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### Penalized Regression Stanford

Quadratic regression, or regression with second order polynomial, is given by the following equation:. Pantic, “Fast Adaptation of Personalized Deep Learning for Facial Action Unit Detection,” in Int’l Joint Conference on Artificial Intelligence IJCNN, Workshop on Affective Computing, Macao, China, 2019. Furthermore, the ultimate goal is often to build a prediction model with these features that accurately assesses risk for future subjects. In this video i'm implementing a binary-class logistic regression classifier to determine whether a student is accepted or not given his marks at two exams, the algorithm was implemented using. Magdon-Ismail CSCI 4100/6100. We also provide causal estimates of two policies aimed at reducing the child penalty. Some studies have concluded that the threat of capital punishment deters murders,. The settings of. “L1-Penalized Quantile Regression in High-Dimensional Sparse Models,” with A. Group-penalized Multiresponse and Multinomial Regression Noah Simon Stanford University Jerome Friedman Stanford University Trevor Hastie Stanford University Abstract In this paper we purpose a blockwise descent algorithm for group-penalized multire-sponse regression. an initial estimator to reduce bias for large regression coe cients An alternative single-stage approach is to use a penalty that tapers o as becomes larger in absolute value Unlike the absolute value penalty employed by the lasso, a tapering penalty cannot be convex Patrick Breheny High-Dimensional Data Analysis (BIOS 7600) 9/34. Linear regression analysis, based on the concept of a regression function, was introduced by F. Penalization is a powerful method for attribute selection and improving the accuracy of predictive models. The usual L 1 penalty yields a convex criterion, but we ﬁnd that it fails to deliver a robust estimator. Ng Computer Science Department Stanford University Stanford, CA 94305 Abstract L1 regularized logistic regression is now a workhorse of machine learning: it is widely used for many classiﬁca-tion problems, particularly ones with many features. It is similar to forward selection, but only enters 'as much' of the β estimate as necessary. The results are compared in Table 5. In logistic regression we assumed that the labels were binary: y^{(i)} \in \{0,1\}. Univariate regression. Simon N (2017). If the group's write-up earns p points, the student receives a final score of. If you want to see examples of recent work in machine learning, start by taking a look at the conferences NIPS(all old NIPS papers are online) and ICML. We introduce the idea of a loss function to quantify our unhappiness with a model's predictions, and discuss two commonly used loss. 9-3 uploaded to CRAN (with intercept option) This update adds an intercept option (by popular request) - now one can fit a model without an intercept Glmnet is a package that fits the regularization path for a number of generalized linear models, with with "elastic net" regularization (tunable mixture of L1 and L2 penalties). Shirshendu - Writing a business proposal every time you Tulshi - Your data will be safe even after uploading Samsons - Anyone can design the company logo to be used. Regression splines operate on a xed set of knots, and there is a substantial literature on knot placement for this problem (see Chapter 9. We show that classification accuracy can be improved when penalized regression techniques are employed, as. com - Read for FREE. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underlying signal in a set of data. Penalized regression methods can perform model selection by shrinking the size of the coefficients, driving the coefficients of markers with little or no apparent effect down toward zero. a penalized matrix decomposition, and its applications a dissertation submitted to the department of statistics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy daniela m. veins were tested in logistic regression (Firth penalized logistic regression in nonconvergence) with PLINK2 15 and adjusted for age, sex, and the first 15 principal components of ancestry. Classiﬁcation of gene microarrays by penalized logistic regression JI ZHU† Department of Statistics, University of Michigan, Ann Arbor, MI 48109, USA jizhu@umich. Thus, members of non-fundamentalist faith groups should oppose capital. Hence, for any machine learning model, be it classification and regression, finding the parameters by maximizing MLE (or minimizing cross entropy) has a statistical significance, whereas minimizing the quadratic cost for logistic regression doesn't have any (although it does for linear regression, as stated before). penalized regressions. It is an attempt to facilitate research on gun violence in the US by making raw data more accessible. I understand the the first two formulas and the penalization part in the likelihood formula, but I cannot figure out how can be the first part of the formula derived. Regression Analysis of Multiple Protein Structures Thomas D. Wavelet regression and additive models for irregularly spaced data. Concentrates on recognizing and solving convex optimization problems that arise in engineering. (in review) 6. Buntine and Weigend (1991) introduce this idea first in penalized regression and logistic regression problems. Multivariate normal distribution. Each late day used after the first two will result in a 25% penalty. Also, it tries to assign same coefficients to variables which are in the same group. edu Robert Tibshirani DepartmentofStatistics SequoiaHall,StanfordUniversity,CA,94305. and then including the new variable Y in your linear mixed model. The LASSO is based on minimizing Mean Squared Error, which is based on balancing the opposing factors of bias and variance to build the most predictive model. Review of random variables, some linear algebra, law of averages (LLN and CLT). We recommend testing alphas at a rate of of 3 times the next smallest value (i. Lasso penalized regression is particularly advantageous when the number of predictors far exceeds the number of observations. Jamie Zeitzer is part of Stanford Profiles, official site for faculty, postdocs, students and staff information (Expertise, Bio, Research, Publications, and more). Elsie Gyang Ross is part of Stanford Profiles, official site for faculty, postdocs, students and staff information (Expertise, Bio, Research, Publications, and more). In lasso regression, the coefficients are estimated by minimizing: RSS subject to. user age and eligibility. The Minnesota PP didn’t have many chances over the first few games of the season, but the Wild have had 22 opportunities over their last four games. The site facilitates research and collaboration in academic endeavors. Manning Stanford University Stanford, CA 94305, USA fjtibs, manningg@cs. edu Simon Dae-oong Kim Stanford University simonkim@stanford. Statistical learning suffers from the so-called curse-of-dimensionality'' when applied to high-dimensional data: The proportion of the training data that lie in a fixed-radius neighborhood of a point decreases to zero at an exponential rate with increasing dimension of the input space. Penalization is a powerful method for attribute selection and improving the accuracy of predictive models. Convergence Rates of Nonparametric Penalized Regression under Misspecified Smoothness (in review) Refereed Book Chapters 1. The ℓ0 penalty can be thought of as a measure of complexity of the model. Complete the following steps for this exercise: Fill in the linear_regression. Our results demonstrate that penalized regression is a promising method for examining associations between neural predictors and clinically relevant traits or behaviors. We are stlil working to complete this list. ture patients. 2 Ridge Regression Solution to the ℓ2 problem Data Augmentation Approach Bayesian Interpretation The SVD and Ridge Regression 3 Cross Validation K-Fold Cross Validation Generalized CV 4 The LASSO 5 Model Selection, Oracles, and the Dantzig Selector 6 References Statistics 305: Autumn Quarter 2006/2007 Regularization: Ridge Regression and the. Glmnet is a package that fits a generalized linear model via penalized maximum likelihood. 5 — Logistic Regression | Simplified Cost Function And Gradient Descent — [ Andrew Ng] Lecture 6. edu Abstract Topic Models such as Latent Dirichlet Allocation (LDA) have been successfully. The limitations of the lasso. Haris A+, Shojaie A, Simon N. This penalty is particularly useful in the p ≫ N situation, or any situation where there are many correlated predictor variables. m' which will be used for logistic regression. We describe this notion of decompos-able in Section 2 and then develop a general framework for analyzing the consistency and model selection consistency of M-estimators with geometrically decomposable penalties. A Statistical View of Some Chemometrics Regression Tools lldiko E. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Arxiv 2009, Annals of Statistics 2011, with A. We propose penalized logistic regression (PLR) as an alternative to the SVM for the microarray cancer diagnosis problem. A value of P<5×10−8 was considered to be statistically sig-nificant. Under the logistic regression model, we use the bridge penalty with γ = 1/2 for penalized gene selection. Hi-- I highly recommend the glmnet package. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underlying signal in a set of data. Penalized Likelihoods: Fast Algorithms and Risk Bounds. 754 STANFORD LAW REVIEW [Vol. Courses offered by the Department of Statistics are listed under the subject code STATS on the Stanford Bulletin's ExploreCourses web site. stanford reserves the right to modify these terms at any time and will publish notice of any modifications on-line at this site. 0 Dec 29, 2013 Abstract The least absolute shrinkage and selection operator (lasso) and ridge regression produce usually dif-ferent estimates although input, loss function and parameterization of the penalty are identical. Generalized Linear Modeling with H2O by Tomas Nykodym, Tom Kraljevic, Amy Wang & Wendy Wong with assistance from Nadine Hussami & Ariel Rao Edited by: Angela Bartz. Provides Stanford University credit that may later be applied towards a graduate degree or certificate. Second, we propose adding a clustering step to l1 penalized regression, to build customized training sets of observations that are similar to the test set. Linear regression is an excellent method for predicting re-sults, using a hypothesis that is computed from training data. By introducing a complexity-based penalty term, we can identify irrelevant features and remove. The Minnesota PP didn’t have many chances over the first few games of the season, but the Wild have had 22 opportunities over their last four games. Hazlett and Xu(2018) instead outline a promising approach for improving SCM estimation by rst using a kernel approach to transform the raw lagged outcomes. Penalized Likelihoods: Fast Algorithms and Risk Bounds. 1 penalized regression, linear regression, or K-nearest neighbors regression, as this these are algorithms that have been perfected, but rather in the construction of meaningful features. the HLM software package will give you PQL and Laplace estimates for multilevel logistic models. a penalized matrix decomposition, and its applications a dissertation submitted to the department of statistics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy daniela m. Outline 9×more penalty for over-estimating Regression. Ridge Regression (from scratch) The heuristics about Lasso regression is the following graph. Thus, members of non-fundamentalist faith groups should oppose capital. An algorithm of this kind has been proposed for the L1-penalized regression (lasso) in the literature, but it seems to have been largely ignored. Email this Article Multivariate adaptive regression splines. Time permitting, the types of models we will study include: Simple Linear Regression Multiple Linear Regression Polynomial Regression Model Selection for Mupltiple Linear Models. Schmidler Section on Medical Informatics Stanford University School of Medicine Stanford, CA 94305, USA Trevor Hastie Dept. In this article, you are going to learn, how the random forest algorithm works in machine learning for the classification task. ridge is appro-priate. A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E. Regression is the analysis of the relation between one variable and some other variable(s), assuming a linear relation. m file to compute J(\theta) for the linear regression problem as defined earlier. 