This talk was Dedicated to Einstein's miracle year at his 26
以此次讲座,致敬当年爱因斯坦26岁时的几篇牛文之一,对布朗运动的研究。
对随机游走的研究,已经取得了很深入的进展,本次讲座从布朗运动模型入手,逐步深入,引入分数阶布朗运动,levy随机飞行等概念
这些模型在各种复杂系统中非常常见,比如金融市场,网络交通流量等等,
会简略介绍这些模型在金融系统的应用,以及分析基于布朗运动随机游走的金融模型的弊端
给大家一个随机游走世界的全景
(1) The document discusses linear perturbations of the metric around a Schwarzschild black hole. It derives the Regge-Wheeler equation, which governs axial perturbations and takes the form of a wave equation with an effective potential.
(2) It shows that the Regge-Wheeler potential has a maximum just outside the event horizon. This allows it to be considered as a scattering potential barrier for wave packets.
(3) It concludes that Schwarzschild black holes are stable under smooth, compactly supported exterior perturbations, as these perturbations will remain bounded for all times according to properties of the Regge-Wheeler equation and solutions to the Schrodinger equation.
Nonclassical Properties of Even and Odd Semi-Coherent StatesIOSRJAP
Even and odd semi-coherent states have been introduced. Some of the nonclasscial properties of the states are studied in terms of the quadrature squeezing as well as sub-Poissonian photon statistics. The Husimi– Kano Q-function and the phase distribution in the framework of Pegg and Barnett formalism, are also discussed.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
Basic Boundary Conditions in OpenFOAM v2.4Fumiya Nozaki
This document discusses different types of boundary conditions in OpenFOAM including:
1) Steady and time-varying boundary conditions such as Dirichlet, Neumann, and Robin conditions.
2) Periodic and symmetry boundary conditions that can be used for repeating geometries.
3) Mixed boundary conditions that are a weighted combination of fixed value and fixed gradient conditions, controlled by a weighting parameter.
4) Direction mixed boundary conditions for vector fields that apply different conditions in different directions using a weighting tensor.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
A smooth-exit the-phase-transition-to-slow-roll-eternal-inflationmirgytoo
This document summarizes research on the phase transition to eternal inflation. It begins by introducing the concept of eternal inflation occurring when quantum fluctuations dominate over classical drift. The authors argue that even in the eternal inflation regime, perturbations of the geometry and interactions remain perturbative, allowing quantitative analysis. They aim to precisely define the critical condition for eternal inflation and calculate statistics of the reheating volume to understand the phase transition.
1998 characterisation of multilayers by x ray reflectionpmloscholte
This document presents a theoretical model for characterizing multilayers using X-ray reflection. The model includes refraction effects and describes diffuse scattering from multilayers with roughened interfaces, including islands and miscut-induced steps. The model calculates X-ray intensity profiles that can be compared to experimental data to deduce the morphology of interfaces, such as mean island size and average step height. The model is applied to experimental data from a Si/Ge multilayer and results in values consistent with AFM images.
(1) The document discusses linear perturbations of the metric around a Schwarzschild black hole. It derives the Regge-Wheeler equation, which governs axial perturbations and takes the form of a wave equation with an effective potential.
(2) It shows that the Regge-Wheeler potential has a maximum just outside the event horizon. This allows it to be considered as a scattering potential barrier for wave packets.
(3) It concludes that Schwarzschild black holes are stable under smooth, compactly supported exterior perturbations, as these perturbations will remain bounded for all times according to properties of the Regge-Wheeler equation and solutions to the Schrodinger equation.
Nonclassical Properties of Even and Odd Semi-Coherent StatesIOSRJAP
Even and odd semi-coherent states have been introduced. Some of the nonclasscial properties of the states are studied in terms of the quadrature squeezing as well as sub-Poissonian photon statistics. The Husimi– Kano Q-function and the phase distribution in the framework of Pegg and Barnett formalism, are also discussed.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
Basic Boundary Conditions in OpenFOAM v2.4Fumiya Nozaki
This document discusses different types of boundary conditions in OpenFOAM including:
1) Steady and time-varying boundary conditions such as Dirichlet, Neumann, and Robin conditions.
