Perturbations, Deformations, and Variations (and "Near-Misses") in Geometry, Physics, and Number Theory

Mathematic

Growth of Selmer Rank in Nonabelian Extensions of Number Fields

Let $p$ be an odd prime number, let E be an elliptic curve over a number field $k$, and let $F/k$ be a Galois extension of degree twice a power of p. We study the $Z_p$-corank $rk_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for $rk_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions. If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G = Gal(F/K)$, if $G+$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if $rk_p(E/K)$ is odd, then we show that (under mild hypotheses) $rkp(E/F)\ge[G:G+]$. As a very specific example of this, suppose that $A$ is an elliptic curve over $Q$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $Q$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $F\frac{x}{p}$ and such that the negative of the conductor of $E$ is not a square modulo $p$, then there is a positive constant $B$ depending on $A$ but not on $E$ or $n$ such that $rk_p(E/Q(A[p^n]))/geBp^{2n}$ for every $n$.Mathematic

Nearly Ordinary Galois Deformations over Arbitrary Number Fields

Let $K$ be an arbitrary number field, and let $\rho: Gal(K \bar/K) \rightarrow GL_2(E)$ be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of $\rho$. When $K$ is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to "automorphic" Galois representations. We conjecture that if $K$ is not totally real, then this is never the case, except in three exceptional cases, corresponding to (1) "base change", (2) "CM" forms, and (3) "Even" representations. The latter case conjecturally can only occur if the image of $\rho$ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's conjecture. Second, when $K$ is an imaginary quadratic field, we prove an unconditional result that implies the existence of "many" positive dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about "$p$-adic functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal$(Q \bar/Q)$ when $n \lt 2$.Mathematic

On Mathematics, Imagination and the Beauty of Numbers

Mathematic

Twisting Commutative Algebraic Groups

If $V$ is a commutative algebraic group over a field $k$, [View the MathML] source is a commutative ring that acts on $V$, and View the MathML source is a finitely generated free View the MathML source-module with a right action of the absolute Galois group of $k$, then there is a commutative algebraic group [View the MathML source] over $k$, which is a twist of a power of $V$. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.Mathematic

Computation of p-Adic Heights and Log Convergence

This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent.Mathematic

Implementing an innovative consent form: the PREDICT experience

<p>Abstract</p> <p>Background</p> <p>In the setting of coronary angiography, generic consent forms permit highly variable communication between patients and physicians. Even with the existence of multiple risk models, clinicians have been unable to readily access them and thus provide patients with vague estimations regarding risks of the procedure.</p> <p>Methods</p> <p>We created a web-based vehicle, PREDICT, for embedding patient-specific estimates of risk from validated multivariable models into individualized consent documents at the point-of-care. Beginning August 2006, outpatients undergoing coronary angiography at the Mid America Heart Institute received individualized consent documents generated by PREDICT. In February 2007 this approach was expanded to all patients undergoing coronary angiography within the four Kansas City hospitals of the Saint Luke's Health System. Qualitative research methods were used to identify the implementation challenges and successes with incorporating PREDICT-enhanced consent documents into routine clinical care from multiple perspectives: administration, information systems, nurses, physicians, and patients.</p> <p>Results</p> <p>Most clinicians found usefulness in the tool (providing clarity and educational value for patients) and satisfaction with the altered processes of care, although a few cardiologists cited delayed patient flow and excessive patient questions. The responses from administration and patients were uniformly positive. The key barrier was related to informatics.</p> <p>Conclusion</p> <p>This preliminary experience suggests that successful change in clinical processes and organizational culture can be accomplished through multidisciplinary collaboration. A randomized trial of PREDICT consent, leveraging the accumulated knowledge from this first experience, is needed to further evaluate its impact on medical decision-making, patient compliance, and clinical outcomes.</p

Intraneuronal Aβ immunoreactivity is not a predictor of brain amyloidosis-β or neurofibrillary degeneration

