This technical memorandum defines a generalized error function to handle the normal probability distribution in n dimensions. For even dimensions n, the error function can be written in closed form involving exponential and factorial terms. For odd dimensions, explicit integral representations are derived. Some properties of the generalized error functions are proved, including recursion formulas and relationships between different dimensional cases. Graphs of the functions are also presented to illustrate their behavior.
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
Quickselect Under Yaroslavskiy's Dual Pivoting AlgorithmSebastian Wild
I gave this talk at the 24th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2013) on Menorca (Spain).
A paper covering the analyses of this talk (and some more!) has been submitted.
Also, in the talk, I refer to the previous speaker at the conference, my advisor Markus Nebel - corresponding results can be found in an earlier talk of mine:
slideshare.net/sebawild/average-case-analysis-of-java-7s-dual-pivot-quicksort
Check my website for preprints of papers and my other talks:
wwwagak.cs.uni-kl.de/sebastian-wild.html
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko
1) The document describes a method called Multilevel Monte Carlo (MLMC) to efficiently compute electromagnetic fields scattered from dielectric objects of uncertain shapes. MLMC balances statistical errors from random sampling and numerical errors from geometry discretization to reduce computational time.
2) A surface integral equation solver is used to model scattering from dielectric objects. Random geometries are generated by perturbing surfaces with random fields defined by spherical harmonics.
3) MLMC is shown to estimate scattering cross sections accurately while requiring fewer overall computations compared to traditional Monte Carlo methods. This is achieved by optimally allocating samples across discretization levels.
This document summarizes the derivation of the EM algorithm for parameter estimation in a mixed normal model. It begins by presenting the log-likelihood function and derives update equations for the mean (μk) and covariance (Σk) parameters of each normal component. An experimental design is then described to statistically analyze the performance of the EM algorithm under different conditions. The results show that the EM estimates are most accurate when the normal components have distinct means and covariances, and when more training data is available. Interactions between factors are also examined.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
Quickselect Under Yaroslavskiy's Dual Pivoting AlgorithmSebastian Wild
I gave this talk at the 24th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2013) on Menorca (Spain).
A paper covering the analyses of this talk (and some more!) has been submitted.
Also, in the talk, I refer to the previous speaker at the conference, my advisor Markus Nebel - corresponding results can be found in an earlier talk of mine:
slideshare.net/sebawild/average-case-analysis-of-java-7s-dual-pivot-quicksort
Check my website for preprints of papers and my other talks:
wwwagak.cs.uni-kl.de/sebastian-wild.html
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko
1) The document describes a method called Multilevel Monte Carlo (MLMC) to efficiently compute electromagnetic fields scattered from dielectric objects of uncertain shapes. MLMC balances statistical errors from random sampling and numerical errors from geometry discretization to reduce computational time.
2) A surface integral equation solver is used to model scattering from dielectric objects. Random geometries are generated by perturbing surfaces with random fields defined by spherical harmonics.
3) MLMC is shown to estimate scattering cross sections accurately while requiring fewer overall computations compared to traditional Monte Carlo methods. This is achieved by optimally allocating samples across discretization levels.
This document summarizes the derivation of the EM algorithm for parameter estimation in a mixed normal model. It begins by presenting the log-likelihood function and derives update equations for the mean (μk) and covariance (Σk) parameters of each normal component. An experimental design is then described to statistically analyze the performance of the EM algorithm under different conditions. The results show that the EM estimates are most accurate when the normal components have distinct means and covariances, and when more training data is available. Interactions between factors are also examined.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
The document appears to be a blueprint for a mathematics exam for class 12. It lists various topics that could be covered in the exam such as functions, derivatives, integrals, differential equations, 3-dimensional geometry, and matrices. For each topic it indicates the number and type of questions that may be asked, such as very short answer (1 mark), short answer (4 marks), and long answer (6 marks). The total number of questions is 29 with 10 short answer questions worth 1 mark each, 12 questions worth 4 marks each, and 7 questions worth 6 marks each. The document also includes sample questions that cover the listed topics as examples of what may be asked on the exam.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Large variance and fat tail of damage by natural disasterHang-Hyun Jo
In order to account for large variance and fat tail of damage by natural disaster, we study a simple model by combining distributions of disaster and population/property with their spatial correlation. We assume fat-tailed or power-law distributions for disaster and population/property exposed to the disaster, and a constant vulnerability for exposed population/property. Our model suggests that the fat tail property of damage can be determined either by that of disaster or by those of population/property depending on which tail is fatter. It is also found that the spatial correlations of population/property can enhance or reduce the variance of damage depending on how fat the tails of population/property are. In case of tornadoes in the United States, we show that the damage does have fat tail property. Our results support that the standard cost-benefit analysis would not be reliable for social investment in vulnerability reduction and disaster prevention.