2) Periodic and symmetry boundary conditions that can be used for repeating geometries.
3) Mixed boundary conditions that are a weighted combination of fixed value and fixed gradient conditions, controlled by a weighting parameter.
4) Direction mixed boundary conditions for vector fields that apply different conditions in different directions using a weighting tensor.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
A smooth-exit the-phase-transition-to-slow-roll-eternal-inflationmirgytoo
This document summarizes research on the phase transition to eternal inflation. It begins by introducing the concept of eternal inflation occurring when quantum fluctuations dominate over classical drift. The authors argue that even in the eternal inflation regime, perturbations of the geometry and interactions remain perturbative, allowing quantitative analysis. They aim to precisely define the critical condition for eternal inflation and calculate statistics of the reheating volume to understand the phase transition.
1998 characterisation of multilayers by x ray reflectionpmloscholte
This document presents a theoretical model for characterizing multilayers using X-ray reflection. The model includes refraction effects and describes diffuse scattering from multilayers with roughened interfaces, including islands and miscut-induced steps. The model calculates X-ray intensity profiles that can be compared to experimental data to deduce the morphology of interfaces, such as mean island size and average step height. The model is applied to experimental data from a Si/Ge multilayer and results in values consistent with AFM images.
- The document discusses issues with achieving a true free-free boundary condition when experimentally testing structures. A true free-free condition allows all parts of the structure to move freely without constraints.
- It analyzes a beam model supported on springs at both ends to simulate a free-free condition. Having very soft springs results in rigid body modes of motion that match a true free-free condition. Stiffer springs distort the modes away from free-free.
- Measured frequencies from the beam model match theoretical free-free frequencies when the ratio of the first elastic mode frequency to highest rigid body frequency is above 10. Below this, the first elastic mode is sensitive to the boundary conditions imposed by the springs.
This document discusses phase and phase difference in waves. It provides definitions of standing waves, which have nodes where the wave disturbance is minimal, and traveling waves, which do not have nodes. The displacement equation for traveling waves includes a term for velocity (v) to account for waves moving in the positive or negative x direction. Phase describes the wave's position and time, while phase difference is the difference in phases between two waves at the same time. A worked example shows waves A and B with a phase difference of 3π/2, and calculates their distance apart as 3λ/4, where λ is the wavelength.
Constraints restrict the motion of a system to fewer than three independent coordinates. Holonomic constraints can be expressed as mathematical equations involving the coordinates and time, while non-holonomic constraints cannot. Constraints introduce difficulties by making coordinates dependent and introducing constraint forces. For holonomic constraints, generalized coordinates can be used to eliminate dependent coordinates and reduce the number of degrees of freedom. D'Alembert's principle states that the sum of the work done by actual and constraint forces during virtual displacements is zero, and can be used to derive the Lagrangian equations of motion for constrained systems.
Momentum flux in the electromagnetic fieldSergio Prats
This article shows how to get the flux of momentum in the electromagnetic field from the Maxwell stress tensor in the scope of classical electromagnetism.
This document proposes a new distribution model for earthquake magnitudes and intensities that addresses limitations of the traditional Gutenberg-Richter distribution model. Specifically, it introduces the generalized exponential distribution, which allows for an upper bound on magnitudes. The distribution is determined by analyzing both the overall distribution of magnitudes/intensities as well as the distribution of annual maximum values. An example is provided analyzing the intensity data of earthquakes in Zagreb over a 100-year period, finding that the generalized exponential distribution provides a good fit to the data.
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
A Novel Space-time Discontinuous Galerkin Method for Solving of One-dimension...TELKOMNIKA JOURNAL
In this paper we propose a high-order space-time discontinuous Galerkin (STDG) method for
solving of one-dimensional electromagnetic wave propagations in homogeneous medium. The STDG
method uses finite element Discontinuous Galerkin discretizations in spatial and temporal domain
simultaneously with high order piecewise Jacobi polynomial as the basis functions. The algebraic
equations are solved using Block Gauss-Seidel iteratively in each time step. The STDG method is
unconditionally stable, so the CFL number can be chosen arbitrarily. Numerical examples show that the
proposed STDG method is of exponentially accuracy in time.
