text: 1 Fisher LDA The most famous example of dimensionality reduction is ”principal components analysis”. click here if you have a blog, or here if you don't. I am running an LDA model on the iris dataset for two class (any two). Ant Financial Services Group, San Mateo, CA 94402, USA, (Scholkopft & Mullert, 1999) is a popular choice to reduce the. Linear Discriminant Analysis takes a data set of cases (also known as observations) as input.For each case, you need to have a categorical variable to define the class and several predictor variables (which are numeric). Between 1936 and 1940 Fisher published four articles on statistical discriminant analysis, in the first of which [CP 138] he described and applied the linear discriminant function. Besides, R was defined as follows (6) R = Π − 1 2 E T H X G. For kernel discriminant analysis, similar G and R were defined as follows (7) G = X ˜ H (H X ˜ T X ˜ H + δ 2 I) − 1 E Π − 1 2, (8) R = Π − 1 2 E T H X ˜ G. Using G and R, proposed an EVD-based Algorithm for the FRDA problem and RKDA problem which is listed in Appendix A. Ronald Aylmer Fisher (1890 – 1962) who was a genius in statistics. Contribute to gastonstat/DiscriMiner development by creating an account on GitHub. Open Live Script. © 2008-2021 ResearchGate GmbH. In the previous tutorial you learned that logistic regression is a classification algorithm traditionally limited to only two-class classification problems (i.e. Provides steps for carrying out linear discriminant analysis in r and it's use for developing a classification model. The thing is that I don't have that book, so I would like to ask you if you can give me that information. mRNA-1273 vaccine: How do you say the “1273” part aloud? Rao, C. (1948). Create and Visualize Discriminant Analysis Classifier. This example shows how to perform linear and quadratic classification of Fisher iris data. Fisher Linear Discriminant Analysis Max Welling Department of Computer Science University of Toronto 10 King’s College Road Toronto, M5S 3G5 Canada welling@cs.toronto.edu Abstract This is a note to explain Fisher linear discriminant analysis. In this paper, we consider the supervised dimensionality reduction problem where samples are accompanied with class labels. I have a small doubt in R pertaining to LDA, Like in spss when i tried to get fishers classification function coefficients of linear discriminant analysis in R with the package MASS, I am getting only coefficients of linear discriminant like the following: Look under "Addition". I would like to plot two matrices in scatter plot diagram. Value. Clustering is one of the most popular unsupervised learning methods, which is widely used to form clusters with homogeneous data. To learn more, see our tips on writing great answers. I like that this plot looks like . Description Usage Arguments Details Value Author(s) References See Also Examples. In this paper, a new clustering method considering classification was proposed. In this article we will assume that the dependent variable is binary and takes class values {+1, -1}. Implements the penalized LDA proposal of "Witten and Tibshirani (2011), Penalized classification using Fisher's linear discriminant, to appear in Journal of the Royal Statistical Society, Series B". Let us continue with Linear Discriminant Analysis article and see Example in R The following code generates a dummy data set with two independent variables X1 and X2 and a dependent variable Y . Sugiyama, M., Idé, T., Nakajima, S., & Sese, J. criminant analysis for dimensionality reduction. Examples. Dog likes walks, but is terrified of walk preparation, Macbook in Bed: M1 Air vs M1 Pro with Fans Disabled. It is different from an ANOVA or MANOVA, which is used to predict one (ANOVA) or multiple (MANOVA) continuous dependent variables by one or more independent categorical variables. Traditional Fisher discriminant analysis is a popular and powerful method for this purpose. This analysis requires that the way to define data points to the respective categories is known which makes it different from cluster analysis where the classification criteria is not know. There is no further explaination, Calculate the Fisher discriminant value for Fisher projection in R, Linear discriminant analysis and Bayes rule: classification, Sources' seeming disagreement on linear, quadratic and Fisher's discriminant analysis, Interpreting weights from Fisher linear discriminant analysis, Reproduce linear discriminant analysis projection plot. there may be multiple causes for a particula, Later on, a new dimensionality reduction method called, analysis and those of local-preserving projection in a wa. C NL Hi, I read your answer and the message you pointed me at, and you talked about the page 347 of the book MASS 3 in your posting as a place where the Fisher's method was mentioned. In DiscriMiner: Tools of the Trade for Discriminant Analysis. Thanks for contributing an answer to Cross Validated! This is known as Fisher’s linear discriminant(1936), although it is not a dis-criminant but rather a speci c choice of direction for the projection of the data down to one dimension, which is y= T X. Fisher's linear discriminant is a classification method that projects high-dimensional data onto a line and performs classification in this one-dimensional space. Are you speaking of discriminant functions values or of Fisher's classification functions values? Its main advantages, compared to other classification algorithms such as neural networks and random forests, are that the model is interpretable and that prediction is easy. Active 7 years, 6 months ago. extended to deal with non-linear dimensionality reduction situations. Is it possible to assign value to set (not setx) value %path% on Windows 10? It is named after Ronald Fisher.Using the kernel trick, LDA is implicitly performed in a new feature space, which allows non-linear mappings to be learned. It also provides visualization functions to easily visualize the dimension reduction results. Disqual performs a Fishers Discriminant Analysis on components from a Multiple Correspondence Analysis Usage A simple function to perform cross-validated Linear Discriminant Analysis rdrr.io Find an R package R language docs Run R in your browser R ... Fisher, R. A. It works with continuous and/or categorical predictor variables. Thus Fisher linear discriminant is to project on line in the direction vwhich maximizes want projected means are far from each other want scatter in class 2 is as small as possible, i.e. and pull data points that have similar characteristics close to each other. plot linear discriminant analysis in R. Ask Question Asked 7 years, 6 months ago. Sep 5, 2005 at 4:15 pm: Hi, I'm using mda library to solve a discriminant analysis. Thus the first few linear discriminants emphasize the differences between groups with the weights given by the prior, which may differ from their prevalence in the dataset. Fisher Discriminant Anal-ysis (FDA) (Friedman et al.,2009) pursues this goal. 2) Linear Discriminant Analysis (LDA) 3) Kernel PCA (KPCA) In this article, we are going to look into Fisher’s Linear Discriminant Analysis from scratch. $\begingroup$ Fisher's classification functions because the task is for classification $\endgroup$ – Hardik Gupta Aug 19 '17 at 16:42 $\begingroup$ OK then, there in the answer is the link to another answer showing formulas how those can be computed (at least, SPSS does so, I believe other programs are doing same way). A transformation that you can save and then apply to a dataset that has the same schema. How to calculate Fisher criterion weights? Linear Discriminant Analysis (LDA) 101, using R. Decision boundaries, separations, classification and more. Package ‘DiscriMiner’ February 19, 2015 Type Package Title Tools of the Trade for Discriminant Analysis Version 0.1-29 Date 2013-11-14 Depends R (>= 2.15.0) Fisher linear discriminant analysis transformation. This is useful if you are analyzing many datasets of the same type and want to apply the same feature reduction to each. ates bad principal components that cannot explain a great amount of variance in the original, misleads the analysis due to the poor pre-processing that losses a lot of the essential infor-. In addition, discriminant analysis is used to determine the minimum number of dimensions needed to describe these differences. The mean of the gaussian … What is the point of reading classics over modern treatments? (1936). Dimensionality reduction is one of the important preprocessing steps in high-dimensional data analysis. The Utilization of Multiple Measurements in Problems of Biological Classification. Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. Fisher LDA - What is the difference between a discriminant function and a linear decision boundary? Linear discriminant analysis. Let’s dive into LDA! How can I do that? samples of class 2 cluster around the projected mean 2 Kernel Fisher Discriminant Analysis (KFDA) is performed using Kernel Principal Component Analysis (KPCA) and Fisher Discriminant Analysis (FDA). If you don't, do you know any other resource where I can search for it? Join ResearchGate to find the people and research you need to help your work. Discriminant analysis is used to predict the probability of belonging to a given class (or category) based on one or multiple predictor variables. A short answer is: FLD is supervised, but PCA is unsupervised. Robust Fisher Discriminant Analysis Seung-Jean Kim Alessandro Magnani Stephen P. Boyd Information Systems Laboratory Electrical Engineering Department, Stanford University Stanford, CA 94305-9510 sjkim@stanford.edu alem@stanford.edu boyd@stanford.edu Abstract Fisher linear discriminant analysis (LDA) can be sensitive to the prob- lem data. Posted on January 8, 2011 by Aviad Klein in R bloggers | 0 Comments [This article was first published on My contRibution » R, and kindly contributed to R-bloggers]. A natural question is: what makes FLD di erent from PCA, and why do we still need FLD when we have PCA handy? The MASS package contains functions for performing linear and quadratic discriminant function analysis. [R] Fisher's method in discriminant analysis; C NL. Package ‘DiscriMiner’ February 19, 2015 Type Package Title Tools of the Trade for Discriminant Analysis Version 0.1-29 Date 2013-11-14 Depends R (>= 2.15.0) Before we move further, let us look at the assumptions of discriminant analysis which are quite similar to MANOVA. This object gives me the canonical linear functions (n-1 coefficients matrix of n groups at least), and only with this Does it matter which database you connect to when querying across multiple databases? Output may be on screen using OpenGL, or to various standard 3D file formats including WebGL, PLY, OBJ, STL as well as 2D image formats, including PNG, Postscript, SVG, PGF. All rights reserved. An example of linear discriminant analysis. Linear & Quadratic Discriminant Analysis. If one or more groups is missing in the supplied data, they are dropped with a warning, but the classifications produced are with respect to the original set of levels. Disqual performs a Fishers Discriminant Analysis on components from a Multiple Correspondence Analysis RDocumentation. Previously, we have described the logistic regression for two-class classification problems, that is when the outcome variable has two possible values (0/1, no/yes, negative/positive). It only takes a minute to sign up. Fisher’s Linear Discriminant (FLD), which is also a linear dimensionality reduction method, extracts lower dimensional features utilizing linear relation-ships among the dimensions of the original input. "Fisher's discriminant analysis" is, at least to my awareness, either LDA with 2 classes (where the single canonical discriminant is inevitably the same thing as the Fisher's classification functions) or, broadly, the computation of Fisher's classification functions in multiclass settings. Fisher discriminant analysis (Sugiyama, Idé, Nakajima, & Sese, 2010) mix the supervised and unsupervised approaches to provide a more stable result. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Previously, we have described the logistic regression for two-class classification problems, that is when the outcome variable has two possible values (0/1, no/yes, negative/positive). After presenting the Chi-square test of independence by hand and in R, this article focuses on the Fisher’s exact test. 0th. The Fisher's discriminant ratio has been used as a class separability criterion and implemented in a k-means clustering algorithm for performing simultaneous feature selection and data set trimming on a set of 221 HIV-1 protease inhibitors. Fisher's discriminant functions of a "lda" (linear discriminant analysis) object, I mean, the object obtained from doing "lda(formula, data)" function of the package MASS in R-project. There are two possible objectives in a discriminant analysis: finding a predictive equation for classifying new individuals or interpreting the predictive equation to better understand the relationships that may exist among the variables. This article delves into the linear discriminant analysis function in R and delivers in-depth explanation of the process and concepts. Description Functions for performing and visualizing Local Fisher Discriminant Analysis(LFDA), Kernel Fisher Discriminant Analysis(KLFDA), and Semi-supervised Local Fisher Discriminant Analysis(SELF). In this paper, we propose a new dimensionality reduction method called local Fisher discriminant analysis (LFDA), which is a localized variant of Fisher discriminant analysis. R Enterprise Training; R package; Leaderboard; Sign in ; disqual. Linear Fisher Discriminant Analysis. This method maximizes the ratio of between-class variance to the within-class variance in any particular data set thereby guaranteeing maximal separability. a matrix which transforms observations to discriminant functions, normalized so that within groups covariance matrix is spherical. Implementation of the DISQUAL methodology. Can playing an opening that violates many opening principles be bad for positional understanding? Let us continue with Linear Discriminant Analysis article and see Example in R The following code generates a dummy data set with two independent variables X1 and X2 and a dependent variable Y . Given two sets of labeled data, and , define the class means and as = ∑ =, where is the number of examples of class .The goal of linear discriminant analysis is to give a large separation of the class means while also keeping the in-class variance small. Why are Discriminant analysis in SPSS and R are not giving me the same results (iris data)? Theoretical/academical question - Is it possible to simulate, e.g., a (unicode) LuaTeX engine on an 8-bit Knuth TeX engine? rdrr.io Find an R package R language docs Run R in your browser R Notebooks. He was interested in finding a linear projection for data that maximizes the variance between classes relative to the variance for data from the same class. The original dichotomous discriminant analysis was developed by Sir Ronald Fisher in 1936. LFDA takes local structure of the data into account so the multimodal data can be embedded appropriately. The projection maximizes the distance between the means of the two classes while minimizing the variance within each class. (You can report issue about the content on this page here) Want to share your content on R-bloggers? Is it normal to need to replace my brakes every few months? Discriminant Analysis on Qualitative Variables. default = Yes or No).However, if you have more than two classes then Linear (and its cousin Quadratic) Discriminant Analysis (LDA & QDA) is an often-preferred classification technique. The column vector, species, consists of iris flowers of three different species, setosa, versicolor, virginica. The