http://ascelibrary.org/doi/abs/10.1061/9780784413609.277
http://arxiv.org/abs/1407.6209
The document provides a course calendar for a class on Bayesian estimation methods. It lists the dates and topics to be covered over 15 class periods from September to January. The topics progress from basic concepts like Bayes estimation and the Kalman filter, to more modern methods like particle filters, hidden Markov models, Bayesian decision theory, and applications of principal component analysis and independent component analysis. One class is noted as having no class.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
Ch 02 MATLAB Applications in Chemical Engineering_陳奇中教授教學投影片Chyi-Tsong Chen
This document discusses solving nonlinear equations using MATLAB. It begins by introducing nonlinear equations that commonly arise in chemical process analysis and design. It then covers relevant MATLAB commands for solving nonlinear equations of a single variable and systems of nonlinear equations. Examples are provided to demonstrate solving for the volume of a gas using the van der Waals equation and finding equilibrium concentrations in a chemical reaction system. The document also shows how to solve nonlinear equations using the fzero and fsolve commands as well as the Simulink interface.
Measures of different reliability parameters for a complex redundant system u...Alexander Decker
This document summarizes a mathematical model of a complex redundant system consisting of two subsystems (A and B) connected in series. Subsystem A has N non-identical units in series, while subsystem B has 3 identical components in parallel. The model analyzes the system's reliability under a "head-of-line" repair policy where failures follow exponential and repair times follow general distributions. Differential equations are formulated and solved using Laplace transforms to obtain state probabilities and an expression for the expected total cost of the system over time.
Ch 01 MATLAB Applications in Chemical Engineering_陳奇中教授教學投影片Chyi-Tsong Chen
The slides of Chapter 1 of the book entitled "MATALB Applications in Chemical Engineering": Solution of a System of Linear Equations. Author: Prof. Chyi-Tsong Chen (陳奇中 教授); Center for General Education, National Quemoy University; Kinmen, Taiwan; E-mail: chyitsongchen@gmail.com.
Ebook purchase: https://play.google.com/store/books/details/MATLAB_Applications_in_Chemical_Engineering?id=kpxwEAAAQBAJ&hl=en_US&gl=US
This document provides an overview of numerical linear algebra concepts including matrix notation, operations, and solving systems of linear equations using Gaussian elimination. It describes the Gaussian elimination process which involves eliminating variables one by one to obtain an upper triangular system that can then be solved using back substitution. The document notes some pitfalls of naive Gaussian elimination such as division by zero, round-off errors, ill-conditioned systems, and singular systems. It introduces pivoting as a technique to avoid division by zero during the elimination process and calculates the determinant as a byproduct of Gaussian elimination.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A New Approach to Design a Reduced Order ObserverIJERD Editor
This document proposes a new method for designing reduced order observers for linear time-invariant systems. The approach is based on inverting matrices of proper dimensions. It reduces the arbitrariness of previous methods by using pole-placement techniques. The method is applied to design a reduced order observer for a 3rd order system. Simulation results show the observer estimates converge to the true system states.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document contains sample questions from a mathematics exam blueprint and marking scheme for Class 12. It includes:
- A blueprint showing the distribution of questions across different units of the syllabus for very short answer (1 mark), short answer (4 marks) and long answer (6 marks) questions.
- Sample questions from sections A to D with varying marks. The questions cover topics like relations and functions, matrices, calculus, vectors, probability and linear programming.
- A marking scheme providing solutions to the sample questions with marks allocated for each step.
