New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Tau Functions and the Limit of Block Toeplitz Determinants

International audience

Stress-triggered synaptic malfunction: a gate along the path from depressionto dementia

Tese de Doutoramento em Ciências da Saúde.

Unifying Time to Contact Estimation and Collision Avoidance across Species

The -function and the -function are phenomenological models that are widely used in the context of timing interceptive actions and collision avoidance, respectively. Both models were previously considered to be unrelated to each other: is a decreasing function that provides an estimation of time-to-contact (ttc) in the early phase of an object approach; in contrast, has a maximum before ttc. Furthermore, it is not clear how both functions could be implemented at the neuronal level in a biophysically plausible fashion. Here we propose a new framework the corrected modified Tau function capable of predicting both -type ("") and -type ("") responses. The outstanding property of our new framework is its resilience to noise. We show that can be derived from a firing rate equation, and, as , serves to describe the response curves of collision sensitive neurons. Furthermore, we show that predicts the psychophysical performance of subjects determining ttc. Our new framework is thus validated successfully against published and novel experimental data. Within the framework, links between -type and -type neurons are established. Therefore, it could possibly serve as a model for explaining the co-occurrence of such neurons in the brain.

Unifying Time to Contact Estimation and Collision Avoidance across Species

The -function and the -function are phenomenological models that are widely used in the context of timing interceptive actions and collision avoidance, respectively. Both models were previously considered to be unrelated to each other: is a decreasing function that provides an estimation of time-to-contact (ttc) in the early phase of an object approach; in contrast, has a maximum before ttc. Furthermore, it is not clear how both functions could be implemented at the neuronal level in a biophysically plausible fashion. Here we propose a new framework- the corrected modified Tau function- capable of predicting both -type ("") and -type ("") responses. The outstanding property of our new framework is its resilience to noise. We show that can be derived from a firing rate equation, and, as , serves to describe the response curves of collision sensitive neurons. Furthermore, we show that predicts the psychophysical performance of subjects determining ttc. Our new framework is thus validated successfully against published and novel experimental data. Within the framework, links between -type and -type neurons are established. Therefore, it could possibly serve as a model for explaining the co-occurrence of such neurons in the brain.

Backlund-Darboux Transformations in Sato's Grassmannian

We define Bäcklund–Darboux transformations in Sato’s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, tau-functions and spectral algebras.

Evaluating Knowledge of a Function-based Cognitive-Behavioural Treatment for Obsessive Compulsive Behaviour in Autism Spectrum Disorder

Children with Autism Spectrum Disorder (ASD) have restricted and repetitive behaviours (RRBs) which may be similar to obsessions and compulsions in Obsessive Compulsive Disorder (OCD). These behaviours can be intrusive and interfere in the lives of the child and their family. Preliminary studies have shown success in using adapted Cognitive Behavioural Therapy (CBT) to treat these behaviors in children with high functioning ASD. Using a hypothetical vignette, this thesis attempted to examine procedural knowledge that the children and their parents gained while participating in a CBT treatment that was evaluated in a Randomized Controlled Trial. For both parents and children, there was a significant increase in number of strategies generated from pre to post-treatment. Further, children in the experimental group generated significantly more strategies than the treatment as usual (TAU) group post-intervention. There was no significant correlation between number of strategies generated and the child’s treatment success, age, or IQ.

Inflammation of the Hypothalamus Leads to Defective Pancreatic Islet Function

Type 2 diabetes mellitus results from the complex association of insulin resistance and pancreatic beta-cell failure. Obesity is the main risk factor for type 2 diabetes mellitus, and recent studies have shown that, in diet-induced obesity, the hypothalamus becomes inflamed and dysfunctional, resulting in the loss of the perfect coupling between caloric intake and energy expenditure. Because pancreatic beta-cell function is, in part, under the control of the autonomic nervous system, we evaluated the role of hypothalamic inflammation in pancreatic islet function. In diet-induced obesity, the earliest markers of hypothalamic inflammation are present at 8 weeks after the beginning of the high fat diet; similarly, the loss of the first phase of insulin secretion is detected at the same time point and is restored following sympathectomy. Intracerebroventricular injection of a low dose of tumor necrosis factor a leads to a dysfunctional increase in insulin secretion and activates the expression of a number of markers of apoptosis in pancreatic islets. In addition, the injection of stearic acid intracerebroventricularly, which leads to hypothalamic inflammation through the activation of tau-like receptor-4 and endoplasmic reticulum stress, produces an impairment of insulin secretion, accompanied by increased expression of markers of apoptosis. The defective insulin secretion, in this case, is partially dependent on sympathetic signal-induced peroxisome proliferator receptor-gamma coactivator Delta a and uncoupling protein-2 expression and is restored after sympathectomy or following PGC1 alpha expression inhibition by an antisense oligonucleotide. Thus, the autonomic signals generated in concert with hypothalamic inflammation can impair pancreatic islet function, a phenomenon that may explain the early link between obesity and defective insulin secretion.

