This document discusses synchronization of oscillatory systems. It introduces a model where the phase of an oscillator φi evolves according to its natural frequency ωi plus a coupling term dependent on the phase differences between oscillators φi - φj. The coupling is described by a phase response curve Z(φ) which can depend on the system. Synchronization occurs when the oscillators lock to a common frequency and their phases cluster.
1. Gibbs sampling is a technique for drawing samples from probability distributions by iteratively sampling each variable conditioned on the current values of the other variables. It can be used to sample from Markov random fields and Bayesian networks.
2. An Ising model is a Markov random field with binary variables on a grid that are correlated with their neighbors. Gibbs sampling in an Ising model samples each variable based on its neighbors' current values.
3. Boltzmann machines generalize the Ising model to arbitrary graph structures between variables. Restricted Boltzmann machines and Hopfield networks are specific types of Boltzmann machines.
The document discusses calculating and interpreting the Green's function or propagator for systems with polarons. It begins by defining the polaron and stating the quantity of interest is the one-particle retarded propagator. It then provides examples of calculating the propagator exactly for simple non-interacting and impurity models to build intuition. Finally, it outlines discussing the Holstein polaron model in bulk materials and how the polaron may be affected near a surface.
Different Quantum Spectra For The Same Classical Systemvcuesta
The document discusses how the one-dimensional and two-dimensional isotropic harmonic oscillators, which correspond to a single classical system each, can have multiple quantum spectra.
Specifically, it shows that:
1) The one-dimensional oscillator admits two different quantum spectra, obtained by defining creation/annihilation operators in two different ways.
2) Similarly, the two-dimensional oscillator is shown to have four distinct quantum spectra, as the x and y directions can be quantized independently using two different operator definitions each.
3) This demonstrates that the same classical system can lead to multiple quantum descriptions and spectra, contrary to some previous works that obtained a unique spectrum for different classical formulations.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
This document summarizes a talk on Lorentz surfaces in pseudo-Riemannian space forms with horizontal reflector lifts. It introduces examples of Lorentz surfaces with zero mean curvature in these spaces. It also discusses reflector spaces and horizontal reflector lifts, and presents a rigidity theorem stating that if two isometric immersions from a Lorentz surface to a pseudo-Riemannian space form both have horizontal reflector lifts and satisfy certain curvature conditions, then the immersions must differ by an isometry of the target space.
Quantum Transitions And Its Evolution For Systems With Canonical And Noncanon...vcuesta
1. The document studies the quantum transitions and time evolution of the phase space coordinates for the one-dimensional harmonic oscillator with both canonical and noncanonical symplectic structures.
2. For the canonical case, the solutions to the classical equations of motion and the quantum transitions between energy levels are obtained. The time evolution of the transition amplitudes between states is also determined.
3. An analogous analysis is performed for the noncanonical case, where modified commutation relations and modified expressions for the creation/annihilation operators are obtained. The quantum transitions and their time evolution are determined.
This document discusses synchronization of oscillatory systems. It introduces a model where the phase of an oscillator φi evolves according to its natural frequency ωi plus a coupling term dependent on the phase differences between oscillators φi - φj. The coupling is described by a phase response curve Z(φ) which can depend on the system. Synchronization occurs when the oscillators lock to a common frequency and their phases cluster.
1. Gibbs sampling is a technique for drawing samples from probability distributions by iteratively sampling each variable conditioned on the current values of the other variables. It can be used to sample from Markov random fields and Bayesian networks.
2. An Ising model is a Markov random field with binary variables on a grid that are correlated with their neighbors. Gibbs sampling in an Ising model samples each variable based on its neighbors' current values.
3. Boltzmann machines generalize the Ising model to arbitrary graph structures between variables. Restricted Boltzmann machines and Hopfield networks are specific types of Boltzmann machines.
The document discusses calculating and interpreting the Green's function or propagator for systems with polarons. It begins by defining the polaron and stating the quantity of interest is the one-particle retarded propagator. It then provides examples of calculating the propagator exactly for simple non-interacting and impurity models to build intuition. Finally, it outlines discussing the Holstein polaron model in bulk materials and how the polaron may be affected near a surface.
Different Quantum Spectra For The Same Classical Systemvcuesta
The document discusses how the one-dimensional and two-dimensional isotropic harmonic oscillators, which correspond to a single classical system each, can have multiple quantum spectra.
Specifically, it shows that:
1) The one-dimensional oscillator admits two different quantum spectra, obtained by defining creation/annihilation operators in two different ways.