This document discusses techniques for evaluating integrals involving exponential functions. It introduces the formulas for integrating exponentials and differentiating them. Several important definite integrals are evaluated, such as the integral from 0 to infinity of e^-ax dx = 1/a. Graphs are used to visualize these integrals. The document then evaluates the more complex integral from negative infinity to positive infinity of e^-ax^2 dx using a change of variables technique. Finally, it discusses how these integrals can be used in kinetic theory and derives an important ratio and normalization factor for Maxwell's velocity distribution.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
El documento describe los descubrimientos clave que llevaron al entendimiento de que el ADN es el material hereditario, incluyendo el aislamiento del ADN por Miescher en 1868, la experiencia de Griffith en 1928 que mostró la transferencia de características hereditarias entre bacterias, y el descubrimiento de Avery en 1944 de que el factor transformante era el ADN. Finalmente, Watson y Crick propusieron en 1953 el modelo de doble hélice del ADN.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
The document appears to be a blueprint for a mathematics exam for class 12. It lists various topics that could be covered in the exam such as functions, derivatives, integrals, differential equations, 3-dimensional geometry, and matrices. For each topic it indicates the number and type of questions that may be asked, such as very short answer (1 mark), short answer (4 marks), and long answer (6 marks). The total number of questions is 29 with 10 short answer questions worth 1 mark each, 12 questions worth 4 marks each, and 7 questions worth 6 marks each. The document also includes sample questions that cover the listed topics as examples of what may be asked on the exam.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Large variance and fat tail of damage by natural disasterHang-Hyun Jo
In order to account for large variance and fat tail of damage by natural disaster, we study a simple model by combining distributions of disaster and population/property with their spatial correlation. We assume fat-tailed or power-law distributions for disaster and population/property exposed to the disaster, and a constant vulnerability for exposed population/property. Our model suggests that the fat tail property of damage can be determined either by that of disaster or by those of population/property depending on which tail is fatter. It is also found that the spatial correlations of population/property can enhance or reduce the variance of damage depending on how fat the tails of population/property are. In case of tornadoes in the United States, we show that the damage does have fat tail property. Our results support that the standard cost-benefit analysis would not be reliable for social investment in vulnerability reduction and disaster prevention.
http://ascelibrary.org/doi/abs/10.1061/9780784413609.277
http://arxiv.org/abs/1407.6209
The document provides a course calendar for a class on Bayesian estimation methods. It lists the dates and topics to be covered over 15 class periods from September to January. The topics progress from basic concepts like Bayes estimation and the Kalman filter, to more modern methods like particle filters, hidden Markov models, Bayesian decision theory, and applications of principal component analysis and independent component analysis. One class is noted as having no class.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
Ch 02 MATLAB Applications in Chemical Engineering_陳奇中教授教學投影片Chyi-Tsong Chen
This document discusses solving nonlinear equations using MATLAB. It begins by introducing nonlinear equations that commonly arise in chemical process analysis and design. It then covers relevant MATLAB commands for solving nonlinear equations of a single variable and systems of nonlinear equations. Examples are provided to demonstrate solving for the volume of a gas using the van der Waals equation and finding equilibrium concentrations in a chemical reaction system. The document also shows how to solve nonlinear equations using the fzero and fsolve commands as well as the Simulink interface.
Measures of different reliability parameters for a complex redundant system u...Alexander Decker
This document summarizes a mathematical model of a complex redundant system consisting of two subsystems (A and B) connected in series. Subsystem A has N non-identical units in series, while subsystem B has 3 identical components in parallel. The model analyzes the system's reliability under a "head-of-line" repair policy where failures follow exponential and repair times follow general distributions. Differential equations are formulated and solved using Laplace transforms to obtain state probabilities and an expression for the expected total cost of the system over time.
Ch 01 MATLAB Applications in Chemical Engineering_陳奇中教授教學投影片Chyi-Tsong Chen
The slides of Chapter 1 of the book entitled "MATALB Applications in Chemical Engineering": Solution of a System of Linear Equations. Author: Prof. Chyi-Tsong Chen (陳奇中 教授); Center for General Education, National Quemoy University; Kinmen, Taiwan; E-mail: chyitsongchen@gmail.com.
