Implementing Geometric Complexity Theory: On the Separation of Orbit Closures via Symmetries

Understanding the difference between group orbits and their closures is a key difficulty in geometric complexity theory (GCT): While the GCT program is set up to separate certain orbit closures, many beautiful mathematical properties are only known for the group orbits, in particular close relations with symmetry groups and invariant spaces, while the orbit closures seem much more difficult to understand. However, in order to prove lower bounds in algebraic complexity theory, considering group orbits is not enough. In this paper we tighten the relationship between the orbit of the power sum polynomial and its closure, so that we can separate this orbit closure from the orbit closure of the product of variables by just considering the symmetry groups of both polynomials and their representation theoretic decomposition coefficients. In a natural way our construction yields a multiplicity obstruction that is neither an occurrence obstruction, nor a so-called vanishing ideal occurrence obstruction. All multiplicity obstructions so far have been of one of these two types. Our paper is the first implementation of the ambitious approach that was originally suggested in the first papers on geometric complexity theory by Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence proofs of obstructions only took into account the symmetry group of one of the two polynomials (or tensors) that were to be separated. In our paper the multiplicity obstruction is obtained by comparing the representation theoretic decomposition coefficients of both symmetry groups. Our proof uses a semi-explicit description of the coordinate ring of the orbit closure of the power sum polynomial in terms of Young tableaux, which enables its comparison to the coordinate ring of the orbit.Comment: 47 page

Multipliers for p-Bessel sequences in Banach spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.Comment: 17 page

A new methodology for anisotropic mesh refinement based upon error gradients

We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered

Transcranial sonography for diagnosis of Parkinson's disease

<p>Abstract</p> <p>Background</p> <p>In idiopathic Parkinson's disease (IPD) transcranial sonography (TCS) represents an alternative diagnostic method to verify clinical diagnosis. Although the phenomenon of an increased echogenicity of the Substantia nigra (SN) is well known this method is still not widly used in the diagnostic workup. Until now reliability of this method is still a matter of debate, partly because data only existed from a few laboratories using the same ultrasound machine. Therefore our study was conducted to test the reliability of this method by using a different ultrasound device and examining a large population of control and IPD subjects by two examiners to calculate interobserver reliability.</p> <p>Method</p> <p>In this study echogenicity of SN was examined in 199 IPD patients and 201 control subjects. All individuals underwent a neurological assessment including Perdue pegboard test and Webster gait test. Using a Sonos 5500 ultrasound device area of SN was measured, echogenicity of raphe, red nuclei, thalamus, caudate and lenticular nuclei, width of third and lateral ventricle were documented.</p> <p>Results</p> <p>We found a highly characteristic enlargement of the SN echogenic signal in IPD. The cut-off value for the SN area was established using a ROC curve with a sensitivity of 95% corresponding to an area of SN of 0.2 cm²and was found to be equivalent to the cut-off values of other studies using different ultrasound devices.</p> <p>Conclusions</p> <p>Our study shows that TCS is a reliable and highly sensitive tool for differentiation of IPD patients from individuals without CNS disorders.</p

Non-Gaussian power grid frequency fluctuations characterized by Levy-stable laws and superstatistics

Multiple types of fluctuations impact the collective dynamics of power grids and thus challenge their robust operation. Fluctuations result from processes as different as dynamically changing demands, energy trading and an increasing share of renewable power feed-in. Here we analyse principles underlying the dynamics and statistics of power grid frequency fluctuations. Considering frequency time series for a range of power grids, including grids in North America, Japan and Europe, we find a strong deviation from Gaussianity best described as Lévy-stable and q-Gaussian distributions. We present a coarse framework to analytically characterize the impact of arbitrary noise distributions, as well as a superstatistical approach that systematically interprets heavy tails and skewed distributions. We identify energy trading as a substantial contribution to today’s frequency fluctuations and effective damping of the grid as a controlling factor enabling reduction of fluctuation risks, with enhanced effects for small power grids

Parkinson’s disease or multiple system atrophy: potential separation by quantitative susceptibility mapping

Background: Due to the absence of robust biomarkers, and the low sensitivity and specificity of routine imaging techniques, the differential diagnosis between Parkinson’s disease (PD) and multiple system atrophy (MSA) is challenging. High-field magnetic resonance imaging (MRI) opened up new possibilities regarding the analysis of pathological alterations associated with neurodegenerative processes. Recently, we have shown that quantitative susceptibility mapping (QSM) enables visualization and quantification of two major histopathologic hallmarks observed in MSA: reduced myelin density and iron accumulation in the basal ganglia of a transgenic murine model of MSA. It is therefore emerging as a promising imaging modality on the differential diagnosis of Parkinsonian syndromes. Objectives: To assess QSM on high-field MRI for the differential diagnosis of PD and MSA. Methods: We assessed 23 patients (nine PDs and 14 MSAs) and nine controls using QSM on 3T and 7T MRI scanners at two academic centers. Results: We observed increased susceptibility in MSA at 3T in prototypical subcortical and brainstem regions. Susceptibility measures of putamen, pallidum, and substantia nigra reached excellent diagnostic accuracy to separate both synucleinopathies. Increase toward 100% sensitivity and specificity was achieved using 7T MRI in a subset of patients. Magnetic susceptibility correlated with age in all groups, but not with disease duration in MSA. Sensitivity and specificity were particularly high for possible MSA, and reached 100% in the putamen. Conclusion: Putaminal susceptibility measures, in particular on ultra-high-field MRI, may distinguish MSA patients from both, PD and controls, allowing an early and sensitive diagnosis of MSA

