This document analyzes the 3D and 1D stochastic Nikolaevskii systems, which model turbulence. It introduces the 3D and 1D stochastic partial differential equations that were obtained by adding small white noise to the original non-stochastic Nikolaevskii systems. The key findings are: 1) Numerical integration of the 1D system is actually solving the stochastic model due to inherent computational noise. 2) Even small noise can significantly impact turbulent modes and change system behavior dramatically. 3) Developed turbulence in physical systems may be characterized as quantum chaos driven by thermal fluctuations based on modeling of the stochastic systems.
Chaos Suppression and Stabilization of Generalized Liu Chaotic Control Systemijtsrd
In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun | Jer-Guang Hsieh "Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20195.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20195/chaos-suppression-and-stabilization-of-generalized-liu-chaotic-control-system/yeong-jeu-sun
The document summarizes an honours project that calculates the pressure of a gluon gas using statistical mechanics and thermal field theory. It introduces the quark-gluon plasma system and describes how both theories predict the same result for an ideal gas case, but field theory provides a simpler way to incorporate interactions. The project uses quantum mechanics path integral formalism to calculate the pressure via the statistical mechanics and field theory approaches, showing they give identical results for an ideal gas.
Computational Complexity Comparison Of Multi-Sensor Single Target Data Fusion...ijccmsjournal
This document compares the computational complexity of four multi-sensor data fusion methods based on the Kalman filter using MATLAB simulations. The four methods are: group-sensor method, sequential-sensor method, inverse covariance form, and track-to-track fusion. The results show that the inverse covariance method has the best computational performance if the number of sensors is above 20. For fewer sensors, other methods like the group sensors method are more appropriate due to lower computational loads when inverting smaller matrices.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document summarizes a study on the effect of parameters of a geometric multigrid method on CPU time for solving one-dimensional problems related to heat transfer and fluid flow. The parameters studied include coarsening ratio of grids, number of inner iterations, number of grid levels, and tolerances. Finite difference methods were used to discretize partial differential equations for problems involving Poisson, advection-diffusion, and heat transfer equations. Comparisons were made between multigrid and single grid methods like Gauss-Seidel and TDMA. Results confirmed some literature findings and presented some new results on the effect of parameters on CPU time.
The klein gordon field in two-dimensional rindler space-timeforssfoxtrot jp R
This document discusses the Klein-Gordon scalar field in a two-dimensional Rindler space-time background. It derives the equation of motion for the scalar field from an action in this background. The equation can be solved exactly using imaginary time, giving an oscillatory solution with a frequency corresponding to an integral number. This suggests the scalar field appears quantized with spatial modes corresponding to integral values.
Chaos Suppression and Stabilization of Generalized Liu Chaotic Control Systemijtsrd
In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun | Jer-Guang Hsieh "Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20195.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20195/chaos-suppression-and-stabilization-of-generalized-liu-chaotic-control-system/yeong-jeu-sun
The document summarizes an honours project that calculates the pressure of a gluon gas using statistical mechanics and thermal field theory. It introduces the quark-gluon plasma system and describes how both theories predict the same result for an ideal gas case, but field theory provides a simpler way to incorporate interactions. The project uses quantum mechanics path integral formalism to calculate the pressure via the statistical mechanics and field theory approaches, showing they give identical results for an ideal gas.
Computational Complexity Comparison Of Multi-Sensor Single Target Data Fusion...ijccmsjournal
This document compares the computational complexity of four multi-sensor data fusion methods based on the Kalman filter using MATLAB simulations. The four methods are: group-sensor method, sequential-sensor method, inverse covariance form, and track-to-track fusion. The results show that the inverse covariance method has the best computational performance if the number of sensors is above 20. For fewer sensors, other methods like the group sensors method are more appropriate due to lower computational loads when inverting smaller matrices.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document summarizes a study on the effect of parameters of a geometric multigrid method on CPU time for solving one-dimensional problems related to heat transfer and fluid flow. The parameters studied include coarsening ratio of grids, number of inner iterations, number of grid levels, and tolerances. Finite difference methods were used to discretize partial differential equations for problems involving Poisson, advection-diffusion, and heat transfer equations. Comparisons were made between multigrid and single grid methods like Gauss-Seidel and TDMA. Results confirmed some literature findings and presented some new results on the effect of parameters on CPU time.
The klein gordon field in two-dimensional rindler space-timeforssfoxtrot jp R
This document discusses the Klein-Gordon scalar field in a two-dimensional Rindler space-time background. It derives the equation of motion for the scalar field from an action in this background. The equation can be solved exactly using imaginary time, giving an oscillatory solution with a frequency corresponding to an integral number. This suggests the scalar field appears quantized with spatial modes corresponding to integral values.
This document summarizes a study that analyzes magnetohydrodynamic (MHD) flow of Newtonian and non-Newtonian nanofluids passing over a magnetic sphere. Nanofluids containing alumina or copper nanoparticles in water or oil bases were examined. Governing equations for continuity, momentum, and energy were derived and non-dimensionalized. The equations were transformed into similarity equations using a stream function and solved numerically. Results showed that increasing the magnetic parameter decreases velocity and temperature. Newtonian nanofluid velocity and temperature were higher than non-Newtonian. Copper-water nanofluid also had higher values than alumina-water.
The klein gordon field in two-dimensional rindler space-time - smcprtfoxtrot jp R
This document discusses the Klein-Gordon scalar field in two-dimensional Rindler space-time. It derives a two-dimensional action for the Klein-Gordon scalar with the Rindler space-time background. The equation of motion obtained from this action is solved exactly, with the solution being oscillatory in imaginary time. The angular frequency of the oscillatory solution corresponds to an integral number, suggesting the scalar field appears quantized in terms of its frequency or has spatial modes corresponding to integral values.
The klein gordon field in two-dimensional rindler space-time 200920ver-displayfoxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define a two-dimensional action for the scalar field and derive from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
The klein gordon field in two-dimensional rindler space-time 14072020foxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define an informal two-dimensional action for the scalar field and obtain from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
The klein gordon field in two-dimensional rindler space-time 28072020ver-drft...foxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define an informal two-dimensional action for the scalar field and obtain from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
THE RESEARCH OF QUANTUM PHASE ESTIMATION ALGORITHMIJCSEA Journal
This document discusses phase estimation in quantum computing. It begins by introducing quantum Fourier transforms and how they are important for algorithms like Shor's algorithm. It then describes the phase estimation algorithm in detail, including how it uses two registers to estimate the phase of a quantum state and how the inverse quantum Fourier transform improves this estimate. Simulation results are presented that show the probability distribution of the estimated phase converging to the true value and how the probability of success increases with more qubits while computational costs rise polynomially. The paper concludes that the optimal number of qubits balances high success probability and low costs for phase estimation.
The klein gordon field in two-dimensional rindler space-time 23052020-sqrdfoxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in the Rindler space-time.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time 04232020updtsfoxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in this background.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time 16052020foxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in this background.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time -sqrdupdt41220foxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in the Rindler space-time.
3) The equation has an exact series solution that is oscillatory in imaginary time, with the oscillation frequency corresponding to an integral number related to the surface gravity of the space-time.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document is an internship report submitted by Yiteng Dang to the École Normale Supérieure on applying mean-field theory to study charge density waves in rare-earth nickelates. Chapter 1 provides theoretical background, discussing concepts like density of states calculations, the nearly free electron model, mean-field theory applied to ferromagnetism and antiferromagnetism, and Green's functions. Chapter 2 focuses on nickelates, introducing a low-energy two-orbital Hamiltonian and applying mean-field theory to obtain results like a phase diagram at half-filling and quarter-filling. Numerical methods are used throughout to solve problems in condensed matter theory.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This document summarizes an academic research paper that analyzes an optimal N-policy for a Bernoulli feedback Mx/G/1 machining system with general setup times. The paper develops a mathematical model of the system using supplementary variable technique to obtain the probability generating function of the system queue size distribution and mean number of failed units. It also derives the Laplace-Stieltjes transform of the waiting time and evaluates the mean waiting time. Finally, it formulates the total operational cost function to determine the optimal value of N that minimizes costs.
The document discusses the Vainshtein mechanism, which is a screening mechanism that allows a scalar field coupled to matter to have a negligible effect as a fundamental force on matter within a specific scale. This is important for explaining the cosmological constant problem, which is that the observed acceleration of the universe requires a cosmological constant that is much smaller than predicted by quantum field theory. The Vainshtein mechanism introduces a scalar field while maintaining the accuracy of general relativity at solar system scales by making the new force negligible at those scales. The document explores how such a screening mechanism for a general scalar field could maintain Newtonian gravity results within the solar system and potentially explain the observed acceleration of the universe.
Kinetic pathways to the isotropic-nematic phase transformation: a mean field ...Amit Bhattacharjee
Here we illustrate the classic Ginzburg-Landau-de Gennes theory of isotropic nematic phase transition and show how fluctuations as well as deterministic kinetics can lead to phase equilibria.
No Cloning Theorem with essential Mathematics and PhysicsRitajit Majumdar
This is the first project report at my University. This report describes No Cloning Theorem, an introductory topic of Quantum Computation and Quantum Information Theory. The report also covers the necessary mathematics and physics.
Unsteady Free Convection MHD Flow of an Incompressible Electrically Conductin...IJERA Editor
In this paper we investigate unsteady free convection MHD flow of an incompressible viscous electrically
conducting fluid through porous medium under the influence of uniform transverse magnetic field between two
heated vertical plate with one plate is adiabatic. The governing equations of velocity and temperature fields with
appropriate boundary conditions are solved by the Integral Transform Technique. The obtained results of
velocity and temperature distributions are shown graphically and are discussed on the basis of it. The effects of
Hartmann number, Darcy parameter, Prandtl number and the decay factor, and effects of adiabatic plate on the
velocity and temperature fields are discussed.
New Scenario for Transition to Slow 3-D Turbulence Part I.Slow 1-D Turbulence...ijrap
Analyticalnon-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, ispresented. Theequation has a threshold of short-waveinstability and symmetry, providing
longwavedynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of the Nikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
New Scenario for Transition to Slow 3-D Turbulence Part I.Slow 1-D Turbulence...ijrap
Analytical non-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, is presented. Theequation has a threshold of short-wave instability and symmetry, providing
long wave dynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of theNikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
Simple Exponential Observer Design for the Generalized Liu Chaotic Systemijtsrd
In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the time-domain approach with differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be precisely calculated. Finally, numerical simulations are presented to exhibit the effectiveness and feasibility of the obtained results. Yeong-Jeu Sun"Simple Exponential Observer Design for the Generalized Liu Chaotic System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd7126.pdf http://www.ijtsrd.com/engineering/engineering-maths/7126/simple-exponential-observer-design-for-the-generalized-liu-chaotic-system/yeong-jeu-sun
This document summarizes Emmy Noether's method for obtaining the infinitesimal point symmetries of Lagrangians using the Noether current. It presents Noether's theorem in the Lanczos approach to construct the first integral associated with each symmetry. Several examples of Lagrangians are analyzed using this method, including those studied by Rothe, Henneaux, and Torres del Castillo. For each Lagrangian, the Noether current is derived and the resulting Killing equations are solved to obtain the point symmetries and associated first integrals.