The usual theory of inflation breaks down in eternal inflation. We derive a
dual description of eternal inflation in terms of a deformed Euclidean CFT located at the
threshold of eternal inflation. The partition function gives the amplitude of different geometries
of the threshold surface in the no-boundary state. Its local and global behavior
in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal
to the round three-sphere and essentially zero for surfaces with negative curvature.
Based on this we conjecture that the exit from eternal inflation does not produce an infinite
fractal-like multiverse, but is finite and reasonably smooth
The document discusses the line joining two bodies and the gravitational constant. The line joining two bodies M1 and M2 is equal to r(M1M2) divided by r squared. The gravitational constant is described as a universal constant.
Divergence is a mathematical operation on a vector field that calculates the density of the vector field's source at a given point. It measures the flux of a vector field out of an infinitesimally small volume element at that point. Divergence equals flux per unit volume, so a positive divergence means flux is spreading out from the point and a negative divergence means flux is converging at the point. Divergence can be represented in Cartesian, cylindrical, or spherical coordinate systems using partial derivatives and the del operator.
The document shows a calculation of the torque equilibrium for a rod with forces applied at different distances from the fulcrum. The perpendicular force F⊥ required for equilibrium is calculated to be 8.7 N based on a 40 N force applied 2.6 cm from the fulcrum and the distance to the fulcrum being 12 cm.
This document discusses calculating the total strain or change in length of bars of different lengths and diameters that are subjected to the same loading. It explains that the total change in length is the sum of the changes in each bar, which can be calculated using simple equations involving the force, length, diameter, modulus of elasticity, and cross-sectional area of each bar. Free body diagrams and dividing the load based on equilibrium conditions are also recommended.
The document discusses Michael Faraday's experiments with static electric fields and induced electromotive forces using concentric metallic spheres. It then summarizes Gauss's law and how it can be applied to determine the electric flux density D for various charge distributions like a point charge or line charge. It shows that a cylindrical surface is suitable for applying Gauss's law to a line charge, as the radial component of D is the only non-zero component and it is everywhere normal to the cylindrical surface. Finally, it derives one of Maxwell's equations from Gauss's law by taking the divergence of the electric flux density D.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary DimensionsMichaelRabinovich
This document summarizes research on constructing solutions to Einstein's equations in arbitrary dimensions d that are dual to fluid dynamics on the boundary. The key points are:
1. Solutions are constructed perturbatively to second order in a boundary derivative expansion and are parameterized by a boundary velocity and temperature field obeying the Navier-Stokes equations.
2. The bulk metric dual to an arbitrary fluid flow on a weakly curved boundary is computed explicitly to second order.
3. The boundary stress tensor dual to these solutions is also computed and expressed in a manifestly Weyl-covariant form involving the velocity, temperature, and their derivatives.
4. Properties of the solutions like the event horizon location and an
Vibration advance longi and trans equation.Khalis Karim
1. A longitudinal disturbance along a stretched string or bar can propagate as a wave.
2. The wave equation derived to describe this propagation is analogous to the wave equation for transverse waves on a string.
3. The speed of the longitudinal wave is determined by the Young's modulus and linear density of the material, similar to the speed of transverse waves on a string.
- The document derives the second order Friedmann equations from the quantum corrected Raychaudhuri equation (QRE), which includes quantum corrections terms.
- One correction term can be interpreted as dark energy/cosmological constant with the observed density value, providing an explanation for the coincidence problem.
- The other correction term can be interpreted as a radiation term in the early universe that prevents the formation of a big bang singularity and predicts an infinite age for the universe by avoiding a divergence in the Hubble parameter or its derivative at any finite time in the past.
In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
(1) The document derives an effective Einstein-Hilbert action from the original Einstein-Hilbert action by splitting it into two terms, one of which is a total divergence that can be ignored. (2) It then uses the resulting effective action to derive Einstein's field equations. (3) The derivation makes use of properties of the metric tensor, Christoffel symbols, Ricci tensor, and Riemann tensor to rewrite terms and arrive at the effective action and field equations.