2) Similarly, the two-dimensional oscillator is shown to have four distinct quantum spectra, as the x and y directions can be quantized independently using two different operator definitions each.
3) This demonstrates that the same classical system can lead to multiple quantum descriptions and spectra, contrary to some previous works that obtained a unique spectrum for different classical formulations.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
This document summarizes a talk on Lorentz surfaces in pseudo-Riemannian space forms with horizontal reflector lifts. It introduces examples of Lorentz surfaces with zero mean curvature in these spaces. It also discusses reflector spaces and horizontal reflector lifts, and presents a rigidity theorem stating that if two isometric immersions from a Lorentz surface to a pseudo-Riemannian space form both have horizontal reflector lifts and satisfy certain curvature conditions, then the immersions must differ by an isometry of the target space.
Quantum Transitions And Its Evolution For Systems With Canonical And Noncanon...vcuesta
1. The document studies the quantum transitions and time evolution of the phase space coordinates for the one-dimensional harmonic oscillator with both canonical and noncanonical symplectic structures.
2. For the canonical case, the solutions to the classical equations of motion and the quantum transitions between energy levels are obtained. The time evolution of the transition amplitudes between states is also determined.
3. An analogous analysis is performed for the noncanonical case, where modified commutation relations and modified expressions for the creation/annihilation operators are obtained. The quantum transitions and their time evolution are determined.
This document summarizes Ja-Keoung Koo's presentation on structure from motion. It discusses image formation, the structure from motion pipeline with calibrated cameras, and the 8-point algorithm. The key points are:
1. Image formation maps 3D world points to 2D image points using a camera's intrinsic and extrinsic parameters.
2. Structure from motion with calibrated cameras recovers 3D structure and camera motion from 2D correspondences using the essential matrix and 8-point algorithm.
3. The 8-point algorithm finds the essential matrix from point correspondences, decomposes it to recover the rotation and translation between views.
The document outlines research on developing optimal finite difference grids for solving elliptic and parabolic partial differential equations (PDEs). It introduces the motivation to accurately compute Neumann-to-Dirichlet (NtD) maps. It then summarizes the formulation and discretization of model elliptic and parabolic PDE problems, including deriving the discrete NtD map. It presents results on optimal grid design and the spectral accuracy achieved. Future work is proposed on extending the NtD map approach to non-uniformly spaced boundary data.
The document describes the support vector machine (SVM) algorithm for classification. It discusses how SVM finds the optimal separating hyperplane between two classes by maximizing the margin between them. It introduces the concepts of support vectors, Lagrange multipliers, and kernels. The sequential minimal optimization (SMO) algorithm is also summarized, which breaks the quadratic optimization problem of SVM training into smaller subproblems to optimize two Lagrange multipliers at a time.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
A crash coarse in stochastic Lyapunov theory for Markov processes (emphasis is on continuous time)
See also the survey for models in discrete time,
https://netfiles.uiuc.edu/meyn/www/spm_files/MarkovTutorial/MarkovTutorialUCSB2010.html
The document discusses dual gravitons in AdS4/CFT3 and proposes that the holographic Cotton tensor can play the role of the dual operator to the metric in the boundary CFT.
The Cotton tensor is a symmetric, traceless and conserved tensor that can be constructed from the linearized metric. It is the stress-energy tensor of gravitational Chern-Simons theory. The document proposes that the Cotton tensor and stress-energy tensor form a dual pair under the holographic duality, with the Cotton tensor acting as the operator corresponding to fluctuations of the metric in the bulk. This is motivated by properties of gravitational instantons and self-dual solutions in AdS.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
- Semi Regular Meshes can be subdivided using regular 1:4 subdivision or represented as Spherical Geometry Images mapped to the unit sphere.
- Subdivision Surfaces are generated by applying local interpolators repeatedly to refine a coarse control mesh. Common subdivision schemes include Linear, Butterfly, and Loop which are demonstrated in examples.
- Biorthogonal Wavelets can be constructed on meshes using a Lifting Scheme to create wavelet coefficients with vanishing moments, allowing for compression of mesh signals. Invariant neighborhoods are used to analyze the refinement of meshes across scales.
The document discusses support vector machines (SVM) for classification. It begins by introducing the concepts of maximum margin hyperplane and soft margin. It then formulates the SVM optimization problem to find the maximum margin hyperplane using Lagrange multipliers. The optimization problem is solved using Kuhn-Tucker conditions to obtain the dual formulation only in terms of the support vectors. Kernel tricks are introduced to handle non-linear decision boundaries. The formulation is extended to allow for misclassification errors by introducing slack variables ξ and a penalty parameter C.