2 Topics covered Simple linear regression. @JasonGregor ( Jason Gregor ) posted 7 hours ago Sending positive vibes to Mark Letestu, one of the nicest men I've met in hockey. Logistic regression is a popular model in statistics and machine learning to fit binary outcomes and assess the statistical significance of explanatory variables. Multivariate regression model. dealing with noisy features and samples by only selecting a subset of them for training [20]. The site facilitates research and collaboration in academic endeavors. Here, we use the same idea in a robust lasso regression. Ridge regression is known to shrink the coefficients of correlated predictors towards each other, allowing them to borrow strength from each other. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Causes of unplanned readmission to the hospital are multiple with some being better targets for intervention than others. The perturbed residual is where we are given ,,, and we seek to find and. Driscoll, Xavier Gabaix, and David Laibson ∗ July 9, 2006 Abstract We show that in ten diﬀerent contexts —three kinds of credit card fee payments, credit card. In this video i'm implementing a binary-class logistic regression classifier to determine whether a student is accepted or not given his marks at two exams, the algorithm was implemented using. 2! and implemented at Stanford University Through a regression. A curated database containing nearly all published HIV RT and protease sequences: a resource designed for researchers studying evolutionary and drug-related variation in the molecular targets of anti-HIV therapy. The course is intended to be a (non-exhaustive) survey of regression techniques from both a theoretical and applied perspective. Navigation menu. a penalized matrix decomposition, and its applications a dissertation submitted to the department of statistics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy daniela m. The regression model adds one mean shift parameter for each of the ndata points. And, high dimensional data are quickly growing in many areas due to the development of technological advances which helps collect data with a large number of variables. Most penalties are enforced by sending the offending player to a penalty box for a set number of minutes. Sinsheimer , Kenneth Lange. This, together with the result in Tian and Taylor [2015b] allows selective inference with arbitrary data generating distribution and general model selection through penalized regression. Nashville has been awful when opponents have an extra skaterthwarting just 69% of powe-play opportunities. SVM or margin-based censored regression (proposed in this paper). If you want to see examples of recent work in machine learning, start by taking a look at the conferences NIPS(all old NIPS papers are online) and ICML. Adam has 5 jobs listed on their profile. This paper studies a generic sparse regression problem with a customizable sparsity pattern matrix, motivated by, but not limited to, a supervised gene clustering problem in microarray data analysis. First, we propose adding a penalty on the nuclear norm of the regression coefficient matrix in multinomial regression to learn which outcomes are similar. edu Computer Science Department, Stanford University, CA 94305 USA Abstract Many applications of supervised learning re-quire good generalization from limited la-beled data. regression (Lockhart et al. She is the reigning champion in women's singles at the Australian Open. Preliminaries Some Foundations Recent Results Regression with ‘1 penalty Summary Minimum Description Length Principle for Statistics Two-stage Code Redundancy and Resolvability Statistical Risk of MDL Estimator. 3 and so on). This basic method is already interesting and useful in its own right (see Homework #3). This sometimes results in the elimination of some coefficients altogether, which can yield. The site facilitates research and collaboration in academic endeavors. Matthew Harding & Carlos Lamarche, 2017. We then apply a regularization favoring a sparse vector of mean shift parameters. Simon N, Shojaie A. Courses offered by the Department of Statistics are listed under the subject code STATS on the Stanford Bulletin's ExploreCourses web site. From this perspective, augmented SCM is analogous to augmented IPW in cross-sectional settings (Robins et al. Witten * Department of Statistics, Stanford University, Stanford, CA 94305, USA. Your job will be to fill in logistic_regression. Chapter: 4 Panel Studies. The nice thing about penalized models is that they deal well with collinearity, and they tend to generalize better than simple linear models. Nuclear penalized multinomial regression with an application to predicting at bat outcomes in baseball Scott Powers 1, Trevor Hastie , and Robert Tibshirani1 1 Department of Statistics, Stanford University, Stanford, CA, USA. We show that. The procedure models the regression surface as a sum of general smooth functions of linear combinations of the predictor variables in an iterative manner. , Sobel E, Lange K. Text as Data Justin Grimmer Associate Professor Department of Political Science Stanford University November 10th, 2014 Justin Grimmer (Stanford University) Text as Data November 10th, 2014 1 / 40. A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E. The first comes up when the number of variables in the linear system exceeds the number of observations. I am trying to understand how penalized logistic regression works and I got stuck with negative binomial log-likelihood. a penalized matrix decomposition, and its applications a dissertation submitted to the department of statistics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy daniela m. So, when we modify this algorithm for regularized linear regression, we're going to end up treating theta zero slightly differently. Starting Autumn 2016 there is a \$100 fee per course for courses dropped before the drop deadline. ##### ### Logistic regression and multinomial regression ##### ### Standard estimation of a logistic regression model. fitted a logical variable, which determines whether the (sometimes large) component. We then apply a reg-ularization favoring a sparse vector of mean shift parameters. Jonathan Maltzman is part of Stanford Profiles, official site for faculty, postdocs, students and staff information (Expertise, Bio, Research, Publications, and more). Computer Age Statistical Inference Algorithms, Evidence, and Data Science BRADLEY EFRON Stanford University, California 1. edu Genki Kondo Stanford University genki@stanford. The limitations of the lasso. Penalized regression methods for simultaneous variable selection and coefficient estimation, especially those based on the lasso of Tibshirani (1996), have received a great deal of attention in recent years, mostly through frequentist models. In regression, the outcome is continuous. Nuclear penalized multinomial regression 29 Table 6: Visualization of tted regression coecient matrices from NPMR applied to the vowel data. Method: The present paper evaluates the performance of lasso penalized logistic regression in case-control disease gene mapping with a large number of SNP (single nucleotide polymorphisms) predictors. ridge function. Even small Initial discriminatory ifferences in wages may cumulate to large ones as men and women make decisions bout human capital investments and time locatton In the market and the home on the. The usual L 1 penalty yields a convex criterion, but fails to deliver a robust estimator. ■ Generalized one sample problem: penalize large values of. Belloni, The Annals of Statistics, 2011. Get a cup of coffee before you begin, As this going to be a long article 😛 We begin with the table of. glmnet, is a package for penalized regression. Dirk Ormoneit Trevor Hastie Volker Tresp Data Mining. Jamie Zeitzer is part of Stanford Profiles, official site for faculty, postdocs, students and staff information (Expertise, Bio, Research, Publications, and more). punishment as a result of their situational attribution styles (Grasmick and McGill, 1994). For instance, the least absolute shrinkage and selection operator (LASSO) can deal with the high-dimensional situations (p ≫ n) by forcing certain regression coefficients to be zero (Tibshirani, 1996). Joe Hilbe ===== i don't know any commands in stata that will give you the estimates you want (other than -gllamm- and -xtlogit-). The clustered lasso method is proposed with the l 1 -type penalties imposed on both the coefficients and their pairwise differences. Our regression model adds one mean shift parameter for each of the ndata points. We usually have ridge regression aka the L2 norm and the lasso regression, aka the L1 norm. Penalized Regression for Neuroimaging. regression penalty (α = 0) and the lasso penalty (α = 1). edu Abstract Annotation errors can signiﬁcantly hurt classiﬁer performance, yet datasets are only growing noisier with the increased use of Amazon Mechanical Turk and tech-. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Arxiv 2009, Annals of Statistics 2011, with A. Poisson regression , the free encyclopedia en m wikipedia wiki Poisson regression AM Poisson regression In statistics, Poisson regression is a form of regression analysis used to for some positive constant This technique, similar to ridge regression, can reduce overfitting PDF& Penalized Regressions The Bridge Versus the Lasso Aliquote aliquote. (in review) 6. Time permitting, the types of models we will study include: Simple Linear Regression Multiple Linear Regression Polynomial Regression Model Selection for Mupltiple Linear Models. comMay 2013Dan SteinbergMikhail GolovnyaSalford SystemsSalford Systems ©2013 1. The present study investigates the linkages among race, religion, and attitudes toward capital. statsmodels. The models include linear regression, two-class logistic regression, and multi-nomial regression problems while the penalties include ' 1 (the lasso), '. The PLUS computes multiple exact local minimizers of a possibly nonconvex penalized loss function in a certain main branch of the graph of critical points of the penalized loss. When multicollinearity occurs, least squares estimates are unbiased, but their variances are large so they may be far from the true value. @JasonGregor ( Jason Gregor ) posted 7 hours ago Sending positive vibes to Mark Letestu, one of the nicest men I've met in hockey. SportSource Analytics is the industry leader when it comes to college football data. Green tones are restricted to areas of the image with foliage, and there seems to be a slight amount of color tinting in the sky. In the regression model, we add one mean shift parameter for each of the n data points. Least Angle Regression (LAR) Similar to ridge regression, LAR is a shrinkage estimator. This is a really good question. method based on penalized regression and convolutional neural networks, and in each case, our method improves upon existing state-of-the-art. This is an "applied" machine learning class, and we emphasize the intuitions and know-how needed to get learning algorithms to work in practice, rather than the mathematical derivations. The perturbed residual is where we are given ,,, and we seek to find and. Penalized regression is a promising and underutilized alternative to OLS regression. The parametric. detection using nonconvex penalized regression mean shift parameter sparse vector soft thresholding tuning parameter convex criterion large data data-dependent choice hard test problem paper study robust estimator tuned ipod show outstanding performance various situation usual l1 penalty iteration cost data point detection problem coefficient. The code calls minFunc with the logistic_regression. Constructing Informative Priors using Transfer Learning Rajat Raina rajatr@cs. The SVD and Ridge Regression. Ridge regression may improve over ordinary least squares by inducing a mild bias while decreasing the variance. coursera-stanford / machine_learning / lecture / week_2 / iv_linear_regression_with_multiple_variables_week_2 / Quiz - Linear Regression with Multiple Variables. It actually fits penalized multinomial logit models, but it's a very clean and elegant package. Fernandez-Val, The Review of Economic Studies, 2011 24. Example: a student had one free late day remaining but his/her group uses two late days on a Problem Set. 754 STANFORD LAW REVIEW [Vol. Anderson Stanford Dissertation Award in Theoretical Statistics (2016) Outstanding Overseas Chinese Student Scholarship (2015) General Wang Yaowu Stanford Graduate Fellowship (2011–2015) Best in Potential Award in Bicoastal Columbia–Stanford DataFest Competition (2013) Shing-Tung Yau College Student Mathematics Contests in China. The regression network outputs are somewhat reason- able. It is a curated set of spatial and temporal data about mass shootings in America, taken from online media sources. Dirk Ormoneit Trevor Hastie Volker Tresp Data Mining. ncvreg-package Regularization paths for SCAD- and MCP-penalized regression mod-els Description Efﬁcient algorithms for ﬁtting regularization paths for a variety of regression models (linear, logis-tic, Poisson, survival) penalized by MCP or SCAD, with optional additional L2 penalty. Github repo for the Course: Stanford Machine Learning (Coursera) Quiz Needs to be viewed here at the repo (because the image solutions cant be viewed as part of a gist). The present study assessed the ability of these two traits, along with indices of disordered social media use, alcohol use, negative mood, and demographic factors, to predict Facebook self-disclosure in a hierarchical regression model. It also includes a helper function named 'map_feature. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. 5 — Logistic Regression | Simplified Cost Function And Gradient Descent — [ Andrew Ng] Lecture 6. Haris A+, Shojaie A, Simon N. I am trying to understand how penalized logistic regression works and I got stuck with negative binomial log-likelihood. We show that when using the same set of genes, PLR and the SVM perform similarly in cancer classification, but PLR has the advantage of additionally providing an estimate of the underlying probability. 6 — Logistic Regression Stanford University. Abramsa, Vinod Menona,d a Department of Psychiatry and Behavioral Sciences, Stanford University School of Medicine, Stanford, CA 94305, USA. Ridge Regression As with least squares, ridge regression seeks coefficient estimates that fit the data well, by making the RSS small. Given that a-priori solutions to these. CART ― Classification and Regression Trees (CART), commonly known as decision trees, can be represented as binary trees. Review of random vectors, independence. These are all variants of Lasso, and provide the entire sequence of coefficients and fits, starting from zero, to the least squares fit. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. edu Isaac Caswell Department of Computer Science Stanford University icaswell@stanford. We use the Lasso estimate as weight in the adaptive Lasso. It is a shrinkage method. Heqiao has 3 jobs listed on their profile. Its output is a continuous piecewise linear path encompassing from the origin for infinite penalty to a least squares solution for zero penalty. Stanford University Midterm Examination 180 minutes Problem Full Points Your Score 1 Multiple Choice Questions 13 2 Short Answer Questions 23 3 Convolutional Neural Networks 28 4 Adversarial Attacks 10 5 Loss comparisons 23 6 The Optimizer 20 Total 117 The exam contains29pages including this cover page. EDU found on Yumpu. 306 Employment and Unemployment since the Early Seventies by T. MULTIPLE REGRESSION IN COMPARATIVE RESEARCH Michael Shalev This paper criticizes the use of multiple regression (MR) in the ﬁelds of comparative social policy and political economy and proposes alternative methods of numerical analysis. Ng Computer Science Department Stanford University Stanford, CA 94305 Abstract L1 regularized logistic regression is now a workhorse of machine learning: it is widely used for many classiﬁca-tion problems, particularly ones with many features. Includes access to online course materials and videos for the duration of the academic quarter. Data: Here is the UCI Machine learning repository, which contains a large collection of standard datasets for testing learning algorithms. Simon N, Shojaie A. penalty functions have several limitations: the LASSO selects at most n variables before it saturates and if there is a group of highly correlated variables the LASSO tends to select one representative from a group and ignore the other components in the group. We present a new method for post‐selection inference for (lasso)'penalized likelihood models, including generalized regression models. Lasso penalized regression is particularly advantageous when the number of predictors far exceeds the number of observations. In statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. This should lead to “multivariate” shrinkage of the vector. Univariate regression. Morever, we described the k-Nearest Neighbor (kNN) classifier which labels images by comparing them to (annotated) images from the training set. edu Abstract Dropout and other feature noising schemes control overﬁtting by artiﬁcially cor-rupting the training data. detection using nonconvex penalized regression mean shift parameter sparse vector soft thresholding tuning parameter convex criterion large data data-dependent choice hard test problem paper study robust estimator tuned ipod show outstanding performance various situation usual l1 penalty iteration cost data point detection problem coefficient. Penalized regression yields more stable estimates and sparser models than OLS regression in situations with large numbers of highly correlated neural predictors. It is a shrinkage method. can be expressed as a sum of support functions over convex sets. Course Description. Penalized regression methods such as the lasso (Tibshirani, 1996), ridge regression (Hoerl and Kennard, 1970), and bridge regression (Frank and Friedman, 1993; Fu, 1998) have become popular alternatives to ordinary least squares (OLS). Notice that the regularization function is not a function of the data, it is only based on the weights. In this paper, we develop a method for de-biasing penalized regression adjustments to allow sparse regression methods like the lasso to be used for √n-consistent inference of average treatment effects in high-dimensional linear models. Bayesian regression modeling with INLA in SearchWorks catalog Skip to search Skip to main content. 1 billion originated loans. Now Assistant Professor at Brown University, Department of Biostatistics. However, they require. Multivariate normal distribution. This basic method is already interesting and useful in its own right (see Homework #3). dealing with noisy features and samples by only selecting a subset of them for training [20]. Package ‘lars’ February 20, 2015 Type Package Version 1. The softmax regression function alone did not fit the training set well, an example of underfitting. Most penalties are similar versions of complexity. The number of regressors, , may be large and possibly greater than the number of observations,. SUMMARY A sample size calculation for logistic regression involves complicated formulae. Includes access to online course materials and videos for the duration of the academic quarter. This can considerably reduce the burden of enforcement. We introduce the idea of a loss function to quantify our unhappiness with a model’s predictions, and discuss two commonly used loss. fit_regularized (start_params=None, method='l1', alpha=0, ceps=0. MULTIPLE REGRESSION IN COMPARATIVE RESEARCH Michael Shalev This paper criticizes the use of multiple regression (MR) in the ﬁelds of comparative social policy and political economy and proposes alternative methods of numerical analysis. edu Simon Dae-oong Kim Stanford University simonkim@stanford. Our results demonstrate that penalized regression is a promising method for examining associations between neural predictors and clinically relevant traits or behaviors. High-Dimensional Regression. We show that when using the same set of genes, PLR and the SVM perform similarly in cancer classification, but PLR has the advantage of additionally providing an estimate of the underlying probability. Time permitting, the types of models we will study include: Simple Linear Regression Multiple Linear Regression Polynomial Regression Model Selection for Mupltiple Linear Models. He lost a couple of fumbles, and the Lions charitably dropped. m file as objective function. Regularization Paths for Conditional Logistic Regression: The clogitL1 Package Stephen Reid Stanford University Rob Tibshirani Stanford University Abstract We apply the cyclic coordinate descent algorithm ofFriedman, Hastie, and Tibshirani (2010) to the tting of a conditional logistic regression model with lasso (‘ 1) and elastic net penalties. Recently proposed method estimate the sparse precision matrix by optimizing penalized likelihood [20, 4, 1, 16, 5, 14, 22] or through neighborhood selection [10, 12, 6, 19], where the structure of the graph is estimated by estimating the neighborhood of each 1. Penalized regression yields more stable estimates and sparser models than OLS regression in situations with large numbers of highly correlated neural predictors. 3 of [17], for example). Complete the following steps for this exercise: Fill in the linear_regression. problem of linear regression ( tting the best hyperplane to a set of data points). Review of random variables, some linear algebra, law of averages (LLN and CLT). Part II: Ridge Regression 1. What the Stanford MSA is. Efﬁcient L1 Regularized Logistic Regression Su-In Lee, Honglak Lee, Pieter Abbeel and Andrew Y. PAC is an extremely general method which encompasses many statistical approaches, such as the fused lasso, monotone curve estimation and the generalized lasso. Shirshendu - Writing a business proposal every time you Tulshi - Your data will be safe even after uploading Samsons - Anyone can design the company logo to be used. The softmax regression function alone did not fit the training set well, an example of underfitting. We then apply a regularization favoring a sparse vector of mean shift parameters. On the right, τ= 0. In the separable case, L 1. 5 — Logistic Regression | Simplified Cost Function And Gradient Descent — [ Andrew Ng] Lecture 6. Outline 9×more penalty for over-estimating Regression. Stanford University | SU In this paper we purpose a blockwise descent algorithm for group-penalized multiresponse regression. No explanation is required. Ridge regression can, in principle, be carried out using lm or lm. Complete the following steps for this exercise: Fill in the linear_regression. Logistic regression is a popular model in statistics and machine learning to fit binary outcomes and assess the statistical significance of explanatory variables. The Stanford MIP-Vienna School of International Studies Academic Exchange is an Autumn Quarter exchange program between the Ford Dorsey Master's in International Policy and the Diplomatische Akademie Wien – Vienna School of International Studies (DA). Please note that these are not Stanford for-credit courses. penalty functions have several limitations: the LASSO selects at most n variables before it saturates and if there is a group of highly correlated variables the LASSO tends to select one representative from a group and ignore the other components in the group. NET Numerics - Numerical foundation of the Math. R packages maintained by Stanford R community members. ridge is appro-priate. With the aid of the new al-gorithms, the SCAD penalized estimators are able to enjoy computational eﬃciency comparable to that of the LASSO penalized estimator. edu Abstract Annotation errors can signiﬁcantly hurt classiﬁer performance, yet datasets are only growing noisier with the increased use of Amazon Mechanical Turk and tech-. In this paper, we develop a method for de-biasing penalized regression adjustments to allow sparse regression methods like the lasso to be used for √n-consistent inference of average treatment effects in high-dimensional linear models. 6 Indeed, they contend that under such conditions, capital punishment should be considered not merely morally. An alternative solution is to apply a penalized regression method. Convex sets, functions, and optimization problems. Random Walk Features for Network-aware Topic Models Ahmed Hefny, Geoffrey Gordon, Katia Sycara School of Computer Science Carnegie Mellon University 5000 Forbes Ave Pittsburgh, PA, United States fahefny,ggordon,ksycarag@cs. tion path for linear regression, logistic and multinomial regression models, Poisson regres-sion and the Cox model. penalized regressions. Make sure that this function's m-file is placed in the same working directory where you plan to write your code. The models include linear regression, twoclass logistic regression, and multinomial regression problems while the penalties include 1 (the lasso), 2 (ridge regression) and mixtures of the two (the elastic net). On the right, τ= 0. Logistic + bridge. Yale University.