This document summarizes a study that analyzes magnetohydrodynamic (MHD) flow of Newtonian and non-Newtonian nanofluids passing over a magnetic sphere. Nanofluids containing alumina or copper nanoparticles in water or oil bases were examined. Governing equations for continuity, momentum, and energy were derived and non-dimensionalized. The equations were transformed into similarity equations using a stream function and solved numerically. Results showed that increasing the magnetic parameter decreases velocity and temperature. Newtonian nanofluid velocity and temperature were higher than non-Newtonian. Copper-water nanofluid also had higher values than alumina-water.
The klein gordon field in two-dimensional rindler space-time - smcprtfoxtrot jp R
This document discusses the Klein-Gordon scalar field in two-dimensional Rindler space-time. It derives a two-dimensional action for the Klein-Gordon scalar with the Rindler space-time background. The equation of motion obtained from this action is solved exactly, with the solution being oscillatory in imaginary time. The angular frequency of the oscillatory solution corresponds to an integral number, suggesting the scalar field appears quantized in terms of its frequency or has spatial modes corresponding to integral values.
The klein gordon field in two-dimensional rindler space-time 200920ver-displayfoxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define a two-dimensional action for the scalar field and derive from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
The klein gordon field in two-dimensional rindler space-time 14072020foxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define an informal two-dimensional action for the scalar field and obtain from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
The klein gordon field in two-dimensional rindler space-time 28072020ver-drft...foxtrot jp R
- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define an informal two-dimensional action for the scalar field and obtain from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
THE RESEARCH OF QUANTUM PHASE ESTIMATION ALGORITHMIJCSEA Journal
This document discusses phase estimation in quantum computing. It begins by introducing quantum Fourier transforms and how they are important for algorithms like Shor's algorithm. It then describes the phase estimation algorithm in detail, including how it uses two registers to estimate the phase of a quantum state and how the inverse quantum Fourier transform improves this estimate. Simulation results are presented that show the probability distribution of the estimated phase converging to the true value and how the probability of success increases with more qubits while computational costs rise polynomially. The paper concludes that the optimal number of qubits balances high success probability and low costs for phase estimation.
The klein gordon field in two-dimensional rindler space-time 23052020-sqrdfoxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in the Rindler space-time.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time 04232020updtsfoxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in this background.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time 16052020foxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in this background.
3) The equation has an exact series solution that terminates at an integral number of terms, suggesting the scalar field frequencies correspond to integer values in imaginary time.
The klein gordon field in two-dimensional rindler space-time -sqrdupdt41220foxtrot jp R
1) The document considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
2) It defines a two-dimensional action for the scalar field and obtains from it the Klein-Gordon equation of motion in the Rindler space-time.
3) The equation has an exact series solution that is oscillatory in imaginary time, with the oscillation frequency corresponding to an integral number related to the surface gravity of the space-time.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document is an internship report submitted by Yiteng Dang to the École Normale Supérieure on applying mean-field theory to study charge density waves in rare-earth nickelates. Chapter 1 provides theoretical background, discussing concepts like density of states calculations, the nearly free electron model, mean-field theory applied to ferromagnetism and antiferromagnetism, and Green's functions. Chapter 2 focuses on nickelates, introducing a low-energy two-orbital Hamiltonian and applying mean-field theory to obtain results like a phase diagram at half-filling and quarter-filling. Numerical methods are used throughout to solve problems in condensed matter theory.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This document summarizes an academic research paper that analyzes an optimal N-policy for a Bernoulli feedback Mx/G/1 machining system with general setup times. The paper develops a mathematical model of the system using supplementary variable technique to obtain the probability generating function of the system queue size distribution and mean number of failed units. It also derives the Laplace-Stieltjes transform of the waiting time and evaluates the mean waiting time. Finally, it formulates the total operational cost function to determine the optimal value of N that minimizes costs.
The document discusses the Vainshtein mechanism, which is a screening mechanism that allows a scalar field coupled to matter to have a negligible effect as a fundamental force on matter within a specific scale. This is important for explaining the cosmological constant problem, which is that the observed acceleration of the universe requires a cosmological constant that is much smaller than predicted by quantum field theory. The Vainshtein mechanism introduces a scalar field while maintaining the accuracy of general relativity at solar system scales by making the new force negligible at those scales. The document explores how such a screening mechanism for a general scalar field could maintain Newtonian gravity results within the solar system and potentially explain the observed acceleration of the universe.
Kinetic pathways to the isotropic-nematic phase transformation: a mean field ...Amit Bhattacharjee
Here we illustrate the classic Ginzburg-Landau-de Gennes theory of isotropic nematic phase transition and show how fluctuations as well as deterministic kinetics can lead to phase equilibria.
No Cloning Theorem with essential Mathematics and PhysicsRitajit Majumdar
This is the first project report at my University. This report describes No Cloning Theorem, an introductory topic of Quantum Computation and Quantum Information Theory. The report also covers the necessary mathematics and physics.
Unsteady Free Convection MHD Flow of an Incompressible Electrically Conductin...IJERA Editor
In this paper we investigate unsteady free convection MHD flow of an incompressible viscous electrically
conducting fluid through porous medium under the influence of uniform transverse magnetic field between two
heated vertical plate with one plate is adiabatic. The governing equations of velocity and temperature fields with
appropriate boundary conditions are solved by the Integral Transform Technique. The obtained results of
velocity and temperature distributions are shown graphically and are discussed on the basis of it. The effects of
Hartmann number, Darcy parameter, Prandtl number and the decay factor, and effects of adiabatic plate on the
velocity and temperature fields are discussed.
New Scenario for Transition to Slow 3-D Turbulence Part I.Slow 1-D Turbulence...ijrap
Analyticalnon-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, ispresented. Theequation has a threshold of short-waveinstability and symmetry, providing
longwavedynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of the Nikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
New Scenario for Transition to Slow 3-D Turbulence Part I.Slow 1-D Turbulence...ijrap
Analytical non-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, is presented. Theequation has a threshold of short-wave instability and symmetry, providing
long wave dynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of theNikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
Simple Exponential Observer Design for the Generalized Liu Chaotic Systemijtsrd
In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the time-domain approach with differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be precisely calculated. Finally, numerical simulations are presented to exhibit the effectiveness and feasibility of the obtained results. Yeong-Jeu Sun"Simple Exponential Observer Design for the Generalized Liu Chaotic System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd7126.pdf http://www.ijtsrd.com/engineering/engineering-maths/7126/simple-exponential-observer-design-for-the-generalized-liu-chaotic-system/yeong-jeu-sun
This document summarizes Emmy Noether's method for obtaining the infinitesimal point symmetries of Lagrangians using the Noether current. It presents Noether's theorem in the Lanczos approach to construct the first integral associated with each symmetry. Several examples of Lagrangians are analyzed using this method, including those studied by Rothe, Henneaux, and Torres del Castillo. For each Lagrangian, the Noether current is derived and the resulting Killing equations are solved to obtain the point symmetries and associated first integrals.
Higher-Order Squeezing of a Generic Quadratically-Coupled Optomechanical SystemIOSRJAP
Using short-time dynamics and analytical solution of Heisenberg equation of motion for the Hamiltonian of quadratically-coupled optomechanical system for different field modes, we have investigated the existence of higher-order single mode squeezing, sum squeezing and difference squeezing in absence of driving and dissipation. Depth of squeezing increases with order number for higher-order single mode squeezing. Squeezing factor exhibits a series of revival-collapse phenomena for single mode, which becomes more pronounced as order number increases. In case of sum squeezing amounts of squeezing is greater than single mode higher-order squeezing (n = 2). It is also greater than from difference squeezing for same set of interaction parameters. Sum squeezing is prominently better for extracting information regarding squeezing.
Chaotic system and its Application in CryptographyMuhammad Hamid
A seminar on Chaotic System and Its application in cryptography specially in image encryption. Slide covers
Introduction
Bifurcation Diagram
Lyapnove Exponent
Spatio-Temporal Characterization with Wavelet Coherence: Anexus between Envir...ijsc
Identifying spatio-temporal synchrony in a complex, interacting and oscillatory coupled-system is a challenge. In particular, the characterization of statistical relationships between environmental or biophysical variables with the multivariate data of pandemic is a difficult process because of the intrinsic variability and non-stationary nature of the time-series in space and time. This paper presents a methodology to address these issues by examining the bivariate relationship between Covid-19 and temperature time-series in the time-localized frequency domain by using Singular Value Decomposition (SVD) and continuous cross-wavelet analysis. First, the dominant spatio-temporal trends are derived by using the eigen decomposition of SVD. The Covid-19 incidence data and the temperature data of the corresponding period are transformed into significant eigen-state vectors for each spatial unit. The Morlet Wavelet transformation is performed to analyse and compare the frequency structure of the dominant trends derived by the SVD. The result provides cross-wavelet transform and wavelet coherence measures in the ranges of time period for the corresponding spatial units. Additionally, wavelet power spectrum and paired wavelet coherence statistics and phase difference are estimated. The result suggests statistically significant coherency at various frequencies providing insight into spatio-temporal dynamics. Moreover, it provides information about the complex conjugate dynamic relationships in terms phases and phase
differences.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
This document discusses strategies for parallelizing spectral methods. Spectral methods are global in nature due to their use of global basis functions, making them challenging to parallelize on fine-grained architectures. However, the document finds that spectral methods can be effectively parallelized. The main computational steps in spectral methods are the calculation of differential operators on functions and solving linear systems, both of which can exploit parallelism. Domain decomposition techniques may also help parallelize computations over non-Cartesian domains.
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
The document presents an exponential state observer design for uncertain chaotic and non-chaotic systems. It introduces a class of uncertain nonlinear systems and explores the state observation problem for such systems. Using an approach with integral and differential equalities, an exponential state observer is established that guarantees global exponential stability of the error system. Numerical simulations exhibit the feasibility and effectiveness of the observer design.