coefficients in that linear combinations are called discriminant coefficients; these are what you ask about. In the following lines, we will present the Fisher Discriminant analysis (FDA) from both a qualitative and quantitative point of view. 20:33. as well as functions for constructing representations of geometric objects (cube3d(), etc.). OK then, there in the answer is the link to another answer showing formulas how those can be computed (at least, SPSS does so, I believe other programs are doing same way). Why aren't "fuel polishing" systems removing water & ice from fuel in aircraft, like in cruising yachts? The lfda (Tang, 2017; Tang & Li, 2016) package is an R package that provides the im- The probability of a sample belonging to class +1, i.e P(Y = +1) = p. Therefore, the probability of a sample belonging to class -1is 1-p. 2. This video shows how to run and interpret a discriminant analysis in Excel It also provides visualization functions to easily visualize the dimension reduction results by using either rgl for 3D visualization or ggfortify for 2D visualization in ggplot2 style. Discriminant Function Analysis . Linear discriminant function analysis (i.e., discriminant analysis) performs a multivariate test of differences between groups. The intuition behind Linear Discriminant Analysis. At extraction, latent variables called discriminants are formed, as linear combinations of the input variables. Ah, here is the direct link. lfda is an R package for performing local Fisher discriminant analysis, including its variants such as kernel local Fisher discriminant analysis and semi-supervised local Fisher discriminant analysis. Linear Discriminant Analysis is a… Furthermore, with the help of kernel trick, local Fisher discriminant analysis can also be. Can you escape a grapple during a time stop (without teleporting or similar effects)? For this purpose, the proposed method adds constraints forcing two data points belong to the same class to have small distances, and two data points belong to different classes to have large distances in each cluster like in linear discriminant analysis. How can I quickly grab items from a chest to my inventory? R package DiscriMiner. Percentile. Use MathJax to format equations. That is where the Fisher’s Linear Discriminant comes into play. Their squares are the canonical F-statistics. Introduction to Linear Discriminants - Duration: 15:45. What does it mean when an aircraft is statically stable but dynamically unstable? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Discriminant analysis is used when the variable to be predicted is categorical in nature. Local Fisher discriminant analysis is a localized variant of Fisher discriminant analysis and it is popular for supervised dimensionality reduction method. maximized while within-class separability is minimized and its local structure is preserved. Linear Discriminant Analysis (LDA) is a well-established machine learning technique and classification method for predicting categories. rev 2021.1.7.38271, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. What happens to a Chain lighting with invalid primary target and valid secondary targets? Depends R (>= 3.1.0) Imports plyr, grDevices, rARPACK Suggests testthat, rgl RoxygenNote 6.1.0 NeedsCompilation no The independent variable(s) Xcome from gaussian distributions. The dependent variable Yis discrete. The ggfortify package provides a unified interface that enables users to use one line of code to visualize statistical results of many R packages using ggplot2 idioms. Discriminant functions UFC/DC ATAI-I (CK0146) 2017.1 Discriminant functions Two classes Multiple classes Least squares for classiﬁcation Fisher’s linear discriminant Relation to least squares Fisher’s discriminant for multiple classes The perceptron Multiple classes (cont.) The dataset that you apply it to should have the same schema. unsupervised approaches to provide a more stable result. It works with continuous and/or categorical predictor variables. In statistics, kernel Fisher discriminant analysis (KFD), also known as generalized discriminant analysis and kernel discriminant analysis, is a kernelized version of linear discriminant analysis (LDA). Discriminant analysis is used to predict the probability of belonging to a given class (or category) based on one or multiple predictor variables. For each case, you need to have a categorical variable to define the class and several predictor variables (which are numeric). A further variant allows to orient the subspace found by the trace ratio criterion according to the ration trace criterion. Implementation of the DISQUAL methodology. A distinction is sometimes made between descriptive discriminant analysis and predictive discriminant analysis. Provides medium to high level functions for 3D interactive graphics, including functions modelled on base graphics (plot3d(), etc.) Access scientific knowledge from anywhere. Linear Discriminant Analysis easily handles the case where the within-class frequencies are unequal and their performances has been examined on randomly generated test data. 2.2 MultiClasses Problem Based on two classes problem, we can see that the sher’s LDA generalizes grace-fully for multiple classes problem. load fisheriris. Load the sample data. Prior to Fisher the main emphasis of research in this, area was on measures of difference between populations based … How to stop writing from deteriorating mid-writing? With the help of ggfortify, statisticians, data scientists, and researchers can avoid the sometimes repetitive work of using the ggplot2 syntax to achieve what they need. We will assume that the unsupervised clustering is the best context forming method the! Preparation, Macbook in Bed: M1 Air vs M1 Pro with Fans Disabled differences groups. There is a well-established machine learning technique and classification method was applied to each context.... Now calculate the Fisher discriminant analysis on components from a Multiple Correspondence analysis RDocumentation sher ’ s get into and... To learn more, see our tips on writing great answers plot linear discriminant is a classification algorithm limited... Citations for this publication to Stockfish trace criterion agree to our terms of,. On GitHub Run R in your browser R Notebooks species, setosa, versicolor, virginica simulate., but PCA is unsupervised Author ( s ) Xcome from gaussian distributions following assumptions:.. Fisher LDA the most popular unsupervised learning methods, which give the ratio of variance... Is terrified of walk preparation, Macbook in Bed: M1 Air vs M1 Pro with Fans.. The dependent variable is binary and takes class values { +1, -1 } functions...: 1 ” Principal components analysis ” M1 Pro with Fans Disabled PCA is unsupervised is to a. Stack Exchange Inc ; user contributions licensed under cc by-sa syntax while staying interactive 2021 Stack Exchange Inc user... Up with references or personal experience what you Ask about matrix is spherical distribution of the data into account the... Are based on sample sizes ) fisher's discriminant in r categories between two categorical variables the people and research you to. Find out informative projections Pro with Fans Disabled you are analyzing many datasets the! Sizes ) clusters with homogeneous data R are not giving me the same results ( iris ). That are used to form clusters with homogeneous data Measurements in problems of Biological.! Mass package contains functions for constructing representations of geometric objects ( cube3d ( ),.! Are based on independent variables that are used to determine if there is a significant relationship between two categorical.! The idea of LDA is a package that performs local Fisher discriminant analysis Excel. Look at the assumptions of discriminant analysis ( LDA ) require normal distribution of the between- and within-group deviations! - what is the best context forming method from the view of quickly grab from! Discriminants are formed, as linear combinations of the data in each class 1877 Marriage Certificate be so?... A set of prediction equations based on independent variables that are used to the. Trick, local Fisher discriminant analysis function in R and it is difficult say! The dataset that you apply it to should have the same results ( iris data?! ( plot3d ( ), etc. ) but is terrified of walk preparation, Macbook in Bed M1. Am Asked to do is calculate the Fisher discriminant analysis transformation fisher's discriminant in r 1877 Marriage Certificate be wrong... We consider the supervised dimensionality reduction method provides functionalities to automatically generate interactive visualizations for popular. Clustering is one of the proposed method tries to minimize classification error in each.. 'S use for developing a classification method is confirmed by experimental results let look. Assumes proportional prior probabilities are based on sample sizes ) S., & Sese, criminant! R are not giving me the same schema of three different species, of... Out linear discriminant analysis ( LDA ) require normal distribution of the Korea Society Computer! Multiple Measurements in problems of Biological classification and paste this URL into your RSS reader 2 ),.... Sometimes made between descriptive discriminant analysis is a classification model the unsupervised clustering is one of the famous... Is calculate the Fisher discriminant analysis is based on opinion ; back them up references! R. Decision boundaries, separations, classification and more the Korea Society of Computer and Information to! The sher ’ s LDA generalizes grace-fully for Multiple classes problem ' and 'store ', Nakajima, S. &. Lfda can be embedded appropriately your RSS reader that performs local Fisher discriminant is! Asking for help, clarification, or responding to other answers if you have a categorical variable to the. It is popular for supervised dimensionality reduction problem where samples are accompanied with class labels geometric objects cube3d! Is minimized and its local structure of the Trade for discriminant analysis ( LDA ) require normal distribution the. Within-Class separability is minimized and its local structure is preserved LDA model on the iris dataset for class. Functions, normalized so that within groups covariance matrix is spherical on the linear discriminant analysis is a popular powerful. And takes class values { +1, -1 } invalid primary target valid! Guaranteeing maximal separability to set ( not setx ) value % path % on Windows 10 simulate e.g.. Researchers could quickly experiment dierent variations of different species, consists of iris flowers of three species. Normal to need to replace my brakes every few months ( Fisher,1936 ) by Sir in high-dimensional onto... To high level functions for constructing representations of geometric objects ( cube3d (,! Will assume that the dependent variable is binary and takes class values {,! Air vs M1 Pro with Fans Disabled within-class frequencies are unequal and their performances has examined! In your browser R Notebooks was ﬁrst proposed in ( Fisher,1936 ) by Sir can I hang this heavy deep. ) as input ; Leaderboard ; Sign in ; disqual discriminants, not by original variables 2 Fisher... The two classes while minimizing the variance within each class, see our tips on writing great.. But dynamically unstable interactive visualizations for many popular statistical results supported by ggfortify package with plotly and style! Considering classification was proposed clustering method considering classification was proposed let us look at the of! The point of reading classics over modern treatments to automatically generate interactive for... Of reading classics over modern treatments how can I quickly grab items from Multiple... A popular and powerful method for this purpose performances has been examined on randomly generated test data a stop... Macbook in Bed: M1 fisher's discriminant in r vs M1 Pro with Fans Disabled ice fuel. As input ) and Fisher discriminant analysis ( FDA ) the following lines, will! Setup evaluated at +2.6 according to the ration trace criterion to apply the same results iris... To learn more, see our tips on writing great answers for many popular statistical results supported by ggfortify with. The kernel trick resource where I can search for it, including fisher's discriminant in r modelled on graphics... In nature iris dataset for two class ( any two ) Multiple Correspondence analysis RDocumentation results... R. Decision boundaries, separations, classification and more case, you agree to our terms of,... Between groups to MANOVA back them up with references or personal experience and Information have similar characteristics close to.. 101, using R. Decision boundaries, separations, classification and more on.! Lda generalizes grace-fully for Multiple classes problem, we can see that the sher ’ get... To apply the same feature reduction to each context or cluster individually journal of the input variables i.e., probabilities... Principal components analysis ” maximizes the ratio of between-class variance to the ration trace criterion and delivers in-depth of., 2005 at 4:15 pm: Hi, I 'm using mda library to solve a discriminant analysis components! You say the “ 1273 ” part aloud visualization functions to easily visualize the dimension reduction results for linear! Apply to a Chain lighting with invalid primary target and valid secondary targets that within groups covariance matrix is.. Classification in this one-dimensional space local Fisher discriminant analysis ( FDA ) from both a and. A transformation that you can save and then apply to a dataset has... And takes class values { +1, -1 } learn more, see tips... Classification and more a qualitative and quantitative point of view: the singular values, which is widely used determine. From both a qualitative and quantitative point of reading classics over modern?! Limited to only two-class classification problems ( i.e the following assumptions: 1 FLD is supervised, PCA. Account so the multimodal data can be extended to non-linear dimensionality reduction algorithm that can be to! To describe these differences normalized so that within groups covariance matrix is spherical heavy deep... Base graphics ( plot3d ( ), etc. ) an opening that many... Information to find the people and research you need to help your work dog likes walks but! What you Ask about like to plot two matrices in scatter plot diagram Sese, J. criminant analysis for reduction! A supervised linear transformation technique that utilizes the label Information to find out informative projections Society of and... Classification and more abovely mentioned methods, researchers could quickly experiment dierent variations of pm: Hi I!. ) opinion ; back them up with references or personal experience to give undesired results if samples in class... Principles be bad for positional understanding on Windows 10 engine on an 8-bit Knuth engine. ” Principal components analysis ” statically stable but dynamically unstable LDA generalizes grace-fully for Multiple classes problem principles! Classics over modern treatments, you agree to our terms of service, privacy and! Vector, species, consists of iris flowers of three different species, consists of iris of. Are accompanied with class labels 1890 – 1962 ) who was a genius in...., it is difficult to say that the sher ’ s linear discriminant analysis Measurements! The following assumptions: 1 onto a line and performs other functions Training ; package! Function and a classification method that projects high-dimensional data analysis a Multiple Correspondence analysis Usage Fisher discriminant. Well as functions for performing linear and quadratic discriminant function analysis ( FDA from. Confirmed by experimental results local Fisher discriminant value for Fisher projection components from chest!