The document discusses triangular norm (t-norm) based kernel functions and their application to kernel k-means clustering. It introduces common kernel functions and describes how t-norms can be used to create new kernel functions. Several parameterized and non-parameterized t-norm based kernel functions are presented. The document then details experiments applying various kernel functions including t-norm kernels to four datasets, evaluating the results using adjusted rand index scores. The best performing kernels for each dataset are identified, with some t-norm kernels performing comparably or better than traditional kernels.
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
This document discusses and compares several different probabilistic models for sequence labeling tasks, including Hidden Markov Models (HMMs), Maximum Entropy Markov Models (MEMMs), and Conditional Random Fields (CRFs).
It provides mathematical formulations of HMMs, describing how to calculate the most likely label sequence using the Viterbi algorithm. It then introduces MEMMs, which address some limitations of HMMs by incorporating arbitrary, overlapping features. CRFs are presented as an improvement over MEMMs that models the conditional probability of labels given observations, avoiding the label bias problem of MEMMs. The document concludes by describing how to train CRF models using generalized iterative scaling.
This document summarizes some statistical models used for calibrating imperfect mathematical models. It discusses three main approaches:
1. Gaussian stochastic process (GaSP) calibration, which models bias as a Gaussian process. This is commonly used but can produce inconsistent parameter estimates.
2. L2 calibration, which estimates reality separately from the model before estimating parameters. However, it does not use model information.
3. Scaled Gaussian stochastic process (S-GaSP) calibration, which constrains the GaSP to have a fixed L2 norm. This satisfies predicting reality and calibrated parameters. The S-GaSP is equivalent to penalized kernel ridge regression.
The document analyzes the nonparametric regression setting
There are three possible ROC's:
1. Outside all poles (a, b, c)
2. Between innermost and outermost pole
3. Inside all poles
So the possible ROC's are:
1. Outside circle through a, b, c
2. Annular region between a, c
3. Inside circle through a, b, c
a b c Re
The z-Transform
Important z-Transform Pairs
Important z-Transform Pairs
1. Unit Impulse: δ(n)
1, if n = 0
δ(n) = 0, otherwise
1
X(z) =
2.
D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid...dishii
An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. Interval-based computation of hybrid systems is often difficult, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations.We formulate the problem of detecting a discrete change in hybrid systems as a hybrid constraint system (HCS), consisting of a flow constraint on trajectories (i.e. continuous functions over time) and a guard constraint on states causing discrete changes. We also propose a technique for solving HCSs by coordinating (i) interval-based solving of nonlinear ODEs, and (ii) a constraint programming technique for reducing interval enclosures of solutions. The proposed technique reliably solves HCSs with nonlinear constraints. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures, while guaranteeing that the enclosure contains a solution.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
This document summarizes Ja-Keoung Koo's presentation on structure from motion. It discusses image formation, the structure from motion pipeline with calibrated cameras, and the 8-point algorithm. The key points are:
1. Image formation maps 3D world points to 2D image points using a camera's intrinsic and extrinsic parameters.
2. Structure from motion with calibrated cameras recovers 3D structure and camera motion from 2D correspondences using the essential matrix and 8-point algorithm.
3. The 8-point algorithm finds the essential matrix from point correspondences, decomposes it to recover the rotation and translation between views.
The document outlines research on developing optimal finite difference grids for solving elliptic and parabolic partial differential equations (PDEs). It introduces the motivation to accurately compute Neumann-to-Dirichlet (NtD) maps. It then summarizes the formulation and discretization of model elliptic and parabolic PDE problems, including deriving the discrete NtD map. It presents results on optimal grid design and the spectral accuracy achieved. Future work is proposed on extending the NtD map approach to non-uniformly spaced boundary data.
The document describes the support vector machine (SVM) algorithm for classification. It discusses how SVM finds the optimal separating hyperplane between two classes by maximizing the margin between them. It introduces the concepts of support vectors, Lagrange multipliers, and kernels. The sequential minimal optimization (SMO) algorithm is also summarized, which breaks the quadratic optimization problem of SVM training into smaller subproblems to optimize two Lagrange multipliers at a time.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
A crash coarse in stochastic Lyapunov theory for Markov processes (emphasis is on continuous time)
See also the survey for models in discrete time,
https://netfiles.uiuc.edu/meyn/www/spm_files/MarkovTutorial/MarkovTutorialUCSB2010.html
The document discusses dual gravitons in AdS4/CFT3 and proposes that the holographic Cotton tensor can play the role of the dual operator to the metric in the boundary CFT.