Foundation and Synchronization of the Dynamic Output Dual Systemsijtsrd
In this paper, the synchronization problem of the dynamic output dual systems is firstly introduced and investigated. Based on the time domain approach, the state variables synchronization of such dual systems can be verified. Meanwhile, the guaranteed exponential convergence rate can be accurately estimated. Finally, some numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result. Yeong-Jeu Sun "Foundation and Synchronization of the Dynamic Output Dual Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29256.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29256/foundation-and-synchronization-of-the-dynamic-output-dual-systems/yeong-jeu-sun
Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Syste...CSEIJJournal
This paper investigates the global chaos synchronization of identical hyperchaotic Wang systems, identical
hyperchaotic Pang systems, and non-identical hyperchaotic Wang and hyperchaotic Pang systems via
adaptive control method. Hyperchaotic Pang system (Pang and Liu, 2011) and hyperchaotic Wang system
(Wang and Liu, 2006) are recently discovered hyperchaotic systems. Adaptive control method is deployed
in this paper for the general case when the system parameters are unknown. Sufficient conditions for global
chaos synchronization of identical hyperchaotic Pang systems, identical hyperchaotic Wang systems and
non-identical hyperchaotic Pang and Wang systems are derived via adaptive control theory and Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control
method is very convenient for the global chaos synchronization of the hyperchaotic systems discussed in
this paper. Numerical simulations are presented to validate and demonstrate the effectiveness of the
proposed synchronization schemes.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Two Types of Novel Discrete Time Chaotic Systemsijtsrd
In this paper, two types of one dimensional discrete time systems are firstly proposed and the chaos behaviors are numerically discussed. Based on the time domain approach, an invariant set and equilibrium points of such discrete time systems are presented. Besides, the stability of equilibrium points will be analyzed in detail. Finally, Lyapunov exponent plots as well as state response and Fourier amplitudes of the proposed discrete time systems are given to verify and demonstrate the chaos behaviors. Yeong-Jeu Sun ""Two Types of Novel Discrete-Time Chaotic Systems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29853.pdf
Paper Url : https://www.ijtsrd.com/engineering/electrical-engineering/29853/two-types-of-novel-discrete-time-chaotic-systems/yeong-jeu-sun
This document describes numerical simulations of subsonic and supersonic flow through a choked nozzle using various schemes to solve the quasi-1D Euler equations. The author compares the steady state solutions from the Jameson-Schmidt-Turkel (JST) scheme (3, 4, and 5 stages), first-order Steger flux-vector splitting, and an analytic solution. The JST scheme converges quickly but is overly diffusive, while Steger converges slower but is less diffuse and more accurate near shocks. Higher-order JST stages and adjusting diffusion coefficients improve accuracy versus the analytic solution.
A new two-scroll chaotic system with two nonlinearities: dynamical analysis a...TELKOMNIKA JOURNAL
Chaos theory has several applications in science and engineering. In this work, we announce a
new two-scroll chaotic system with two nonlinearities. The dynamical properties of the system such as
dissipativity, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension and bifurcation diagram are
explored in detail. The presence of coexisting chaotic attractors, coexisting chaotic and periodic attractors
in the system is also investigated. In addition, the offset boosting of a variable in the new chaotic system is
achieved by adding a single controlled constant. It is shown that the new chaotic system has rotation
symmetry about the z-axis. An electronic circuit simulation of the new two-scroll chaotic system is built
using Multisim to check the feasibility of the theoretical model.
Development, Optimization, and Analysis of Cellular Automaton Algorithms to S...IRJET Journal
This document summarizes research on using cellular automaton algorithms to solve stochastic partial differential equations (SPDEs). It proposes a finite-difference method to approximate an SPDE modeling a random walk with angular diffusion. A Monte Carlo algorithm is also developed for comparison. Analysis finds a moderate correlation between the two methods, suggesting the finite-difference approach is reasonably accurate. It also identifies an inverse-square relationship between variables, linking to a foundational stochastic analysis concept. The research concludes the finite-difference method shows promise for approximating SPDEs while considering boundary conditions.
SEQUENTIAL CLUSTERING-BASED EVENT DETECTION FOR NONINTRUSIVE LOAD MONITORINGcscpconf
The problem of change-point detection has been well studied and adopted in many signal processing applications. In such applications, the informative segments of the signal are the stationary ones before and after the change-point. However, for some novel signal processing and machine learning applications such as Non-Intrusive Load Monitoring (NILM), the information contained in the non-stationary transient intervals is of equal or even more importance to the recognition process. In this paper, we introduce a novel clustering-based sequential detection of abrupt changes in an aggregate electricity consumption profile with the accurate decomposition of the input signal into stationary and non-stationary segments. We also introduce various event models in the context of clustering analysis. The proposed algorithm is applied to building-level energy profiles with promising results for the residential BLUED power dataset.
SEQUENTIAL CLUSTERING-BASED EVENT DETECTION FOR NONINTRUSIVE LOAD MONITORINGcsandit
The problem of change-point detection has been well studied and adopted in many signal processing applications. In such applications, the informative segments of the signal are the
stationary ones before and after the change-point. However, for some novel signal processing and machine learning applications such as Non-Intrusive Load Monitoring (NILM), the information contained in the non-stationary transient intervals is of equal or even more importance to the recognition process. In this paper, we introduce a novel clustering-based sequential detection of abrupt changes in an aggregate electricity consumption profile with
accurate decomposition of the input signal into stationary and non-stationary segments. We also introduce various event models in the context of clustering analysis. The proposed algorithm is applied to building-level energy profiles with promising results for the residential BLUED power dataset.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Similar to NEW SCENARIO FOR TRANSITION TO SLOW 3-D TURBULENCE PART I.SLOW 1-D TURBULENCE IN NIKOLAEVSKII SYSTEM. (20)
New Thermodynamics: A Superior Fit Revised Kinetic Theoryijrap
The accepted kinetic theory forms a basis for modern thermodynamics and is mathematically based upon equipartition and degrees of freedom. It remains plagued with the necessity of numerous degrees of freedom exceptions for it to explain both empirically determined heat capacities and adiabatic indexes. Furthermore, assuming kT/2 per degree of freedom is to accept that a gas molecule possesses a specified energy without providing any clarity concerning that energy’s origins. Energy without an origin contravenes the first law of thermodynamics. This author’s previously published superior fit kinetic theory will be clarified and elaborated upon. This includes showing that this revised kinetic theory is a superior fit to both known heat capacities and adiabatic indexes. Not only is it a superior fit that does not rely upon any exceptions, this author’s kinetic theory also provides insight into the actual sources of a gas molecule’s energy. Furthermore, clarity concerning the difference between isometric (isochoric) and isobaric heat capacities in terms of sensible work will be discussed, along withits likely empirical verification.
On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
This paper supports Descartes' idea of a constant quantity of motion, modernized by Leibniz. Unlike Leibniz, the paper emphasizes that the idea is not realized by forms of energy, but by energy itself. It remains constant regardless of the form, type, or speed of motion, even that of light. Through force, energy is only transformed. Here it is proved that force is its derivative. It exists even at rest, representing the object's minimal energy state. With speed, we achieve its multiplication up to the maximum energy state, from which a maximum force is derived from the object. From this point, corresponding to Planck's Length, we find the value of the force wherever we want. Achieving this removes the differences between various natural forces. The new idea eliminates infinite magnitudes. The process allows the laws to transition from simple to complex forms and vice versa, through differentiation-integration. For this paper, this means achieving the Unification Theory.
Gravity Also Redshifts Light – the Missing Phenomenon That Could Resolve Most...ijrap
In this paper I discover that gravity also redshifts light like the velocity of its source does. When light travels towards a supermassive object, its waves (or photons) undergo continuous stretching, thereby shifting towards lower frequencies. Gravity redshifts light irrespective of whether its source is in motion or static with respect to its observer. An equation is derived for gravitational redshift, and a formula for combined redshift is presented by considering both the velocity, and gravity redshifts. Also explained is how frequencies of electromagnetic spectrum continuously downgrade as a light beam of mix frequencies passes towards a black hole. Further, a clear methodology is provided to figure out whether expansion of the universe is accelerating or decelerating, or alternatively, the universe is contracting.
In this paper I present a new theory that explains as to when and how dark energy is created as mass is destroyed. The theory extends Einstein’s mass energy equation to a more generic form in order to make it work even in high gravity conditions. It also explains why dark energy is created. Further, it is proved Einstein’s mass energy equation holds good only when the destroyed mass has no supermassive object in its close vicinity. The relationship between dark energy and dark matter is unveiled. An extended mathematical form of Einstein’s mass energy equation is derived, based on which the conditions leading to dark energy creation are explained. Three new physical parameters called dark energy discriminant, dark energy radius and dark energy boundary are introduced to facilitate easy understanding of the theory. It is explained in detail that an extremely superdense object has two dark energy boundaries, outer and inner. Mass destroyed only between these two boundaries creates dark energy. Dark energy space, the space between the two aforementioned boundaries, shrouds visible matter in obscurity from optical and electromagnetic telescopes. This theory identifies Gargantuan as a superdense black hole currently creating fresh dark energy, which could be the subject of interest for the astronomical research community having access to sophisticated telescopes, and working on dark energy. It also upholds dark energy and denies the existence of dark matter. Dark matter is nothing but the well-known visible matter positioned in dark energy space. An important relationship is derived between a photon’s frequency and its distance from a black hole to demonstrate the effect of gravity on light. Another important fact revealed by this theory is gravity stretches out light, thereby causing redshift, which is unaccounted in the computation of velocities of outer galaxies. Whether the universe is undergoing accelerated or decelerated expansion, or accelerated contraction can precisely be determined only after accounting for the redshift caused by gravity
International Journal on Soft Computing, Artificial Intelligence and Applicat...ijrap
International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI)
is an open access peer-reviewed journal that provides an excellent international forum for sharing
knowledge and results in theory, methodology and applications of Artificial Intelligence, Soft
Computing. The Journal looks for significant contributions to all major fields of the Artificial
Intelligence, Soft Computing in theoretical and practical aspects. The aim of the Journal is to
provide a platform to the researchers and practitioners from both academia as well as industry to
meet and share cutting-edge development in the field.
Authors are solicited to contribute to the journal by submitting articles that illustrate research
results, projects, surveying works and industrial experiences that describe significant advances in
the areas of Database management systems.
SOME THEORETICAL ASPECTS OF HYDROGEN DIFFUSION IN BCC METALS AT LOW TEMPERATURESijrap
Purpose of the work is to discuss some theoretical aspects of the diffusion of hydrogen atoms in the crystal
lattice of BCC metals at low temperatures using the methods of statistical thermodynamics. The values of
the statistical model calculations of H diffusion coefficients in α-Fe, V, Ta, Nb, K are in good agreement
with the experimental data. The statistical model can also explain deviations from the Arrhenius equation
at temperatures 300-100 K in α-Fe, V, Nb and K. It was suggested that thermally activated fast tunnelling
transition of hydrogen atoms through the potential barrier at a temperature below 300 K provides an
almost free movement of H atoms in the α-Fe and V lattice at these temperatures. The results show that
quantum-statistical effects play a decisive role in the H diffusion in BCC metals at low temperatures. Using
the statistical model allows for the prediction of the diffusion coefficient for H in BCC metals at low
temperatures, where it’s necessary to consider quantum effects.
MASSIVE PHOTON HYPOTHESIS OPENS DOORS TO NEW FIELDS OF RESEARCHijrap
1) A massive photon hypothesis is proposed, where the photon mass is directly calculated from kinetic gas theory to be 1.25605 x 10-39 kg.
2) This photon mass explains various experiments like light deflection near the Sun and the gravitational redshift.
3) The photon gas is found to behave as a perfect blackbody and ideal gas, with photons having 6 degrees of freedom.
4) The thermal de Broglie wavelength of this photon gas is calculated to be 1.75967 x 10-3 m, matching the wavelength of the cosmic microwave background radiation.