Déclaration sur l’approche par l’archéologie sociale du changement climatique

Manifiesto sobre Arqueología Social del Cambio Climático aprobado en la Cumbre SACC celebrada en Kiel. Aprobado y firmado el 6 de septiembre de 2021.[ES] El SACC es un grupo independiente, constituido en Kiel, de investigadores e investigadoras que trabajan sobre cambio climático. El objetivo de SACC es reunir científicos y científicas internacionales y representantes de importantes organizaciones internacionales de las áreas de arqueología, paleoecología y gestión del patrimonio para con el fin de discutir y evaluar la contribución de la investigación arqueológica y paleo-ecológica para comprender la interrelación entre el cambio social, el cultural, el ecológico y el climático. Pretendemos resaltar cómo la arqueología, a través del estudio de la conducta adaptativa en el pasado, es capaz de reforzar tanto la resiliencia socio-ecológica de nuestras sociedades, como su capacidad adaptativa ante el actual cambio climático. Además, pretendemos contribuir a la comprensión del impacto del cambio climático en los yacimientos y sitios arqueológicos y patrimoniales, así como en los paisajes culturales, los museos, las colecciones y archivos patrimoniales. SACC celebrará cumbres cada dos años y emitirá una declaración o manifiesto al término de cada una de ellas. S ACC está organizada por un comité interino presidido por las personas convocantes del SACC 1.[EN] SACC is an independent group of researchers working on climate change in the past constituted in Kiel. The aim of SACC is to bring together international scientists and representatives of important international organisations in the fields of archaeology, paleoecology and heritage management to discuss and evaluate the contribution of archaeological and paleo-ecological research to understand the link between social, cultural, ecological and climatic change; and to highlight how archaeology, through the study of past adaptive behaviour, is able to enhance socio-ecological resilience of societies as well as their adaptive capacity to current climate change; furthermore, to contribute to the understanding of the impact of climate change on archaeological and heritage sites as well as on cultural landscapes, museums, collections, and archives. SACC will hold its summit every second year with a declaration at the end of each summit. SACC is organized by a steering committee chaired by the SACC 1 organisers.[FR] Le SACC est un groupe indépendant de chercheurs travaillant sur le changement climatique dans le passé, qui s’est formé à Kiel. L’objectif du S ACC est de réunir des scientifiques internationaux et des représentants d’organisations internationales importantes dans les domaines de l’archéologie, de la paléoécologie et de la gestion du patrimoine. Il a pour objectif de discuter et d’évaluer la contribution de la recherche archéologique et paléo-écologique à la compréhension du lien entre les changements sociaux, culturels, écologiques et climatiques et de souligner comment l’archéologie, par l’étude du comportement adaptatif du passé, est capable d’améliorer la résilience socioécologique des sociétés ainsi que leur capacité d’adaptation au changement climatique actuel. En outre, il contribue à la compréhension de l’impact du changement climatique sur les sites archéologiques et patrimoniaux ainsi que sur les paysages culturels, les musées, les collections et les archives. Le SACC tiendra son conseil tous les deux ans avec une déclaration à la fin de chaque conseil. Il est organisé par un comité de pilotage présidé par les organisateurs de SACC 1.Peer reviewe

Deep histopathology genotype–phenotype analysis of focal cortical dysplasia type II differentiates between the GATOR1-altered autophagocytic subtype IIa and MTOR-altered migration deficient subtype IIb

AbstractFocal cortical dysplasia type II (FCDII) is the most common cause of drug-resistant focal epilepsy in children. Herein, we performed a deep histopathology-based genotype–phenotype analysis to further elucidate the clinico-pathological and genetic presentation of FCDIIa compared to FCDIIb. Seventeen individuals with histopathologically confirmed diagnosis of FCD ILAE Type II and a pathogenic variant detected in brain derived DNA whole-exome sequencing or mTOR gene panel sequencing were included in this study. Clinical data were directly available from each contributing centre. Histopathological analyses were performed from formalin-fixed, paraffin-embedded tissue samples using haematoxylin–eosin and immunohistochemistry for NF-SMI32, NeuN, pS6, p62, and vimentin. Ten individuals carried loss-of-function variants in the GATOR1 complex encoding genes DEPDC5 (n = 7) and NPRL3 (n = 3), or gain-of-function variants in MTOR (n = 7). Whereas individuals with GATOR1 variants only presented with FCDIIa, i.e., lack of balloon cells, individuals with MTOR variants presented with both histopathology subtypes, FCDIIa and FCDIIb. Interestingly, 50% of GATOR1-positive cases showed a unique and predominantly vacuolizing phenotype with p62 immunofluorescent aggregates in autophagosomes. All cases with GATOR1 alterations had neurosurgery in the frontal lobe and the majority was confined to the cortical ribbon not affecting the white matter. This pattern was reflected by subtle or negative MRI findings in seven individuals with GATOR1 variants. Nonetheless, all individuals were seizure-free after surgery except four individuals carrying a DEPDC5 variant. We describe a yet underrecognized genotype–phenotype correlation of GATOR1 variants with FCDIIa in the frontal lobe. These lesions were histopathologically characterized by abnormally vacuolizing cells suggestive of an autophagy-altered phenotype. In contrast, individuals with FCDIIb and brain somatic MTOR variants showed larger lesions on MRI including the white matter, suggesting compromised neural cell migration.</jats:p