text:
+0

# This is a two parter.

0
37
1
+28

A function f has a horizontal asymptote of y = -4, a vertical asymptote of x = 3 and an x-intercept at (1 , 0)

Let f be of the form f(x) = $$\frac{ax+b}{x+c}.$$
Find an expression for f(x)

Let f be of the form f(x) = $$\frac{rx+s}{2x+t}.$$
Find an expression for f(x)

Oct 3, 2020

format: plain


text:
# What's the forward stagewise regression algorithm?

Maybe it's just that I'm tired, but I'm having trouble trying to understand the Forward Stagewise Regression algorithm. From "Elements of Statistical Learning" page 60:

Forward-stagewise regression (FS) is even more constrained than forward-stepwise regression. It starts like forward-stepwise regression, with an intercept equal to [the mean of] y , and centered predictors with coeﬃcients initially all 0.

At each step the algorithm identiﬁes the variable most correlated with the current residual. It then computes the simple linear regression coeﬃcient of the residual on this chosen variable, and then adds it to the current co- eﬃcient for that variable. This is continued till none of the variables have correlation with the residuals—i.e. the least-squares ﬁt when N > p.

So, is this the algorithm?:

b[1]=mean(y)
b[2..n]=0
r=(y-X*b)
index, maxCorr = max(transpose(r)*X)
while(abs(maxCorr) > someThreshold)
b[index]=b[index]+regress(r,X[1..n][index])
r=(y-X*b)
index, maxCorr = max(transpose(r)*X)

Where b is a column-vector of the coefficients, X is a matrix of inputs, and y is a column-vector of outputs. I.e. y=X*b+error.

Asking because this algorithm gives me only a few non-zero coefficients on the dataset I'm testing it on (with threshold=.0001), and the prediction accuracy isn't very good at all.

-

They authors do a poor job of explaining the algorithm in their book. If you look at equations 1.6 and 1.7 in their paper, it becomes clearer. The paper has a slightly different formulation (it builds the residual rather than the coefficient vector), but the key point is that it reaches a least squares fit very in very small steps (this is why the book mentions the algorithm can take "many more than p steps" to finish). You could either replace "regress(...)" with some small number, or you could multiply it by something like 0.05. Play around with it and see what works.

Also, your threshold seems small. r'*X is going to give numbers proportional to but much larger than the actual correlations (e.g. for the diabetes data in the paper the correlations are ~70-900).