5) This links the CMB radiation to being continuously generated by the photon gas permeating space, rather than being a relic of
PHENOMENOLOGICAL METHOD REGARDING A THIRD THEORY OF PHYSICS “THE EVENT:THE TH...ijrap
The quest for a third theory uniting macro-cosmos (relativity) and micro-cosmos (quantum mechanics) has coexisted with the denial of feminine/subjective polarity to masculine/objective. The dismissal of electromagnetism as the tension of opposites in quest of inner/outer unity is sourced in the denial of the feminine qualia -- the negative force field attributed to dark energy/dark matter. However, a conversion philosophy sourced in the hieros gamos and signified by the Mobius strip has formulated an integral consciousness methodology producing quantum objects by means of embracing the shadow haunting contemporary physics. This Self-reflecting process integrating subject/object comprises an ontology of kairos as the “quantum leap.” An interdisciplinary quest to create a phenomenological narrative is disclosed via a holistic apparatus of hermeneutics manifesting image/text of a contemporary grail journey. Reflected in this Third space is the sacred reality of autonomous number unifying polarities of feminine/subjective (quality) and objective/masculine (quantity) as new measurement apparatus for the quantum wave collapse.
3rd International Conference on Integrating Technology in Education (ITE 2022)ijrap
3rd International Conference on Integrating Technology in Education (ITE 2022) This forum also aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals.
A SPECIAL RELATIONSHIP BETWEEN MATTER, ENERGY, INFORMATION, AND CONSCIOUSNESSijrap
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and consciousness.
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and
consciousness.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for theall-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, whileothers posit that time is only a social or mental construct. The author presents an African thought systemon space and time conception, focusing on the African (Bantu) view of space and time. The author arguesthat before the advent of the Western linear view of space and time, Africans had their own visionregarding these two concepts. Their conception of time appears to be holistic, highly philosophical, non-linear, and thought-provoking. The author hopes that exploring these two concepts from an African perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest forthe ToE
Learning to Pronounce as Measuring Cross Lingual Joint Orthography Phonology ...ijrap
Machine learning models allow us to compare languages by showing how hard a task in each language might be to learn and perform well on. Following this line of investigation, we explore what makes a language “hard to pronounce” by modelling the task of grapheme-to-phoneme (g2p) transliteration. By training a character-level transformer model on this task across 22 languages and measuring the model’s proficiency against its grapheme and phoneme inventories, we show that certain characteristics emerge that separate easier and harder languages with respect to learning to pronounce. Namely the complexity of a language's pronunciation from its orthography is due to the expressive or simplicity of its grapheme-to phoneme mapping. Further discussion illustrates how future studies should consider relative data sparsity per language to design fairer cross-lingual comparison tasks.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for the all-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, while others posit that time is only a social or mental construct. The author presents an African thought system on space and time conception, focusing on the African (Bantu) view of space and time. The author argues
that before the advent of the Western linear view of space and time, Africans had their own vision
regarding these two concepts. Their conception of time appears to be holistic, highly philosophical, nonlinear, and thought-provoking. The author hopes that exploring these two concepts from an African
perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
The Concept of Space and Time: An African Perspectiveijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for the all-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, while others posit that time is only a social or mental construct. The author presents an African thought system on space and time conception, focusing on the African (Bantu) view of space and time. The author argues that before the advent of the Western linear view of space and time, Africans had their own vision regarding these two concepts. Their conception of time appears to be holistic, highly philosophical, nonlinear, and thought-provoking. The author hopes that exploring these two concepts from an African perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
The majority of physicists take it for granted that the universe is made up of matter. In turn, matter is composed of atoms; atoms are made up of particles such as electrons, protons, neutrons, etc. Also, protons
and neutrons are composed of quarks, etc. Furthermore, that everything in nature is governed by the known laws of physics and chemistry. The author only partially shares this view. He argues that many phenomena in the universe may depend on rules or factors as yet incorporated by the physical sciences.
The last few years have led him to reflect on the many unsolved physics problems, such as the quest for the theory of everything (ToE), the arrow of time, the interpretation of quantum mechanics, the fine-tuned
universe, etc. to mention just a few. The author posits that a field carries information, performs various mathematical and computational operations, and behaves as an intelligent entity embedded with consciousness.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
The International Journal of Recent Advances in Physics (IJRAP) is a peer-reviewed open access journal that addresses impacts and challenges in the field of physics. It covers theoretical and practical results across many areas of physics including advanced functional materials, applied optics, condensed matter physics, nuclear physics, quantum physics, and more. Authors are invited to submit papers by email before October 30, 2021. Notifications of acceptance will be provided by November 25, 2021.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
How to Get CNIC Information System with Paksim Ga.pptxdanishmna97
Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
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20240605 QFM017 Machine Intelligence Reading List May 2024
NEW SCENARIO FOR TRANSITION TO SLOW 3-D TURBULENCE PART I.SLOW 1-D TURBULENCE IN NIKOLAEVSKII SYSTEM.
1. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
DOI : 10.14810/ijrap.2015.4102 21
NEW SCENARIO FOR TRANSITION TO SLOW
3-D TURBULENCE PART I.SLOW 1-D
TURBULENCE IN NIKOLAEVSKII SYSTEM.
J. Foukzon
Israel Institute of Technology,Haifa, Israel
Abstract:
Analyticalnon-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, ispresented. Theequation has a threshold of short-waveinstability and symmetry, providing
longwavedynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of theNikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
Keywords:
3D turbulence, chaos, quantum chaos,additive thermal noise,Nikolaevskii system.
1.Introduction
In the present work a non-perturbative analyticalapproach to the studying of problemof quantum chaos in
dynamical systems withinfinite number of degrees of freedom isproposed.Statistical descriptions of
dynamical chaos and investigations of noise effects on chaoticregimes arestudied.Proposed approach also
allows estimate the influence of additive (thermal)fluctuations on the processes of formation ofdeveloped
turbulence modes in essentially nonlinearprocesses like electro-convection andother. A principal rolethe
influence ofthermalfluctuations on thedynamics of some types of dissipative systems inthe approximate
environs of derivation rapid of ashort-wave instability was ascertained. Impotentphysicalresults follows
from Theorem 2, is illustrated by example of 3D stochastic model system:
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݐ
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝሻ
+ ቈߜଵ
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݔଵ
+ ߜଶ
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݔଶ
+ ߜଷ
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݔଷ
ݑఎሺ,ݔ ,ݐ ߝሻ +
+݂ሺ,ݔ ݐሻ − ඥߟݓሺ,ݔ ݐሻ = 0, ݔ ∈ ℝଷ
(1.1)
ݑఎሺ,ݔ 0, ߝሻ = 0, ݓሺ,ݔ ݐሻ =
డరௐሺ௫,௧ሻ
డ௫భడ௫మడ௫యడ௧
, ߟ ≪ 1,0 < ߜ, ݆ = 1,2,3, (1.2)
2. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
22
which was obtained from thenon-stochastic3ܦNikolaevskiimodel:
డ௨ሺ௫,௧,ఌሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑሺ,ݔ ,ݐ ߝሻ + ቂߜଵ
డ௨ሺ௫,௧,ఌሻ
డ௫భ
+ ߜଶ
డ௨ሺ௫,௧,ఌሻ
డ௫మ
+ ߜଷ
డ௨ሺ௫,௧,ఌሻ
డ௫య
ቃ ݑሺ,ݔ ,ݐ ߝሻ + ݂ሺ,ݔ ݐሻ (1.3)
which is perturbed by additive “small” white noise ඥߟݓሺ,ݔ ݐሻ. And analytical result also illustrated by
exampleof1ܦstochasticmodel system
డ௨ആሺ௫,௧,ఌሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝሻ + ߜ
డ௨ആሺ௫,௧,ఌሻ
డ௫భ
ݑఎሺ,ݔ ,ݐ ߝሻ + ݂ሺ,ݔ ݐሻ − ඥߟݓሺ,ݔ ݐሻ = 0, ݔ ∈ ℝ (1.4)
ݑఎሺ,ݔ 0, ߝሻ = 0, ݓሺ,ݔ ݐሻ =
డమௐሺ௫,௧ሻ
డ௫డ௧
, ߟ ≪ 1,0 < ߜ, (1.5)
which was obtained from thenon-stochastic 1DNikolaevskiismodel:
డ௨ሺ௫,௧,ఌሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑሺ,ݔ ,ݐ ߝሻ + ߜ
డ௨ሺ௫,௧,ఌሻ
డ௫భ
ݑሺ,ݔ ,ݐ ߝሻ + ݂ሺ,ݔ ݐሻ = 0, ݑሺ,ݔ 0, ߝሻ = 0, ݔ ∈ ℝ, (1.6)
ݑሺ,ݔ 0, ߝሻ = 0.(1.7)
Systematic study of a different type of chaos at onset ‘‘soft-mode turbulence’’based onnumerical
integration of the simplest 1DNikolaevskii model (1.7)has been executed by many authors [2]-[7].There is
an erroneous belief that such numerical integration gives a powerful analysisismeans of the processes of
turbulence conception, based on the classical theory ofchaos of the finite-dimensional classical systems
[8]-[11].
Remark1.1.However, as it well known, such approximations correct only in a phase ofturbulence
conception, when relatively small number of the degrees of freedom excites. In general case, when a very
large number of the degrees of freedom excites, well known phenomena of thenumerically induced chaos,
can to spoils in the uncontrollable wayany numerical integration[12]-[15]
Remark1.2.Other non trivial problem stays from noise roundoff error in computer computation using
floatingpoint arithmetic [16]-[20].In any computer simulation the numerical solution is fraught with
truncation by roundoff errors introduced by finite-precision calculation of trajectories of dynamical systems,
where roundoff errors or other noise can introduce new behavior and this problem is a very more
pronounced in the case of chaotic dynamical systems, because the trajectories of such systems
exhibitextensivedependence on initial conditions. As a result, a small random truncation or roundoff error,
made computational error at any step of computation will tend to be large magnified by future
computationalof the system[17].
Remark1.3.As it well known, if the digitized or rounded quantity is allowed to occupy the nearest of a
large number of levels whose smallest separation is ܧ, then, provided that the
original quantity is large compared to ܧ and is reasonably well behaved, theeffect of the quantization or
rounding may betreated as additive random noise [18].Bennett has shown that such additive noise is nearly
white, withmean squared value of ܧ
ଶ
/12[19].However the complete uniform white-noise model to be valid
in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices
of realization of the system in the state space satisfy certain nonresonance conditions and the
finite-dimensional distributions of the input signal are absolutely continuous[19].
The method deprived of these essential lacks in general case has been offered by the author in papers
3. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
23
[23]-[27].
Remark1.4.Thus from consideration above it is clear thatnumerical integration procedure ofthe1D
Nikolaevskii model (1.6)-(1.7) executed in papers [2]-[7]in fact dealing withstochastic model
(1.4)-(1.5).There is an erroneous the point of view,that a white noise with enough small intensity does not
bringanysignificant contributions in turbulent modes, see for example [3]. By this wrong assumptions the
results of the numerical integration procedure ofthe1D Nikolaevskii model (1.6)-(1.7) were mistakenly
considered and interpretedas a veryexact modeling the slow turbulence within purely non stochastic
Nikolaevskii model (1.6)-(1.7).Accordingly wrongconclusionsabout that temperature noisesdoes not
influence slowturbulence have been proposed in [3].However in [27] has shown non-perturbativelythat that
a white noise with enough small intensity can to bring significant contributions in turbulent modes and
even to change this modes dramatically.