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1

## Contact

2023-01-10

IP Status

No known IP claims.

Interactions and External Dependencies
• This extension interacts with VK_KHR_dynamic_rendering

• This extension interacts with VK_QCOM_render_pass_transform

Contributors
• Jeff Leger, Qualcomm

• Jonathan Tinkham, Qualcomm

• Jonathan Wicks, Qualcomm

## Description

Certain use-cases (e.g., side-by-side VR rendering) use multiview and render to distinct regions of the framebuffer for each view. On some implementations, there may be a performance benefit for providing per-view render areas to the implementation. Such per-view render areas can be used by the implementation to reduce the pixels that are affected by attachment load, store, and multisample resolve operations.

The extension enables a multiview render pass instance to define per-view render areas. For each view of a multiview render pass instance, only those pixels in the per-view render area are affected by load, store and resolve operations.

## New Enum Constants

• VK_QCOM_MULTIVIEW_PER_VIEW_RENDER_AREAS_EXTENSION_NAME

• VK_QCOM_MULTIVIEW_PER_VIEW_RENDER_AREAS_SPEC_VERSION

• Extending VkStructureType:

• VK_STRUCTURE_TYPE_MULTIVIEW_PER_VIEW_RENDER_AREAS_RENDER_PASS_BEGIN_INFO_QCOM

• VK_STRUCTURE_TYPE_PHYSICAL_DEVICE_MULTIVIEW_PER_VIEW_RENDER_AREAS_FEATURES_QCOM

## Issues

1) Do the per-view renderAreas interact with vkGetRenderAreaGranularity ?

RESOLVED: There is no change. The granularity returned by vkGetRenderAreaGranularity also applies to the per-view renderAreas.

2) How does this extension interact with VK_QCOM_render_pass_transform?

RESOLVED: When VK_QCOM_render_pass_transform is enabled, the application provides render area in non-rotated coordinates which is rotated by the implementation to the rotated coordinate system. When this extension is used in combination with VK_QCOM_render_pass_transform, then the renderArea provided in VkRenderingInfo::renderArea, VkRenderPassBeginInfo::renderArea, or VkCommandBufferInheritanceRenderPassTransformInfoQCOM::renderArea is rotated by the implementation. The per-view render areas are not rotated.

3) How does this extension interact with VK_QCOM_multiview_per_view_viewports

RESOLVED: There is no direct interaction. The per-view viewports and the per-view renderAreas are orthogonal features.

4) When a per-view renderArea is specified, must multiview rendering for each view of a multiview render pass be contained within the per-view renderArea?

RESOLVED: Yes, and the VK_QCOM_multiview_per_view_viewports may help here since it provides per-view scissors.

5) When per-view render areas are specified, what purpose if any do VkRenderPassBeginInfo::renderArea and VkRenderingInfo::renderArea serve?

RESOLVED: The per-view renderArea effectively overrides the per-renderpass renderArea. The per-view renderArea defines the regions of the attachments that are effected by load, store, and multisample resolve operations. A valid implementation could ignore the per-renderpass renderArea. However, as an aid to the implementation, the application must set the per-renderpass renderArea to an area that is at least as large as the union of all the per-view render areas. Pixels that are within the per-renderpass renderArea but not within any per-view render area must not be affected by load, store, or multisample resolve operations.

## Version History

• Revision 1, 2023-01-10 (Jeff Leger)

format: plain


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Note on Angle of Contact and Capillary

• Note
• Things to remember

Angle of Contact

Angle of contact is the angle which is tangent to a liquid surface at the point of contact makes with the solid surface inside the liquid.

It's value is acute (90o) for liquid which do notwet the walls of container.

Capillary Tube

A glass tube of very fine bare (hair like) is known as capillary tube. Liquid having concave meniscus (θ o) depresses below the free surface inside a capillary tube. When capillary tube open at both ends when dipped in a liquid.

This rise or fall of a liquid in a capillary tube is known as capillary action or capillarity.

Measurement of Surface Tension of a liquid by Capillarity

Let us consider a capillary tube opened at both ends of radius is dipped in a liquid of density (Ρ) having concave meniscus.

Let h be the height of liquid rises in the tube above free surface as shown in the figure.Suppose the liquid molecules at the point of contact with the tube. The force due to the surface tension (T) acts on it along the tangent to the meniscus inside the liquid.

According to Newton's third law of motion, the tube also exerts on equal and opposite reaction force (R) can be resolved into two compounds i.e Tcosθ which acts vertically upward and Tsinθ which acts horizontally. The horizontal components Tsinθ acting on molecules at the point of contact cancel each other whereas vertical component added up. Hence a liquid experiences a net upward force on the liquid $$F = T\cos\theta \times 2\pi r \dots (i)$$

In equilibrium,

\begin{align*} \text {Net force on the surface of the liquid} (F) &= \text {height of the liquid in the tube} (w) \\ \text {Now volume of liquid in the tube (v)} &= \text {volume of cylinder of radius 'r' and height} (h) -\text {volume of radius} (v) \\ V &= \pi r^2r + \pi r^2r + \frac 23 \pi r^3 \dots (iii) \\ \end{align*}

Now, equation (i) becomes

\begin{align*} T\cos\theta \times 2\pi r &= V\rho g\\ T\cos\theta \times 2\pi r &= (\pi r^2 h +\frac 13 \pi r^3)\rho g \\ 2T\cos\theta &= r^2 \left ( h + \frac r3 \right ) \rho g \\ T &= \frac {r \left ( h + \frac r3\right)\rho g}{2\cos\theta}\end{align*}

Here $$\frac r3 << h$$ so neglect $$\frac r3$$

\begin{align*} T &= \frac {rh\rho g}{2\cos \theta} \\ h &= \frac {2\cos\theta}{r\rho g} \dots (vi) \\ h \propto \frac 1r \\ \end{align*}

Equation (vi) is known ascent formula

For liquid having $$\theta > 90^o, \cos \theta$$ becomes negative such liquid depresses below the free surface the corresponding formula is known as descent formula.

Rise of Liquid in a Tube of Insufficient Length

Let us consider a capillary tube opened at both ends of radius is dipped in a liquid of density (Ρ) having concave meniscus.

Let h be the height of liquid rises in the tube above free surface as shown in the figure.Suppose the liquid molecules at the point of contact with the tube. The force due to the surface tension (T) acts on it along the tangent to the meniscus inside the liquid.

As we know that the height of a liquid that rises in a capillary tube of radius 'r' is given by

$$h =\frac {2T\cos \theta}{r \rho g} \dots (i)$$

From figure

\begin{align*} \cos\theta = \frac rR \\ \therefore r = R\cos\theta \dots (ii) \\ \end{align*}

Where R is the radius of curvature of the concave meniscus. Equation (i) becomes

\begin{align*} h = \frac{2T\cos \theta}{R\cos \theta \rho g}\\ hR = \frac {2T}{\rho g} =\text {constant} \dots (iii) \end{align*}

If a capillary tube if insufficient height 'h' is dipped in the same liquid then equation (iii) becomes

$$h'R' = \frac {2T}{\rho g} \dots (iv)$$

Fro, equation (iii) and (iv)

$$hR = h'R'$$

As $$h' < h \: \text {so} \: R' > R$$

liquid will not spill out from the tube but the radius of curvature of its meniscus will increase.

1In equilibrium,

\begin{align*} \text {Net force on the surface of the liquid} (F) &= \text {height of the liquid in the tube} (w) \\ \text {Now volume of liquid in the tube (v)} &= \text {volume of cylinder of radius 'r' and height} (h) -\text {volume of radius} (v) \\ V &= \pi r^2r + \pi r^2r + \frac 23 \pi

.

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# PS3 May Be In Trouble = A Cheaper SP3

Homework Helper
Since it's in trouble with respect to profits do any of you predict a cheaper PS3? Could they actually make a profit with a cheaper than $400 PS3? Considering the fact that Blu Ray players themselves are$ 300 ... probably not.

http://www.yahoo.com/s/956782 [Broken]

Last edited by a moderator:

Maybe, but then the profit may not be from the sale of the actual console...

NO kidding. they could give away the consoles and probably not see a dent in their bottom line. Considering that more people would be buying games

Sony has been in alot of trouble even since before the PS3 came out. They decided to amp up the processing power of the system fairly high making it rather expensive considering the components alone. This was supposed to outdo XBox, which it technically does, but the problem is developers may or may not be able to take full advantage of the system's full capabilities. So long as the games are only on par with XBox and occasionally somewhat better what's the reason to purchase the more expensive console? Fewer game developers, particularly larger companies, are willing to maintain exclusive contracts now aswell so there is less likelihood there will be many popular games you can only get on one system. BluRay was a big gamble for them too since they had no idea whether or not it would become a standard or die out. So far I dont think it has caught on strongly enough.

XBox 360 came out long before PS3 and microsoft has already made the money they needed off the box and can afford to continually bring down the price and undercut Sony. I'm pretty sure their box is much cheaper component-wise aswell.

I was pretty psyched about the PS3, but it didnt really catch on. One reason around here is that you can get a lot of pirated game cd's for the XBox 360 and you cant get those for the PS3. Its easier to stomach a cd for about $1 than a$40-\$90 cd.

Kurdt
Staff Emeritus
Gold Member
I was under the impression that Sony having won the format wars will be in quite a good position financially. I still prefer it and it is worth the hefty price tag. Compared to the other competition it is something quite other worldly.

BluRay was a big gamble for them too since they had no idea whether or not it would become a standard or die out. So far I dont think it has caught on strongly enough.

Also, https://www.amazon.com/dp/B000PALZE0/?tag=pfamazon01-20.

I think we'll start seeing a lot more deals like this in the coming months across the industry to try and increase BD's market share.

