scholarly journals A non-local reduction principle for cocycles in Hilbert spaces

2020 ◽  
Vol 269 (9) ◽  
pp. 6699-6731
Author(s):  
Mikhail Anikushin
Keyword(s):  
Hilbert Spaces ◽  
Reduction Principle ◽  
Local Reduction ◽  
Non Local

scholarly journals On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem

2020 ◽  
Vol 65 (4) ◽  
pp. 622-635
Author(s):  
Mikhail M. Anikushin ◽  
Keyword(s):  
Hilbert Spaces ◽  
Adjoint Problem ◽  
Operator Inequalities ◽  
Operator Solution ◽  
Frequency Theorem ◽  
Regularity Properties ◽  
Local Reduction ◽  
Lyapunov Inequalities ◽  
Non Local ◽  
Autonomous Dynamical Systems

We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.

The Method of Non-Local Reduction

1992 ◽  
pp. 89-127
Author(s):  
Gennadij A. Leonov ◽  
Volker Reitmann ◽  
Vera B. Smirnova
Keyword(s):  
Local Reduction ◽  
Non Local

Approximate controllability of semilinear impulsive functional differential systems with non-local conditions

2020 ◽  
Vol 37 (4) ◽  
pp. 1070-1088 ◽  
Author(s):  
Sumit Arora ◽  
Soniya Singh ◽  
Jaydev Dabas ◽  
Manil T Mohan
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Approximate Controllability ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Local Conditions ◽  
Functional Differential Systems ◽  
Non Local ◽  
Functional Differential

Abstract This paper is concerned with the approximate controllability of semilinear impulsive functional differential systems in Hilbert spaces with non-local conditions. We establish sufficient conditions for approximate controllability of such systems via resolvent operator and Schauder’s fixed point theorem. An application involving the impulse effect associated with delay and non-local conditions is presented to verify our claimed results.

scholarly journals Existence, uniqueness and stability results for semilinear integrodifferential non-local evolution equations with Random impulse

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6615-6626
Author(s):  
B. Radhakrishnan ◽  
M. Tamilarasi ◽  
P. Anukokila
Keyword(s):  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Local Conditions ◽  
Fixed Point Approach ◽  
Non Local ◽  
The Stability ◽  
Random Impulse ◽  
Stability Results

In this paper, authors investigated the existence and uniqueness of random impulsive semilinear integrodifferential evolution equations with non-local conditions in Hilbert spaces. Also the stability results for the same evolution equation has been studied. The results are derived by using the semigroup theory and fixed point approach. An application is provided to illustrate the theory.

scholarly journals Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

2022 ◽  
Author(s):  
Frederic Weber ◽  
Rico Zacher
Keyword(s):  
Heat Kernel ◽  
General Framework ◽  
Important Application ◽  
Reduction Principle ◽  
Non Local ◽  
Beta 2 ◽  
Local Diffusion ◽  
Continuous Setting ◽  
Discrete Setting ◽  
Diffusion Problems

AbstractWe establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Elastic Energy Relaxation in Buried Quantum Dots

2006 ◽  
Vol 959 ◽  
Author(s):  
Vladimir V Chaldyshev ◽  
Anna L. Kolesnikova ◽  
Alexei E. Romanov
Keyword(s):  
Quantum Dots ◽  
Elastic Energy ◽  
Energy Relaxation ◽  
The Other ◽  
Matrix Interface ◽  
Dislocation Loops ◽  
The Third ◽  
Plastic Distortion ◽  
Local Reduction ◽  
Non Local

ABSTRACTWe theoretically analyze three models, which correspond to three different ways of the elastic energy relaxation in buried quantum dots. The first model considers formation of a pair of prismatic dislocation loops. One of them lies on dot/matrix interface, whereas the other is a satellite and locates in the adjacent matrix. The second model also includes the satellite loop and differs from the first one by non-local reduction of the dot plastic distortion. The origin of the satellite loop is the materials conservation requirement. The third model considers the case when this requirement is violated and only the misfit dislocation loop is formed. We determine the critical radii of the dots and loops, as well as the dependence of the satellite loop size on the dot size. The model calculation are compared to the relevant experimental data.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

Chromatic aberration and low-voltage SEM

1987 ◽  
Vol 45 ◽  
pp. 546-547
Author(s):  
Zhifeng Shao ◽  
A.V. Crewe
Keyword(s):  
Electron Energy ◽  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Primary Electron ◽  
Probe Size ◽  
Non Local ◽  
Optical Treatment ◽  
Electron Microscopes ◽  
Scanning Electron Microscopes

For scanning electron microscopes, it is plausible that by lowering the primary electron energy, one can decrease the volume of interaction and improve resolution. As shown by Crewe /1/, at V0 =5kV a 10Å resolution (including non-local effects) is possible. To achieve this, we would need a probe size about 5Å. However, at low voltages, the chromatic aberration becomes the major concern even for field emission sources. In this case, δV/V = 0.1 V/5kV = 2x10-5. As a rough estimate, it has been shown that /2/ the chromatic aberration δC should be less than ⅓ of δ0 the probe size determined by diffraction and spherical aberration in order to neglect its effect. But this did not take into account the distribution of electron energy. We will show that by using a wave optical treatment, the tolerance on the chromatic aberration is much larger than we expected.

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