scholarly journals Growth–fragmentation–coagulation equations with unbounded coagulation kernels

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190612
Author(s):  
J. Banasiak ◽  
W. Lamb
Keyword(s):  
Linear Growth ◽  
Main Tool ◽  
High Order ◽  
Theme Issue ◽  
Continuous Growth ◽  
Coagulation Equations ◽  
Coagulation Equation ◽  
Large Clusters ◽  
Finite Moments

In this paper, we prove the global in time solvability of the continuous growth–fragmentation–coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth–fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue ‘Semigroup applications everywhere’.

Richtmyer–Meshkov instability on a quasi-single-mode interface

2019 ◽  
Vol 872 ◽  
pp. 729-751 ◽  
Author(s):  
Yu Liang ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Xisheng Luo
Keyword(s):  
Nonlinear Model ◽  
Small Amplitude ◽  
Linear Growth ◽  
Single Mode ◽  
High Order ◽  
Initial Amplitude ◽  
Previous Theory ◽  
Weakly Nonlinear ◽  
Nonlinear Stage ◽  
Amplitude Growth

Experiments on Richtmyer–Meshkov instability of quasi-single-mode interfaces are performed. Four quasi-single-mode air/$\text{SF}_{6}$ interfaces with different deviations from the single-mode one are generated by the soap film technique to evaluate the effects of high-order modes on amplitude growth in the linear and weakly nonlinear stages. For each case, two different initial amplitudes are considered to highlight the high-amplitude effect. For the single-mode and saw-tooth interfaces with high initial amplitude, a cavity is observed at the spike head, providing experimental evidence for the previous numerical results for the first time. For the quasi-single-mode interfaces, the fundamental mode is the dominant one such that it determines the amplitude linear growth, and subsequently the impulsive theory gives a reasonable prediction of the experiments by introducing a reduction factor. The discrepancy in linear growth rates between the experiment and the prediction is amplified as the quasi-single-mode interface deviates more severely from the single-mode one. In the weakly nonlinear stage, the nonlinear model valid for a single-mode interface with small amplitude loses efficacy, which indicates that the effects of high-order modes on amplitude growth must be considered. For the saw-tooth interface with small amplitude, the amplitudes of the first three harmonics are extracted from the experiment and compared with the previous theory. The comparison proves that each initial mode develops independently in the linear and weakly nonlinear stages. A nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) is then modified by considering the effects of high-order modes. The modified model is proved to be valid in the weakly nonlinear stage even for the cases with high initial amplitude. More high-order modes are needed to match the experiment for the interfaces with a more severe deviation from the single-mode one.

scholarly journals Random matrix theory for complexity growth and black hole interiors

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Arjun Kar ◽  
Lampros Lamprou ◽  
Moshe Rozali ◽  
James Sully
Keyword(s):  
Black Hole ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Linear Growth ◽  
Matrix Theory ◽  
Main Tool ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Maximal Volume ◽  
Quantum Theories

Abstract We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy — a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.

scholarly journals Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

2019 ◽  
Vol 150 (4) ◽  
pp. 1805-1825 ◽  
Author(s):  
Prasanta Kumar Barik ◽  
Ankik Kumar Giri ◽  
Philippe Laurençot
Keyword(s):  
Weak Solutions ◽  
Linear Growth ◽  
Weak Compactness ◽  
Singular Kernel ◽  
Initial Condition ◽  
Existence Proof ◽  
Global Weak Solutions ◽  
Smoluchowski Coagulation Equation ◽  
Coagulation Equation ◽  
Coagulation Kernel

AbstractGlobal weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Chromospheric activity as a function of age in main-sequence stars

1966 ◽  
Vol 24 ◽  
pp. 40-43
Author(s):  
O. C. Wilson ◽  
A. Skumanich
Keyword(s):  
Main Sequence ◽  
The Sun ◽  
Main Sequence Stars ◽  
Tentative Conclusion ◽  
Chromospheric Activity ◽  
General Field ◽  
Large Clusters

Evidence previously presented by one of the authors (1) suggests strongly that chromospheric activity decreases with age in main sequence stars. This tentative conclusion rests principally upon a comparison of the members of large clusters (Hyades, Praesepe, Pleiades) with non-cluster objects in the general field, including the Sun. It is at least conceivable, however, that cluster and non-cluster stars might differ in some fundamental fashion which could influence the degree of chromospheric activity, and that the observed differences in chromospheric activity would then be attributable to the circumstances of stellar origin rather than to age.

Triple Fixation For Electron Microscopy

1968 ◽  
Vol 26 ◽  
pp. 90-91 ◽  
Author(s):  
M. A. Hayat
Keyword(s):  
Electron Microscopy ◽  
Endoplasmic Reticulum ◽  
Electron Beam ◽  
Plasma Membrane ◽  
Myelin Sheath ◽  
Good Deal ◽  
Granular Structure ◽  
Oxidizing Agent ◽  
Fixation Technique ◽  
Large Clusters

Potassium permanganate has been successfully employed to study membranous structures such as endoplasmic reticulum, Golgi, plastids, plasma membrane and myelin sheath. Since KMnO4 is a strong oxidizing agent, deposition of manganese or its oxides account for some of the observed contrast in the lipoprotein membranes, but a good deal of it is due to the removal of background proteins either by dehydration agents or by volatalization under the electron beam. Tissues fixed with KMnO4 exhibit somewhat granular structure because of the deposition of large clusters of stain molecules. The gross arrangement of membranes can also be modified. Since the aim of a good fixation technique is to preserve satisfactorily the cell as a whole and not the best preservation of only a small part of it, a combination of a mixture of glutaraldehyde and acrolein to obtain general preservation and KMnO4 to enhance contrast was employed to fix plant embryos, green algae and fungi.

Determination of the Burgers vector of a dislocation in a Fe-Mn alloy by weak-beam image in HVEM

1978 ◽  
Vol 36 (1) ◽  
pp. 330-331
Author(s):  
Y. Ishida ◽  
H. Ishida ◽  
K. Kohra ◽  
H. Ichinose
Keyword(s):  
High Order ◽  
Simulation Technique ◽  
The Other ◽  
Image Simulation ◽  
Diffraction Vector ◽  
Burgers Vector ◽  
Weak Beam ◽  
Beam Image ◽  
Edge Component

IntroductionA simple and accurate technique to determine the Burgers vector of a dislocation has become feasible with the advent of HVEM. The conventional image vanishing technique(1) using Bragg conditions with the diffraction vector perpendicular to the Burgers vector suffers from various drawbacks; The dislocation image appears even when the g.b = 0 criterion is satisfied, if the edge component of the dislocation is large. On the other hand, the image disappears for certain high order diffractions even when g.b ≠ 0. Furthermore, the determination of the magnitude of the Burgers vector is not easy with the criterion. Recent image simulation technique is free from the ambiguities but require too many parameters for the computation. The weak-beam “fringe counting” technique investigated in the present study is immune from the problems. Even the magnitude of the Burgers vector is determined from the number of the terminating thickness fringes at the exit of the dislocation in wedge shaped foil surfaces.

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