The document discusses generalized distributions and the central limit theorem. It begins by covering the normal distribution and how the central limit theorem explains why it is so common. It then discusses power law distributions and how they are more common in complex systems. The generalized central limit theorem states that the sum of random variables with power law distributions will tend towards a stable distribution. Stable distributions include the normal distribution as a special case and exhibit power law behavior for large values.
1) The document discusses the relationship between transforming entangled quantum states via local operations and classical communication (LOCC) and the theory of majorization from linear algebra.
2) Nielsen's theorem states that one entangled state can be transformed into another via LOCC if and only if the vector of eigenvalues of one state is majorized by the vector of eigenvalues of the other state.
3) The proof of Nielsen's theorem relies on five key properties, including that any two-way classical communication in an LOCC protocol can be simulated by a one-way communication protocol.
- The document discusses issues with achieving a true free-free boundary condition when experimentally testing structures. A true free-free condition allows all parts of the structure to move freely without constraints.
- It analyzes a beam model supported on springs at both ends to simulate a free-free condition. Having very soft springs results in rigid body modes of motion that match a true free-free condition. Stiffer springs distort the modes away from free-free.
- Measured frequencies from the beam model match theoretical free-free frequencies when the ratio of the first elastic mode frequency to highest rigid body frequency is above 10. Below this, the first elastic mode is sensitive to the boundary conditions imposed by the springs.
This document discusses phase and phase difference in waves. It provides definitions of standing waves, which have nodes where the wave disturbance is minimal, and traveling waves, which do not have nodes. The displacement equation for traveling waves includes a term for velocity (v) to account for waves moving in the positive or negative x direction. Phase describes the wave's position and time, while phase difference is the difference in phases between two waves at the same time. A worked example shows waves A and B with a phase difference of 3π/2, and calculates their distance apart as 3λ/4, where λ is the wavelength.
Constraints restrict the motion of a system to fewer than three independent coordinates. Holonomic constraints can be expressed as mathematical equations involving the coordinates and time, while non-holonomic constraints cannot. Constraints introduce difficulties by making coordinates dependent and introducing constraint forces. For holonomic constraints, generalized coordinates can be used to eliminate dependent coordinates and reduce the number of degrees of freedom. D'Alembert's principle states that the sum of the work done by actual and constraint forces during virtual displacements is zero, and can be used to derive the Lagrangian equations of motion for constrained systems.
Momentum flux in the electromagnetic fieldSergio Prats
This article shows how to get the flux of momentum in the electromagnetic field from the Maxwell stress tensor in the scope of classical electromagnetism.
This document proposes a new distribution model for earthquake magnitudes and intensities that addresses limitations of the traditional Gutenberg-Richter distribution model. Specifically, it introduces the generalized exponential distribution, which allows for an upper bound on magnitudes. The distribution is determined by analyzing both the overall distribution of magnitudes/intensities as well as the distribution of annual maximum values. An example is provided analyzing the intensity data of earthquakes in Zagreb over a 100-year period, finding that the generalized exponential distribution provides a good fit to the data.
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
A Novel Space-time Discontinuous Galerkin Method for Solving of One-dimension...TELKOMNIKA JOURNAL
In this paper we propose a high-order space-time discontinuous Galerkin (STDG) method for
solving of one-dimensional electromagnetic wave propagations in homogeneous medium. The STDG
method uses finite element Discontinuous Galerkin discretizations in spatial and temporal domain
simultaneously with high order piecewise Jacobi polynomial as the basis functions. The algebraic
equations are solved using Block Gauss-Seidel iteratively in each time step. The STDG method is
unconditionally stable, so the CFL number can be chosen arbitrarily. Numerical examples show that the
proposed STDG method is of exponentially accuracy in time.
The usual theory of inflation breaks down in eternal inflation. We derive a
dual description of eternal inflation in terms of a deformed Euclidean CFT located at the
threshold of eternal inflation. The partition function gives the amplitude of different geometries
of the threshold surface in the no-boundary state. Its local and global behavior
in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal
to the round three-sphere and essentially zero for surfaces with negative curvature.