Ebook purchase: https://play.google.com/store/books/details/MATLAB_Applications_in_Chemical_Engineering?id=kpxwEAAAQBAJ&hl=en_US&gl=US
This document provides an overview of numerical linear algebra concepts including matrix notation, operations, and solving systems of linear equations using Gaussian elimination. It describes the Gaussian elimination process which involves eliminating variables one by one to obtain an upper triangular system that can then be solved using back substitution. The document notes some pitfalls of naive Gaussian elimination such as division by zero, round-off errors, ill-conditioned systems, and singular systems. It introduces pivoting as a technique to avoid division by zero during the elimination process and calculates the determinant as a byproduct of Gaussian elimination.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A New Approach to Design a Reduced Order ObserverIJERD Editor
This document proposes a new method for designing reduced order observers for linear time-invariant systems. The approach is based on inverting matrices of proper dimensions. It reduces the arbitrariness of previous methods by using pole-placement techniques. The method is applied to design a reduced order observer for a 3rd order system. Simulation results show the observer estimates converge to the true system states.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document contains sample questions from a mathematics exam blueprint and marking scheme for Class 12. It includes:
- A blueprint showing the distribution of questions across different units of the syllabus for very short answer (1 mark), short answer (4 marks) and long answer (6 marks) questions.
- Sample questions from sections A to D with varying marks. The questions cover topics like relations and functions, matrices, calculus, vectors, probability and linear programming.
- A marking scheme providing solutions to the sample questions with marks allocated for each step.
This document discusses techniques for evaluating integrals involving exponential functions. It introduces the formulas for integrating exponentials and differentiating them. Several important definite integrals are evaluated, such as the integral from 0 to infinity of e^-ax dx = 1/a. Graphs are used to visualize these integrals. The document then evaluates the more complex integral from negative infinity to positive infinity of e^-ax^2 dx using a change of variables technique. Finally, it discusses how these integrals can be used in kinetic theory and derives an important ratio and normalization factor for Maxwell's velocity distribution.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
El documento describe los descubrimientos clave que llevaron al entendimiento de que el ADN es el material hereditario, incluyendo el aislamiento del ADN por Miescher en 1868, la experiencia de Griffith en 1928 que mostró la transferencia de características hereditarias entre bacterias, y el descubrimiento de Avery en 1944 de que el factor transformante era el ADN. Finalmente, Watson y Crick propusieron en 1953 el modelo de doble hélice del ADN.
El documento presenta las primeras ideas evolutivas en la antigüedad. Anaximandro y Empédocles propusieron que los seres vivos no son fijos y pueden cambiar, y Anaximandro sugirió que los humanos evolucionaron de otros animales como peces. Estas ideas anticiparon conceptos modernos de evolución aunque usaron mecanismos distintos a la selección natural.
Este documento presenta el seminario "Gerencia de Centros de I&D" que se orienta a suministrar información sobre la gerencia en centros de investigación y desarrollo. El seminario cubre temas como el proceso administrativo en centros de I&D desde una perspectiva de sistemas, la planificación del desarrollo organizacional, las estructuras organizacionales, y el liderazgo, toma de decisiones, control y seguimiento en estos centros. El contenido se divide en cuatro unidades temáticas y se utilizarán estrategias como clases
Este documento describe un foro sobre planificación operativa y gestión de calidad en la gerencia pública que se llevará a cabo del 21 al 25 de noviembre de 2011. Incluye la coordinadora, los oradores, el tema general del foro y una breve introducción sobre la planificación operativa y sus características. También incluye información sobre los tipos de planificación y la organización del gobierno como sistema.
El documento describe los diferentes niveles de estudio de la lengua, incluyendo el fonológico, morfológico, sintáctico, semántico y textual. Explica las partes constituyentes del signo lingüístico y los elementos básicos de la comunicación como el emisor, receptor, mensaje y canal. También resume las características de las diferentes categorías gramaticales como sustantivos, verbos, adjetivos, pronombres, preposiciones y conjunciones.
El documento discute los problemas asociados con el crecimiento de las ciudades y el impacto ambiental de la construcción. Menciona que la construcción consume grandes cantidades de recursos naturales y emite muchos contaminantes. También describe varias técnicas de construcción con tierra, como adobe y COB, que son más sostenibles y tienen menos impacto ambiental que los materiales convencionales como el cemento.
Este documento establece lineamientos políticos y estratégicos para la educación secundaria obligatoria en Argentina. Define las finalidades de la educación secundaria como habilitar a adolescentes y jóvenes para el ejercicio de la ciudadanía, el trabajo y la continuación de estudios. También establece que la educación secundaria debe ofrecer una propuesta formativa inclusiva en condiciones pedagógicas y materiales adecuadas, superando las desigualdades sociales que dificultan el acceso a la educación. Propone estrategias
El documento describe los descubrimientos clave que llevaron al entendimiento de que el ADN es el material hereditario, incluyendo el aislamiento del ADN por Miescher en 1868, la experiencia de Griffith en 1928 que mostró la transferencia de características hereditarias entre bacterias, y la identificación del ADN como el factor transformante por Avery en 1944. Finalmente, Watson y Crick propusieron en 1953 el modelo de doble hélice del ADN que explicó cómo almacena y transmite la información genética.
