Diophantine Geometry as Galois Theory in the Mathematics of Serge Lang

A remark about the role of Galois theory in Diophantine geometry as reflected in the work of Serge Lang. An entry in `The mathematical contributions of Serge Lang.

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Theory of Fermionic superfluid with SU(2)xSU(6) symmetry

We study theoretically interspecies Cooper pairing in a fermionic system with SU(2)xSU(6) sym- metry. We show that, with suitable unitary transformations, the order parameter for the ground state can be reduced to only two non-vanishing complex components. The ground state has a large degeneracy. We find that while some Goldstone modes have linear dispersions, others are quadratic at low frequencies. We compare our results with the case of SU(N).Comment: extended version; include comparison with SU(N

Observations of the relationship between sprite morphology and in-cloud lightning processes

[1] During a thunderstorm on 23 July 2003, 15 sprites were captured by a LLTV camera mounted at the observatory on Pic du Midi in the French Pyrénées. Simultaneous observations of cloud-to-ground (CG) and intracloud (IC) lightning activity from two independent lightning detection systems and a broadband ELF/VLF receiver allow a detailed study of the relationship between electrical activity in a thunderstorm and the sprites generated in the mesosphere above. Results suggest that positive CG and IC lightning differ for the two types of sprites most frequently observed, the carrot- and column-shaped sprites. Column sprites occur after a short delay (<30 ms) from the causative +CG and are associated with little VHF activity, suggesting no direct IC action on the charge transfer process. On the other hand, carrot sprites are delayed up to about 200 ms relative to their causative +CG stroke and are accompanied by a burst of VHF activity starting 25–75 ms before the CG stroke. While column sprites associate with short-lasting (less than 30 ms) ELF/VLF sferics, carrot sprites associate with bursts of sferics initiating at the time of the causative +CG discharge and persisting for 50 to 250 ms, indicating extensive in-cloud activity. One carrot event was found to be preceded by vigorous IC activity and a strong, long-lived cluster of ELF/VLF sferics but lacking a +CG. The observations of ELF/VLF sferic clusters associated with lightning and sprites form the basis for a discussion of the reliability of lightning detection systems based on VHF interferometry.Peer ReviewedPostprint (published version

Workplace-Related Interpersonal Group Psychotherapy to Improve Life at Work in Individuals With Major Depressive Disorders: A Randomized Interventional Pilot Study

Individuals suffering from major depressive disorder (MDD) often report workplace-related stress as the major cause of their disorder. Accordingly, workplace-related stress was established as a fifth psychosocial problem area of Interpersonal Psychotherapy (workplace-related Interpersonal Psychotherapy, W-IPT). The aim of the study was to investigate the influence of W-IPT on depressive symptoms and on workplace-related issues in individuals with MDD compared to a treatment-as-usual (TAU) condition.; A total of 27 individuals with MDD (mean age = 43 years, 48% males) were randomly assigned either to eight weekly group sessions of W-IPT or to the TAU condition. At baseline, 8 weeks later at the end of the intervention, and 20 weeks later at follow-up, the Hamilton Rating Scale for Depression was conducted. In addition, the participants completed the Beck Depression Inventory, the Work Ability Index (WAI), the Return to Work Attitude (RTW-SE), and the Insomnia Severity Index (ISI).; Symptoms of depression in experts' ratings as well as in self-rated ratings decreased over time, but more so in the W-IPT condition compared to the TAU condition [experts rating: large effect size (; d; = 1.25) and self-assessment: large effect sizes (; d; = 0.94)]. The subjective ability to work (WAI) [medium effect size (; d; = 0.68)], self-efficacy to returning to work RTW-SE [medium effect size (; d; = 0.57)], and subjective symptoms of insomnia (ISI) [large effect size (; d; = 1.15)] increased over time, but again more so in the W-IPT condition compared to the TAU condition. The effects of the intervention remained stable from the end of the intervention to follow-up.; The pattern of results of this pilot study suggests that a newly established fifth IPT focus on workplace-related stress appeared to be particularly efficient in individuals with MDD due to work-related stress in reducing depressive symptoms and reducing sleep complaints as well as in improving occupational outcomes

Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

Short-term outcomes of physical activity counseling in in-patients with Major Depressive Disorder: Results from the PACINPAT randomized controlled trial

IntroductionA physical activity counseling intervention based on a motivation-volition model was developed and delivered to in-patients with Major Depressive Disorders with the aim of increasing lifestyle physical activity. The aim of this study is to evaluate the short-term outcomes of this intervention.MethodsA multi-center randomized controlled trial was conducted in four Swiss psychiatric clinics. Adults who were initially insufficiently physically active and were diagnosed with Major Depressive Disorder according to ICD-10 were recruited. The sample consisted of 113 participants in the intervention group (Mage = 42 years, 56% women) and 107 in the control group (Mage = 40 years, 49% women). Motivation and volition determinants of physical activity were assessed with questionnaires. Implicit attitudes were assessed with an Implicit Association Test. Physical activity was self-reported and measured with hip-worn accelerometers over 7 consecutive days starting on the day following the data collection.ResultsAccording to accelerometer measures, step count decreased on average 1,323 steps less per day (95% CI = −2,215 to −431, p &lt; 0.01) over time in the intervention group compared to the control group. A trend was recognized indicating that moderate-to-vigorous physical activity decreased on average 8.37 min less per day (95% CI = −16.98 to 0.23, p &lt; 0.06) over time in the intervention group compared to the control group. The initial phase of the intervention does not seem to have affected motivational and volitional determinants of and implicit attitudes toward physical activity.ConclusionPhysical activity counseling may be considered an important factor in the transition from in-patient treatment. Methods to optimize the intervention during this period could be further explored to fulfill the potential of this opportunity.Clinical trial registrationhttps://www.isrctn.com/ISRCTN10469580, identifier ISRCTN10469580

Eisenstein Series of Weight One, q-Averages of the 0-Logarithm and Periods of Elliptic Curves

For any elliptic curve E over k ⊂ R with E(C) = C^×/q^Z, q = e^(2πiz),Im(z) >, we study the q-average D_(0,q), defined on E(C), of the function D_0(z) = Im(z/(1−z)). Let Ω+(E) denote the real period of E. We show that there is a rational function R ∈ Q(X_1(N)) such that for any non-cuspidal real point s ∈ X_1(N) (which defines an elliptic curve E(s) over R together with a point P(s) of order N), πD_(0,q)(P(s)) equals Ω+(E(s))R(s). In particular, if s is Q-rational point of X_1(N), a rare occurrence according to Mazur, R(s) is a rational number

Fecal Microbiota Transplantation (FMT) as an Adjunctive Therapy for Depression-Case Report

Depression is a debilitating disorder, and at least one third of patients do not respond to therapy. Associations between gut microbiota and depression have been observed in recent years, opening novel treatment avenues. Here, we present the first two patients with major depressive disorder ever treated with fecal microbiota transplantation as add-on therapy. Both improved their depressive symptoms 4 weeks after the transplantation. Effects lasted up to 8 weeks in one patient. Gastrointestinal symptoms, constipation in particular, were reflected in microbiome changes and improved in one patient. This report suggests further FMT studies in depression could be worth pursuing and adds to awareness as well as safety assurance, both crucial in determining the potential of FMT in depression treatment