Based on this we conjecture that the exit from eternal inflation does not produce an infinite
fractal-like multiverse, but is finite and reasonably smooth
The document discusses the line joining two bodies and the gravitational constant. The line joining two bodies M1 and M2 is equal to r(M1M2) divided by r squared. The gravitational constant is described as a universal constant.
Divergence is a mathematical operation on a vector field that calculates the density of the vector field's source at a given point. It measures the flux of a vector field out of an infinitesimally small volume element at that point. Divergence equals flux per unit volume, so a positive divergence means flux is spreading out from the point and a negative divergence means flux is converging at the point. Divergence can be represented in Cartesian, cylindrical, or spherical coordinate systems using partial derivatives and the del operator.
The document shows a calculation of the torque equilibrium for a rod with forces applied at different distances from the fulcrum. The perpendicular force F⊥ required for equilibrium is calculated to be 8.7 N based on a 40 N force applied 2.6 cm from the fulcrum and the distance to the fulcrum being 12 cm.
This document discusses calculating the total strain or change in length of bars of different lengths and diameters that are subjected to the same loading. It explains that the total change in length is the sum of the changes in each bar, which can be calculated using simple equations involving the force, length, diameter, modulus of elasticity, and cross-sectional area of each bar. Free body diagrams and dividing the load based on equilibrium conditions are also recommended.
The document discusses Michael Faraday's experiments with static electric fields and induced electromotive forces using concentric metallic spheres. It then summarizes Gauss's law and how it can be applied to determine the electric flux density D for various charge distributions like a point charge or line charge. It shows that a cylindrical surface is suitable for applying Gauss's law to a line charge, as the radial component of D is the only non-zero component and it is everywhere normal to the cylindrical surface. Finally, it derives one of Maxwell's equations from Gauss's law by taking the divergence of the electric flux density D.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary DimensionsMichaelRabinovich
This document summarizes research on constructing solutions to Einstein's equations in arbitrary dimensions d that are dual to fluid dynamics on the boundary. The key points are:
1. Solutions are constructed perturbatively to second order in a boundary derivative expansion and are parameterized by a boundary velocity and temperature field obeying the Navier-Stokes equations.
2. The bulk metric dual to an arbitrary fluid flow on a weakly curved boundary is computed explicitly to second order.
3. The boundary stress tensor dual to these solutions is also computed and expressed in a manifestly Weyl-covariant form involving the velocity, temperature, and their derivatives.
4. Properties of the solutions like the event horizon location and an
Vibration advance longi and trans equation.Khalis Karim
1. A longitudinal disturbance along a stretched string or bar can propagate as a wave.
2. The wave equation derived to describe this propagation is analogous to the wave equation for transverse waves on a string.
3. The speed of the longitudinal wave is determined by the Young's modulus and linear density of the material, similar to the speed of transverse waves on a string.
- The document derives the second order Friedmann equations from the quantum corrected Raychaudhuri equation (QRE), which includes quantum corrections terms.
- One correction term can be interpreted as dark energy/cosmological constant with the observed density value, providing an explanation for the coincidence problem.
- The other correction term can be interpreted as a radiation term in the early universe that prevents the formation of a big bang singularity and predicts an infinite age for the universe by avoiding a divergence in the Hubble parameter or its derivative at any finite time in the past.
In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
(1) The document derives an effective Einstein-Hilbert action from the original Einstein-Hilbert action by splitting it into two terms, one of which is a total divergence that can be ignored. (2) It then uses the resulting effective action to derive Einstein's field equations. (3) The derivation makes use of properties of the metric tensor, Christoffel symbols, Ricci tensor, and Riemann tensor to rewrite terms and arrive at the effective action and field equations.
The document discusses generalized distributions and the central limit theorem. It begins by covering the normal distribution and how the central limit theorem explains why it is so common. It then discusses power law distributions and how they are more common in complex systems. The generalized central limit theorem states that the sum of random variables with power law distributions will tend towards a stable distribution. Stable distributions include the normal distribution as a special case and exhibit power law behavior for large values.