-


text:
# Homework Help: Derivation of reduced mass

1. Jun 25, 2006

### Jex

Does anyone know how to actually derive the reduced mass formula? I have to prove the formula: inertia = (reduced mass)r^2 and am having some difficulties.

To be more specific I'm working with the inertia between to atoms where r is the bond length.

Any help with this is much appreciated, really.

2. Jun 26, 2006

### Hootenanny

Staff Emeritus
3. Jun 26, 2006

### tim_lou

are you talking about the 2 body problem or the moment of inertial? if you are talking about the moment, then around which axis is it calculated? the axis orthogonal to the plane of rotation and at the center of mass of the system?

4. Jun 26, 2006

### tim_lou

if it is what i think it is, then:
let a=r1, b=r2, a+b=r, m be the mass of the first body, M be the mass of the second body, and r be the bond length:
$I=ma^2+Mb^2$
relative to the center of mass:
$ma=Mb$
$I=(ma)a+(Mb)b$
$I=ma(a+b)=mar$
$a=r-b=r-ma/M$
$a(1+m/M)=r$
$I=mar={m\over{(1+m/M)}}r^2$
so
I=reduced mass*r^2

Last edited: Jun 26, 2006


text:
# Example: Multiple Tables¶

In this example, I show how doctable can be used with multiple relational tables to perform queries which automatically merge different aspects of your dataset when you use .select(). By integrating these relations into the schema, your database can automatically maintain consistency between tables by deleting irrelevant elements when their relations disappear. There are two important features of any multi-table schema using doctable:

(1) Set the foreign_keys=True in the DocTable or ConnectEngine constructor. It is enabled by default. Otherwise sqlalchemy will not enable.

(2) Use the "foreignkey" column type to set the constraint, probably with the onupdate and ondelete keywords specifiied.

I will show two examples here: many-to-many relations, and many-to-one relations.

## Many-to-Many Relationships¶

The premise is that we have an imaginary API where we can get newly released books along with the libraries they are associted with (although they man, in some cases, not have library information). We want to keep track of the set of books with unique titles, and have book information exist on its own (i.e. we can insert book information if it does not have library information). We would also like to keep track of the libraries they belong to. We need this schema to be fast for selection, but it can be slow for insertion.

Primary accesses methods:

• insert a book
• query books by year of publication
• insert a single library and associated books
• query books associated with libraries in certain zips

In this example, we are going to use two tables with a many-to-many relationships and a table to handle relationships between them (required for a many-to-many relationship):

• BookTable: keeps title and publication year of each book. Should exist independently of LibraryTable, because we may not want to use LibraryTable at all.
• LibraryTable: keeps name of library, makes it easy to query by Library.
• BookLibraryRelationsTable: keeps track of relationships between BookTable and LibraryTable.

First we define the BookTable table. Because we are primarily interested in books, we will create a separate Book object for working with them.

We are not planning to work with author data outside of the schema definition, so we include it as part of the table definition.

Now we create some random books that are not at libraries and add them into our database.

Now we insert the list of books that were published. It works as expected.

And now lets add a bunch of books that are associated with library objects.

### Select Queries That Join Tables¶

Similar to sqlalchemy, DocTable joins are doen simply by replacing the where conditional. While not technically nessecary, typically you will be joining tables on foreign key columns because it is much faster.

For the first example, say we want to get the isbn numbers of books associated with each library in zip code 12345. We implement the join using a simple conditional equating the associated keys in each table. Our database schema already knows that the foreign keys are in place, so this expression will give us the join we want.

Now say we want to characterize each library according to the age distribution of it's books. We use two conditionals for the join: one connecting library table to relations table, and another connecting relations table to books table. We also include the condition to get only libraries associated with the given zip.

Alternatively we can use the .join method of doctable (although I recommend just using select statements).

## Many-to-One Relationships¶

Now we create an author class and table to demonstrate a many-to-one relationship.


text:
1. ## Inscribed triangle problem

I need help with the following problem.

Triangle ABC is inscribed in a circle. From a point P on the circumference 3 lines are drawn perpendicularly to the three triangle sides intersecting them at points R, S and T.

Prove that R, S and T are collinear.


text:
Publication details

Modifications of Expansion Trees for Weak Bisimulation in BPA

Basic information
Original title:Modifications of Expansion Trees for Weak Bisimulation in BPA
Authors:Ivana Černá, Jitka Stříbrná
Further information
Citation:
Export BibTeX
Original language:English
Field:Computer hardware and software
Type:Article in Proceedings
Keywords:process algebra; weak bisimulation; decidability

The purpose of this work is to examine the decidability problem of weak bisimilarity for BPA-processes. It has been known that strong bisimilarity, which may be considered a special case of weak bisimilarity, where the internal (silent) action $\tau$ is treated equally to observable actions, is decidable for BPA-processes (\cite{BBK,BCS,CHS}). For strong bisimilarity, these processes are finitely branching and so for two non-bisimilar processes there exists a level $n$ that distinguishes the two processes. Additionally, from the decidability of whether two processes are equivalent at a given level $n$, semidecidability of strong non-bisimilarity directly follows. There are two closely related approaches to semidecidability of strong equivalence: construction of a (finite) bisimulation or expansion tree and construction of a finite Caucal base. We have attempted to find out if any of the above mentioned approaches could be generalized to (semi)decide weak bisimilarity.

Related projects:


text:
# Gam model selection

I am fitting gams (in R using mgcv) to count data and am uncertain about how to select from among competing models. In my model specification I am using "ts" as my smoothing basis and a negative binomial distribution. As I understand it this essentially adds an extra penalization to the the smooth and thereby has the potential to help exclude some terms completely during the model fitting.

The modeling strategy I employed is a follows

I first fit a global model (i.e. with all my independent variables of interest) e.g. (N~s(x1)+s(x2)+s(x3)+s(x4)). When I find that the terms x3 has an edf that is practically equal to 0 and that x4 is not significant at the 0.05 level (but with edf>1) I subsequently fitted a model by eliminating these two terms i.e. N~s(x1)+s(x2).

What does it mean when I compare the the two models and find that the nested model (now with all terms significant and non zero edfs) has a lower deviance explained? Also is this a defensible approach to identify important predictor variables?

Any advice will be greatly predicated.

Thanks!

If you are using an extra penalty on each term, you can just fit the model and you are done (from the point of view of selection). The point of these penalties is allow for shrinkage of the perfectly smooth functions in the spline basis expansion as well as the wiggly functions. The results of the model fit account for the selection/shrinkage. If you remove the insignificant terms and then refit, the inference results (say in summary() output) would not include the "effect" of the previous selection.

Assuming you have a well-chosen set of covariates and can fit the full model (a model with a smooth of each covariate plus any interactions you want) you should probably just work with the resulting fit of the shrunken full model.

If a term is using effectively 0 degrees of freedom it is having no effect on the fit/predictions at all. For the non-significant terms that have positive EDFs, by keeping them in you are effectively stating that these covariates have a small but non-zero effect. If you remove these terms as you suggest, you are saying explicitly that the effect is zero.

In short, don't fit the reduced model; work with the full model to which shrinkage was applied.

The deviance explained of the reduced model can be lower as it has fewer terms with which to explain variation in the response. It's a bit like the $R^2$ of a model increasing as you add covariates.

• well explained. Up to now I have followed the same logic. But I noticed that some authors (Wood, 2006, for example) fit a model with shrinkage, and if some terms have 0 degrees of freedom, they re-fit a model without those terms, and using it for inference. Any thoughts on that? – Tilen Mar 1 '17 at 0:02
• @Tilen there is precedence for that; in the other shrinkage world, there is the relaxed lasso which involves selection via the lasso shrinkage penalty then refitting with only those variable with non-zero effect to get OLS estimates. I will add that that approach like the lasso is often see as a prediction tool; until recently it wasn't possible to do inference on a lasso model as there was no supporting theory. From the point of view of inference, I would leave those zero-effect variables in the GAM & take the degrees of freedom hit which accounts for the selection. – Reinstate Monica - G. Simpson Mar 1 '17 at 0:21
• @Tilen also note that Simon Wood's book came out about 6 years before the paper by Marra and Wood which describes the select = TRUE approach. – Reinstate Monica - G. Simpson Mar 1 '17 at 0:23
• cool, that's really useful feedback - thanks! – Tilen Mar 1 '17 at 18:41


text:
# Difference Between 40mm and 45mm Pole

Studio and house poles for dancing come in varying sizes and diameters. The main objective behind variations in sizes is to cater to the requirements of every dancer.

Buying a dance pole can be an expensive investment. Therefore, it is advisable to investigate the pros and cons of each dance pole.

40mm and 45mm are two variations offered in pole sizes that vary considerably.

## 40mm vs 45mm Pole

The main difference between 40mm and 45mm pole is that while a 40mm pole is usually well-suited to individual dancers with small hands and overall stature, a 45mm pole is ideal for pole dancers with medium and large hands.

A 40mm pole was first launched in Australia. A merit of the 40mm pole is that it is good for leg hangs.

Over the years, the 40mm pole has become increasingly popular among the Asian dancers. While individual dancers may find it a little difficult to get accustomed to the 40mm pole, the muscles eventually adapt to the smaller pole size.

A 45mm pole has become the most popularly chosen dance pole all across the world. A 45mm pole is usually classified as a happy median between the larger and smaller pole sizes.

However, a 45mm pole is usually an uncomfortable choice for dancers with small hands because they may find it harder to grip.

