Some weil group representations motivated by algebraic topology

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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Disrupted metabolic connectivity in dopaminergic and cholinergic networks at different stages of dementia from 18F-FDG PET brain persistent homology network

Scientific Reports ◽

10.1038/s41598-021-84722-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Tun-Wei Hsu ◽

Jong-Ling Fuh ◽

Da-Wei Wang ◽

Li-Fen Chen ◽

Chia-Jung Chang ◽

...

Keyword(s):

Algebraic Topology ◽

Fdg Pet ◽

Persistent Homology ◽

Network Connectivity ◽

Brain Regions ◽

Imaging Data ◽

Cholinergic Pathways ◽

Positron Emission ◽

Metabolic Connectivity ◽

Neurochemical Changes

AbstractDementia is related to the cellular accumulation of β-amyloid plaques, tau aggregates, or α-synuclein aggregates, or to neurotransmitter deficiencies in the dopaminergic and cholinergic pathways. Cellular and neurochemical changes are both involved in dementia pathology. However, the role of dopaminergic and cholinergic networks in metabolic connectivity at different stages of dementia remains unclear. The altered network organisation of the human brain characteristic of many neuropsychiatric and neurodegenerative disorders can be detected using persistent homology network (PHN) analysis and algebraic topology. We used 18F-fluorodeoxyglucose positron emission tomography (18F-FDG PET) imaging data to construct dopaminergic and cholinergic metabolism networks, and used PHN analysis to track the evolution of these networks in patients with different stages of dementia. The sums of the network distances revealed significant differences between the network connectivity evident in the Alzheimer’s disease and mild cognitive impairment cohorts. A larger distance between brain regions can indicate poorer efficiency in the integration of information. PHN analysis revealed the structural properties of and changes in the dopaminergic and cholinergic metabolism networks in patients with different stages of dementia at a range of thresholds. This method was thus able to identify dysregulation of dopaminergic and cholinergic networks in the pathology of dementia.

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Crystallography: Symmetry groups and group representations

EPJ Web of Conferences ◽

10.1051/epjconf/20122200006 ◽

2012 ◽

Vol 22 ◽

pp. 00006

Author(s):

B. Grenier ◽

R. Ballou

Keyword(s):

Symmetry Groups ◽

Group Representations

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Probing charge pumping and relaxation of the chiral anomaly in a Dirac semimetal

Science Advances ◽

10.1126/sciadv.abg0914 ◽

2021 ◽

Vol 7 (16) ◽

pp. eabg0914

Author(s):

Bing Cheng ◽

Timo Schumann ◽

Susanne Stemmer ◽

N. P. Armitage

Keyword(s):

Terahertz Spectroscopy ◽

Point Group ◽

Spectral Weight ◽

Chiral Anomaly ◽

Group Representations ◽

Field Independence ◽

Transport Channel ◽

Scattering Rate ◽

Chiral Magnetic Effect ◽

A Charge

The linear band crossings of 3D Dirac and Weyl semimetals are characterized by a charge chirality, the parallel or antiparallel locking of electron spin to its momentum. These materials are believed to exhibit an E · B chiral magnetic effect that is associated with the near conservation of chiral charge. Here, we use magneto-terahertz spectroscopy to study epitaxial Cd3As2 films and extract their conductivities σ(ω) as a function of E · B. As field is applied, we observe a markedly sharp Drude response that rises out of the broader background. Its appearance is a definitive signature of a new transport channel and consistent with the chiral response, with its spectral weight a measure of the net chiral charge and width a measure of the scattering rate between chiral species. The field independence of the chiral relaxation establishes that it is set by the approximate conservation of the isospin that labels the crystalline point-group representations.

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Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

Entropy ◽

10.3390/e23010125 ◽

2021 ◽

Vol 23 (1) ◽

pp. 125

Author(s):

Tobias Gulden ◽

Alex Kamenev

Keyword(s):

Quantum Mechanics ◽

Algebraic Topology ◽

Riemann Surfaces ◽

Surrounding Medium ◽

Coulomb Systems ◽

Dielectric Constants ◽

Ion Interactions ◽

Multivalent Ions ◽

Electric Filed ◽

Entropic Effects

We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.

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