1) The document discusses the relationship between transforming entangled quantum states via local operations and classical communication (LOCC) and the theory of majorization from linear algebra.
2) Nielsen's theorem states that one entangled state can be transformed into another via LOCC if and only if the vector of eigenvalues of one state is majorized by the vector of eigenvalues of the other state.
3) The proof of Nielsen's theorem relies on five key properties, including that any two-way classical communication in an LOCC protocol can be simulated by a one-way communication protocol.
The document discusses the normal distribution, also called the Gaussian distribution, which is a very commonly used probability distribution in statistics. It has two parameters: the mean μ, which is the expected value, and the standard deviation σ. The normal distribution is symmetric around the mean and bell-shaped. It is useful because of the central limit theorem and is applied when variables are expected to be the sum of many independent processes.
Geometric properties for parabolic and elliptic pdeSpringer
This document discusses recent advances in fractional Laplacian operators and related problems in partial differential equations and geometric measure theory. Specifically, it addresses three key topics:
1. Symmetry problems for solutions of the fractional Allen-Cahn equation and whether solutions only depend on one variable like in the classical case. The answer is known to be positive for some dimensions and fractional exponents but remains open in general.
2. The Γ-convergence of functionals involving the fractional Laplacian as the small parameter ε approaches zero. This characterizes the asymptotic behavior and relates to fractional notions of perimeter.
3. Regularity of interfaces as the fractional exponent s approaches 1/2 from above, which corresponds to a critical threshold
Existence of Hopf-Bifurcations on the Nonlinear FKN ModelIJMER
This document discusses Hopf bifurcations, which occur when the stability of an equilibrium point in a nonlinear dynamical system changes as a parameter is varied, resulting in the emergence of periodic solutions. It first provides background on limit cycles and the Hopf bifurcation theorem. It then determines the indicator k, whose sign indicates whether a Hopf bifurcation is supercritical (k<0) or subcritical (k>0). The analysis is extended to three-dimensional systems by reducing them to a two-dimensional system near the equilibrium point. Finally, the document applies this analysis to the Field-Körös-Noyen (FKN) chemical reaction model to determine its supercritical and subcritical Hopf bifurcations.
The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.
This document reviews research on the convergence of perturbation series in quantum field theory. It discusses Dyson's argument that perturbation series in quantum electrodynamics (QED) have zero radius of convergence due to vacuum instability when the coupling constant is negative. Large-order estimates show that perturbation series coefficients grow factorially fast in quantum mechanics and field theories. Finally, it describes the method of Borel summation, which may allow extracting the exact physical quantity from a divergent perturbation series through a unique mapping.
On elements of deterministic chaos and cross links in non- linear dynamical s...iosrjce
In this paper we examine the existing definitions of deterministic chaos and the characterisation of
its various ingredients. We then make use of some classical examples to provide cross links between the
different chaotic behaviour of some simple but interesting maps which are then explained in a precise manner.
This document is Scott Shermer's master's thesis on instantons and perturbation theory in the 1-D quantum mechanical quartic oscillator. It begins by reviewing the harmonic oscillator and perturbation theory. It then discusses non-perturbative phenomena like instantons and Borel resummation. The focus is on obtaining the ground state energy of the quartic oscillator Hamiltonian using both perturbative and non-perturbative techniques, and addressing ambiguities that arise for negative coupling.
This document provides an overview of key concepts in probability and statistics that are used in population genetics. It defines probability, random variables, and common probability distributions including the binomial, Poisson, normal, chi-square, gamma, and beta distributions. It also discusses concepts such as expectation, mean, and variance that are used to characterize probability distributions. The summary focuses on defining key terms and concepts rather than providing details about specific equations.
Logistic Regression, Linear and Quadratic Discriminant Analyses, and KNN Tarek Dib
A summary of the classification methods: Logistic regression, Linear Discriminant Analysis, Quadratic Discriminant Analysis and a comparison of these three methods with K-Nearest Neighbors algorithm.