Este documento presenta una breve historia de las tabernas en Mérida y describe algunas tabernas históricas de la ciudad, incluyendo su ubicación y especialidades. Se mencionan tabernas como Casa Cañero, Casa Carrasco, Casa Curro, Casa Dionisio, Casa Enrique, Casa Felipe y Casa Gallega. El autor también proporciona algunos antecedentes históricos sobre el origen y papel social de las tabernas.
Este documento presenta información sobre tres temas principales:
1. Un resumen de la historia del Humilladero de Santa Eulalia en Mérida, incluyendo detalles sobre su ubicación original y las inscripciones que contiene.
2. Una experiencia positiva de una voluntaria en el taller de francés del Hogar, donde aprendió y enseñó a otros.
3. Una descripción del taller de encuadernación del Hogar, dirigido durante 11 años por un monitor voluntario que ha enseñado
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La Web 2.0 permitió la participación e interacción de los usuarios al publicar opiniones, comentarios y correcciones en las páginas web, a diferencia de la Internet original que solo permitía que una pequeña élite editara la información. Esto trajo ventajas como acelerar la integración de información mundial a nivel local, permitir el acceso sin conocimientos avanzados de programación, y expresar e intercambiar ideas. Para la educación, la Web 2.0 mejora la calidad de información al enriquecerse con opiniones, facilita el acceso
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In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
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with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
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This document discusses using neuro-fuzzy networks to identify parameters for mathematical models of geofields. It proposes a new technique using fuzzy neural networks that can be applied even when data is limited and uncertain in the early stages of modeling. A numerical example is provided to demonstrate the identification of parameters for a regression equation model of a geofield using a fuzzy neural network structure. The network is trained on experimental fuzzy statistical data to determine values for the regression coefficients that satisfy the data. The technique is concluded to have advantages over traditional statistical methods as it can be applied regardless of the parameter distribution and is well-suited for cases with insufficient data in early modeling stages.
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Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document appears to be an exam for a third semester engineering course covering topics in transforms and partial differential equations. It contains 15 multiple choice and numerical response questions testing knowledge of various transforms (Fourier, Laplace, Z) and partial differential equations (heat, wave, etc). The questions cover topics such as singular integrals, Fourier series behavior at discontinuities, sketching even/odd extensions, solving PDEs using separation of variables, convolution theorems, and applications of transforms.
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field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
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Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
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This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
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aziz sancar nobel prize winner: from mardin to nobel
4017
1. Technical Memorandum No. NMC-TM-63-8
C
I• A GENERALIZED ERROR FUNCTION
J INnDIMENSIONS
"By
M.BROWN
Theoretical Analysis Division
S,. 12 April 1963 7 .
SAPR 19 1963
C%4
QUALIFIED REQUESTERS MAY OBTAIN COPIES
OF THIS REPORT DIRECT FROM ASTIA.
U. S. NAVAL MISSILE CENTER
Point Mugu, California
2. U. S. NAVAL MISSILE CENTER
AN ACTIVITY OF THE BUREAU OF NAVAL WEAPONS
K. C. CHILDERS, JR., CAPT USN
Commander
Mr. It. G. McCarty, Head, Theoretical Analysis Division; Mr. J. J. O'Brien, Head, Advanced
Programs Department; CDlI It. K. Engle, Director, Astronautics; and Mr. T. E. Hlanes, Consultant
to the Technical Director and Commander NMC, have reviewed this report for publication.
Approved by:
D. F. SULLIVAN
7echnical Di~rector
THIS REPORT HAS IIEEN PREPARED PRIMARILY FOR TIMELY PRESENTATION OF INFORMA-
TION. ALTHOUGH CARE HAS BEEN TAKEN IN THE PREPARATION OF THE TECHNICAL
MATERIAL PRESENTED. CONCLUSIONS DRAWN ARE NOT NECESSARILY FINAL AND MAY BE
SUBJECT TO REVISION.