At the present time it is generally recognized thatturbulence in its developed phase has essentiallysingular
spatially-temporal structure. Such asingular conduct is impossible to describeadequately by the means of
some model system of equations of a finite dimensionality. In thispoint a classical theory of chaos is able
todescribe only small part of turbulencephenomenon in liquid and another analogous sof dynamical
systems. The results of non-perturbative modeling ofsuper-chaotic modes, obtained in the present paper
allow us to put out a quite probablehypothesis: developed turbulence in the realphysical systems with
infinite number of degreesof freedom is a quantum super-chaos, at that the quantitative characteristics of
this super-chaos, iscompletely determined by non-perturbativecontribution of additive (thermal)
fluctuations inthe corresponding classical system dynamics [18]-[20].
2.Main Theoretical Results
We study the stochastic -ݎdimensionaldifferential equation analogous proposed byNikolaevskii [1] to
describe longitudinal seismicwaves:
డ௨ആሺ௫,௧,ఌ,ఠሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ + ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ ∑ ߜ
డ௨ആሺ௫,௧,ఌ,ఠሻ
డ௫
ୀଵ + ݂ሺ,ݔ ݐሻ −
ඥߟݓሺ,ݔ ,ݐ ߱ሻ = 0, (2.1)
ݔ ∈ ℝ
, ݑఎሺ,ݔ 0, ߝ, ߱ሻ = 0, ݓሺ,ݔ ݐሻ =
డೝశభௐሺ௫,௧ሻ
డ௫భడ௫మ…డ௫ೝడ௧
, 0 < ߜ, ݆ = 1, … , )2.2(.ݎ
The main difficulty with the stochasticNikolaevskii equationis that the solutions do not take values in an
function space but in generalized functionspace. Thus it is necessary to give meaning to the non-linear
terms ߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߱ሻ, ݆ = 1, … , ݎ because the usual product makes no sense for arbitrary distributions.
We deal with product of distributions via regularizations, i.e., we approximate the distributions by
appropriate way and pass to the limit.In this paper we use the approximation of the distributions by
approach of Colombeaugeneralized functions [28].
Notation 2.1.We denote byࣞሺℝ
× ℝାሻthe space of the infinitely differentiable functionswith compact
supportin ℝ
× ℝାandbyࣞ′ሺℝ
× ℝାሻ its dual space.Let ℭ = ሺΩ, Σ, µሻbe a probability space. We denote
by۲the space of allfunctionsܶ: Ω → ࣞ′ሺℝ
× ℝାሻ such that 〈ܶ, ߮〉is a random variable for all߮ ∈
ࣞሺℝ
× ℝାሻ.Theelements of۲are called random generalized functions.
Definition 2.1.[29].We say that a random field ሼℜሺ,ݔ ݐሻ|ݐ ∈ ℝା, ݔ ∈ ℝሽ isa spatiallydependent
semimartingale if for each ݔ ∈ ℝ
, ሼℜሺ,ݔ ݐሻ|ݐ ∈ ℝାሽ is asemimartingale in relation to the same
filtration ሼℱ௧|ݐ ∈ ℝାሽ. If ℜሺ,ݔ ݐሻ is a ܥ∞
-function of ݔ and continuous inalmost everywhere,it is called
aܥ∞
-semimartingale.
4. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
24
Definition 2.2.We say that that ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ ∈ ۲ is a strong generalized solution(SGS) of the
Eq.(2.1)-(2.2) if there exists asequence ofܥ∞
-semimartingalesݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߳ ∈ ሺ0,1ሿ such that there
exists
ሺiሻ ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ =ୢୣ limఢ→ݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻinࣞ′ሺℝ
× ℝାሻ almost surely for ߱ ∈ Ω,
ሺiiሻ ߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ =ୢୣ limఢ→߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ݆ = 1, … , ݎalmost surely for ߱ ∈ Ω,
ሺiiiሻfor all ߮ ∈ ࣞሺℝ
× ℝାሻ
ሺivሻ〈߲௧ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ, ߮〉 − 〈∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ, ߮〉 −
ߜ
2
〈߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߱ሻ, ߮〉
ୀଵ
− 〈݂ሺ,ݔ ݐሻ, ߮〉 +
+ඥߟ ݀ݔℝೝ ߮ሺ,ݐ ݔሻ
∞
ܹ݀௧ሺ,ݔ ݐሻ = 0, ݐ ∈ ℝାalmost surely for ߱ ∈ Ω,
and where ܹ௧ሺ,ݔ ݐሻ =
డೝௐሺ௫,௧ሻ
డ௫భడ௫మ…డ௫ೝ
,
ሺvሻݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ = 0almost surely for ߱ ∈ Ω.
However in this paper we use the solutionsofstochastic Nikolaevskii equation only in the sense of
Colombeaugeneralized functions [30].
Remark2.1.Note that from Definition 2.2it is clear that any strong generalized solutionݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ of
the Eq.(2.1)-(2.2) one can to recognized as Colombeaugeneralized function such that
ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ =ୢୣ ቀݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻቁ
ఢ
ሺ#ሻ
By formula ሺ#ሻone can todefine appropriate generalized solutionof the Eq.(2.1)-(2.2) even if a strong
generalized solutionof the Eq.(2.1)-(2.2) does not exist.
Definition 2.3.Assumethata strong generalized solution of the Eq.(2.1)-(2.2) does not exist.We shall say
that:
(I)Colombeaugeneralized stochastic process ቀݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻቁ
ఢ
is a weak generalized solution (WGS) of
the Eq.(2.1)-(2.2) orColombeausolutionof the Eq.(2.1)-(2.2) iffor all ߮ ∈ ࣞሺℝ
× ℝାሻandfor all߳ ∈ ሺ0,1ሿ
ሺiሻ〈ݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߲௧߮〉 − 〈∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߮〉 −
ߜ
2
〈߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߮〉
ୀଵ
+
+ 〈݂ሺ,ݔ ݐሻ, ߮〉 + ඥߟ ݀ݔℝೝ ߮ሺ,ݐ ݔሻ
∞
ܹ݀௧ሺ,ݔ ݐሻ = 0, ݐ ∈ ℝାalmost surely for ߱ ∈ Ω,
ሺiiሻݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ = 0almost surely for ߱ ∈ Ω.
5. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
25
(II)Colombeaugeneralized stochastic process ቀݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻቁ
ఢ
is aColombeau-Ito’ssolutionof the
Eq.(2.1)-(2.2) if for all ߮ ∈ ࣞሺℝሻand for all߳ ∈ ሺ0,1ሿ
ሺiሻ〈߲௧ݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߮〉 + 〈∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߮〉
ߜ
2
〈߲௫ೕ
ݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ, ߮〉
ୀଵ
−
−〈݂ሺ,ݔ ݐሻ, ߮〉 − ඥߟ ߮ሺݔሻℝೝ ݓሺ,ݔ ݐሻ݀ݔ = 0, ݐ ∈ ℝାalmost surely for ߱ ∈ Ω,
ሺiiሻݑఎሺ,ݔ ,ݐ ߝ, ߳, ߱ሻ = 0almost surely for ߱ ∈ Ω.
Notation 2.2.[30]. Thealgebra of moderate element we denote byℰெሾℝሿ.The Colombeau algebra of the
Colombeau generalized functionwe denote by ࣡ሺℝሻ.
Notation 2.3.[30].We shall use the following designations. If ܷ ∈ ࣡ሺℝሻit representatives will be
denoted byܴ, their values on ߮ = ൫߮ఢሺݔሻ൯,߳ ∈ ሺ0,1ሿwill be denoted by ܴሺ߮ሻ and it pointvalues at
ݔ ∈ ℝ
will be denoted ܴሺ߮, ݔሻ.
Definition 2.4.[30].Let ܣ = ܣሺℝሻ be the set of all ߮ ∈ ܦሺℝሻ such that ߮ሺݔሻ ݀ݔ = 1.
Let ℭ = ሺΩ, Σ, µሻbe a probability space.Colombeau random generalized function this is a
map ܷ: Ω → ࣡ሺℝሻ such that there is representing functionܴ: ܣ × ℝ
× Ω with the properties:
(i) for fixed߮ ∈ ܣሺℝሻ the function ሺ,ݔ ߱ሻ → ܴሺ߮, ,ݔ ߱ሻ is a jointly measurable on ℝ
× Ω;
(ii) almost surely in ߱ ∈ Ω,the function ߮ → ܴሺ߮, . , ߱ሻbelongs to ℰெሾℝሿ and is a
representative of ܷ;
Notation 2.3.[30]. TheColombeaualgebra ofColombeau random generalized functionis denoted
by࣡Ωሺℝሻ.
Definition 2.5.Let ℭ = ሺΩ, Σ, µሻbe a probability space. Classically, a generalizedstochastic process on
ℝ
is a weakly measurable mapܸ: Ω → ܦ′ሺℝሻ denoted by ܸ ∈ ܦΩ
′
ሺℝሻ. If ߮ ∈ ܣሺℝሻ, then
(iii) ܸሺ߱ሻ ∗ ߮ሺݔሻ = 〈ܸሺ߱ሻ, ߮ሺ.−ݔ ሻ〉 is a measurable with respect to߱ ∈ Ω and
(iv) smooth with respect toݔ ∈ ℝ
and hence jointly measurable.
(v) Also ൫ܸሺ߱ሻ ∗ ߮ሺݔሻ൯ ∈ ℰெሾℝሿ.
(vi) Therefore ܴሺ߮, ,ݔ ߱ሻ = ܸሺ߱ሻ ∗ ߮ሺݔሻ qualifies as an representing function for an element
of࣡Ωሺℝሻ.
(vii) In this way we have an imbedding ࣞ′ሺℝሻ → ࣡Ωሺℝሻ.
Definition 2.6.Denote by ܵሺܶሻ = ܵሺℝାଵሻ ↾ ܶ the space of rapidly decreasing smooth functions on
ܶ = ℝ
× ሾ0, ∞ሻ. Letℭ = ሺΩ, Σ, µሻwith (i) Ω = ܵ′ሺܶሻ, ሺiiሻ Σ- the Borelߪ-algebra generated by the weak
topology. Therefore there is unique probability measure ߤ on ሺΩ, Σሻ such that
න ݀ߤሺ߱ሻexp ሾ݅〈߱, ߮〉ሿ = exp ൬−
1
2
‖φ‖మሺ்ሻ
ଶ
൰
for all ߮ ∈ ܵሺܶሻ. White noiseݓሺ߱ሻ with the support in ܶ is the generalized process ݓሺ߱ሻ: Ω →
ࣞ′ሺℝାଵሻ such that: (i) ݓሺ߮ሻ = 〈ݓሺ߱ሻ, ߮〉 = 〈߱, ߮ ↾ ܶ〉 (ii) ۳ሾݓሺ߮ሻሿ = 0, (iii) ۳ሾݓଶሺ߮ሻሿ =
‖φ‖మሺ்ሻ
ଶ
.Viewed as a Colombeau random generalized function, it has a representative (denoting on
variables in ℝାଵ
by ሺ,ݔ ݐሻ): ܴ௪ሺ߮, ,ݔ ,ݐ ߱ሻ = 〈߱, ߮ሺ,−ݔ ݐ −ሻ ↾ ܶ〉, which vanishes if ݐ is less than
minus the diameter of the support of߮.Thereforeݓ is a zero on ℝ
× ሺ−∞, 0ሻ in ࣡Ωሺℝାଵሻ. Note that its
variance is the Colombeau constant:۳ሾܴ௪
ଶ ሺ߮, ,ݔ ,ݐ ߱ሻሿ = ݀ݕ |߮ሺݔ − ,ݕ ݐ − ݏሻ|ଶ
݀ݏ
∞
ℝೝ .
7. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
27
Proof.The proof based on Strong large deviations principles(SLDP-Theorem) for Colombeau-Ito’ssolution
of the Colombeau-Ito’s SDE, see [26],theorem 6.BySLDP-Theorem one obtain directlythe differential
master equation (see [26],Eq.(90)) for Colombeau-Ito’s SDE(2.5)-(2.7):
݀ ቀܷఢ,ሺݔ, ,ݐ ߝሻቁ
ఢ
݀ݐ
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ ቀܷఢ,ሺݔ, ,ݐ ߝሻቁ
ఢ
+
+ߣ ∑ ߜ ቆܨఢ ቀ
ച,శభ,൫௫శ,,௧,ఌ൯ିച,൫௫,,௧,ఌ൯ା
ಿ
ቁቇ
ఢ
ୀଵ + ൫݂ఢሺݔ, ݐሻ൯ఢ
+ ሺ߳ሻ = 0,(2.11)
ܷఢ,ሺݔ, 0, ߝሻ = −ߣ. (2.12)
We set now ߣ ≡ ߣ ∈ ℝ. Then from Eq.(2.13)-Eq.(2.14) we obtain
݀ ቀܷఢ,ሺݔ, ,ݐ ߝ, ߣሻቁ
ఢ
݀ݐ
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ ቀܷఢ,ሺݔ, ,ݐ ߝ, ߣሻቁ
ఢ
+
+ߣ ∑ ߜ ቆܨఢ ቀ
ച,శభ,൫௫శ,,௧,ఌ,ఒ൯ିച,൫௫,,௧,ఌ,ఒ൯
ಿ
ቁቇ
ఢ
ୀଵ + ൫݂ఢሺݔ, ݐሻ൯ఢ
+ ܱሺ߳ሻ = 0,(2.13)
ܷఢ,ሺݔ, 0, ߝ, ߣሻ = −ߣ. (2.14)
From Eq.(2.5)-Eq.(2.7) and Eq.(2.13)-Eq.(2.14) by SLDP-Theorem (see see[26], inequality(89)) we obtain
the inequality
liminfఢ→۳ ቂหݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻ − ߣห
ଶ
ቃ ≤ ܷఢ,ሺݔ, ,ݐ ߝ, ߣሻ, ߳ ∈ ሺ0,1ሿ.(2.15)
Let us consider now the identity
|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
= หൣݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻ൧ + ൣݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻ − ߣ൧ห
ଶ
. (2.16)
Fromtheidentity (2.16) bythetriangle inequality we obtaintheinequality
|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ หݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻห
ଶ
+ หݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻ − ߣห
ଶ
. (2.17)
From theidentity (2.17) by integration we obtain theinequality
۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ ۳หݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻห
ଶ
+ หݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻ − ߣห
ଶ
. (2.18)
From theidentity (2.18) by theidentity (2.15) for all ߳ ∈ ሺ0,1ሿwe obtain theinequality
۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ ۳หݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻห
ଶ
+ ܷఢ,ሺݔ, ,ݐ ߝ, ߣሻ.(2.19)
8. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
28
In the limit ܰ → ∞ fromthe inequality we obtain the inequality
۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ limsupே→∞۳หݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻห
ଶ
+ limsupே→∞ܷఢ,ሺݔ, ,ݐ ߝ, ߣሻ. (2.20)
Wenote that
limsupே→∞۳หݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ݑఢ,ሺݔ, ,ݐ ߝ, ߟ, ߱ሻห
ଶ
= 0. (2.21)
Therefore from (2.20) and (2.21) we obtain the inequality
۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ limsupே→∞ܷఢ,ሺݔ, ,ݐ ߝ, ߣሻ(2.22)
In the limit ܰ → ∞ from Eq.(2.13)-Eq.(2.14) for any fixed ߳ ≠ 0, ߳ ≪ 1, we obtain the differential master
equationforColombeau-Ito’s SPDE (2.3)-(2.4)
݀൫ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ൯ఢ
݀ݐ
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ൫ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ൯ఢ
+
+ߣ ∑ ߜ ቆܨఢ ቀ
డചሺ௫,௧,ఌ,ఒሻ
డ௫
ቁቇ
ఢ
ୀଵ + ൫݂ఢሺ,ݔ ݐሻ൯ఢ
+ ܱሺ߳ሻ = 0,(2.23)
ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ = −ߣ. (2.24)
Therefore from the inequality (2.22) followsthe inequality
۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ. (2.25)
In the limit ߳ → 0from differential equation (2.23)-(2.24)we obtain the differential equation (2.8)-(2.9)and
it is easy to see that
limఢ→ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ = ℜሺ,ݔ ,ݐ ߝ, ߣሻ. (2.26)
From the inequality (2.25) one obtainthe inequality
liminfఢ→۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ limఢ→ܷఢሺ,ݔ ,ݐ ߝ, ߣሻ = ℜሺ,ݔ ,ݐ ߝ, ߣሻ.(2.27)
From the inequality (2.27) and Eq.(2.26) finally we obtainthe inequality
liminfఢ→۳|ݑఢሺ,ݔ ,ݐ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ ℜሺ,ݔ ,ݐ ߝ, ߣሻ. (2.28)
The inequality (2.28) finalized the proof.
Definition 2.7.(TheDifferential Master Equation)The linear PDE:
డℜሺ௫,௧,ఌ,ఒሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿℜሺ,ݔ ,ݐ ߝ, ߣሻ + ߣ ∑ ߜ
డℜሺ௫,௧,ఌ,ఒሻ
డ௫
ୀଵ − ݂ሺ,ݔ ݐሻ = 0, ߣ ∈ ℝ,(2.29)
ℜሺ,ݔ 0, ߝ, ߣሻ = 0 (2.30),
We will call asthe differential master equation.
9. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
29
Definition 2.8.(TheTranscendental Master Equation)Thetranscendental equation
ℜ൫,ݔ ,ݐ ߝ, ߣሺ,ݔ ,ݐ ߝሻ൯ = 0, (2.31)
wewill call asthe transcendentalmaster equation.
Remark2.2.We note that concrete structure of the Nikolaevskii chaos is determined by the
solutionsߣሺ,ݔ ,ݐ ߝሻvariety bytranscendentalmaster equation(2.31).Master equation (2.31)is determines by the
only way some many-valued functionߣሺ,ݔ ,ݐ ߝሻ which is the main constructive object, determining
thecharacteristics of quantum chaos in the corresponding model of Euclidian quantum fieldtheory.
3.Criterion of the existence quantum chaos in Euclidian quantum
N-model.
Definition3.1.Let ݑఎሺ,ݔ ,ݐ ߝ, ߱ሻbe the solution of the Eq.(2.1). Assume that for almost all pointsሺ,ݔ ݐሻ ∈
ℝ
× ℝା(in the sense of Lebesgue–measureonℝ
× ℝା), there exist a function ݑሺ,ݔ ݐሻ such that
limఎ→۳ ቀݑఎሺ,ݔ ,ݐ ߝ, ߱ሻ − ݑሺ,ݔ ݐሻቁ
ଶ
൨ = 0. (3.1)
Then we will say that afunction ݑሺ,ݔ ݐሻis a quasi-determined solution (QD-solution of the Eq.(2.).
Definition3.2. Assume that there exist a setℌ ⊂ ℝ
× ℝାthat is positive
Lebesgue–measure, i.e.,ߤሺℌሻ > 0 and
∀ሺ,ݔ ݐሻ൛ሺ,ݔ ݐሻ ∈ ℌ → ¬∃limఎ→۳ൣݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߱ሻ൧ൟ,(3.2)
i.e., ሺ,ݔ ݐሻ ∈ ℌ imply that the limit: limఎ→۳ൣݑఎ
ଶሺ,ݔ ,ݐ ߝ, ߱ሻ൧does not exist.
Then we will say thatEuclidian quantum N-model has thequasi-determined Euclidian quantum chaos
(QD-quantum chaos).
Definition3.3.For each pointሺ,ݔ ݐሻ ∈ ℝ
× ℝାwe define a set ൛ℜ෩ሺ,ݔ ,ݐ ߝሻൟ ⊂ ℝ by the condition:
∀ߣൣߣ ∈ ൛ℜ෩ሺ,ݔ ,ݐ ߝሻൟ ⟺ ℜሺ,ݔ ,ݐ ߝ, ߣሻ = 0൧.(3.3)
Definition3.4.Assume that Euclidian quantum N-model(2.1) has the Euclidian QD-quantum chaos.
For each point ሺ,ݔ ݐሻ ∈ ℝ
× ℝା we define a set-valued functionℜ෩ሺ,ݔ ݐሻ: ℝ
× ℝା → 2ℝ
by the condition:
ℜ෩ሺ,ݔ ,ݐ ߝሻ = ൛ℜ෩ሺ,ݔ ,ݐ ߝሻൟ(3.4)
We will say thattheset -valued functionℜ෩ሺ,ݔ ,ݐ ߝሻis a quasi-determinedchaotic solution(QD-chaotic
solution)of the quantum N-model.
10. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Pic.3.1.Evolution of
at pointݔ
Pic.3.2.The spatial structure of
instant
Theorem3.1.Assume that݂ሺ,ݔ ݐሻ
such that ݎ ∈ Գ, ߜ ∈ ℝା, ݆ = 1, …
QD-chaotic solutions.
Definition3.5.For each point ሺ,ݔ
(i) ݑାሺ,ݔ ,ݐ ߝሻ = limsupఎ→
(ii) ݑିሺ,ݔ ,ݐ ߝሻ = liminfఎ→۳
(iii) ݑ௪ሺ,ݔ ,ݐ ߝሻ = ݑାሺ,ݔ ,ݐ ߝ,
Definition3.7.
(i) Function ݑାሺ,ݔ ,ݐ ߝሻis called
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Evolution of QD-chaotic solutionℜ෩ሺ,ݔ ,ݐ ߝሻin timeݐ ∈ ሾ0,10ሿ
ݔ = 3. ݐ ∈ ሾ0,10ሿ, ߝ = −10ିଶ
, ߪ = 10ଷ
, = 1.1.
spatial structure ofQD-chaoticsolutionℜ෩ሺ,ݔ ,ݐ ߝሻ at
instant ݐ = 3, ߝ = −10ିଶ
, ߪ = 10ଷ
, = 1.1.