The Cotton tensor is a symmetric, traceless and conserved tensor that can be constructed from the linearized metric. It is the stress-energy tensor of gravitational Chern-Simons theory. The document proposes that the Cotton tensor and stress-energy tensor form a dual pair under the holographic duality, with the Cotton tensor acting as the operator corresponding to fluctuations of the metric in the bulk. This is motivated by properties of gravitational instantons and self-dual solutions in AdS.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
- Semi Regular Meshes can be subdivided using regular 1:4 subdivision or represented as Spherical Geometry Images mapped to the unit sphere.
- Subdivision Surfaces are generated by applying local interpolators repeatedly to refine a coarse control mesh. Common subdivision schemes include Linear, Butterfly, and Loop which are demonstrated in examples.
- Biorthogonal Wavelets can be constructed on meshes using a Lifting Scheme to create wavelet coefficients with vanishing moments, allowing for compression of mesh signals. Invariant neighborhoods are used to analyze the refinement of meshes across scales.
The document discusses support vector machines (SVM) for classification. It begins by introducing the concepts of maximum margin hyperplane and soft margin. It then formulates the SVM optimization problem to find the maximum margin hyperplane using Lagrange multipliers. The optimization problem is solved using Kuhn-Tucker conditions to obtain the dual formulation only in terms of the support vectors. Kernel tricks are introduced to handle non-linear decision boundaries. The formulation is extended to allow for misclassification errors by introducing slack variables ξ and a penalty parameter C.
The document discusses triangular norm (t-norm) based kernel functions and their application to kernel k-means clustering. It introduces common kernel functions and describes how t-norms can be used to create new kernel functions. Several parameterized and non-parameterized t-norm based kernel functions are presented. The document then details experiments applying various kernel functions including t-norm kernels to four datasets, evaluating the results using adjusted rand index scores. The best performing kernels for each dataset are identified, with some t-norm kernels performing comparably or better than traditional kernels.
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
This document discusses and compares several different probabilistic models for sequence labeling tasks, including Hidden Markov Models (HMMs), Maximum Entropy Markov Models (MEMMs), and Conditional Random Fields (CRFs).
It provides mathematical formulations of HMMs, describing how to calculate the most likely label sequence using the Viterbi algorithm. It then introduces MEMMs, which address some limitations of HMMs by incorporating arbitrary, overlapping features. CRFs are presented as an improvement over MEMMs that models the conditional probability of labels given observations, avoiding the label bias problem of MEMMs. The document concludes by describing how to train CRF models using generalized iterative scaling.
This document summarizes some statistical models used for calibrating imperfect mathematical models. It discusses three main approaches:
1. Gaussian stochastic process (GaSP) calibration, which models bias as a Gaussian process. This is commonly used but can produce inconsistent parameter estimates.
2. L2 calibration, which estimates reality separately from the model before estimating parameters. However, it does not use model information.
3. Scaled Gaussian stochastic process (S-GaSP) calibration, which constrains the GaSP to have a fixed L2 norm. This satisfies predicting reality and calibrated parameters. The S-GaSP is equivalent to penalized kernel ridge regression.
The document analyzes the nonparametric regression setting
There are three possible ROC's:
1. Outside all poles (a, b, c)
2. Between innermost and outermost pole
3. Inside all poles
So the possible ROC's are:
1. Outside circle through a, b, c
2. Annular region between a, c
3. Inside circle through a, b, c
a b c Re
The z-Transform
Important z-Transform Pairs
Important z-Transform Pairs
1. Unit Impulse: δ(n)
1, if n = 0
δ(n) = 0, otherwise
1
X(z) =
2.
D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid...dishii
An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. Interval-based computation of hybrid systems is often difficult, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations.We formulate the problem of detecting a discrete change in hybrid systems as a hybrid constraint system (HCS), consisting of a flow constraint on trajectories (i.e. continuous functions over time) and a guard constraint on states causing discrete changes. We also propose a technique for solving HCSs by coordinating (i) interval-based solving of nonlinear ODEs, and (ii) a constraint programming technique for reducing interval enclosures of solutions. The proposed technique reliably solves HCSs with nonlinear constraints. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures, while guaranteeing that the enclosure contains a solution.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The document summarizes key concepts in social network analysis including metrics like degree distribution, path lengths, transitivity, and clustering coefficients. It also discusses models of network growth and structure like random graphs, small-world networks, and preferential attachment. Computational aspects of analyzing large networks like calculating shortest paths and the diameter are also covered.