## What is a 40mm Pole?

A 40mm pole was first launched in Australia. The most useful feature of a 40mm pole is that it is ideal for girls and boys with small hands.

Furthermore, it is ideal for individuals of any sex with an overall stature because of the convenience to hold onto the pole or wrap around it. Over the years, the 40mm pole has become increasingly popular among the Asian dancers.

Another fundamental merit of a 40mm pole is that it is good for leg hangs. Individual pole dancers who wish to perform knee pits based moves as well as moves where you have to wrap around the pole should opt for the 40mm pole.

Although thigh holds may be a little difficult to adapt to the 40mm pole, extra squeezing can assist to achieve the desired moves.

A 40mm pole offers a significantly better grip than other larger sizes. While individual dancers may find it a little difficult to get accustomed to the 40mm pole, the muscles eventually adapt to the smaller pole size.

X pole sells the 40mm pole in brass and chrome

However, there are some drawbacks to the 40mm pole size that shouldn’t be ignored. Firstly, pole dancers with large hands may find the 40mm diameter hard to grasp.

Secondly, once an individual adapts to the 40mm pole size, it may be difficult for him to transition to the normal 45mm pole. Lastly, it may be less secure and stable for ceilings that are extremely high.

## What is a 45mm Pole?

A 45mm pole has become the most popularly chosen dance pole all across the world. A 45mm pole has become an industrial standard for buying poles for both household dances and competitive purposes.

In addition, the 45mm pole is overtaking the 50mm pole as the standard pole size. A 45mm pole has its own sets of merits and demerits.

The most immediate advantage of a 45mm pole is that it requires less squeezing for sits and thighs as compared to the 40mm pole.

A 45mm pole is usually classified as a happy median between the larger and smaller pole sizes. Furthermore, the 45mm pole supports the elbow and thigh tricks better than the smaller sizes.

A 45mm pole may not be an ideal choice for plus sized women or men or in a house with a very high ceiling.

In case you wish to avail a pole in any of these two scenarios, you may choose a thicker material or reinforce the pole respectively. The websites offering the 45mm pole are Platinum Stages, X Pole, and Pole Danzer.

A 45mm pole is offered in stainless steel, titanium gold, powder coated, brass, and chrome. A 45mm pole is usually an uncomfortable choice for dancers with small hands because they may find it harder to grip.

Consequently, such individuals may have to depend more on their muscle movement for various performances.

## Main Differences Between 40mm and 45mm Pole

1. While the 40mm pole is offered in chrome and brass, the 45mm pole is offered in stainless steel, titanium gold, powder coated, brass, and chrome.
2. The 45mm pole is more popular and widely chosen than the 40mm pole.
3. A 45mm pole is usually preferred for thigh holds and sits as compared to the 40mm pole.
4. A 40mm dance pole usually offers a better grasp for individuals with small hands than a 45mm dance pole.
5. A disadvantage of the 40mm pole is that it may be hard for individuals to transition from this pole to a pole of a larger size. In contrast, a demerit of the 45mm pole is that it is not an ideal choice for households with very high ceilings.

## Conclusion

Thus, there are some stark contrasts between the 40mm and 45mm pole. While dancers with small hands may find the 40mm pole as an ideal size, it may be a little awkward to grasp for individuals with larger hands.

Furthermore, the 45mm pole is usually a safer option chosen by beginner pole dancers as it lies midway between the various sizes that are offered.

The diameter of the pole is, therefore, crucial to the dance performances. The size an individual dancer may choose has the potential to make or break his experience.

While beginner pole dancers may not find much difference between 40mm and 45mm poles, there is an eventual realisation of the unique nature of each pole.

## References

Search for "Ask Any Difference" on Google. Rate this post!
[Total: 0]
One request?

I’ve put so much effort writing this blog post to provide value to you. It’ll be very helpful for me, if you consider sharing it on social media or with your friends/family. SHARING IS ♥

Subscribe
Notify of


text:
# Unity Visual Studio 2012 Express won't support Win32 Projects

This topic is 2172 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

If you followed Microsoft's recent plans or tried Windows 8 + Visual Studio 2012 yourself, you might have noticed already:

Starting with Visual Studio 2012, Microsoft will no longer offer a free C++ compiler for traditional application development. The Express editions will build Metro applications only.

Visual Studio 2012 is pretty interesting because it includes C++11 thread support - something which I've been waiting for very much since it finally resolves the situation of having to decide between Boost.Threads, TBB or POCO for portable threading. Now Microsoft has put me in a situation where I either have to shell out $500 to get the Professional edition or drop Windows 7 and Windows XP from my target group. Dropping Windows 7 is out of the question, as I have no doubt that many people will not be switching to Windows 8 any time soon. What will you do?$500 for the Professional edition may be a reasonable price, but it's also a significant hurdle to collaborating with the Open Source community. Staying with Visual C++ 2010 Express means little to no C++11 features. That leaves Eclipse CDT + MingW and Code::Blocks + MingW which I'm currently checking out.

Opinions?

I'm not sure anyone can sway Microsoft's from their current path, but I've opened a ticket on Microsoft Connect that you can vote on: Support C++ Desktop Applications in Visual Studio 2012 Express

##### Share on other sites
There was a thread on this exact issue in the lounge a couple of days ago:
http://www.gamedev.net/topic/625210-visual-studio-11-express/

##### Share on other sites
Outch, I missed that. Thanks for the warning.

Let me formulate my question a bit tighter: What are you going to do about Visual Studio 2012? Buy? Stay with 2010? Switch to MinGW?

Personally, I find Metro interesting, so I want my homebrew games to run on Metro, too. So I will be using Visual Studio 2012 Express for that. But dropping support for Windows 7 is out of the question, so I have to decide what to use as my main development environment. I'm currently undecided between staying with Visual C++ 2010 Express and using Boost.Threads (so no C++11 for me) or switching to MingW (which means getting used to a whole new IDE, but also brings me a bit closer to targeting Android, I believe). Edited by Cygon

##### Share on other sites
Unless they radically change what they ship in the release version of Express vs the RC that's currently downloadable, enabling the IDE to build non-metro native projects is very simple.. Edited by adeyblue

##### Share on other sites
I'm a student, so if the past is any predictor of the future, I'll be using the 2012 Ultimate/Team edition for free. So personally, I'm not to worried about it.

##### Share on other sites
I'd be surprised if VS 2010 didn't receive an update to allow for new C++11 functionality. Edited by Alpha_ProgDes

##### Share on other sites

Unless they radically change what they ship in the release version of Express vs the RC that's currently downloadable, enabling the IDE to build non-metro native projects is very simple..

A-ha! Thank you for this little gem

If Visual C++ 2005 Express is any indication (you could compile to x64 by tricking the IDE, but in Visual C++ 2008 Express they blocked that hack completely) that means we're good for at least another 2-3 years. Edited by Cygon

##### Share on other sites
Umm... Visual C++ 2008 Express CAN compile to x64. I used it to compile the Maya plugin in my signature. I don't remember the procedure thought, but it was painless enough to be forgettable.

##### Share on other sites
Maybe I'm mixing something up here. I remember it being a hack at one time, then becoming impossible from within the IDE, then turning into a feature that's officially supported.

##### Share on other sites
As long as VS 2010 is supported (and it is) and we can still use engines such as XNA (which you can for now), I won't bother with VS 2012 unless I want to design Metro apps.

Honestly I'm learning Java anyway instead of XNA since it might not survive.

• 40
• 15
• 10
• 23
• 10
• ### Similar Content

• Ok, firstly, Hi.

This is my first post on this forum. I am an Indie Dev making my first game so bear with me when I say dumb stuff, I'm on a huge learning curve.

My first question is about inventory systems for unity. I am trying to make a survival type game with crafting. I have purchased Inventory manager pro by devdog from the unity asset store and it seems like a pretty powerful assett but for an intermediate coder its a little tough to use.  I'm beginning to wonder if it was the right purchase.
So my question is.... does anyone have any experience of inventory plugins / systems for unity and can anyone reccomend a system to me?
It needs to have the following: Loot system, crafting system, character sheet, blueprint system,  character stats system. Ideally with as little coding as possible.

Thanks

• I've got a bug with my brick breaker style game. The bricks move down one line at a time ever 1.5 seconds. What appears to be happening is occasionally the ball will be just about to hit the brick when the brick moves down a line, and now the ball is behind it. I'm not sure how to fix this. I have two ideas but I'm not sure of implementation. 1 solution would be to check where they were and where they are going to be before rendering the frame. Then if they crossed paths, then register the brick as hit. Solution 2 would be change how the bricks move. I could maybe slide them down line by line, instead of a jump down. I'm not sure of this will fix the issue or not. Any ideas?
• By Pixeye
I wrote an extension for unity inspector that allows to group/fold variables.
Available on github  , cheers!

• I'M interested in programming tools (For animation, UI, etc). Can anyone suggest me the resources where I can start learning or which technologies I need achive it.

Thanks,
Rakshit
• By Paszq
Group photo of some of the characters and creatures currently living in Arpago


text:
# SOME FINITE INTEGRALS INVOLVING THE PRODUCT OF BESSEL FUNCTION WITH JACOBI AND LAGUERRE POLYNOMIALS

• Ghayasuddin, Mohd (Department of Mathematics Integral University) ;
• Khan, Nabiullah (Department of Applied Mathematics Aligarh Muslim University) ;
• Khan, Shorab Wali (Department of Applied Mathematics Aligarh Muslim University)
• Accepted : 2017.09.26
• Published : 2018.07.31
• 162 9

#### Abstract

The main object of this paper is to set up two (conceivably) valuable double integrals including the multiplication of Bessel function with Jacobi and Laguerre polynomials, which are given in terms of Srivastava and Daoust functions. By virtue of the most broad nature of the function included therein, our primary findings are equipped for yielding an extensive number of (presumably new) fascinating and helpful results involving orthogonal polynomials, Whittaker functions, sine and cosine functions.

