseline phosphorus (P) concentrations inferred by diatom-P transfer functions and export coefficient models at 62 lakes in Great Britain to assess whether the techniques produce similar estimates of historical nutrient status. 2. There was a strong linear relationship between the two sets of values over the whole total P (TP) gradient (2-200 mu g TP L-1). However, a systematic bias was observed with the diatom model producing the higher values in 46 lakes (of which values differed by more than 10 mu g TP L-1 in 21). The export coefficient model gave the higher values in 10 lakes (of which the values differed by more than 10 mu g TP L-1 in only 4). 3. The difference between baseline and present-day TP concentrations was calculated to compare the extent of eutrophication inferred by the two sets of model output. There was generally poor agreement between the amounts of change estimated by the two approaches. The discrepancy in both the baseline values and the degree of change inferred by the models was greatest in the shallow and more productive sites. 4. Both approaches were applied to two lakes in the English Lake District where long-term P data exist, to assess how well the models track measured P concentrations since approximately 1850. There was good agreement between the pre-enrichment TP concentrations generated by the models. The diatom model paralleled the steeper rise in maximum soluble reactive P (SRP) more closely than the gradual increase in annual mean TP in both lakes. The export coefficient model produced a closer fit to observed annual mean TP concentrations for both sites, tracking the changes in total external nutrient loading. 5. A combined approach is recommended, with the diatom model employed to reflect the nature and timing of the in-lake response to changes in nutrient loading, and the export coefficient model used to establish the origins and extent of changes in the external load and to assess potential reduction in loading under different management scenarios. 6. However, caution must be exercised when applying these models to shallow lakes where the export coefficient model TP estimate will not include internal P loading from lake sediments and where the diatom TP inferences may over-estimate TP concentrations because of the high abundance of benthic taxa, many of which are poor indicators of trophic state.

Sources of sulphur in gypsiferous sediments and crusts and pathways of gypsum redistribution in southern Tunisia

Southern Tunisia contains one of the most extensive gypsum accumulations in Africa comprising Triassic, Cretaceous, Eocene and Mio-Pliocene marine evaporites, spring deposits, playa sediments, aeolian sands and gypsum crusts. Sulphur isotope analysis (delta(34)S) of bedrock samples, groundwater, playa brines, playa sediments, and gypsiferous crusts provides insight into the sources of gypsum in the region and sheds light on the processes that lead to gypsum crust formation. Results Suggest that recycling of marine gypsum is the most likely source of the sulphate in the groundwater, playa sediments and crusts. The low PS values found in Eocene and Mio-Pliocene samples suggest that this recycling has been going on for millions of years. Though bedrock appears to be the ultimate source of the gypsum in the crusts, transport of this sulphate to playas, concentration therein, and subsequent dispersal across the landscape by aeolian processes provides the most likely pathway for surticial gypsum crust formation. Comparison of these results with those from Australia, Chile and Namibia suggests that, although the source of the sulphur varies from region to region, the processes of surficial crust formation appear to be similar. Copyright (C) 2004 John Wiley Sons, Ltd.

PIGOK: linking protein identity to gene ontology and function

Here we introduce a computer database that allows for the rapid retrieval of physicochemical properties, Gene Ontology, and Kyoto Encyclopedia of Genes and Genomes information about a protein or a list of proteins. We applied PIGOK analyzing Schizosaccharomyces pombe proteins displaying differential expression under oxidative stress and identified their biological functions and pathways. The database is available on the Internet at http://pc4-133.ludwig.ucl.ac.uk/pigok.html.

Right hemisphere language functions and schizophrenia: the forgotten hemisphere?

This review highlights the importance of right hemisphere language functions for successful social communication and advances the hypothesis that the core deficit in psychosis is a failure of segregation of right from left hemisphere functions. Lesion studies of stroke patients and dichotic listening and functional imaging studies of healthy people have shown that some language functions are mediated by the right hemisphere rather than the left. These functions include discourse planning/comprehension, understanding humour, sarcasm, metaphors and indirect requests, and the generation/comprehension of emotional prosody. Behavioural evidence indicates that patients with typical schizophrenic illnesses perform poorly on tests of these functions, and aspects of these functions are disturbed in schizo-affective and affective psychoses. The higher order language functions mediated by the right hemisphere are essential to an accurate understanding of someone's communicative intent, and the deficits displayed by patients with schizophrenia may make a significant contribution to their social interaction deficits. We outline a bi-hemispheric theory of the neural basis of language that emphasizes the role of the sapiens-specific cerebral torque in determining the four-chambered nature of the human brain in relation to the origins of language and the symptoms of schizophrenia. Future studies of abnormal lateralization of left hemisphere language functions need to take account of the consequences of a failure of lateralization of language functions to the right as well as the left hemisphere.

Prevalence of disorders of the autism spectrum in a population cohort of children in South Thames: the special needs and autism project (SNAP)

Background Recent reports have suggested that the prevalence of autism and related spectrum disorders (ASDs) is substantially higher than previously recognised. We sought to quantify prevalence of ASDs in children in South Thames, UK. Methods Within a total population cohort of 56946 children aged 9-10 years, we screened all those with a current clinical diagnosis of ASD (n=255) or those judged to be at risk for being an undetected case (n=1515). A stratified subsample (n=255) received a comprehensive diagnostic assessment, including standardised clinical observation, and parent interview assessments of autistic symptoms, language, and intelligence quotient (IQ). Clinical consensus diagnoses of childhood autism and other ASDs were derived. We used a sample weighting procedure to estimate prevalence. Findings The prevalence of childhood autism was 38.9 per 10000 (95% CI 29.9-47.8) and that of other ASDs was 77.2 per 10000 (52.1-102.3), making the total prevalence of all AS Ds 116.1 per 10000 (90.4-141.8). A narrower definition of childhood autism, which combined clinical consensus with instrument criteria for past and current presentation, provided a prevalence of 24.8 per 10 000 (17.6-32.0). The rate of previous local identification was lowest for children of less educated parents. Interpretation Prevalence of autism and related ASDs is substantially greater than previously recognised. Whether the increase is due to better ascertainment, broadening diagnostic criteria, or increased incidence is unclear. Services in health, education, and social care will need to recognise the needs of children with some form of ASD, who constitute 1% of the child population.