ሺ ሻ = ߪ sinሺ ∙ ݔሻThen for all values of parameters,ݎ ߝ, ߪ
… , ,ݎ ߝ ∈ ሾ−1,1ሿ, ∈ ℝ
, ߪ ≠ 0, quantum N-model (2.1) has the
ሺ ݐሻ ∈ ℝ
× ℝା we define the functions such that:
۳ሾݑఎሺ,ݔ ,ݐ ߝ, ߱ሻሿ,
۳ሾݑఎሺ,ݔ ,ݐ ߝ, ߱ሻሿ,
߱ሻ − ݑିሺ,ݔ ,ݐ ߝ, ߱ሻ.
is calledupper boundof the QD-quantum chaosat point ሺ,ݔ ݐሻ
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
30
ሿ
ߪ, ߜ, ݆ = 1, … , ݎ
model (2.1) has the
ሺ ሻ.
11. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
(ii) Function ݑିሺ,ݔ ,ݐ ߝሻis called lower bound of the
(iii) Function ݑ௪ሺ,ݔ ,ݐ ߝሻis called
Definition3.8. Assume now that
limsup௧→∞ݑ௪ሺ,ݔ ,ݐ ߝሻ = ݑ௪ሺ,ݔ ߝሻ ൏
Then we will say thatEuclidian quantum N
finitewidthat pointݔ ∈ ℝ
.
Definition3.9.Assume now that
limsup௧→∞ݑ௪ሺ,ݔ ,ݐ ߝሻ = ݑ௪ሺ,ݔ ߝሻ =
Then we will say that Euclidian quantum N
width at point ݔ ∈ ℝ
.
Pic.3.3.TheQD-quantum chaos of the asymptotically infinite width at point
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
is called lower bound of the QD-quantum chaosat point ሺ,ݔ ݐ
ሻis called width oftheQD-quantum chaosat pointሺ,ݔ ݐሻ.
that
ሻ ൏ ∞.(3.5)
Then we will say thatEuclidian quantum N-model has QD-quantum chaos of the asymptotically
ሻ = ∞. (3.6)
Then we will say that Euclidian quantum N-model has QD-quantum chaos of the asymptotically
quantum chaos of the asymptotically infinite width at point ݔ = 3. ߝ = 0.1
10ଷ
, = 1.
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
31
ሺ ݐሻ.
quantum chaos of the asymptotically
quantum chaos of the asymptotically infinite
1, ߜ = 10, ߪ =
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32
Pic.3.4.The fine structure of the QD-quantum chaos of the asymptotically infinite width at point ݔ =
3, ߝ = 0, ߜ = 10, ߪ = 10ସ
, = 1, ݐ ∈ ሾ10ସ
, 10ସ
+ 10ିଵሿ, ߣ ∈ ሾ−0.676,0.676ሿ.
Definition3.10. For each point ሺ,ݔ ݐሻ ∈ ℝ
× ℝା we define the functions such that:
(i) ℜ෩ାሺ,ݔ ,ݐ ߝሻ = sup൛ℜ෩ሺ,ݔ ,ݐ ߝሻൟ,
(ii) ℜ෩ିሺ,ݔ ,ݐ ߝሻ = inf൛ℜ෩ሺ,ݔ ,ݐ ߝሻൟ,
(iii) ℜ෩௪ሺ,ݔ ,ݐ ߝሻ = ℜ෩ାሺ,ݔ ,ݐ ߝሻ − ℜ෩ିሺ,ݔ ,ݐ ߝሻ.
Theorem3.2. For each point ሺ,ݔ ݐሻ ∈ ℝ
× ℝାis satisfiedtheinequality
ℜ෩௪ሺ,ݔ ,ݐ ߝሻ ≤ ݑ௪ሺ,ݔ ,ݐ ߝሻ.(3.7)
Proof. Immediately follows by Theorem2.1 and Definitions 3.5, 3.10.
Theorem3.3. (Criterion of QD-quantum chaos in Euclidian quantum N-model)
Assume that
mes൛ሺ,ݔ ݐሻ|ℜ෩௪ሺ,ݔ ,ݐ ߝሻ > 0ൟ > 0.(3.8)
Then Euclidian quantum N-model has QD-quantum chaos.
Proof. Immediately follows bytheinequality(3.7)and Definition3.2.
4. Quasi-determined quantum chaos and physical turbulencenature.
In generally accepted at the present time hypothesiswhatphysical turbulencein the dynamical systems with
an infinite number of degrees of freedom really is, thephysical turbulence is associated with a
strangeattractors, on which the phase trajectories of dynamical system reveal the knownproperties of
stochasticity: a very high dependence on the initial conditions, whichis associated with exponential
dispersion of the initially close trajectories and bringsto their non-reproduction; everywhere the density on
the attractor almost of all thetrajectories a very fast decreaseoflocal auto-correlation function[2]-[9]
Φሺx, τሻ = 〈ݑሺ,ݔ ݐሻݑሺ,ݔ ߬ + ݐሻ〉,(4.1)
Here
ݑሺ,ݔ ݐሻ = ݑሺ,ݔ ݐሻ − 〈ݑሺ,ݔ ݐሻ〉, 〈݂ሺݐሻ〉 = lim்→∞〈݂ሺݐሻ〉், 〈݂ሺݐሻ〉் =
1
ܶ
න ݂
்
ሺݐሻ.
In contrast with canonical numerical simulation, by using Theorem2.1 it is possible to study
non-perturbativelythe influence of thermal additive fluctuationson classical dynamics, which in the
consideredcase is described by equation (4.1).
The physicalnature of quasi-determined chaosis simple andmathematically is associated
withdiscontinuously of the trajectories of the stochastic processݑఎሺ,ݔ ,ݐ ߝ, ߱ሻon parameter ߟ.
In order to obtain thecharacteristics of this turbulence, which is a very similarlytolocal auto-correlation
function (3.1) we define bellowsomeappropriatefunctions.
13. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
33
Definition 4.1.The numbering functionܰሺ,ݐ ݔሻ of quantum chaos in Euclidian quantum N-modelis
defined by
ܰሺ,ݔ ݐሻ = card൛ℜ෩ሺ,ݔ ݐሻൟ.(4.2)
Here by cardሼܺሽ we denote the cardinality of a finite setܺ,i.e., the number of its elements.
Definition 4.2.Assume now that a set ൛ℜ෩ሺ,ݔ ݐሻൟis ordered be increase of its elements. We introduce the
functionℜ෩ሺ,ݔ ݐሻ, ݅ = 1, … , ܰሺ,ݔ ݐሻwhich value at pointሺ,ݔ ݐሻ, equals the ݅-th element ofa set ൛ℜ෩ሺ,ݔ ݐሻൟ.
Definition 3.3.The mean value functionݑሺ,ݔ ݐሻ ofthe chaotic solution ℜ෩ሺ,ݔ ݐሻat point ሺ,ݔ ݐሻis defined by
ݑሺ,ݔ ݐሻ = ൫ܰሺ,ݔ ݐሻ൯
ିଵ
∑ ℜ෩ሺ,ݔ ݐሻேሺ௫,௧ሻ
ୀଵ .(4.3)
Definition 3.4.The turbulent pulsations function ݑ∗ሺ,ݔ ݐሻ of the chaotic solution ℜ෩ሺ,ݔ ݐሻat point ሺ,ݔ ݐሻis
defined by
ݑ∗ሺ,ݔ ݐሻ = ට൫ܰሺ,ݔ ݐሻ൯
ିଵ
∑ หℜ෩ሺ,ݔ ݐሻ − ݑሺ,ݔ ݐሻหேሺ௫,௧ሻ
ୀଵ .(4.4)
Definition3.5.Thelocal auto-correlation function is definedby
Φሺ,ݔ ߬ሻ = lim்→∞〈ݑሺ,ݔ ߬ሻݑሺ,ݔ ߬ + ݐሻ〉் = lim்→∞
ଵ
்
ݑ
்
ሺ,ݔ ݐሻݑሺ,ݔ ߬ + ݐሻ݀)5.4(,ݐ
ݑሺ,ݔ ݐሻ = ݑሺ,ݔ ݐሻ − ݑුሺݔሻ, ݑුሺݔሻ = lim்→∞
ଵ
்
ݑ
்
ሺ,ݔ ݐሻ݀)6.4(.ݐ
Definition 3.5.Thenormalized local auto-correlation function is defined by
Φ୬ሺ,ݔ ߬ሻ =
Φሺ௫,ఛሻ
Φሺ୶,ሻ
.(4.7)
Let us consider now 1DEuclidian quantum N-model corresponding to classical dynamics
డమ
డ௫మ ߝ − ቀ1 +
డమ
డ௫మቁ
ଶ
൨ ݑሺ,ݔ ߝሻ + ߜ
డ௨ሺ௫,ఌሻ
డ௫
ݑሺ,ݔ ߝሻ − ߪ sinሺ ∙ ݔሻ = 0,(4.8)
Corresponding Langevin equation are [34]-[35]:
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݐ
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿݑఎሺ,ݔ ,ݐ ߝሻ + ߜ
߲ݑఎሺ,ݔ ,ݐ ߝሻ
߲ݔ
ݑఎሺ,ݔ ,ݐ ߝሻ −
−ߪ sinሺݔሻ = ඥߟݓሺ,ݔ ݐሻ, ߜ > 0∆=
డమ
డ௫మ,(4.9)
ݑఎሺ,ݔ 0, ߝሻ = 0, ݓሺ,ݔ ݐሻ =
డమௐሺ௫,௧ሻ
డ௫డ௧
.(4.10)
Corresponding differential master equation are
14. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
డℜሺ௫,௧,ఌ,ఒሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿℜሺ,ݔ ,ݐ
ℜሺ,ݔ 0, ߝ, ߣሻ = −ߣ.(4.12)
Corresponding transcendental master equation
ሼୡ୭ୱሺ∙௫ሻିୣ୶୮ሾ௧∙ఞሺሻሿୡ୭ୱሾሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఒ∙ఋ∙
ఞమሺሻାఒమ∙ఋమ∙మ
߯ሺሻ = ଶሾߝ − ሺଶ
− 1ሻଶሿ.(4.14)
We assume now that ߯ሺሻ = 0.Then from Eq.(4.13) for a
ሼୡ୭ୱሺ∙௫ሻିୡ୭ୱሾሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఒ∙ఋ∙
ఒమ∙ఋమ∙మ +
ఒ
ఙ
= 0
ሼcosሺ ∙ ݔሻ − cosሾሺݔ − ߣ ∙ ߜ ∙ ݐሻሿሽ
The result of calculation using transcendental
is presented by Pic.4.1 and Pic.4.2.