NMC Technical Memorandum NMC-TM-63-8
Published by ........................................... Editorial Division
Technical Information Department
First printing ............................. ............. ..... . 65 copies
Security classification ...................................... UNCLASSIFIED
3. INITIAL DISTRIBUTION
EXTERNAL Copies INTERNAL Copies
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Code DLI-31 ................... 4 Code N21 .................... 1
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Commander Code N212 ................... 2
Armed Services Technical Code N2124
Information Agency Mr. M. Brown ................ 10
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Arlington 12, Va................. 10 Code 01-2 .................... 1
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Code N03022 .................. 7
4. TABLE OF CONTENTS
Page
SUMMARY ....................................................... 1
INTRODUCTION .................................................. 3
THE SPECIAL CASE OF TWO DIMENSIONS ................................ 4
THE GENERAL CASE OF n DIMENSIONS .................................. 5
SOME FURTHER ANALYSIS ........................................... 7
APPLICATION TO HYPERELLIPSOIDAL DISTRIBUTIONS ...................... 10
RECAPITULATION ................................................ 11
ILLUSTRATION
Figure 1. Generalized Error FUnctions, erfn (x) ........................... 9
5. SWSUMMARY
The error function, which occurs in much of the literature of mathematica, physics, and
'w engineering, Is generalised to handle the normal probability distribution In n dimensions.
Explicit integral representations for these functions arn found to be of two general forms,
depending upon whether n is even or odd.
Some readily established recursion formulas and other relationships are derived for these
functions.
S•
"4I
6. INTRODUCTION
The mathematical theory of probability and the related techniques of statistics are being used
by an increasing number of workers in many diverse fields embracing the sciences and engineering,
as well as mathematics.
Of particular importance, therefore, to engineers, to operations and systems analysts, and to the
designers of experiments are certain standard probability distributions. The most widely employed
of the continuous distributions is undoubtedly the "Gaussian," or "normal" distribution, which is
of enormous theoretical, historical, and practical importance.
The normal distribution, with zero mean, is given by
12
p e-- (1)
and is completely specified by the parameter a, called the "standard deviation."
This distribution has been generalized to n dimensions. The most general n-dimensional nor-
mal distribution contains parameters to account for nonzero means for the n independent variables,
for correlations among the variables, and for unequal standard deviations with respect to each of
the n variables.
This report will concern itself with a very special case of the n-dimensional distribution. In
particular, the means will all be assumed zero, the correlations will all be assumed zero, and the
standard deviations will be assumed equal. The resulting probability distribution is then given by
1-12 + Kn+ 2
p(xl, x2 .. x) n e 2,2 (2)
S~(2 n' "
In the one-dimensional case, a commonly occurring expression containing p (x) is that for the
probability with which
xIx< a
This expression is (for X_ 0)
Probl- < x < 4=X p(x) dx (3)
2
eI fe2adx (3)
2
S2 e d. (since the integrand is even)
17,
3
7. This expression cannot be evaluated in closed form for an arbitrary upper limit, but occurs so
frequently that its values have been tabulated, and the expression itself has been given the name
of "error function."
The customary definition of the error function is
erf (x) = 2 e-Y2 dy (4)
In terms of this definition, it follows that
Prob I x I S erf(t2) (5)
The analog of the problem in n dimensions presents no new problems in rectangular coordinates,
since
Prob Ix x< I x&'x 2 a2. xn an
erf /.....Lrf /. L ... erf / (6)
as a consequence of the appropriate integral separating into a product of integrals with respect to
one variable.
Something new does arise, however, from regarding r = (x1 , . . . , xn) as a vector in n dimen-
sions, and asking what is
Prob Ijr' t
This is a very natural question to ask for the cases n --2, 3.
It is the purpose of the present paper to investigate this question. The result, as will be seen,
is to define an "error function" generalized to n dimensions, in terms of which the required proba-
bility can be written in a manner analogous to equation (5).
Some simple properties of these generalized error functions are proved, and graphs of these
functions are presented.
THE SPECIAL CASE OF TWO DIMENSIONS
As a natural way of introducing generalized error functions it is instructive to consider the
case of two dimensions. This case is well known among the users of probability theory because
of its frequent occurrence and the fact that the mathematics fortuitously permits a solution in
closed form.
The two-dimensional treatment, moreover, is capable of a direct generalization to n dimen-
sions, as will be seen later, and it is therefore profitable to dwell at some length upon this
special case. This will now be done.
