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## Finite-Dimensional Reduction of Systems of Nonlinear Diffusion Equations

Mathematical Notes. 2023. Vol. 113. No. 2. P. 267-273.

We consider a class of one-dimensional systems of nonlinear parabolic equations whose phase dynamics at large time can be described by ODE with a Lipschitz vector field in R^n. In the case of the Dirichlet boundary-value problem considered here, sufficient conditions for the finite-dimensional reduction turn out to be essentially wider than the known similar conditions in the periodic situation.

A.V. Romanov, Finite-dimensional reduction of systems of nonlinear diffusion equations / Cornell University. Series arXiv "math". 2022. No. 2210.00499.

We present a class of one-dimensional systems of nonlinear parabolic equations for which long-time phase dynamics can be described by an ODE with a Lipschitz vector field in R^n. In the considered case of the Dirichlet boundary value problem sufficient conditions for a finite-dimensional reduction turn out to be much wider than the known conditions ...

Added: October 5, 2022

Aleksandr V. Romanov, AIMS MATHEMATICS 2021 Vol. 6 No. 12 P. 13407-13422

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in RN. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated ...

Added: September 18, 2021

А. В. Романов, Математические заметки 2023 Т. 113 № 2 С. 265-272

Мы предъявляем класс одномерных систем нелинейных параболических уравнений, для которых фазовая динамика при большом времени может быть описана ОДУ с липшицевым векторным полем в R𝑛. В рассматриваемом случае краевой задачи Дирихле достаточные условия конечномерной редукции оказываются существенно шире известных условий такого рода для периодической ситуации. ...

Added: January 15, 2023

Romanov A., Dynamics of Partial Differential Equations 2016 Vol. 13 No. 3 P. 263-272

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an ...

Added: June 26, 2016

A. V. Romanov, Final Dynamics of Systems of Nonlinear Parabolic Equations on the Circle / Cornell University. Series arXiv "math". 2020. No. 2011.01822.

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamicsof a system can be described by an ODE with Lipschitz vector field in RN. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property ...

Added: November 6, 2020

Kondratieva L. A., A.V. Romanov, Electronic Journal of Qualitative Theory of Differential Equations 2019 No. 96 P. 1-11

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincaré–Bendixson theory. In the case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of ...

Added: December 22, 2019

Romanov A., Mathematical notes 2014 Vol. 96 No. 4 P. 548-555

A family of parabolic integro-differential equations with nonlocal diffusion on the circle which have no smooth inertial manifold is presented. ...

Added: September 15, 2014

A.V. Romanov, Kondratieva L. A., Inertial Manifolds and Limit Cycles of Dynamical Systems in Rn / Cornell University. Series math "arxiv.org". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019

Alexander V. Romanov, Parabolic Equation with Nonlocal Diffusion without a Smooth Inertial Manifold / Cornell University. Series math "arxiv.org". 2013. No. 1306.4249.

We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier. ...

Added: November 18, 2013

Kondratieva L. A., Inertial Manifolds and Limit Cycles of Dynamical Systems in Rn / Cornell University Library. Series math.RT "arXiv:1808.06395 [math.RT]". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019

Romanov A., On the Hyperbolicity Properties of Inertial Manifolds of Reaction–Diffusion Equations / Cornell University. Series math "arxiv.org". 2016. No. 1602.08953.

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an ...

Added: June 26, 2016

A.V. Romanov, Kondratieva L. A., Inertial Manifolds and Limit Cycles of Dynamical Systems in Rn / Cornell University. Series math "arxiv.org". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019

Romanov A., Математические заметки 2014 Т. 96 № 4 С. 578-587

We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have smooth inertial manifold in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier. ...

Added: August 19, 2014

Shaposhnikov S., Манита О. А., Доклады Академии наук 2012 Т. 447 № 6 С. 610-614

Получены достаточные условия существования вероятностного решения нелинейного параболического уравнения. ...

Added: October 14, 2014

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Litvin Y. V., Абрамов И. В., Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019

Buchstaber V., Limonchenko I., Embeddings of moment-angle manifolds and sequences of Massey products / Cornell University. Series math "arxiv.org". 2018. No. 1808.08851.

We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes. ...

Added: September 29, 2019

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017

Arzhantsev I., Journal of Lie Theory 2000 Vol. 10 No. 2 P. 345-357

Added: July 8, 2014

Decrouez G. G., Hall P., Bernoulli: a journal of mathematical statistics and probability 2013 Vol. 19 No. 4 P. 1268-1293

Motivated by a problem arising when analysing data from quarantine searches, we explore properties of distributions of sums of independent means of independent lattice-valued random variables. The aim is to determine the extent to which approximations to those sums require continuity corrections. We show that, in cases where there are only two different means, the ...

Added: September 29, 2014

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Bagrov A. N., Gordin V. A., Bykov P. L., Russian Meteorology and Hydrology 2014 No. 5 P. 283-291

The evaluations of the forecasts of surface air temperature and precipitation for the period July 2010 - June 2013 are presented. The forecasting of surface air temperature at 5 days and precipitation at 3 days are considered. Our complex statistical scheme uses the results of the best foreign global schemes, regional scheme COSMO-RU7. The joint ...

Added: December 7, 2013