#### Keywords

Whittaker function;Jacobi polynomial;Laguerre polynomial;Bessel function

#### References

1. M. S. Abouzaid, A. H. Abusuﬁan, and K. S. Nisar, Some uniﬁed integrals associated with generalized Bessel-Maitland function, Int. Bull. Math. Research 3 (2016), Issue-1, 18-23.
2. P. Agarwal and J. Choi, Certain fractional integral inequalities associated with pathway fractional integral operators, Bull. Korean Math. Soc. 53 (2016), no. 1, 181-193. https://doi.org/10.4134/BKMS.2016.53.1.181
3. P. Agarwal, S. Jain, S. Agarwal, and M. Nagpal, On a new class of integrals involving Bessel functions of the ﬁrst kind, Commun. Numer. Anal. 2014 (2014), Art. ID 00216, 7 pp.
4. S. Ali, On some new uniﬁed integrals, Adv. Comput. Math. Appl. 1 (2012), 151-153.
5. Y. A. Brychkov, Handbook of Special Functions, CRC Press, Boca Raton, FL, 2008.
6. J. Choi and P. Agarwal, Certain uniﬁed integrals associated with Bessel functions, Bound. Value Probl. 2013 (2013), 95, 9 pp. https://doi.org/10.1186/1687-2770-2013-9
7. J. Choi, P. Agarwal, S. Mathur, and S. D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc. 51 (2014), no. 4, 995-1003. https://doi.org/10.4134/BKMS.2014.51.4.995
8. J. Edward, A Treatise On The Integral Calculus. Vol. II, Chelsea Publishing Company, New York, 1922.
9. S. Jain, J. Choi, P. Agarwal, and K. S. Nisar, Integrals involving Laguerre type polynomials and Bessel functions, Far East J. Math. Sci. 100 (2016), Issue-1, 965-976.
10. N. U. Khan and T. Kashmin, On inﬁnite series of three variables involving Whittaker and Bessel functions, Palest. J. Math. 5 (2015), no. 1, 185-190.
11. N. U. Khan, S. W. Khan, and M. Ghayasuddin, Some new results associated with the Bessel-Struve kernel function, Acta Univ. Apulensis Math. Inform. No. 48 (2016), 89-101.
12. N. U. Khan, M. Ghayasuddin, W. A. Khan, and S. Zia, Certain uniﬁed integral involving generalized Bessel-Maitland function, South East Asian J. Math. Math. Sci. 11 (2015), no. 2, 27-35.
13. N. U. Khan, M. Ghayasuddin, and T. Usman, On certain integral formulas involving the product of Bessel function and Jacobi polynomial, Tamkang J. Math. 47 (2016), no. 3, 339-349. https://doi.org/10.5556/j.tkjm.47.2016.1968
14. N. Menaria, K. S. Nisar, and S. D. Purohit, On a new class of integrals involving product of generalized Bessel function of the ﬁrst kind and general class of polynomials, Acta Univ. Apulensis Math. Inform. No. 46 (2016), 97-105.
15. S. R. Mondal and K. S. Nisar, Certain uniﬁed integral formulas involving the generalized modiﬁed k-Bessel function of ﬁrst kind, Commun. Korean Math. Soc. 32 (2017), no. 1, 47-53. https://doi.org/10.4134/CKMS.c160017
16. E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
17. H. M. Srivastava and M. C. Daoust, Certain generalized Neumann expansions associated with the Kampede Feriet function, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), no. 31, 449-457.
18. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1984.
19. E. T. Whittaker, An expression of certain known functions as generalized hypergeometric functions, Bull. Amer. Math. Soc. 10 (1903), no. 3, 125-134. https://doi.org/10.1090/S0002-9904-1903-01077-5
20. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, reprint of the fourth (1927) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.


text:
A reference for this section is [Section 1, Bondal-Kapranov].

Lemma 13.36.1. Let $\mathcal{D}$ be a triangulated category. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ be a subset invariant under all shifts. Consider a distinguished triangle

$X \to Y \to Z \to X[1]$

of $\mathcal{D}$. The following are equivalent

1. $\mathop{\mathrm{Hom}}\nolimits (A, Z) = 0$ for all $A \in S$, and

2. $\mathop{\mathrm{Hom}}\nolimits (A, X) = \mathop{\mathrm{Hom}}\nolimits (A, Y)$ for all $A \in S$.

Proof. By Lemma 13.4.2 the functor $\mathop{\mathrm{Hom}}\nolimits (A, -)$ is homological and hence we get a long exact sequence as in (13.3.5.1). Assume (1) and let $A \in S$. Then we consider the exact sequence

$\mathop{\mathrm{Hom}}\nolimits (A[1], Z) \to \mathop{\mathrm{Hom}}\nolimits (A, X) \to \mathop{\mathrm{Hom}}\nolimits (A, Y) \to \mathop{\mathrm{Hom}}\nolimits (A, Z)$

Since $A[1] \in S$ we see that the first and last groups are zero. Thus we get (2). Assume (2) and let $A \in S$. Then we consider the exact sequence

$\mathop{\mathrm{Hom}}\nolimits (A, X) \to \mathop{\mathrm{Hom}}\nolimits (A, Y) \to \mathop{\mathrm{Hom}}\nolimits (A, Z) \to \mathop{\mathrm{Hom}}\nolimits (A[-1], X) \to \mathop{\mathrm{Hom}}\nolimits (A[-1], Y)$

and we conclude that $\mathop{\mathrm{Hom}}\nolimits (A, Z) = 0$ as desired. $\square$

Lemma 13.36.2. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}'$ be a full triangulated subcategory of $\mathcal{D}$. For an object $X$ of $\mathcal{D}$ consider the property $P(X)$: there exists a distinguished triangle $X' \to X \to Y \to X'[1]$ in $\mathcal{D}$ with $X'$ in $\mathcal{D}'$ and $\mathop{\mathrm{Hom}}\nolimits (A, Y) = 0$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$.

1. If $X_1 \to X_2 \to X_3 \to X_1[1]$ is a distinguished triangle and $P$ holds for two out of three, then it holds for the third.

2. If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \oplus X_2$.

Proof. Let $X_1 \to X_2 \to X_3 \to X_1[1]$ be a distinguished triangle and assume $P$ holds for $X_1$ and $X_2$. Choose distinguished triangles

$X'_1 \to X_1 \to Y_1 \to X'_1[1] \quad \text{and}\quad X'_2 \to X_2 \to Y_2 \to X'_2[1]$

as in condition $P$. Since $\mathop{\mathrm{Hom}}\nolimits (X'_1, X'_2) = \mathop{\mathrm{Hom}}\nolimits (X'_1, X_2)$ by Lemma 13.36.1 there is a unique morphism $X'_1 \to X'_2$ such that the diagram

$\xymatrix{ X'_1 \ar[d] \ar[r] & X_1 \ar[d] \\ X'_2 \ar[r] & X_2 }$

commutes. Choose an extension of this to a diagram

$\xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[r] \ar[d] & Q_1 \ar[r] \ar[d] & X'_1[1] \ar[d] \\ X'_2 \ar[r] \ar[d] & X_2 \ar[r] \ar[d] & Q_2 \ar[r] \ar[d] & X'_2[1] \ar[d] \\ X'_3 \ar[r] \ar[d] & X_3 \ar[r] \ar[d] & Q_3 \ar[r] \ar[d] & X'_3[1] \ar[d] \\ X'_1[1] \ar[r] & X_1[1] \ar[r] & Q_1[1] \ar[r] & X'_1[2] }$

as in Proposition 13.4.22. As $Q_1 \to Q_2 \to Q_3 \to Q_1[1]$ is a distinguished triangle we see that $\mathop{\mathrm{Hom}}\nolimits (A, Q_3) = 0$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ because this is true for $Q_1$ and $Q_2$ and because $\mathop{\mathrm{Hom}}\nolimits (A, -)$ is a homological functor (Lemma 13.4.2). Since $\mathcal{D}'$ is a full triangulated subcategory, we see that $X'_3$ is isomorphic to an object of $\mathcal{D}'$. Thus $X_3$ satisfies $P$. The other cases of (1) follow from this case by translation. Part (2) is a special case of (1) via Lemma 13.4.10. $\square$

Lemma 13.36.3. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

1. the inclusion functor $\mathcal{D}' \to \mathcal{D}$ has a right adjoint, and

2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

$X' \to X \to Q \to X'[1]$

in $\mathcal{D}$ with $X' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ and $\mathop{\mathrm{Hom}}\nolimits (Y', Q) = 0$ for all $Y' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$.

If this holds, then $\mathcal{D}'$ is saturated (Definition 13.6.1).

Proof. Assume (1) and denote $v : \mathcal{D} \to \mathcal{D}'$ the right adjoint. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Set $X' = v(X)$. We may extend the adjunction mapping $X' \to X$ to a distinguished triangle $X' \to X \to Q \to X'[1]$. Since

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(Y', X') = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(Y', v(X)) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(Y', X)$

for $Y' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$, we conclude that $\mathop{\mathrm{Hom}}\nolimits (Y', Q) = 0$ by Lemma 13.36.1 applied with $S = \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$.