Amyloid β (Aβ) immunoreactivity in neurons was examined in brains of 32 control subjects, 31 people with Down syndrome, and 36 patients with sporadic Alzheimer’s disease to determine if intraneuronal Aβ immunoreactivity is an early manifestation of Alzheimer-type pathology leading to fibrillar plaque formation and/or neurofibrillary degeneration. The appearance of Aβ immunoreactivity in neurons in infants and stable neuron-type specific Aβ immunoreactivity in a majority of brain structures during late childhood, adulthood, and normal aging does not support this hypothesis. The absence or detection of only traces of reaction with antibodies against 4–13 aa and 8–17 aa of Aβ in neurons indicated that intraneuronal Aβ was mainly a product of α- and γ-secretases (Aβ(17–40/42)). The presence of N-terminally truncated Aβ(17–40) and Aβ(17–42) in the control brains was confirmed by Western blotting and the identity of Aβ(17–40) was confirmed by mass spectrometry. The prevalence of products of α- and γ -secretases in neurons and β- and γ-secretases in plaques argues against major contribution of Aβ-immunopositive material detected in neuronal soma to amyloid deposit in plaques. The strongest intraneuronal Aβ(17–42) immunoreactivity was observed in structures with low susceptibility to fibrillar Aβ deposition, neurofibrillary degeneration, and neuronal loss compared to areas more vulnerable to Alzheimer-type pathology. These observations indicate that the intraneuronal Aβ immunoreactivity detected in this study is not a predictor of brain amyloidosis or neurofibrillary degeneration. The constant level of Aβ immunoreactivity in structures free from neuronal pathology during essentially the entire life span suggests that intraneuronal amino-terminally truncated Aβ represents a product of normal neuronal metabolism

Effectiveness of student response systems in terms of learning environment, attitudes and achievement

In order to investigate the effectiveness of using Student Response Systems (SRS) among grade 7 and 8 science students in New York, the How Do You Feel About This Class? (HDYFATC) questionnaire was administered to 1097 students (532 students did use SRS and 565 students who did not use SRS). Data analyses attested to the sound factorial validity and internal consistency reliability of the HDYFATC, as well as its ability to differentiate between the perceptions of students in different classrooms. Very large differences between users and non-users of SRS, ranging from 1.17 to 2.45 standard deviations for various learning environment scales, attitudes and achievement, supported the efficacy of using SRS

Abnormal Intracellular Accumulation and Extracellular Aβ Deposition in Idiopathic and Dup15q11.2-q13 Autism Spectrum Disorders

<div><h3>Background</h3><p>It has been shown that amyloid ß (Aβ), a product of proteolytic cleavage of the amyloid β precursor protein (APP), accumulates in neuronal cytoplasm in non-affected individuals in a cell type–specific amount.</p> <h3>Methodology/Principal Findings</h3><p>In the present study, we found that the percentage of amyloid-positive neurons increases in subjects diagnosed with idiopathic autism and subjects diagnosed with duplication 15q11.2-q13 (dup15) and autism spectrum disorder (ASD). In spite of interindividual differences within each examined group, levels of intraneuronal Aβ load were significantly greater in the dup(15) autism group than in either the control or the idiopathic autism group in 11 of 12 examined regions (p<0.0001 for all comparisons; Kruskall-Wallis test). In eight regions, intraneuronal Aβ load differed significantly between idiopathic autism and control groups (p<0.0001). The intraneuronal Aβ was mainly N-terminally truncated. Increased intraneuronal accumulation of Aβ_17–40/42 in children and adults suggests a life-long enhancement of APP processing with α-secretase in autistic subjects. Aβ accumulation in neuronal endosomes, autophagic vacuoles, Lamp1-positive lysosomes and lipofuscin, as revealed by confocal microscopy, indicates that products of enhanced α-secretase processing accumulate in organelles involved in proteolysis and storage of metabolic remnants. Diffuse plaques containing Aβ_1–40/42 detected in three subjects with ASD, 39 to 52 years of age, suggest that there is an age-associated risk of alterations of APP processing with an intraneuronal accumulation of a short form of Aβ and an extracellular deposition of full-length Aβ in nonfibrillar plaques.</p> <h3>Conclusions/Significance</h3><p>The higher prevalence of excessive Aβ accumulation in neurons in individuals with early onset of intractable seizures, and with a high risk of sudden unexpected death in epilepsy in autistic subjects with dup(15) compared to subjects with idiopathic ASD, supports the concept of mechanistic and functional links between autism, epilepsy and alterations of APP processing leading to neuronal and astrocytic Aβ accumulation and diffuse plaque formation.</p> </div