This document provides a summary of a project report on bifurcation analysis and its applications. It discusses key concepts in nonlinear systems such as equilibrium points, stability, linearization, and bifurcations including saddle node, transcritical, pitchfork and Hopf bifurcations. Examples are given to illustrate each type of bifurcation. Population models involving competition and prey-predator interactions are also discussed. The document outlines the contents which cover preliminary remarks, local theory of nonlinear systems, different types of bifurcations, and applications to population models.
The document discusses optical waveguide analysis using the Beam Propagation Method (BPM). BPM is a numerical technique to determine electromagnetic fields inside complicated waveguide structures like Y-couplers. It works by decomposing the optical mode into plane waves that are propagated through the structure and then recombined. The author implemented BPM in MATLAB to simulate double slit diffraction and a Gaussian beam. Code examples are provided for the Gaussian beam, BPM algorithm, and double slit simulation.
Stochastic Optimal Control & Information Theoretic DualitiesHaruki Nishimura
In this series of two presentations we discuss two different, yet related, approaches to general nonlinear stochastic control: stochastic optimal control and information theoretic control. This week is the first half and the main focus is on the stochastic optimal control. We begin our discussion by reviewing the deterministic case and see that Bellman's principle of optimality leads to the Hamilton-Jacobi-Bellman equation. We then learn how the problem can be extended to handle stochasticity in the system dynamics, where the Wiener noise affects the resulting value function. Difficulties around solving the stochastic dynamic program are presented, leading to the information theoretic control that is based on another notion of optimality.
Regression is a statistical technique for modeling the relationship between variables. Simple linear regression fits a straight line to the data to predict a dependent variable from an independent variable. Multiple linear regression uses two or more independent variables to predict the dependent variable. The output of regression includes coefficients, R-squared, standard error, and an equation to make predictions with new data.
The document summarizes research on spacey random walks, which are a type of stochastic process that can model higher-order Markov chains. Key points:
1. Spacey random walks generalize higher-order Markov chains by forgetting history but pretending to remember a random previous state, with the stationary distribution given by a tensor eigenvector of the transition tensor.
2. This connects higher-order Markov chains to tensor eigenvectors and provides a stochastic interpretation of tensor eigenvectors as stationary distributions.
3. The dynamics of spacey random walks can be modeled as an ordinary differential equation, allowing tensor eigenvectors to be computed by numerically integrating the dynamical system.
The usual theory of inflation breaks down in eternal inflation. We derive a dual description of eternal inflation in terms of a deformed Euclidean CFT located at the threshold of eternal inflation. The partition function gives the amplitude of different geometries of the threshold surface in the no-boundary state. Its local and global behavior in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal to the round three-sphere and essentially zero for surfaces with negative curvature. Based on this we conjecture that the exit from eternal inflation does not produce an infinite fractal-like multiverse, but is finite and reasonably smooth.
A brief history of generative models for power law and lognormal ...sugeladi
This document provides a brief history of generative models for power law and lognormal distributions. It discusses how:
1) Models of preferential attachment in dynamically growing networks like the web graph can lead to power law distributions for properties like node degree.
2) Lognormal and power law distributions are intrinsically connected, as small variations in basic generative models can produce either distribution.
3) Debates over whether empirical observations follow power laws or lognormals have occurred across many fields, as these distributions can appear similar. Understanding the history provides useful context for current issues.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is large.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is sufficiently large.
Similar to Above under and beyond brownian motion talk (20)
This document discusses attractors and dynamical systems through examples like number games and the Lorenz system. It begins by using a 3-digit number game to illustrate the concept of attractors, showing how different starting numbers eventually converge to the fixed point attractors of 495 and 000. The document then discusses how this idea can be applied in other contexts like numerical analysis, economics, and system identification. It also introduces different types of attractors like fixed points, periodic attractors, and strange attractors. Finally, it summarizes recent work extending the Lorenz system to a one-parameter family and exploring relationships between different 3D autonomous systems.