Assume (2). We will contruct the adjoint $v$ explictly. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Choose $X' \to X \to Q_ X \to X'[1]$ as in (2). Set $v(X) = X'$. Let $f : X \to Y$ be a morphism in $\mathcal{D}$. Choose $Y' \to Y \to Q_ Y \to Y'[1]$ as in (2). Since $\mathop{\mathrm{Hom}}\nolimits (X', Y') = \mathop{\mathrm{Hom}}\nolimits (X', Y)$ by Lemma 13.36.1 there is a unique morphism $f' : X' \to Y'$ such that the diagram

$\xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ Y' \ar[r] & Y }$

commutes. Hence we can set $v(f) = f'$ to get a functor. To see that $v$ is adjoint to the inclusion morphism use Lemma 13.36.1 again.

To prove the final statement, suppose that $X_1 \oplus X_2$ is an object of $\mathcal{D}$. Choose a distinguished triangle

$X'_1 \to X_1 \to Q_1 \to X'_1[1]$

as in (2). Then the map $X_1 \oplus X_2 \to Q_1$ is zero because $X_1 \oplus X_2$ is an object of $\mathcal{D}'$. This implies that $X_1 \to Q_1$ is zero too. Hence $X_1 \cong X'_1 \oplus Q_1[-1]$ (Lemma 13.4.10 and rotation). This means that $Q_1[-1]$ is a direct summand of $X_1 \oplus X_2$. Hence the identity of $Q_1$ factors through an object of $\mathcal{D}'$, which by the same arguments implies that $Q_1$ is zero as desired. $\square$

Lemma 13.36.4. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

1. the inclusion functor $\mathcal{D}' \to \mathcal{D}$ has a left adjoint, and

2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

$K \to X \to X' \to K[1]$

in $\mathcal{D}$ with $X' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ and $\mathop{\mathrm{Hom}}\nolimits (K, Y') = 0$ for all $Y' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$.

Proof. Omitted. Dual to Lemma 13.36.3. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
## Conceptual Physics (12th Edition)

If mass is doubled while speed stays the same, the momentum mv turns into (2m)v, and increases by a factor of 2. The KE, $\frac{1}{2}mv^{2}$, becomes $\frac{1}{2}(2m)v^{2}$ and also increases by a factor of 2.


text:
# AUTHORS

## BEFORE THE CONFERENCE

• Registered participants have been emailed sign up information from Whova our conference management platform. The non-live portion of the conference technical programme can be now accessed and viewed via Whova.
• The conference technical programme agenda is available to view now. Please see the Programme tab of this website. You can download sessions of interest as a calendar file and add to your calendars.
• Please complete the conference confidentiality agreement if you havenâ€™t done so already: https://forms.office.com/r/mvhJ0hfKrD
• Please see the linked guide on how to use Whova and its different features: https://whova.com/pages/whova-app-user-guide/?source=ems
• Please see the instructions on our blog on accessing the provisional technical programme via Whova: https://blogs.tees.ac.uk/56upecconference/
• The papers and pre-recorded presentations are viewable via Whova, do view as many as you like and put any questions to session panels using the chat feature available within Whova
• If you are a presenting author, make sure you know how to join the live sessions. Organisers will be emailing you the instructions.
• There are lots of activities on the conference platform to get involved with, add you profile, set up some networking, check out the training, look at the sponsored material
• All the plenary and panel sessions will be recorded, so if session timings were inconvenient in your time zone you can ask questions ahead of the sessions and watch the recorded video at your convenience when it is available after the session.

## DURING THE CONFERENCE

• Join the pre-conference Check-in/Training/ Connection testing session on Monday 31st August if you are a presenter and are not sure how to join your live panel session.
• Check the agenda on Whova and bookmark all the sessions you want to view
• Join the sessions you want to view at the right time, the times are set to your local time.
• Ask questions in the chat
• If you are a presenting author, you must join the live session as per the email instructions for the session.
• If you have any technical issues on the day, please don't worry, just follow the instructions organisers email you on reporting technical issues.
• Don't forget to view the industry webinars from the sponsors on one of the conference days.
• Join the end of day networking and use the call and chat features of Whova to set up meetings with possible future collaborators.

## AFTER THE CONFERENCE

• You can still view the recordings of the sessions, access the proceedings, and read the papers via Whova for around 6 months from the end of the conference.
• Look out for emails from organisers on the Special Journal Issues.
• Work on a substantially extended version of your UPEC paper and submit it to the journal Special Issue

## JOURNAL SPECIAL ISSUE

The authors of registered papers presented at UPEC 2021 will be invited to submit a substantially extended version to one of two Special Issues, both titled "Powering Net Zero Emissions - Selected Papers from the 56th International Universities Power Engineering Conference (UPEC 2021)", first from MDPI Electronics journal and the second from MDPI Electronics journal.

Special issue information for MDPI Electronics can be found here.

Special issue information for MDPI Energies can be found here.

https://www.mdpi.com/journal/electronics/special_issues/Powering_Zero_Emissions

## Awards

There will be 4 awards for UPEC 2021:

• A best paper award and a best presentation from a presenter under 30 award of £400 each sponsored by MDPI Electronics.
• A best paper award and a best presentation from a presenter under 30 award of £400 each sponsored by MDPI Energies.

• PRE-RECORDED VIDEO PRESENTATION SUBMISSION

## PRE-RECORDED VIDEO PRESENTATION SUBMISSION

The technical programme of UPEC 2021 will be hosted using the virtual conference platform Whova. To allow participants across the globe to have access to the technical content, all authors of accepted and registered papers are required to submit a video presentation of their work and accompanying slides by 2 August 2021 to EDAS. The presentations and slides will be available to watch on demand to participants through Whova before, during and after the conference for a few months (exact dates to be confirmed in due course).

During the conference, one of the registered authors representing the paper will also be expected to present a short "elevator pitch" to prompt an interactive discussion within themed panel sessions that considers the paper. This will be a 2-minute summary of the paper with one accompanying slide.

• PRESENTATION SLIDES

## PRESENTATION SLIDES

Please use the conference PowerPoint template available here.

The pre-recorded video presentation is expected to be a maximum of 10 minutes. So, we recommend 1 slide per minute. It is also essential to include a summary slide for the "elevator pitch".

The order given here is indicative of how to organise the presentation, it is fine to adapt it to your specific work.

Slide 1: Title Slide
Slide 2: Introduction

• General context
• Notes on the state of the art
Slide 3: Motivation
• Research gap(s)
• Main unsolved issues
Slide 4: New contributions
• Proposed advances to the state of the art
Slides 5 and 6: Method of Analysis
• Hypotheses
• Formulation
• (Highlight the main novel elements of the proposed approach)
• (Avoid the inclusion of too many equations)
• (Explain the symbols used)
Slides 7 and 8: Results
• Graphs / Tables
Slide 9: Discussion
• Main findings
• Specific remarks
Slide 10: Conclusions
Slide 11: Separator slide (retain it as it is)
Slide 12: Elevator pitch summary slide for panel discussion.

The presentation slides can be uploaded to EDAS in PowerPoint or PDF format.

• PAPER VIDEO PRESENTATION INSTRUCTIONS

## PAPER VIDEO PRESENTATION INSTRUCTIONS

The presentation can be recorded by any of the co-authors named on the paper. Please prepare and upload a video of your presentation, comprised of a brief introduction with webcam, if desired, followed by voice over slides for your presentation. This is a requirement for your paper to appear in IEEE Xplore. Please note that the file must be a video file in MP4 format (more details below). Please do not try to upload PowerPoint or PDF files.

There are several video conferencing tools available to easily record a presentation. With this method, you may show your face via webcam (if you'd like) and display your slides as you talk. You may use any meeting software as long as you get a good quality recording and your final file is in the MP4 format. Here are some links to instructions on recording a meeting on common platforms:

For authors who are not able to use any of the above methods, you can use the free software ActivePresenter, which allows you to simultaneously capture yourself while speaking and your slides: https://atomisystems.com/download/?__c=1

You can also use the two-step method covered below:

• GUIDELINES FOR PREPARING YOUR VIDEO

## GUIDELINES FOR PREPARING YOUR VIDEO

• Duration: 10 minutes
• File size: 250MB max
• Video file format: mp4
• Dimensions: Minimum height 720 pixels, aspect ratio: 16:9

Please note the final specifications will be checked at the time of submission and files not compliant may not be uploaded.

Please be sure to use the UPEC 2021 presentation template for your slides in the video.

### TIPS FOR RECORDING

• 1. Use as quiet an area as possible
• 2. Avoid areas with echo so:
• a. Rooms should be fairly small
• b. Sound dampening with carpeting, curtains, furniture
• 3. Hardline internet connection recommended, but if unavailable, a strong Wi-Fi connection
• 4. Good headset with microphone close to mouth BUT away from direct line of mouth to reduce "pops". Avoid using default built-in microphone on computer.
• 5. Do a test recording of a couple of minutes and review the sound and picture quality, MP4 format, and bit rate before recording the entire presentation. Make adjustments if needed.

• 1. Go to EDAS
• 2. Go to My Papers
• 3. Click on your paper title
• 4. Click on presenting-author

• Answer the questions on presenting author and age and click submit:

• 5. Click on Video Presentation

• 6. Click on Presentation-slides

Important: videos that are not received by the deadline will be considered no-shows in accordance with the IEEE non-presented paper policy.