4
8. p
Introduce polar coordinates (r, 0). Then
X2 +y2
'ProbII'Hr 4ff 2~ 2 a x 7IV2 2rro2
circle about
origin, radius a
1.ce_T2 rdrd )
0 0 2rra
2
2S~r2I.-. " --
- e 2a2 r dr
a
2
2
erf2 Lm(x 2e 9
fe- 2¢'U2
=0
a2
•.Prob )LIr~•a -e'2" (8)
This is a well known expression which appears very frequently in the literature.
The above result suggests defining an error function in two dimensions by
erf 2 (X ) - 1 - e -X (9 )
0 Then
Probi I _' erf2 (10)
in two dimensions, analogously to equation (5).
The above computational procedure will now be extended to an n-dimensional distribution. It
will be seen later that all the even-dimensional error functions are expressible in closed ;orm,
although not always conveniently so.
THE GENERAL CASE OF n DIMENSIONS
The case of n dimensions, for n :- 2, is carried out analogously to the above two-dimensional
treatment.
It is necessary to evaluate the following expression:
If.f "n2UX x
n 2
Prob• jrI<_ l ]J.] •1 e 202 "dxI ... dxn (11)
hyperaphere (2 n ) 2 (,
.bout origin,
ra(UUm (
5
9. I
The integration is simplified by introducing hyperapherical coordinates, (R, 01, 02, .... Ol0
according to the transformation equations
X1 R in ( 6L, 02 0.2 &4" 0nn1
X2 = R ait 0li.n 02 . . . Oin-2 :-1 n1
x3 = R ain 01 ain 02 . . . 3 '~ On-2
x4 =R ,i 01 &i. 02 . . . •On.-4 (n-3 (12)
.........................
Xn-1 = R a,. 01 C" 02
Xn = R c:oa 01
Equation (11) then becomes
Prob LI, z I e RRno -',,f f d ., (13)
(2ff)T.n O
where d an is an element of hypersolid angle and is independent of R and d R.
In particular,
1=Prob ' tr' I< f e Rn dRf f dIn (14)
(2n) 2
n 0 ...
It follows that
n
Substituting into equation (13),
2• Rn-' dR
Prob I OMC-R 2<d (16)
An obvious change of variable in the integrals gives the result
f'/e-.2 un-' du
Probl Ir" - (17)
f e!"2 un"l d u
0
6
10. Now define a generalized n-dimensional error function, erfn(x), by
fo0x e•"U2 u-'l d u
erfn(x) R
(18)
f e n-' du
Equations (17) and (18) then imply that
Prob r' O_• errtfn .• (19)
which is analogous to equation (5), and is valid for n = 1, 2, 3, 4,
For n = 1, 2, the definition (18) reduces to the definitions (4) and (9), respectively.
SOME FURTHER ANALYSIS
Consider the integral
Sfe'u 2
uk du (20)
0
which is used as a normalizing factor in equation (18).
Integrating by parts, it is readily established that
Ik+2 k
(21)
It follows by mathematical induction that
I2m+1 m!1 ;m 0,1,2 .... (22)
and
12m (2m)i' ;m O , 2 .... (23)
22m m!
but
1=1 e u du
(24)
and
~oo
IO=fo e"-du2d V/u -
(25)
f 2
7
11. so that
I2m÷1 (26)
2
and
(2 m) (27)
S22m+l (27)
The definition (18) can therefor-a•re be written
erf2.+(x) =2 -2m+) . 2
m du ; m =0, 1, 2,... (28)
.F(2 m)! 0
and
erf( 2xO €2 u2 =m-1 du ;m = 1, 2, 3,... (29)er2 (X) = (m -1 )! f
Equations (28) and (29) may, ift A desired, together be taken as the definition of the generalized
error function, rather than equation .A (18).
The error functions will now behwe shown to satisfy certain relationships.
Integrate equation (28) by part--its, The resulting expression simplifies to
(2 X)
2
+em!fm(M2 m 0, 1, 2.... (30)
erf 2m +I(x) - erf2 m+.3(X)
V 7
(2m + 1)!