This document summarizes recent discoveries and challenges in chaos theory presented by Xiong Wang from the Centre for Chaos and Complex Networks at City University of Hong Kong. Some key points discussed include: discovering a chaotic system with one stable equilibrium point, using symmetry to create systems with multiple equilibria whose stability can be tuned, and open questions around reconciling local stability with global chaotic behavior. The document concludes by acknowledging that chaos is a global phenomenon not precluded by local stability near equilibria.
This document is a lecture on the concept of mass in Newtonian mechanics and Lagrangian mechanics. It begins with an introduction that defines physics as the study of matter, space-time and motion. It then discusses mass as a way to model matter. The lecture will cover the concept of mass in Newtonian mechanics, including inertia mass, gravitational mass and others. It will also discuss mass in Lagrangian mechanics. The goal is to see how the understanding of mass has developed over time through different theories.
The document discusses challenging implicit assumptions in physics, such as differentiability. It argues that general relativity is incomplete because it only considers differentiable coordinate transformations, rather than arbitrary transformations. Quantum phenomena like uncertainty and non-commutativity could emerge from spacetime non-differentiability. The author proposes a complete theory with non-differentiable spacetime where physical laws remain valid under differentiable or non-differentiable transformations, reconciling quantum effects.
This document discusses discoveries and challenges in chaos theory. It begins by asking basic questions about the mechanisms that generate chaos and the types of systems that can exhibit chaotic dynamics. It then discusses concepts like equilibria, Jacobian matrices, Hartman-Grobman and Shilnikov theorems as they relate to the stability of equilibria and generation of chaos. Examples of chaotic systems like Lorenz, Chen and Rossler are provided. The document finds a chaotic system with one stable equilibrium, then extends this finding to systems with no equilibria, two equilibria and tunable stability. It concludes that chaos is a global phenomenon while local stability exists near equilibria.
Chen's attractor family and Rossler attractor family are two types of attractors that are being compared in this document. The document notes there is a parallel gradual change process between the two families that is interesting and asks if there are any underlying connections between them or possibilities for other extensions.
Xiong Wang studied the relationship between chaotic dynamical systems and the stability of equilibria. Wang presents a three-dimensional system with one stable equilibrium that exhibits chaotic behavior for certain parameter values, as shown by a positive largest Lyapunov exponent and bifurcation diagram. This contrasts with previous systems that generate chaos via unstable equilibria. Open questions remain about rigorously proving chaos and the coexistence of the stable point attractor and strange attractor.
1. Above and Under
Brownian Motion
Brownian Motion , Fractional Brownian
Motion , Levy Flight, and beyond
Seminar Talk at Beijing Normal University
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong 1
2. Outline
Discrete Time Random walks
Ordinary random walks
Lévy flights
Generalized central limit
theorem
Stable distribution
Continuous time random walks
Ordinary Diffusion Lévy Flights
Fractional Brownian motion
(subdiffusion) Ambivalent processes 2
10. Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 / | x | α + 1 where
0 < α < 2 (and therefore having infinite
variance) will tend to a stable distribution
f(x;α,0,c,0) as the number of variables grows.
10
11. Stable distribution
In probability theory, a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution, up to
location and scale parameters.
The stable distribution family is also
sometimes referred to as the Lévy alpha-
stable distribution.
11
12. Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c, respectively, and two
shape parameters β and α, roughly
corresponding to measures of asymmetry
and concentration, respectively (see the
figures).
C:chaosTalklevyStableDensityFunction.cdf
16. Unified normal and power law
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ; the
skewness parameter β has no effect
The asymptotic behavior is described, for α < 2
16
17. Log-log plot of skewed centered stable distribution PDF's showing the
power law behavior for large x. Again the slope of the linear portions
is equal to -(α+1)
28. Concluding Remarks
The ratio of the exponents α/β resembles the
interplay between sub- and superdiffusion.
For β < 2α the ambivalent CTRW is effectively
superdiffusive,
for β > 2α effectively subdiffusive.
For β = 2α the process exhibits the same
scaling as ordinary Brownian motion, despite
the crucial difference of infinite moments and a
non-Gaussian shape of the pdf W(x, t).
28
29. Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: wangxiong8686@gmail.com
29