Pic.4.1.Evolution of QD-chaotic solution
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
ሺ , ߝ, ߣሻ + ߣߜ
డℜሺ௫,௧,ఌ,ఒሻ
డ௫
− ߪ sinሺݔሻ = 0,(4.11)
Corresponding transcendental master equation (2.29)-(2.30) are
∙
+
ሼୱ୧୬ሺ∙௫ሻିୣ୶୮ሾ௧∙ఞሺሻሿୱ୧୬ሾሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఞሺሻ
ఞమሺሻାఒమ∙ఋమ∙మ +
ఒ
ఙ
= 0,(4.13)
Then from Eq.(4.13) for allݐ ∈ ሾ0, ∞ሻ we obtain
0,or(4.14)
ሻሿሽ ∙ ߪ ∙ ߜିଵ
∙ ିଵ
ߣଶ
ൌ 0.(4.15)
transcendental master equation (4.15) the corresponding function
is presented by Pic.4.1 and Pic.4.2.
chaotic solutionԸ෩ሺ10ଷ
, ,ݐ ߝሻin timeݐ ∈ ሾ0, 10ଷሿ, ∆ݐ ൌ 0.1, ߝ ൌ
ߪ ൌ 10ଶ
, ߜ ൌ 1, , ∆ߣ ൌ 0.01
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
34
master equation (4.15) the corresponding function Ը෩ሺ,ݔ ,ݐ ߝሻ
ൌ 0, ൌ 1,
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Pic.4.2.The spatial structure ofQD
The result of calculation using master equation(4.
presented by Pic.4.3 and Pic.4.4
Pic.4.3.The
EuclidianquantumN-model
Pic.4.4.The development
Euclidian quantum N-model at point
Let us calculate now corresponding
of calculation using Eq.(4.7)-Eq.(4.7)
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
QD-chaoticsolutionԸ෩ሺ,ݔ ,ݐ ߝሻ at instant ݐ ൌ 10ଷ
, ߝ ൌ 0, ൌ
ߜ ൌ 1, ∆ݔ ൌ 0.1, ∆ߣ ൌ 0.01.
calculation using master equation(4.13) the correspondingfunction
4.
Thedevelopment of temporal chaotic regime of1D
model at point ݔ ൌ 1, ݐ ∈ ሾ0, 10ଶሿ. ߝ ൌ 10ି
, ߪ ൌ 10ଶ
, ߜ ൌ 1,
The developmentof temporal chaotic regimeof 1D
model at pointݔ ൌ 1, ݐ ∈ ሾ0, 10ଶሿ, ߝ ൌ 10ି
, ߪ ൌ 5 ∙ 10ହ
, ߜ ൌ 1
correspondingnormalized local auto-correlation functionΦ୬ሺݔ
Eq.(4.7) is presented by Pic.4.5 and Pic.4.6.
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35
= 1, ߪ ൌ 10ଶ
,
unction Ը෪ሺ,ݔ ,ݐ ߝሻis
, ൌ 1.
1, ൌ 1.
ሺ,ݔ ߬ሻ.The result
16. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Pic.4.5. Normalized local auto
ݐ ∈
Pic.4.6.Normalized local auto
ݐ ∈ ሾ0
Inpaper [7]the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg
equation(4.14). Numerical data are compared with experimental results.
డሺ௫,௧ሻ
డ௧
ൌ ݅߱ሺݔሻܽሺ,ݔ ݐሻ
ଵ
ଶ
ሺ1 െ |ܽሺ
߲ܽሺ
߲ݔ
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Normalized local auto-correlation function Φ୬ሺ1, ߬ሻ
∈ ሾ0,50ሿ, ߝ ൌ 10ି
, ߪ ൌ 10ଶ
, ߜ ൌ 1, ൌ 1.
Normalized local auto-correlation functionΦ୬ሺ1, ߬ሻ.
ሾ0,100ሿ, ߝ ൌ 10ି
, ߪ ൌ 5 ∙ 10ହ
, ߜ ൌ 1, ൌ 1.
the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg
. Numerical data are compared with experimental results.
ሺ,ݔ ݐሻ|ଶሻܽሺ,ݔ ݐሻ ݃
డమሺ௫,௧ሻ
డ௫
, (4.14)
ሺ0, ݐሻ
߲ݔ
ൌ 0,
߲ܽሺ݈, ݐሻ
߲ݔ
ൌ 0, ݔ ∈ ሾ0, ݈ሿ, ݈ ൌ 50.
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
36
the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg– Landau
17. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Pic.4.7. Normalized local auto
However as pointed out above (see
for stochastic model
డሺ௫,௧ሻ
డ௧
ൌ ݅߱ሺݔሻܽሺ,ݔ ݐሻ
ଵ
ଶ
ሺ1 െ |ܽሺ
߲ܽሺ
߲ݔ
5.The order of the phase transition
turbulent state at instant
In order to obtain the character of the phase transition
a spatially uniform to a turbulent state
Ը൫,ݔ ,ݐ ߝ, ߣሺ,ݔ ,ݐ ߝሻ൯ ൌ 0. (5.1)
Bydifferentiation the Eq.(5.1) one obtain
ௗԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
ௗఌ
ൌ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఒ
ௗఒሺ௫
ௗఌ
FromEq.(5.2) one obtain
ௗఒሺ௫,௧,ఌሻ
ௗఌ
ൌ െ ቀ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఌ
ቁ ∙ ቀ
డԸ൫
Let us consider now 1DEuclidian quantum N
transcendental master equation Eq.(4.13)
ߣone obtain
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
Normalized local auto-correlation functionΦ୬ሺ25, ߬ሻ[7].
However as pointed out above (see Remark1.1-1.4 ) such numerical simulation in factgives
ሺ,ݔ ݐሻ|ଶሻܽሺ,ݔ ݐሻ ݃
డమሺ௫,௧ሻ
డ௫
√ߝݓሺ,ݔ ݐሻ, ߝ ≪ 1, (4.15)
ሺ0, ݐሻ
߲ݔ
ൌ 0,
߲ܽሺ݈, ݐሻ
߲ݔ
ൌ 0, ݔ ∈ ሾ0, ݈ሿ, ݈ ൌ 50.
phase transitionfrom a spatially uniformstate
at instant ࢚ ൎ .
In order to obtain the character of the phase transition (first-order or second-order on parameter
a spatially uniform to a turbulent stateat instant ݐ ൎ 0one can to use the master equation () of the form
one obtain
൯ ሺ௫,௧,ఌሻ
ௗఌ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఌ
ൌ 0. (5.2)
ቀ
൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఒ
ቁ
ିଵ
.(5.3)
Euclidian quantum N-model given byEq. (4.9)-Eq. (4.10). From corresponding
transcendental master equation Eq.(4.13)by differentiation the equation Eq.(4.13)with respect to
International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
37
givesnumerical data
state to a
on parameters ߝ, ) from
one can to use the master equation () of the form
Eq. (4.10). From corresponding
with respect to variable
19. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
39
ሾℑሺߝ, ,ݔ ݐሻሿ௧ൎ ൌ ቂ
ଵ
௧
ௗఒሺ௫,௧,ఌሻ
ௗఌ
ቃ
௧ൎ
ൎ ߪଶሺଶ
െ 1ሻଶ ୱ୧୬ሺ∙௫ሻ
ఞሺሻ
signሺ ∙ ݔሻ (5.8)
In the limit ݐ → 0 from Eq. (5.8) one obtain
ሺߝ, ݔሻ = lim௧→
ௗఒሺ௫,௧,ఌሻ
௧ௗఌ
= ߪଶሺଶ
− 1ሻଶ ୱ୧୬ሺ∙௫ሻ
ఞሺሻ
signሺ ∙ ݔሻ,(5.9)
and where ߯ሺሻ = ଶሾߝ − ሺଶ
− 1ሻଶሿ = ଶ
ߩሺߝሻ, ߩሺߝሻ = ߝ − ሺଶ
− 1ሻଶ
.
FromEq. (5.9) follows that
limఘሺఌሻ→శ
ሺߝ, ݔሻ = +∞,(5.10)
limఘሺఌሻ→ష
ሺߝ, ݔሻ = −∞.(5.11)
FromEq. (5.10)-(5.11) follows second orderdiscontinuity of the quantity ሺߝ, ,ݔ ݐሻ at instant ݐ = 0.
Therefore the system causing it to make a direct transitionfrom a spatially uniformstateݑఎ≈ሺ,ݔ 0, ߝሻ = 0
to a turbulent statein an analogous fashion to the second-order phase transition inquasi-equilibrium
systems.
6.Chaotic regime generatedby periodical multi-modes external
perturbation.
Assume nowthat external periodical force݂ሺݔሻhas the followingmulti-modes form
݂ሺݔሻ = − ∑ ߪ
ୀଵ sinሺݔሻ.(6.1)
Corresponding transcendental master equation are
ሺ,ݔ ,ݐ ߝ, ߣሻ = ߪ
ሼcosሺ ∙ ݔሻ − expሾݐ ∙ ߯ሺሻሿcosሾሺݔ − ߣ ∙ ߜ ∙ ݐሻሿሽ ∙ ߣ ∙ ߜ ∙
߯ଶሺሻ + ߣଶ ∙ ߜଶ ∙
ଶ
ୀଵ
+
+ ∑ ߪ
ሼୱ୧୬ሺೖ∙௫ሻିୣ୶୮ሾ௧∙ఞሺೖሻሿୱ୧୬ሾೖሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఞሺೖሻ
ఞమሺೖሻାఒమ∙ఋమ∙ೖ
మ
ୀଵ + ߣ = 0, ߯ሺሻ = ଶሾߝ − ሺଶ
− 1ሻଶሿ.(6.2)
Let us consider the examples of QD-chaotic solutions with a periodical force:
݂ሺݔሻ = −ߪ ∑ sin ቀ
௫
ቁ
ୀଵ .(6.3)
20. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
40
Pic.6.1.Evolution of QD-chaotic solution෩ሺ10ଷ
, ,ݐ ߝሻin timeݐ ∈ ሾ7 ∙ 10ଷ
, 10ସሿ,
∆ݐ = 0.1, ݉ = 1, ݊ = 100, ߝ = −1, = 1, ߪ = 10ଶ
, ߜ = 1, ∆ߣ = 0.01.
Pic.6.2.The spatial structure ofQD-chaoticsolution෩ሺ,ݔ ,ݐ ߝሻ at instant ݐ = 10ଷ
,
ݔ ∈ ሾ1.4 ∙ 10ଷ
, 2.5 ∙ 10ଷሿ, ݉ = 1, ݊ = 100, ߝ = −1, = 1, ߪ = 10ଶ
, ߜ = 1, ∆ݔ = 0.1, ∆ߣ = 0.01.
Pic.6.2.The spatial structure ofQD-chaoticsolution෩ሺ,ݔ ,ݐ ߝሻ at instant ݐ = 5 ∙ 10ଷ
,
ݔ ∈ ሾ1.4 ∙ 10ଷ
, 2.5 ∙ 10ଷሿ, ݉ = 1, ݊ = 100, ߝ = −1, = 1, ߪ = 10ଶ
, ߜ = 1,
∆ݔ = 0.1, ∆ߣ = 0.01.
7.Conclusion
A non-perturbative analytical approach to the studying of problemof quantum chaos in dynamical systems
withinfinite number of degrees of freedom isproposed and developed successfully.It is shown that the
additive thermal noise destabilizes dramatically the ground state of the system thus causing it to make a
direct transition from a spatially uniform to a turbulent state.
21. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
41
8.Acknowledgments
A reviewer provided important clarifications.
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