The funniest part of all this lately is the joint marketing campaign for BD by Sony and Microsoft. They're giving away BD disks in the Oct issue of Wired while promoting Vaio and Vista.


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# Zeer pot refrigeration (design)

Status
 This OSAT has been deployed by: Peter Rinker /Movement e.V. at least: >50 (ongoing growth) times in: Burkina Faso.

You can help Appropedia by contributing to the next step in this OSAT's Category:Status

 This page meets the criteria for a highlighted page. If not, remove this notice.

The Zeer pot is a refrigeration device invented in Africa which operates by the principle of evaporative cooling. According to Science in Africa, each device can store 12kg of vegetables, keeping them fresh for up to 20 days while costing less than 2USD to produce[1]. This article will showcase how to build a pot-in-pot refrigerator, discuss various aspects of the technology and present the results of a parametric analysis.

# Overview of Technology

Figure 1: Conceptual Flow of Energy and Water in Zeer Refrigerator

In harsh dry climates, food preservation plays a vital role in maximizing both economic and nutritional yield from the rare opportunity of a good harvest. The dry heat significantly reduces the life of produce and as a result there is a high level of wasted crop[2].

The following shelf-life comparison has been adapted from a case study performed by the Practical Action Organization and can be directly accessed here.

Table 1: Shelf of common produce with Zeer pot refrigerator[3]
Produce Shelf-Life without Zeer Shelf-Life with Zeer
Tomatoes 2 days 20 days
Guavas 2 days 20 days
Okra 4 days 17 days
Carrots 4 days 20 days
Arugula 1 day 5 days

The impact of the pot-in-pot refrigerator is immediately realized as the shelf life for most produce is extended by 5 to 10 times. This means that farmers are not only able sell their produce beyond the initial days following a harvest, but will also be able to safely consume the unsold goods because of their extended shelf life.

## Operating Principle

When evaporation occurs from a surface, there is an energy associated with the phase change known as the latent heat of vaporization. In a given system, as a gas species flows over the wet surface, evaporation and condensation continuously occur to maintain steady-state conditions.

In order to sustain evaporation, there must be a draw of internal energy in the liquid, which would result in a temperature reduction. This cooling effect is known as evaporative cooling and is most effective in dry climates due to the lack of moisture content (relative humidity) in the air.[4]

In the case of the Zeer refrigerator, water evaporates out of the sand through the surface of the outer clay pot and from the whole top surface of the moist sand exposed to the solar radiation, removing energy from the system. Figure 1 is a graphical representation of the water and energy flow within a Zeer refrigerator.

Additional information on evaporative cooling technologies can be accessed here.

## How to make a Pot-in-Pot Refrigerator

The document File:ClayTech.pdf has been made available by the Practical Action Organization. Within the document, pages 15 to 19 outline in detail how to make a pot-in-pot refrigerator. The remainder of the document discusses other clay based technologies that may be of interest. The steps for making a pot-in-pot refrigerator are outlined below in case the reader is unable to open PDF files[5]. You can find an alternative manual based on the experience of Movement e.V. in Burkina Faso [6]

Step
 Figure 2: A stone is used to shape the inside of the mould into a bowl shape
 Figure 3: Dimensions for Mould

Making the Moulds

• Make a small hole in the ground and cover with a mat. Use a small amount of wood chippings on the mat to prevent sticking
• Mix and knead an even mixture of mud, dung and water into a ball
• Repeatedly press a stone into the mixture to eventually form a bowl shape. Continue to do this, adding more material if necessary, until the mould reaches the dimensions given in Figure 2
• Once the mould has been made it must be left to dry in the sun for approximately 30 minutes
 Figure 4: Using water and a stone the surface of the mixture is smoothened
 Figure 5: The mould is turned over and the clay is pressed on top of it

Making the Clay Pots

• Make a small hole in the ground and cover with a a mat. Use a small amount of wood chippings on the mat to prevent sticking
• Knead the clay into a dough like mixture
• Roll the clay flat and place on top of an upside down mould
• Spread the mixture over the mould, maintaining a thickness of approximately 10mm
• Use a flat stone and some water to smooth the surface
• Once the round shape has been formed the mould can be removed
• The pot walls can be extended to the desired height
• Add an extra thick layer to the rim (approximately 20mm)
• This completes the first pot
• Form the larger second pot in the same manner, using the appropriate mould
• Extend the height of this pot as needed
• Form the rim for the larger pot, using a thickness of approximately 30mm
• The final part of this process is to add decoration around the outside of the pot. This is done by rolling the mixture into a long sausage shape and sticking it around the outside of the pot about two thirds of the way up. The pattern is made by pressing fingers into it. The large pot is now complete.
• Both pots are now left in the sun to dry. This normally takes between two to four days depending on the ambient temperature and sunlight
 Figure 6: The pots are burned under a pile of rocks and cow dung

Preparing the Clay Pots

• Identify an area that is cleared and make a circle with rocks
• Cover the ground within the circle with cow dung
• Place as many clay pots as possible inside the circle
• Completely cover the pots with wooden sticks and more cow dung
• Light the fire and let it burn for 24 hours
• To make this process more efficient it is better to burn as many pots as possible at once
 Figure 7: Assembled pot-in-pot refrigerator

Assembly and Operation

• Sand should be placed in the bottom of the large pot forming a layer of approximately 5cm in depth
• Place the smaller pot on top of the sand and centre it in the large pot (the pots should now be level)
• Fill the remaining space between the pots with sand
• If possible place the assembled device on a stand to maximize air flow
• The small pot should be covered with a lid (clay or cloth) to prevent hot air from entering the storage chamber
• The pot-in-pot refrigerator operates passively as long as the sand remains moist
• Check the sand twice a day and add water as needed

#  Analysis of Technology

The success of the pot-in-pot refrigerator is heavily dependent on the surrounding conditions. Due to the device's reliance on natural evaporative cooling it can only be viewed as an appropriate technology for regions that demonstrate a suitably low relative humidity and a sufficient level of air flow. In order to maximize the effectiveness of the pot-in-pot refrigeration device, the rate of evaporation must be increased. To properly investigate the technology it is important to quantifying the effects of:

• Relative Humidity
• Permeability
• Flow Characteristics
• Flow Velocity
• Laminar vs Turbulent Flow
• Boundary Layer Considerations
• Area available for Evaporation

It should be noted that the values presented in the following sections indicate the maximum cooling effect for a given set of parameters. The actual cooling will likely be lower due to irregularities in parameters, such as wind speed and relative humidity.

## Relative Humidity

Relative humidity is a measure of the content of water than can be held in the air at a specific temperature. An environment with a low relative humidity in comparison to one with a high relative humidity will evaporate moisture more readily and to a greater extent. For this reason, the pot-in-pot refrigeration device is only effective in environments with a low relative humidity (Figure 8). Northern central regions of Mexico (such as Chihuahua) and Africa (such as Sudan) are the most appropriate locations for use of this technology.

In some cases, the technology may prove to be suitable for specific times of the year when the humidity level is low. For a set of given conditions, information regarding humidity can be measured directly off of a psychrometric chart. Additionally, local weather centers typically keep record of humidity levels.

Figure 8: Cooling Effect vs. Wind Speed for varying Relative Humidity Levels (Device Radius = 0.25m, Permeability Correction Factor = 0.3, Ambient Temperature = 35 degrees Celsius, Turbulent Flow)

A relative humidity of 0.3 will has been used for all calculations where this value is held constant. At this relative humidity and a typical wind speed of 2.5 m/s, a cooling effect of 4.46kW is observed under turbulent flow conditions. The relationship established above, in which the device is expected to perform better in low humidity environments, is clearly indicated by this graph.

## Permeability

While evaporation undoubtedly occurs through the outer clay pot, the permeability of this layer plays a significant role in determining the actual rate at which water is evaporated. Earthenware is the type of clay that is used to construct this device. This type of clay is relatively porous and permeable in relation to other forms of clay, such as porcelain and stoneware[7]. In addition to permeating through the outer clay pot, water must travel through the sand to continuously replace moisture that has passed through the clay and evaporated. The cooling effect will be limited both by this diffusion rate and the permeability of the clay.

With these considerations, a permeability correction factor of 0.3 is incorporated into the performance calculations (as detailed in Appendix A). Additionally, a graph has been produced (Figure 12) varying only this correction factor in order to provide a general insight as to how this number effects the findings presented.

Figure 12: Cooling Effect vs. Permeability Correction Factor for varying Wind Speeds (Device Radius = 0.25m, Ambient Temperature = 35 degrees Celsius, Relative Humidity = 0.3, Turbulent Flow)

It is acknowledged that the validity of setting this correction factor to 0.3 may not be representative of the actual limitations created by the mechanisms involved with moisture diffusion and clay permeability within the device. This provides a platform for a future project to investigated this in greater detail, including the identification of methods to increase the permeability of the clay.

## Flow Characteristics

### Flow Velocity

As the water is evaporated into the surrounding air, the local relative humidity increases and thereby reduces the likelihood of further evaporation. Air flow is necessary to replace this moist air with dry air. This is provided by natural winds in the area. A high flow velocity will result in the body of air immediately surrounding the device to continuously remain in a dry state, and therefore will induce a higher rate of evaporation.

An average wind speed of 2.5m/s or 5.6mph has been selected based on a review of weather data for regions of Northern and Central Africa.

### Laminar vs Turbulent Flow

In a given system the existence of turbulence within a fluid flow increases the level of mixing, heat transfer and mass transfer. In this particular case, for a typical setting the flow can be assumed to be turbulent. If we consider air flow over a flat plate (where the air flow is wind and the flat plate is the ground), the distance at which the flow transitions from laminar to turbulent flow can be calculated. This transition occurs for a Reynolds number of approximately 5 x $10^5$.

Reynolds Number is dimensionless parameter that is defined as a measure of the ratio of inertial forces to viscous forces and can be calculated as follows:

${Re} = \frac{{\rho}{V}{x}}{\mu}$   [8]

For an ambient air temperature at 35 degrees Celsius and a wind speed of 2.5m/s, it is found that for the flow transitions into turbulence at a distance of approximately 2.6m.

