On embedding theorems of spaces of functions with mixed logarithmic smoothness

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and spaces with mixed logarithmic smoothness. Equivalent norms of a space with mixed logarithmic smoothness are found and embedding theorems are proved

On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space

We consider the generalized Lorentz space Lψ,τ(Tm) defined by some continuous concave function ψ such that ψ(0) = 0. For two spaces Lψ1,τ1(Tm) and Lψ2,τ2(Tm) such that αψ1 = limt→0ψ1(2t)/ψ1(t) = βψ2 = limt→0ψ2(2t)/ψ2(t), we prove an order-exact inequality of different metrics for multiple trigonometric polynomials. We also prove an auxiliary statement for functions of one variable with monotonically decreasing Fourier coefficients in a trigonometric system. In this statement we establish a two-sided estimate for the norm of the function f ∈ Lψ,τ(T) in terms of the series composed of the Fourier coefficients of this function. © 2019 Mofid University - Center for Human Rights Studies. All rights reserved.Ministry of Education and Science of the Russian Federation, MinobrnaukaUral Federal University, UrFUThis work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Оценки билинейных приближений функций в пространстве Лоренца

В статье рассмотрены пространство Лоренца периодических функций многих переменных и класс Никольского-Бесова. В пространстве Лоренца дано определение билинейного приближения функции и приведена теорема Марцинкевича-Зигмунда для тригонометрического полинома. Установлены оценки наилучших билинейных приближений класса Никольского-Бесова в пространстве Лоренца

Estimations of the best M-term approximations of functions in the Lorentz space with constructive methods

This paper considers the Lorentz space of periodic functions of many variables with the anisotropic norm, of functional Nikol’skii-Besov’s class and of the best M -term approximation of function. We have established sufficient conditions for the function to belong to one of the Lorentz spaces in another. We obtain upper and lower bounds for the best M -member approximations of functions from the Nikol’skii-Besov class in the anisotropic Lorentz space To prove the upper bound, we used a new constructive method developed by V.N. Temlyakov

Some Fourier inequalities for orthogonal systems in Lorentz–Zygmund spaces

A number of classical inequalities and convergence results related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of Lorentz–Zygmund spaces. © 2019, The Author(s).We thank the referees and Professors Dag Lukkasson and Annette Meidell for some good advice which improved the final version of the paper. Moreover, the first author is grateful for the support of this work given by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

ON ESTIMATES OF M-TERM APPROXIMATIONS ON CLASSES OF FUNCTIONS WITH BOUNDED MIXED DERIVATIVE IN THE LORENTZ SPACE

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space Lq,τ(T m) , the class of functions with bounded mixed fractional derivative Wq,τr¯, 1 < q, τ< ∞, and studies the order of the best M-term approximation of a function f∈ Lp,τ(T m) by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class Wq,τ1r¯ in the norm of the space Lp,τ2(Tm) for various relations between the parameters p, q, τ1, τ2. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG

Complex dynamic behaviors of nonequilibrium atmospheric dielectric-barrier discharges

Copyright 2006 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the authors and the American Institute of Physics. This article appeared in the Journal of Applied Physics and may be found at: http://link.aip.org/link/?JAPIAU/100/063304/1In this paper, a one-dimensional fluid model is used to investigate complex dynamic behaviors of a nonequilibrium dielectric-barrier discharge (DBD) in atmospheric helium. By projecting its evolution trajectory in the three-dimensional phase space of gas voltage, discharge current density, and electrode-surface charge density, the atmospheric DBD is shown to undergo a sequence of complex bifurcation processes when the applied voltage is increased from prebreakdown to many times of the breakdown voltage. Once the gas voltage exceeds the breakdown voltage, the discharge plasma is found to acquire negative differential conductivity and as a result its stability is compromised. For atmospheric DBD, however, the resulting low plasma stability is mitigated by a rapid accumulation of surface charges on the electrodes, thus allowing the atmospheric DBD to retain their character as a glow discharge. At certain values of the applied voltage, a highly complex phenomenon of period multiplication is observed in which the period of the discharge current is three times that of the applied voltage. This suggests that nonequilibrium atmospheric DBD may support evolution patterns that are quasiperiodic or even chaotic. These complex dynamic behaviors are likely to be critical to a full understanding of plasma stability of nonequilibrium atmospheric discharges and to the development of their instability control strategies

Cathode fall characteristics in a dc atmospheric pressure glow discharge

Copyright 2003 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the authors and the American Institute of Physics. This article appeared in the Journal of Applied Physics and may be found at: http://link.aip.org/link/?JAPIAU/94/5504/1Atmospheric pressure glow discharges are attractive for a wide range of material-processing applications largely due to their operation flexibility afforded by removal of the vacuum system. These relatively new atmospheric plasmas are nonequilibrium plasmas with gas temperature around 100 °C and electron temperature in the 1–10 eV range. Their appearance is characteristically diffuse and uniform, and their temporal features are repetitive and stable. Of the reported numerical studies of atmospheric glow discharges, most are based on the hydrodynamic approximation in which electrons are assumed to be in equilibrium with the local electric field. Spectroscopic and electrical measurements suggest however that the cathode fall region is fundamentally nonequilibrium. To this end we consider a hybrid model that treats the cathode fall region kinetically but retains a hydrodynamic description for the region between the thin cathode fall layer and the anode. Using this hybrid model, a helium discharge system excited at dc is studied numerically for a very wide current density range that spans from Townsend dark discharge, through normal glow discharge, to abnormal glow discharge. Numerical results confirm many distinct characteristics of glow discharges and compare well with that of low-pressure glow discharges. Generic relationships, such as that between the electric field and the current density, are also established and are in good agreement with experimental data. This hybrid model is simple and insightful as a theoretical tool for atmospheric pressure glow discharges

Plasma–liquid interactions: a review and roadmap

Plasma–liquid interactions represent a growing interdisciplinary area of research involving plasma science, fluid dynamics, heat and mass transfer, photolysis, multiphase chemistry and aerosol science. This review provides an assessment of the state-of-the-art of this multidisciplinary area and identifies the key research challenges. The developments in diagnostics, modeling and further extensions of cross section and reaction rate databases that are necessary to address these challenges are discussed. The review focusses on non-equilibrium plasmas