Search for high-mass resonances decaying into tau-lepton pairs in pp collisions at root s=7 TeV

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Search for neutral Higgs bosons decaying to tau pairs in pp collisions at root s=7 TeV

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Search for Neutral Minimal Supersymmetric Standard Model Higgs Bosons Decaying to Tau Pairs in pp Collisions at root s=7 TeV

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

SEARCH FOR POINT-LIKE SOURCES OF ULTRA-HIGH ENERGY NEUTRINOS AT THE PIERRE AUGER OBSERVATORY AND IMPROVED LIMIT ON THE DIFFUSE FLUX OF TAU NEUTRINOS

The surface detector array of the Pierre Auger Observatory can detect neutrinos with energy E-nu between 10(17) eV and 10(20) eV from point-like sources across the sky south of +55 degrees and north of -65 degrees declinations. A search has been performed for highly inclined extensive air showers produced by the interaction of neutrinos of all flavors in the atmosphere (downward-going neutrinos), and by the decay of tau leptons originating from tau neutrino interactions in Earth's crust (Earth-skimming neutrinos). No candidate neutrinos have been found in data up to 2010 May 31. This corresponds to an equivalent exposure of similar to 3.5 years of a full surface detector array for the Earth-skimming channel and similar to 2 years for the downward-going channel. An improved upper limit on the diffuse flux of tau neutrinos has been derived. Upper limits on the neutrino flux from point-like sources have been derived as a function of the source declination. Assuming a differential neutrino flux k(PS) . E-nu(-2). from a point-like source, 90% confidence level upper limits for k(PS) at the level of approximate to 5x10(-7) and 2.5x10(-6) GeV cm(-2) s(-1) have been obtained over a broad range of declinations from the searches for Earth-skimming and downward-going neutrinos, respectively.

A search for high-mass resonances decaying to tau⁺ tau⁻ in pp collisions at sqrts=7 TeV with the ATLAS detector

This Letter presents a search for high-mass resonances decaying into tau(+)tau(-) final states using proton-proton collisions at root s = 7 TeV produced by the Large Hadron Collider. The data were recorded with the ATLAS detector and correspond to an integrated luminosity of 4.6 fb(-1). No statistically significant excess above the Standard Model expectation is observed; 95% credibility upper limits are set on the cross section times branching fraction of Z' resonances decaying into tau(+)tau(-) pairs as a function of the resonance mass. As a result, Z' bosons of the Sequential Standard Model with masses less than 1.40 TeV are excluded at 95% credibility.

Rapid treadmilling of brain microtubules free of microtubule-associated proteins in vitro and its suppression by tau

We have determined the treadmilling rate of brain microtubules (MTs) free of MT-associated proteins (MAPs) at polymer mass steady state in vitro by using [3H]GTP-exchange. We developed buffer conditions that suppressed dynamic instability behavior by ≈10-fold to minimize the contribution of dynamic instability to total tubulin-GTP exchange. The MTs treadmilled rapidly under the suppressed dynamic instability conditions, at a minimum rate of 0.2 μm/min. Thus, rapid treadmilling is an intrinsic property of MAP-free MTs. Further, we show that tau, an axonal stabilizing MAP involved in Alzheimer’s disease, strongly suppresses the treadmilling rate. These results indicate that tau’s function in axons might involve suppression of axonal MT treadmilling. We describe mathematically how treadmilling and dynamic instability are mechanistically distinct MT behaviors. Finally, we present a model that explains how small changes in the critical tubulin subunit concentration at MT minus ends, caused by intrinsic differences in rate constants or regulatory proteins, could produce large changes in the treadmilling rate.

Relationship Between Orofacial Function, Dentofacial Morphology, And Bite Force In Young Subjects.

The aim was to evaluate the relationship between orofacial function, dentofacial morphology, and bite force in young subjects. Three hundred and sixteen subjects were divided according to dentition stage (early, intermediate, and late mixed and permanent dentition). Orofacial function was screened using the Nordic Orofacial Test-Screening (NOT-S). Orthodontic treatment need, bite force, lateral and frontal craniofacial dimensions and presence of sleep bruxism were also assessed. The results were submitted to descriptive statistics, normality and correlation tests, analysis of variance, and multiple linear regression to test the relationship between NOT-S scores and the studied independent variables. The variance of NOT-S scores between groups was not significant. The evaluation of the variables that significantly contributed to NOT-S scores variation showed that age and presence of bruxism related to higher NOT-S total scores, while the increase in overbite measurement and presence of closed lip posture related to lower scores. Bite force did not show a significant relationship with scores of orofacial dysfunction. No significant correlations between craniofacial dimensions and NOT-S scores were observed. Age and sleep bruxism were related to higher NOT-S scores, while the increase in overbite measurement and closed lip posture contributed to lower scores of orofacial dysfunction.