Herein lies the justification for our turbulent assumption. If we rely on natural wind, it is highly unlikely that the pot-in-pot refrigerator will be located within 2.6m of where the air flow initially interacts with the ground surface. For completeness, a graph has been produced to highlight how performance varies between laminar and turbulent flow (Figure 9).

Figure 9: Cooling Effect vs. Wind Speed for Laminar and Turbulent Flow (Device Radius = 0.25m, Permeability Correction Factor = 0.3, Ambient Temperature = 35 degrees Celsius, Relative Humidity = 0.3)

### Boundary Layer Considerations

Continuing with the flat plate analysis, the existence of a boundary layer must be acknowledged. Within this boundary layer a velocity gradient exists in which there is an increase in velocity as we move away from the ground. The actual wind velocity (free stream velocity) exists only beyond this boundary layer[9]. Due to this velocity gradient, the boundary layer height becomes an important factor that effects the performance of the pot-in-pot refrigerator. This height can be calculated as follows:

$\part_{lam} = {5} \sqrt { \frac { \mu x } { \rho V } }$   [10]

$\part_{turb} = {0.37}{x}{Re_{x}}^{\frac{-1}{5}}$   [11]

Figure 8 provides a graphical representation of how this value increases in the direction of flow. It becomes clear that by increasing the height of the pot-in-pot refrigerator, the device can be exposed to a greater wind velocity. The wind velocity has a significant impact on the performance of the device. Observation of the various graphs presented in this article indicates that their is an increase in the cooling effect as the wind speed increases.

Figure 10: Boundary Layer Development and Velocity Profile for Flow Over a Flate Plate (Free Stream Velocity of 2.5m/s and Air Temperature of 35 degrees Celsius). Image is not to scale.

The existence of a velocity gradient within the boundary layer, and the expected performance gains due to higher velocities suggest that the device should be placed as high as possible and without obstruction of air flow. This can be accomplished by using a frame structure for the device to sit on top off. If possible, the frame itself should be placed on high ground or on top of existing solid structures.

## Area available for Evaporation

Figure 11: Dimensions for Pot-in-Pot Refrigerator Recommended by The Practical Action Organization

The surface area available for evaporation to occur on/through can be approximated for the pot-in-pot system as:

Total Area = Surface Area of Spherical Portion of Outer Pot

+ Surface Area of Cylindrical Portion of Outer Pot

+ Surface Area of Exposed Sand in between Pots

For example, if we use the recommended dimensions given by the Practical Action Organization (Figure 11), the area is found to be:

${Area_{total}}=\frac{1}{2}4 \pi \bar{OR}^2+2 \pi \bar{OR} \left ( \bar{H}- \bar{OR} \right )+ \pi \left ( \left ( \bar{OR}- \bar{TH} \right )^2- \bar{IR}^2 \right ) = 0.773 m^2$

This value has been used for all calculations in which area is held constant.

Figure 12 highlights how performance of the device varies with area. An overview of the section detailing How to make a Pot-in-Pot Refrigerator, will show that the radius is the simplest dimension to vary in the process. As such, the radius of the outer pot has been selected to vary the area available for evaporation.

Figure 12: Cooling Effect vs. Wind Speed for varying Device Radii (Permeability Correction Factor = 0.3, Relative Humidity = 0.3, Ambient Temperature = 35 degrees Celsius, Turbulent Flow)

As expected, there is a noticeable increase in the cooling effect as the surface area is increased. For the chosen wind speed of 2.5m/s, there is a cooling effect of 4.46kW, 6.58kW and 8.85kW for radii of 0.25m, 0.35m and 0.45m respectively.

An increase in the area, however, implies the need to use more clay and mould material for each device, which will drive up the average price of 2USD. Additionally, the device will require more sand and more water to operate. The option to increase the area available for evaporation should only be implemented after consideration of these factors. It may prove to be a viable approach for families wishing to pool resources in order to create a larger and more effective pot-in-pot device. In this case, the inner pot should also be made larger to increase capacity of goods to be stored inside the device.

# Discussion

## Interpretation of Reported Values

The values for the cooling effect have been reported in kilo-Watts(kW) and not in the form of energy. The actual energy removed from the system depends on the existence of these conditions and how long they are present for.

Additionally, energy derived from this number is an indication of the total energy that can be removed from the entire pot-in-pot device. This includes both clay pots, the volume of sand, water content within the sand, the air within the storage chamber and the contents of the storage chamber. The value therefore, should not be interpreted as the energy withdrawn exclusively from the storage chamber. A complex conduction and mass diffusion model would be required to provide a full analysis of the heat transfer mechanisms within the pot-in-pot structure, which falls outside the scope of this article.

## Convection

The pot-in-pot refrigerator will experience both forms of convection. When the air is idle the device will experience natural convection and subsequently, when the air is moving at a given velocity, the device will undergo the effects of forced convection.

In both cases it is likely that the convective process will transfer energy to the system due to the high ambient temperature and cooler device surface temperature. This energy that is transferred into the system will reduce the overall cooling effect. In comparison to the energy associated with a change of phase reaction (evaporation) however, the convective losses/gains are relatively small.

The Zeer pot refrigerator is expected to operate in open conditions with exposure to direct sunlight during the day. This will transfer a significant amount of energy to the system, reliant on the level solar irradiation, surface temperature of the outer pot and material properties of the clay.

## Rough Approximation for Net Energy Balance

This section will attempt to quantify the amount of energy withdrawn from the storage chamber by applying an energy balance including evaporative cooling and radiation.

The following conditions describe the base case scenario used throughout the article:

$Ambient\ Temperature: T_{amb} = 308 K$

$Relative\ Humidity: {RH} = 0.3$

$Wind\ Speed: W = 2.5 m/ s$

$Permeability\ Correction\ Factor: {PCF} = 0.3$

$Surface\ Area\ for\ Evaporation: A = 0.733 m/ s$

To include the effects of radiation let us consider only daylight hours, assumed to be between the 8am and 6pm (10 hours).

The maximum cooling effect for the parameters listed above was reported to be 4.46kW. Over the course of 10 hours, the overall cooling (considering periods of flow stagnation, environmental irregularities, and periods of reduced of moisture content in the sand) has been approximated to be equal to three hours at this maximum value for the prescribed conditions. This gives us an effective energy loss from the system of 13.4kWh.

The following calculation determines the net energy transfered to the system by radiation for the described scenario. Additionally let:

$Effective\ Sky\ Temperature: T_{sky}=263K$

$Device\ Surface\ Temperature: T_s=293K$

$Solar\ Irradiation: G_s=1000 W/ m^2$

$Device\ Surface\ Absorptivity: \alpha_s=0.5$

$Device\ Surface\ Emissivity: \epsilon = 0.8$

Performing an energy balance on the device (only for radiation):

$q''_{rad}=\alpha G_S-\epsilon\sigma\left ({T_s}^4-{T_{sky}}^4 \right )$

$q''_{rad} = \left({500} - {117}\right)W/ m^2$

$q''_{rad}\$ = $383\ W/ m^2$

We can approximate the area exposed for radiative on heat transfer equal to:

$\left(\frac{1}{2}\times Surface\ Area\ Available\ for\ Evaporation \right) + Top\ Surface\ Area\ of\ Small\ Pot$

$= \left(\frac{1}{2}\times {0.773} \right) + {0.11} m^2$

$= {0.50}\ m^2$

Over an exposure period of 10 hours the total net radiative energy transferred to the system is 1.92kWh.

Therefore, the energy flowing out of the device over a period of 10 hours is found to be (13.4 - 1.92) = 11.48kWh

As mentioned above, this value represents the energy loss for the entire pot-in-pot device. Due to the nature of conduction, a temperature gradient within the device must be present to create an out-flux of energy. Each layer of the device therefore must be cooled: the outer pot; volume of sand; water content within the sand; inner pot; air within the storage chamber; and the contents of the storage chamber. Taking this into consideration, it is reasonable to estimate that the only 5% of the total energy out-flux directly impacts the storage chamber.

Therefore, the actual cooling within the storage chamber for the prescribed conditions is found to be roughly 0.57kWh.

## Limitations of Device

Beyond the limitations of the required climatic conditions for the pot-in-pot refrigerator to be successful, there is also a need for a continuous supply of water. For many regions, water may be prioritized for other purposes making the it difficult for communities to adopt the technology. The device also has no proper seal for the storage chamber, reducing it's overall effectiveness since warm ambient air can seep into this chamber and increase the temperature of chilled zone. (however warm air will rise and chilled air is heavier and drops, so the temperature will always be coldest on the bottom)

# Conclusion

A parametric analysis concerning the performance of a Zeer pot-in-pot refrigeration device has been performed. As expected, the device performs well only in climates possessing a low relative humidity. The velocity of the wind and the area available for evaporation to occur on/through are two primary factors that can be addressed to improve the performance of the pot-in-pot refrigerator.

It has been shown that increasing the radius of the outer pot from 0.25m to 0.45m, almost doubles the total cooling effect. The adaptation of this however, is restricted by the increase in cost associated with using more materials. It is suggested that the strategy to make larger pot-in-pot refrigerators be employed only if community members are willing and able to pool their resources to share a device with superior performance.

It is unrealistic to assume that electricity is available to ensure that their is a constant and adequate source of air flow. The device is dependent solely on naturally occurring winds. To maximize air flow, it is recommended that Zeer refrigerator be placed as high above the ground as possible. This can be accomplished by building a simple frame to support the device, and placing them on high ground or on top of buildings.

There remains the potential for future analysis of this device. The development of a detailed conduction model to analyse heat transfer and mass diffusion mechanisms within the various layers would aid in identifying factors limiting performance and how they can be addressed. Additionally, by experimentation, a study could be performed to replace the permeability correction factor used in this analysis with real diffusion rates of moisture through clay.