Proceeding similarly with equ&Aation (29), one gets
2rn
erf 2m+2!-2(X) ;m 1,
2
,
3
,.. (31)
Recalling equation (9), it follc_• .ows by mathematical induction on equation (31) that
erf 2 m(X)=l-eI 1+-.4 -4*..+...+( ;m 0,1,2.... (32)
Equation (32) shows that all Ur the even-dimensional error functions are expressible in closed
form, although for sufficiently larg. e m the closed form expressions become increasingly compli-
cated.
It follows also from equation * (32) that
2
er ' )x 7 (33)1- e erf2 lx) 9• erf4 (x) erf 6 (x) . (
and that for any preassigned x,
8
12. m?•nerf2m(x) = 0 (34)
In a similar manner, mathematical induction applied to equation (30) gives the result
erf2 m,+(x) = er (x)- e- (2 x) 0 +, (2x)3 11 + + (2x) 2
m'l(m'1)- (3S)3 (3Sm-1
m 11, 2, 3, ...
Since erf ,(x) = erf(x) cannot be expressed in closed form, neither can the odd-dimensional
error functions.
However, it is clear from equation (35) that
erf(x) =-erf (x) > erf3 (x) > erf5(x)_> . . . (36)
since x was assumed non-negative in the definition, equation (18), in view of the probability
application with regard to which the generalized error functions were introduced.
From the point of view of the pure mathematician, it is, of course, desirable to use the de-
finition, equation (18), for negative values of x, as well as for positive values. If that is done,
then the odd-dimensional error functions turn out to be odd functions, while the even-dimensional
functions are even.
i.e., erf 2 m(x) = erf2 m(.X)
(37)
erf 2m +,(X) = - erf 2. +1(-x)
if x is permitted to become negative. For negative x, the inequalities (36) will all be reversed.
These error functions are plotted in figure 1 for positive arguments and dimensions up to 10.
to.
0|
06t
04 /
02
0/
0 04 01 I a 1 6 20 2 4 21 12 3.6 4.0
Figure 1. Generalized Error Functions, erfn (x).
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13. APPLICATION TO HYPERELLIPSOIDAL DISTRIBUTIONS
Retaining the assumptions of zero bias and zero correlations, but allowing the standard devia-
tions to take on the unequal values a1, (2,..., ant equation (2) generalizes to
' . . +' 2_..
p (xIS ... #x2 )- 1 e-aiV2) (38)n
(2 Fr)'3 0s. . . an
The loci along which p (x,, . , . , xn) is constant are given by
2 2_+ +... Ln U2, a constant. (39)
a, 2 an2
These loci thus form a family of hyperellipsoids, centered at the origin, with semiaxes equal
to ua,, ua2, .... uan and consequently the distribution given by equation (38) may be thought
of as a hyperellipsoidal distribution.
Consider now the probability that a random vector 7= (x1, . . . xn) lies within the hyper-
ellipsoid
X12 X.2
_ +. +. p2
(40)
a, 2 an2
Probf .. L<I321 ff f p (r') dV (41)
hypwittllp.oid
where dV is an element of hypervolume in the n-dimensional space.
Prob f X.L<32I=ff f e ca2a2 ..d (42)
_i .2t,•,÷
,- -,. (2r)2 °, ... an
hyperttlipsioid
f e"n "2 dX1 ... dX, (43)
. . . (2 ir)2
hyper•phere I..,
2
= p2
where
x
A , i 1, 2,.... n (44)
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14. ""Prob 321 I R Iff-• dD. (45)
introducing hyperspherical polar coordinates, as before.
Again, remembering that
Prbx2< 1l ~Rn- Ie:T
Prob < ff..0
- 2
dRff fd~ln (46)
a12l (2 r)• o ...
it follows that
?<I•2 0(Rn' e- R- dR n-Ie-.n deu
X'2 }- -R d n
fO Rn' dR uf - e-" du
0
i.e.,
Prob I' < ?2- =erfn (47)
Equation (47) is the desired generalization of equation (19) to the case in which the compo-
nents of r have unequal standard deviations.
It is to be noted that the dimensionless quantity ji appearing in equation (47) reduces to the
quantity ;(of equation (19) for the special case in which a1 - f2 - • n 7
RECAPITULATION
The error function, which occurs in much of the literature of mathematics, physics, and engi-
neering, has been generalized to handle the normal probability distribution in n dimensions.
Explicit integral representations for these functions were found to be of two general forms,
depending upon whether n is even or odd.
Some readily established recursion formulas and other relationships were derived for these
functions.
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