locally homogeneous
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2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Alice Lim

Abstract In this paper, we classify the compact locally homogeneous non-gradient m-quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m-quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S1 is the only compact manifold of any dimension which admits a metric which is nontrivially m-quasi Einstein and Einstein.


2021 ◽  
Vol 60 (1) ◽  
pp. 17-22
Author(s):  
Tatiana A. Andreeva ◽  
Dmitry N. Oskorbin ◽  
Evgeny D. Rodionov

Conformally Killing fields play an important role in the theory of Ricci solitons and also generate an important class of locally conformally homogeneous (pseudo) Riemannian manifolds. In the Riemannian case, V. V. Slavsky and E.D. Rodionov proved that such spaces are either conformally flat or conformally equivalent to locally homogeneous Riemannian manifolds. In the pseudo-Riemannian case, the question of their structure remains open. Pseudo-Riemannian symmetric spaces of order k, where k 2, play an important role in research in pseudo-Riemannian geometry. Currently, they have been investigated in cases k=2,3 by D.V. Alekseevsky, A.S. Galaev and others. For arbitrary k, non-trivial examples of such spaces are known: generalized Kachen - Wallach manifolds. In the case of small dimensions, these spaces and Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov, and I.V. Ernst with the helpof systems of computer mathematics. In this paper, using the Sagemath SCM, we investigate conformally Killing vector fields on five-dimensional indecomposable 2- symmetric Lorentzian manifolds, and construct an algorithm for their computation.


2021 ◽  
Author(s):  
Abhirup Bandyopadhyay ◽  
Spase Petkoski ◽  
Viktor Jirsa

Changes in extracellular ion concentrations are known to modulate neuronal excitability and play a major role in controlling the neuronal firing rate, not just during the healthy homeostasis, but also in pathological conditions such as epilepsy. The microscopic molecular mechanisms of field effects are understood, but the precise correspondence between the microscopic mechanisms of ion exchange in the cellular space of neurons and the macroscopic behavior of neuronal populations remains to be established. We derive a mean field model of a population of Hodgkin Huxley type neurons. This model links the neuronal intra- and extra-cellular ion concentrations to the mean membrane potential and the mean synaptic input in terms of the synaptic conductance of the locally homogeneous mesoscopic network and can describe various brain activities including multi-stability at resting states, as well as more pathological spiking and bursting behaviors, and depolarizations. The results from the analytical solution of the mean field model agree with the mean behavior of numerical simulations of large-scale networks of neurons. The mean field model is analytically exact for non-autonomous ion concentration variables and provides a mean field approximation in the thermodynamic limit, for locally homogeneous mesoscopic networks of biophysical neurons driven by an ion exchange mechanism. These results may provide the missing link between high-level neural mass approaches which are used in the brain network modeling and physiological parameters that drive the neuronal dynamics.


2021 ◽  
Vol 248 ◽  
pp. 106517
Author(s):  
Cong Du ◽  
Pengfei Liu ◽  
Quan Liu ◽  
Sabine Leischner ◽  
Yiren Sun ◽  
...  

Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
O.P. Khromova

The problem of restoring a (pseudo)Riemannian manifold  from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds. For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open. This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.


2021 ◽  
Author(s):  
Daniel Gomes Albuquerque ◽  
Gustavo Coelho Abade ◽  
Hanna Pawłowska

<p>Several microphysical processes determine phase partitioning between ice and liquid water in a mixed-phase cloud. Here we investigate the collective growth of ice particles and liquid droplets affected by turbulent fluctuations in temperature and water vapor fields. All cloud particles, including inactivated nuclei (both CCN and IN), are described by Lagrangian super-particles. To account for local variability in the turbulent cloud environment we apply a Lagrangian microphysical scheme, where temperature and vapor mixing ratio are stochastic attributes attached to each super-particle. In addition, a simple linear relaxation scheme models turbulent mixing of the scalar fields probed by each super-particle. The limit of a locally homogeneous growth environment corresponds to an infinitely short turbulent mixing timescale. The impact of our Lagrangian microphysical scheme on phase partitioning is tested in adiabatic cloud parcel simulations. Results are confronted with idealized reference simulations that use bulk microphysics based on an assumed (temperature-dependent) phase partitioning function. Our study suggests that accounting for local variability in a turbulent cloud is important for reproducing steady-state mixed-phase conditions.</p>


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