Olivier Hudry (INFRES-MIC2 Télécom ParisTech)
A Branch and Bound Algorithm to Compute a Median Permutation
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
This document discusses using Gaussian process models for change point detection in atmospheric dispersion problems. It proposes using multiple kernels in a Gaussian process to model different regimes indicated by change points. A two-stage process is used to first estimate the change point (release time) and then estimate the source location. Simulation results show the approach outperforms existing techniques in estimating change points and source locations from concentration sensor measurements. The approach is applied to model real concentration data to estimate a CBRN release scenario.
This document describes the equations of state used to model the phases and phase transitions in neutron stars. It summarizes the relativistic mean field theory used to model the nucleonic phase and parametric equations of state. It also discusses the Maxwell and Glendenning constructions used to model first-order phase transitions from hadronic to quark matter, including the mixed phase region. Key parameters like the bag constant are specified to generate example equations of state with phase transitions.
This document provides definitions and notations for 2-D systems and matrices. It defines how continuous and sampled 2-D signals like images are represented. It introduces some common 2-D functions used in signal processing like the Dirac delta, rectangle, and sinc functions. It describes how 2-D linear systems can be represented by matrices and discusses properties of the 2-D Fourier transform including the frequency response and eigenfunctions. It also introduces concepts of Toeplitz and circulant matrices and provides an example of convolving periodic sequences using circulant matrices. Finally, it defines orthogonal and unitary matrices.
This document summarizes key sections from Chapter 2 (part 3) of the textbook "Pattern Classification" regarding Bayesian decision theory and discriminant functions. [1] It describes how discriminant functions can be derived for the normal density and multivariate normal distributions. [2] Linear discriminant functions and decision boundaries for linear classifiers are discussed. [3] The derivation of discriminant functions is also covered for the case of discrete features, such as binary variables.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document covers key topics in seismic data processing including complex numbers, vectors, matrices, determinants, eigenvalues, singular values, matrix inversion, series, Taylor series, Fourier series, delta functions, and Fourier integrals. It provides examples of using Taylor series to approximate nonlinear systems as linear systems and using Fourier series to approximate periodic functions. The importance of Fourier transforms for spectral analysis and various geophysical applications is also discussed.
CVPR2010: higher order models in computer vision: Part 1, 2zukun
This document discusses tractable higher order models in computer vision using random field models. It introduces Markov random fields (MRFs) and factor graphs as graphical models for computer vision problems. Higher order models that include factors over cliques of more than two variables can model problems more accurately but are generally intractable. The document discusses various inference techniques for higher order models such as relaxation, message passing, and decomposition methods. It provides examples of how higher order and global models can be used in problems like segmentation, stereo matching, reconstruction, and denoising.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
This document discusses linear response theory and time-dependent density functional theory (TDDFT) for calculating absorption spectroscopy. It begins by motivating the use of absorption spectroscopy to study many-body effects. It then outlines how to calculate the response of a system to a perturbation within linear response theory and the Kubo formula. The document discusses using TDDFT to include electron correlation effects beyond the independent particle and time-dependent Hartree approximations. It emphasizes that TDDFT provides an exact framework for calculating neutral excitations if the correct exchange-correlation functional is used.
Why are stochastic networks so hard to simulate?Sean Meyn
http://arxiv.org/abs/0906.4514
Strange behavior of simulation of queues and other "skip free" stochastic models, including the R-W Hastings Metropolis algorithm.
Presented at the Workshop on Markov chains and MCMC, in honor of Persi Diaconis
http://http//pages.cs.aueb.gr/users/yiannisk/AWMCMC.html
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
This document provides an overview of the key points covered in a calculus lecture on derivatives of logarithmic and exponential functions:
1) It discusses the derivatives of exponential functions with any base, as well as the derivatives of logarithmic functions with any base.
2) It covers using the technique of logarithmic differentiation to find derivatives of functions involving products, quotients, and/or exponentials.
3) The document provides examples of finding derivatives of various logarithmic and exponential functions.