# References

1. Elkheir, M., "The Zeer Pot - a Nigerian invention keeps food fresh without electricity", Science in Africa, 2002
2. Elkheir, M., "The Zeer Pot - a Nigerian invention keeps food fresh without electricity", Science in Africa, 2002
3. Practical Action Organization, "How a zeer pot fridge makes food last longer", 2009, http://practicalaction.org/home/zeerpots
4. Committee on Microgravity Research; Commission on Physical Sciences, Mathematics and Applications; "Microgravity Research in Support of Technologies for the Human Exploration and Development of Space and Planetary Bodies", Space Studies Board, National Research Council, 2000
5. Practical Action Organization, "Clay Based Technologies", 2007
6. The Clay Room, Telephone Interview with Representative, 2010
7. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., "Fundamentals of Heat and Mass Transfer", John Wiley and Sons, 2007
8. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., "Fundamentals of Heat and Mass Transfer", John Wiley and Sons, 2007
9. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., "Fundamentals of Heat and Mass Transfer", John Wiley and Sons, 2007
10. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., "Fundamentals of Heat and Mass Transfer", John Wiley and Sons, 2007

#  Appendix A: Calculation Methodology

All equations have been solved used the software package EES (Engineering Equation Solver). The equations used are outlined below.

Given,
Outer_Clay_Pot_Radius $\left( \bar{OR} \right )$ = 0.25 m
Inner_Clay_Pot_Radius $\left ( \bar{IR} \right )$ = 0.185 m
Thickness $\left ( \bar{TH} \right )$ = 0.015 m
Height $\left ( {H} \right )$ = 0.45 m
Ambient Temperature $\left ( {T} \right )$ = 308 K
Ambient Pressure $\left ( {P} \right )$ = 101.3 kPa
Wind Speed $\left ( {W} \right )$ = 2.5 m/s
Relative Humidity $\left ( {x} \right )$ = 0.3
Permeability Correction Factor $\left ( {PCF} \right )$ = 0.3
Fluid Properties,
Density of Air $\left ( {\rho_{air}} \right )$ acquired as a function of T and P
Viscosity of Air $\left ( {\mu_{air}} \right )$ acquired as function of T
Density of Water $\left ( {\rho_{water}} \right )$ acquired as a function of T and P
Density of Water Vapour $\left ( {\rho_{vap}} \right )$ acquired as a function of T, P and RH
Latent Heat of Vaporization for Water $\left ( {h_{fg}} \right )$ = 2270 kJ/kg
Diffusion Coefficient of Water into Air (D) =
$-2.775\times10^{-6} + 4.479\times10^{-8}{T} + 1.656\times10^{-10}{T}^2$   $m^2/s$                 (Curve Fit by Boltz and Tuve 1976)
Manipulations,
Area: $A = \frac{1}{2}4 \pi \bar{OR}^2+2 \pi \bar{OR} \left ( \bar{H}- \bar{OR} \right )+ \pi \left ( \left ( \bar{OR}- \bar{TH} \right )^2- \bar{IR}^2 \right )$   $m^2$
Reynolds_Number: $\bar{Re} = \frac{\rho_{air} W \bar{Lchar}}{\mu_{air}}$
where, $\bar{Lchar} = 2 \left ( \left ( \bar{OR} - \bar{TH} \right ) - \bar{RI} \right ) + 2 \pi \bar{OR}$   $m$
(Lchar for exposed sand + Lchar for outer clay pot)
(Length of exposed sand along diameter of device + Maximum circumference of device)
Schimdt_Number: ${Sc} = \frac{\mu_{air}}{\rho_{air}D}$
Avg_Sherwood_Number(Laminar): $\bar{Sh_{lam}}=0.664 \bar{Re}^{0.5}{Sc}^{0.3}$
Avg_Sherwood_Number(Turbulent): $\bar{Sh_{turb}}=0.037 \bar{Re}^{0.8}{Sc}^{0.3}$
Coeff_MassTransfer: ${h_m}=\frac{\bar{Sh}{D}}{\bar{Lchar}}$   $m/s$                          (Employing Heat and Mass Transfer Analogy)
Evap_Rate: ${Evap_{rate}}= {PCF}\left( {A}{h_m}{\rho_{vap}}\left (1 - x \right ) \right )$   $kg/s$
Evap_Cooling: ${Evap_{cool}} = \left ({Evap_{rate}}{h_fg} \right )$   $kW$
 This page was part of a project for Mech425, a Queen's University class on Engineering for Sustainable Development. Please leave comments using the discussion tab. It is now open edit.


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# Write the following numbers in the standard form : 4,18,25,00,000.

Last updated date: 20th Mar 2023
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Hint: In order to get the answer to it, write a number from 1 to 10 according to the number then write the left numbers, then in powers of 10 write the number you have transferred the decimal number of times.

We can write the given number as, $4,18,25\, \times \,{10^5}$
Then we can write the above number as, $4.1825 \times {10^9}$ by moving decimal beside 4. Since the decimal point has moved to 4 places we can rise to ${10^9}$.
Hence the answer to this question is $4.1825 \times {10^9}$.


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# How to dim an LED with constant current supply?

I have a power RGB led with 350 mA max per color. If I get 3x 300-350 mA constant power supplies will I be able to dim each individual color from 0 to 100% power with Arduino (with NPN transistors)?

I am a bit confused since constant current psu's are the best way to power led's but since they provide constant current can they be dimmed with pwm as one would expect with any power supply?

I am thinking of this PSU.

• This is a good video on how to implement it with a 555 Timer – Dean Dec 4 '12 at 9:39
• I'm very new to electronics so I think I will keep it simple for now and just usePWM of Aruino Uno, at least I know how it works :) – DominicM Dec 4 '12 at 21:02

Yes, you can use PWM to dim LEDs with constant current power supply.

Personally I would go with MOSFETs instead of NPN transistors. If nothing else MOSFET will dissipate less heat.

• Good to know! Why are mosfets better and by how much in my case? I have fairchild BC337 transistors and LM317T, are either of them suitable for the job? – DominicM Dec 4 '12 at 20:45
• MOSFETs have low on resistance and thus they will dissipate less heat when turned on. Drawback is that there is some gate capacitance and, depending on exact frequency, there is possibility of it heating up due to linear operation. Since you will be driving your LEDs at probably atmost at few hundred Hz, I doubt you will have a problem so almost any MOSFET will do. – Josip Medved Dec 5 '12 at 2:49
• Transistor needs voltage typically 0.6V accross junction. That means that you will always dissipate some heat. On other hand they are generally faster because there is no gate capacitance (which is moot point in this case since you don't need fast PWM). There is added complication of taking hfe into account to make sure you can deliver enough current from control pin. BC337 has hfe of 60 at 300mA so you should be ok with driving it from Arduino. – Josip Medved Dec 5 '12 at 2:49
• thanks for clarifying, I may go with single very high power led(15-30w) to avoid additional complexities with multiple led's. Can you think of a good mosfet or transistor to use with such high power led's(I'm not sure what to look for)? – DominicM Dec 5 '12 at 16:23
• That highly depends on what exact voltage and current you intend to use. Just search for MOSFET with rds as low as possible at given current (e.g. 10A). E.g. IPS040N03L G might be a good choice. – Josip Medved Dec 5 '12 at 22:46

I have used this constant current LED driver with success in the past. May be worth checking out. It can be controlled with an analog voltage or a PWM signal.

• Looks interesting but seems very expensive, 20$on mouser.com If I can use a 4.60$ psu to dim that's an easy choice :) I mean I might use 3 led's for more lumens meaning my led moodlight would cost well over 100\$ which is crazy expensive... – DominicM Dec 4 '12 at 20:36

As others have said, yes, it will work with PWM to control brightness. The name "constant current" is kind of misleading, I prefer "current limited". You can let the LED pull on the supply as hard as it wants, but it will only ever get 300 mA. If you shut off the LED the supply won't push harder than its rated voltage to force 300 mA through it. With the supply you linked that is 13 V, so it would just output 13 V and no current.


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# Distribution of half-life from radioactive decay

Suppose that $$X$$ measures the half-life of a radioactive element, with decay rate $$\lambda$$ (per unit of time).

Starting from a population of $$N$$ particle, I believe you can model the number of particles that decay over a unit of time as a $$Poisson(N\lambda)$$, and the time until the first decay is detected is modeled using an $$Exponential(N\lambda)$$.

My question is: How can we model the distribution of the half-life $$X$$, i.e., starting from a population of $$N$$ particles, the amount of time it takes for $$N/2$$ of the particles to have decayed ?

Assuming the decays are independent from one another, it seems an Erlang distribution could be appropriate https://en.wikipedia.org/wiki/Erlang_distribution? Should I then use $$f(x,N/2,N\lambda)$$ (where the notation refers to the Wikipedia article)? If the half-life intuition is correct, it would seem $$f(x,N/2,N\lambda)$$ should then be invariant to the choice of $$N$$, but that's not obvious to me looking at the algebraic for of the Erlang's pdf. Is that correct? If so, there should be a single distribution $$g = f(x,N/2,N\lambda)$$ for all $$N$$ that models half-lifes?

• I am not sure you are correct that the number of particles that decay over a unit of time has a Poisson distribution, partly because a Poisson distribution has unbounded support but you cannot have more than all the particles decay. Apr 22 at 15:29
• @Henry: Good point. In practice, for this kind of application, $N$ is typically very large (number of particles in a batch of matter) and $\lambda$ relatively small. So I am hoping it would still provide a good approximation? Independence issues aside, to be more precise, I guess I could use a $Bi(N, \lambda)$?
– FZS
Apr 22 at 16:08
• If $N$ is large and $\lambda$ is small then it will be a long time until half of the particles have decayed. Initially you will be seeing about $N \lambda$ decays per unit time and close to a Poisson process. The issue is that until this rate of decays starts to slow down noticeably, you will have no good way of estimating the overall half-life of about $\frac{\log_e(2)}{\lambda}$, and once it does slow down it will no longer be a homogeneous Poisson process. Apr 22 at 16:35
• Hmm... I wonder, then, how physicists have managed to measure the half-lives of long-lived isotopes like $U^{238}$ (half life, $4.5\times 10^9$ years) with such accuracy. Answer: any macroscopic mass contains a huge number of particles so even over a short period of time a very large number of decays occur, providing precise estimation of $\lambda.$ @Henry
– whuber
Apr 22 at 16:39
• @whuber $N\lambda$ is easy to measure reasonably accurately if you have enough decays: just count the number of decays in a second or a day or whatever. The question is separating $N$ from $\lambda$ since you are interested in $\lambda$. For long-lived isotopes, you create a pure sample with a known total mass and a known atomic mass, dividing one by the other and multiplying by Avogadro's number then gives you $N$. Apr 22 at 16:55

You ask about the distribution of an order statistic $$X_{(k)}$$ of $$N$$ independent and identically distributed random variables $$X_1, X_2, \ldots, X_N$$ and $$k=N/2$$ when $$N$$ is even and $$k=(N+1)/2$$ when $$N$$ is odd. Because this order statistic is the median when $$N$$ is odd and otherwise is a median, I will refer to it henceforth as the "median."