NIPS2010: optimization algorithms in machine learningzukun
The document summarizes optimization algorithms for machine learning applications. It discusses first-order methods like gradient descent, accelerated methods like Nesterov's algorithm, and non-monotone methods like Barzilai-Borwein. Gradient descent converges at a rate of 1/k, while methods like heavy-ball, conjugate gradient, and Nesterov's algorithm can achieve faster linear or 1/k^2 convergence rates depending on the problem structure. The document provides convergence analysis and rate results for various first-order optimization algorithms applied to machine learning problems.
3. Outline
1. Introduction
–What is a complex ideal chain?
–What is the width of it?
–Why bother?
2. Method
–Principle
–Base functions
–The rest are details
3. Some examples
4. Conclusions
4. (r, n)
∂P (r, n) b 2
2
= ∇ P (r, n)
∂n 6
Ideal Chain Statistics
5. linear star pom-pom
(two-branch point)
Ringed
comb ring 8-shaped theta-shaped
Branched
tadpole Double-headed Double-tailed
tadpole tadpole
manacles
Complex Architecture
15. HOW TO CALCULATE IT?
for an ideal chain but of complex architecture
16. The basic principles
• Isotropy of a polymer chain in
free space
• Identity between one half of
ri
the mean span dimension
and the depletion layer
thickness near a hard wall
ˆ
u Wang et al. JCP, 129, 074904 (2008)
=
X max( ri ⋅ u ) − min(ri ⋅ u )
ˆˆ
i i
• Multiplication rule for
independent events.
30. Arm 1
Loop
o
Arm 2
Connector
x=0 xo x
Multiplication rule
P( A B) = P( A) P( B)
if events A and B are independent
31. Three base functions
Arm 1
Loop
Arm 2
Connector
PArm ( x; n, b) = erf ( px ) 3
x ∈ [0,∞), x' ∈ [0,∞), p = 2
2nb
PLoop ( x; n, b) =exp ( −4 p 2 x 2 )
1−
PConnector ( x, x '; n, b)=
dx '
p
π 1/ 2
{exp[− p ( x − x ') ] − exp[− p ( x + x ') ]} dx '
2 2 2 2
32. Arm 1
Loop
Arm 2
Connector
x=0 xo xp x
∞
P( xo ) = PArm ( xo ; na1 , b) PArm ( xo ; na 2 , b) ∫ PConnector ( xo , x p ; nc , b)PLoop ( x p ; nl , b)d px
0
1 ∞
2
=X ∫ 0
[1 − P ( xo )]dxo
Wang et al. (2010) submitted
34. A linear chain
= PArm ( xo ; n, b) erf ( pxo )
P( xo ) =
∞ 2 8 Nb 2
2∫
X = [1 − P ( xo )]dxo = =
0 π
1/ 2
p 3π
X 2 X 16
= 1/ 2 ≈ 1.12838 = ≈ 1.69765
2 Rg π 2 RH 3π
35. A ring
P( xo ) = ( xo ; n, b) =exp ( −4 p 2 x 2 )
PLoop 1−
∞ π 1/ 2 π Nb 2
2 ∫ [1
X = − P ( xo )]dxo ==
0 2p 6
X π X π
= ≈ 1.25331 = ≈ 1.5708
2 Rg 2 2 RH 2
36. A 3-arm star
[=
PArm ( xo ; n, b)] erf ( pxo )
3 3
P( xo )
∞ 12 2
2∫
X = [1 − P ( xo )]dxo =2 arctan(2−1/ 2 )
0 π 3/ p
X X
≈ 1.22800 ≈ 1.72003
2 Rg 2 RH
37. An f-arm (symmetric) star
= [ = erf ( pxo )
P( xo ) PArm ( xo ; n, b)]
f f
∞
= 2∫ [1 − P( xo )]dxo
X
0
38. An f-arm (symmetric) star
= [ = erf ( pxo )
P( xo ) PArm ( xo ; n, b)]
f f
2.0 X
2R
1.8 = H2 ∞ [1 − P( x )]dx
X ∫ o o
0
1.6
X
1.4 2 Rg
1.2
1.0
0 5 10 15 20
39. Linear PE
3-arm star
2-branch point
comb
Sun et al. Macromolecules 37, 4304 (2004)
40. Conclusions
• A general method is developed for calculating the
width (mean span dimension) of polymer chains
assuming ideal chain statistics.
• The method comes from
– Isotropy of a polymer chain in free space
– Polymer depletion near a hard wall
– Multiplication rule for independent events.
• The method can be routinely applied to any
complicated chain architectures.