When the common distribution function $$F$$ has a density $$F^\prime = f,$$ the order statistic has a density, too, given by

$$f_{k;X}(x) = \binom{N}{k-1;1;N-k}\, F^{k-1}(x) f(x) (1-F(x))^{N-k}.$$

By choosing time units in which the half-life is $$1$$, the exponential distribution function is $$F(x) = 1 - e^{-x},$$ whence $$f(x) = e^{-x}$$ and therefore

$$f_{k;X}(x) = \frac{N!}{(k-1)!(N-k)!} e^{-(N-k+1)x}\left(1 - e^{-x}\right)^{k-1}.$$

One way to understand this is to express it in terms of $$U = e^{-X}$$ (which, incidentally, has a Uniform$$[0,1]$$ distribution). The density of $$U$$ is

$$f_{k;U}(u) \ \propto\ u^{N-k}(1-u)^{k-1},$$

immediately recognizable as that of a Beta$$(N+1-k, k)$$ distribution. Thus, $$X_{(k)}$$ is seen as the (negative) logarithm of a Beta variable.

Here, for example, are histograms of the median of $$N=12$$ Exponential variables (from 40,000 simulations) and its negative exponential on which, in red, graphs of $$f_{k;X}$$ and $$f_{k;U}$$ have been overlaid to demonstrate the correctness of this analysis.

At this point you can apply what is known about the Beta distribution to determine in great detail anything you like about the distribution of $$X_{(k)},$$ so I won't belabor the point. Instead, consult the references, including the following CV threads:


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# Evaluating Coulomb and exchange integrals in practice using cartesian coordinates [closed]

Many software packages provide means to calculate the Coulomb and exchange integrals, for example, the exchange

$$\int \phi^*(1) \, \psi^*(2) \frac{1}{|\mathbf{r_1} - \mathbf{r_2}|} \, \psi(1) \, \phi(2) \, d\mathbf{r_1} \, d\mathbf{r_2}$$

using gaussian type orbitals, either in cartesian or spherical coordinates.

I would like to know how this integral is evaluated in practice, since the Coulomb term $$1/|\mathbf{r_1}-\mathbf{r_2}|$$ should diverge whenever two different electron densities' volume elements overlap. I understand that when you use spherical coordinates, you can get rid of $$1/r_{12}$$ because the spherical volume element becomes something like $$r^2 \cos{\theta} \, dr \, d\theta \, d\phi$$, so you cancel $$r^2$$ in the numerator with $$r$$ in the denominator. But I'm having trouble to see how can this be done for cartesian GTO in cartesian coordinates.

• There are a number of methods, and a number of introductions to them - a quick search shows up esqc.org/lectures/WK4.pdf which looks like a relatively straightforward introduction to the evaluation of integrals with GTOs, and the material in onlinelibrary.wiley.com/doi/book/10.1002/9781119019572 is good as well. Szabo and Ostlund in the appendix show how to evaluate the ssss term, which in practice is all you need as the rest are derived from recurrences starting at this term. Jan 22, 2022 at 9:52
• @IanBush, great, thanks for the references. Yep, old and faithful integral transforms are the answer. Szabo and Ostlund perform a three-dimensional Fourier transform, and the slides from Wim Klopper present an integral transform of $1/r$ that results in the Boys function, now I understand that better. Unfortunately, I don't have access here to the book on Molecular Electronic‐Structure Theory, but will check it later.
– Arc
Jan 22, 2022 at 15:45
• The Molecular Electronic‐Structure Theory book is very good but not cheap - but your institution might give you online access depending where you are, try onlinelibrary.wiley.com/doi/pdf/10.1002/9781119019572.ch9 . And as well as integral transforms the Gaussian Product Theorem is the other tool you need to do this. Later will see f I have time/can write an answer, unfortunately lots of maths needed and my Math jax-fu is not strong. Jan 22, 2022 at 16:13
• Check out joshuagoings.com/2017/04/28/integrals for as simple an explanation as one will find, along with clear Python code. Say hello to the Boys function. Jan 24, 2022 at 1:19
• @IanBush, my MathJax is ok, so I can help you, if you care to write an answer in the following days, I can fix the math for you. I think its worth to have a fine answer, as this question attracted some upvotes.
– Arc
Jan 24, 2022 at 6:01


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# Again Integrate with singularities

Integrate[1/(a x + Sqrt[b x^2 - c]), {x, 1, d}]

where a,b,c are positive values. Mathematica runs this forever.

• In general, fixing assumptions makes timing considerably shorter. – Alexei Boulbitch Jan 12 at 9:11
• @AlexeiBoulubitch: Only assumptions do not help here. – user64494 Jan 12 at 11:03
• Assumptions that force the denominator to be nonzero and force the radicand to be nonnegative can help. I used for example Assumptions -> {a > 0, d > 1, b > 1, 0 < c < 1}; that takes a minute or three to run to completion. – Daniel Lichtblau Jan 12 at 14:57
• @DanielLichblau: Resolve[ForAll[{a, b, c}, Implies[a > 0 && b > 1 && 0 < c < 1, a > 0 && b > 0 && c > 0 && Sqrt[c/b] < 1]], Reals] produces True. You consider a partial case of the conditions from my answer. – user64494 Jan 12 at 15:18

Mathematica can quickly find the antiderivative, so another approach is to evaluate the antiderivative, call it $$\psi$$, between the limits, as

ψ = Integrate[1/(a x + Sqrt[b x^2 - c]), x]

(ψ /. x -> d) - (ψ /. x -> 1) // Simplify

This gives

Making assumptions and no conditions

Integrate[1/(a x + Sqrt[b x^2 - c]),{x, 1, d},Assumptions->d>= 1&&a>0&& b > 0 && c > 0,
GenerateConditions -> False]

, we have

  (* (1/(2 (a^2 - b)))(a Log[1 - a/Sqrt[b - c]] -
a Log[1 + a/Sqrt[b - c]] + 2 Sqrt[b] Log[b + Sqrt[b] Sqrt[b - c]] -
a Log[a^2 - b + c] + a Log[c + (a^2 - b) d^2] -
a Log[1 - (a d)/Sqrt[-c + b d^2]] +
a Log[1 + (a d)/Sqrt[-c + b d^2]] -
2 Sqrt[b] Log[b d + Sqrt[b] Sqrt[-c + b d^2]]) *)

PS. In order to obtain a nonformal result (in version 12.2), one should make an additional assumption which guarantees positivity of b*x^2-c:

Integrate[1/(a x + Sqrt[b x^2 - c]), {x, 1, d},
Assumptions -> d >= 1 && a > 0 && b > 0 && c > 0 && Sqrt[c/b] < 1]

ConditionalExpression[(a*Log[-a + Sqrt[b - c]] - a*Log[a + Sqrt[b - c]] + 2*Sqrt[b]*Log[b + Sqrt[b*(b - c)]] - a*Log[a^2 - b + c] + a*Log[c + (a^2 - b)*d^2] - a*Log[-(a*d) + Sqrt[-c + b*d^2]] + a*Log[a*d + Sqrt[-c + b*d^2]] - 2*Sqrt[b]*Log[b*d + Sqrt[b*(-c + b*d^2)]])/(2*(a^2 - b)), a^2 != b && Sqrt[b - c] <= a && d > 1 && (Sqrt[c]*Re[1/Sqrt[-a^2 + b]] < 1 || d < Sqrt[c]*Re[1/Sqrt[-a^2 + b]] || NotElement[Sqrt[-a^2 + b], Reals]) && ((a^2 + c >= b && ((a^2 < b && Re[(1 + Sqrt[c]/Sqrt[-a^2 + b])/(1 - d)] <= 0) || NotElement[(1 + Sqrt[c]/Sqrt[-a^2 + b])/(1 - d), Reals])) || (a^2 > b && NotElement[(1 + Sqrt[c]/Sqrt[-a^2 + b])/(1 - d), Reals]))]

• @AntoinAntonov: I didn't edit your posts. Do you understand me? – user64494 Jan 12 at 14:58
• Missed a spot, btw. There's a "rollback" button in the edit history that's probably more appropriate. (I prefer Anton's edit though.) – Michael E2 Jan 12 at 15:02

Or you can try Rubi

<< Rubi
Int[1/(a x + Sqrt[b x^2 - c]), {x, 1, d}]

• Nasser (@does not work)Rubi` produces a formal answer. Am I not right? – user64494 Jan 12 at 15:20
• That's not a definite integral. It's an indefinite integral in a pretty dress. – Daniel Lichtblau Jan 12 at 20:35


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My Math Forum Starlike functions proof

 Complex Analysis Complex Analysis Math Forum

 March 11th, 2018, 07:23 AM #1 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 101 Thanks: 2 Starlike functions proof Show the following regions are starlike: The open right half plane $z\in\mathbb{C}:Re(z)>0$ The cut plane $\mathbb{C}=\mathbb{C}/(-\infty,0]$ This seems easy intuitively. However, how would I actually prove that these functions are starlike? Could I create a parameter and show that every arbitrary point from the starcentre is reached? Last edited by skipjack; March 11th, 2018 at 03:41 PM.
 April 25th, 2018, 04:52 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 Well the first thing you should do is choose a point to be the "centre". Then, yes, it should be easy to write the equation of a straight line from that "centre" to any other point, (x, y), in the set.
 April 25th, 2018, 09:49 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 A couple of points: a set (not a function) is "starlike" if there exist a point (not necessarily unique), $\displaystyle z= x_0+ y_0i$, such that the straight line from $\displaystyle z= x_0+ y_0i$ to any point, x+ iy, in the set, that lies entirely in the set. For both of these problems, I think I would choose that point to be z= 1+ 0i. What is the line from 1+ 0i to x+ yi? That will, of course, be a linear function of some parameter, t. Then show every point on that line segment is in the given set.

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