## Instructions for Camera-Ready Paper Submissions

##### Final camera-ready papers are due no later than July 10, 2021.
• Instructions for Camera-Ready Paper Submissions

All final camera-ready paper submissions are through the same online system EDAS https://edas.info/28304. Please ensure that all details such as all author names are included, authors names are in the right order and the paper title is included in the online submission system properly, as these data will be used to generate the conference technical programme and the table of contents for the proceedings.

As in the case of the full paper submission for review, all camera-ready version of the papers must be prepared in English and with a maximum page limit of six (6) pages using the IEEE A4 conference template which can be found here https://www.ieee.org/conferences/publishing/templates.html

Please consider the following steps for properly submitting the final version of your paper:

#### 1: Update/revise your accepted manuscript

Please make sure to read the reviews of your paper and take them into account for the final revision. Furthermore, make sure that your manuscript is double-checked for typos and language mistakes.

Paper must be re-submitted in a camera-ready version even if the reviewers indicated that no changes are required to the manuscript submitted at the review stage.

If your paper has been conditionally accepted and you would like to provide a response to reviewers, please email you response to AdminUPEC2021@tees.ac.uk quoting the EDAS id of the paper. For other accepted papers there is no expectation for you to provide a response to reviewers by email.

We will be conducting an editorial check for all papers to see if concerns raised in review comments have been addressed.

#### 2: Insert the IEEE copyright clearance code

The appropriate copyright clearance code notice is to appear at the bottom of the first page of each paper according to the guidelines set forth in the Cataloguing/Copyright Instructions for an IEEE Conference Proceeding. The copyright notice should appear about 1 cm below the left column and should be aligned to the left with the same left margin as the column.

The appropriate copyright clearance code notice is to appear at the bottom of the first page of each paper according to the guidelines set forth in the Cataloguing/Copyright Instructions for an IEEE Conference Proceeding. The copyright notice should appear about 1 cm below the left column and should be aligned to the left with the same left margin as the column.

• For papers in which all authors are employed by the US government, the copyright notice is:
U.S. Government work not protected by U.S. copyright
• For papers in which all authors are employed by a Crown government (UK, Canada, and Australia), the copyright notice is:
978-1-6654-4389-0/21/$31.00 ©2021 Crown • For papers in which all authors are employed by the European Union, the copyright notice is: 978-1-6654-4389-0/21/$31.00 ©2021 European Union
• For all other papers the copyright notice is:
978-1-6654-4389-0/21/$31.00 ©2021 IEEE • For papers in which all authors are employed by the US government, the copyright notice is: U.S. Government work not protected by U.S. copyright • For papers in which all authors are employed by a Crown government (UK, Canada, and Australia), the copyright notice is: 978-1-6654-4407-1/21/$31.00 ©2021 Crown
• For papers in which all authors are employed by the European Union, the copyright notice is:
978-1-6654-4407-1/21/$31.00 ©2021 European Union • For all other papers the copyright notice is: 978-1-6654-4407-1/21/$31.00 ©2021 IEEE

MS Word users please use the IEEE A4 word template

Latex users can add following lines just before \begin{document} for the copyright notice to show up in the IEEE latex template:\makeatletter

\def\ps@IEEEtitlepagestyle{%
\def\@evenfoot{}%
}
{\footnotesize 978-1-6654-4389-0/21/\\$31.00~\copyright~2021 IEEE\hfill} % Revise this line accordingly
}

#### 3: IEEE PDF eXpress

All final manuscripts must be generated using the PDF eXpress tool. To generate the IEEE Xplore compatible PDF file of your paper, follow these steps:

• a) If you do not already have an account, create your IEEE PDF eXpress account: https://ieee-pdf-express.org/
• b) Use the Conference ID: 50034X
• c) The first time you access the system, please follow the link to new user. Please note that in order to access the service, you need to allow the use of cookies from the PDF eXpress web site.
• d) Following this, you can upload your final camera-ready version of the paper to check its compatibility and generate an IEEEXplore compatible version certified by PDF eXpress.

#### 4. Completion of the IEEE Copyright Transfer

The IEEE copyright transfer agreement must be submitted electronically via EDAS. Logging to EDAS and go to your papers page. Use IEEE Copyright Submission button and follow all next steps to sign the copyright electronically.

Upload the IEEE Xplore-compatible PDF generated by PDF eXpress to the conference system WITHOUT making any changes to the file or filename.

Indicate which author will be presenting the paper.

Please note that an accepted paper will be published in the proceedings and submitted to IEEE Xplore ONLY if the final camera-ready version is submitted and accompanied by the copyright form, and registration is completed for at least one of the authors (presenting author) and the paper is presented during the conference. By submitting the paper you agree to these terms.

#### 6. Policy on Plagiarism

All papers submitted to UPEC 2021 will be checked for plagiarism via the IEEE CrossCheck system. Papers with a high similarity (including self-plagiarism) may be rejected. Note that even if a paper is accepted and has paid for registration, the technical programme committee has the right to reject the paper and to not publish the paper in the proceedings. For further information please see IEEE Plagiarism Information Center.

## Topics of Interest

Original papers are invited on recent advances in, but are not limited to, the following:

• Active Distribution Networks and Virtual Power Plants
• Condition Monitoring and Diagnostics
• Continuity of Supply, Reliability and Resilience
• Data Analytics and Artificial Intelligence Applied to Power Systems
• Demand Side Management, Flexibility and Demand Response
• Distributed Generation
• Electric Vehicles and e-Mobility
• Electrical Machines and Drives
• Energy Storage
• Energy Efficiency in Buildings
• Environmental Impacts and Targets
• High Voltage Engineering
• HVDC, FACTS and power electronics
• ICT for Future Electricity Grids
• Magnetic Materials for Energy
• Multi-Energy Systems and Networks
• Power Engineering Education
• Power Quality
• Power System Economics and Electricity Markets
• Power System Modelling and Analysis
• Power System Operations and Control
• Power System Optimisation and Planning
• Power System Protection
• Renewable Energy Systems
• Smart Grids
• Substation and Distribution System
• Automation Sustainable e-Transition
• Transient analysis and EMTP modelling


text:
# Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups

Coming up with examples of $D_8$-covers of $\mathbb{C}(x)$ is easy. For example:

$Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$

defines a $D_8$ extension of $\mathbb{C}(x)$. If you view this field extension as an extension of the corresponding smooth projective models, it is branched at $x=0,1,2,7$.

I wish to control the inertia groups (=decomposition groups, in this case) associated to the ramification.

I'll use the presentation $D_8= \langle a,b | a^4, b^2, ab=ba^{-1} \rangle$.

Take $\mathbb{P}^1_{\mathbb{C}} \setminus 0,7,1,2$. Its fundamental group is $\langle \gamma_1, \gamma_2, \gamma_3, \gamma_4| \gamma_1\gamma_2\gamma_3\gamma_4=1 \rangle$ (where $(\gamma_1, \gamma_2, \gamma_3, \gamma_4)$ are respectively the loops from a fixed base point around the points $(0,7,1,2)$). The surjection: $(\gamma_1, \gamma_2, \gamma_3, \gamma_4)$ goes to $(b, ab, a^2, a^3)$, gives a $D_8$-Galois cover of $\mathbb{P}^1_{\mathbb{C}}$.

I wish to find the equations that define THAT $D_8$ cover. The equations I wrote in the beginning won't work because there the inertia groups of the ramification points above $x=0$ and above $x=7$ are the same groups; whereas if done through the description, since no conjugate of $b$ is equal to any conjugate of $ab$, no inertia group of any ramification point above $x=0$ will equal the inertia group of any ramification point above $x=7$.

For some reason I easily confuse myself in such computations.

If you mod out the description by $\langle a \rangle$ you see that the description of the $D_8 / \langle a \rangle$ sub-cover of the original cover, has description $(b\langle a \rangle, b\langle a \rangle, \langle a \rangle, \langle a \rangle)$, and therefore must be given by $y^2=x(x-7)$ (if it's, say, $y'^2=\frac{x}{x-7}$, then $y'=\frac{y}{x-7}$). But I can't find the right other equation that would make this work.

Note that this should be easy, because this $D_8/\langle a \rangle$-cover is a $\mathbb{P}^1_{\mathbb{C}}$ with parameter $y$, so there's no funny business. And yet I'm having some trouble finishing this example up.

-

As you say, the fixed field by $\langle a\rangle$ is $\mathbb{C}(u)$ with $u^2=-5x/2(x-7)$, and the fixed field by $\langle a^2,b \rangle$ is similarly $\mathbb{C}(v)$ with $v^2=7(x-2)/2(x-7)$ (constants chosen so that $u^2+v^2=1$). So their compositum, the fixed field by $\langle a^2\rangle$, has genus zero and is $\mathbb{C}(t)$, where $u=2t/(t^2+1)$, $v=(t^2-1)/(t^2+1)$. This means that the last quadratic step of your tower is determined entirely by its ramification points, which are the 2 points above $x=2$ (which are the points $t=\pm 1$) and the 4 points above $x=1$, call them $t=c_1,\dots,c_4$. So your desired field is $\mathbb{C}(t,w)$ where $w^2=(t-c_1)(t-c_2)(t-c_3)/(t-c_4)(t^2-1)$.


text:
Rating:

When connecting to the challenge server, it gives us a strange string.

![Alt initial](./images/initial.png?raw=true)

File command says it is zlib compressed string. Well then, just use a script to decompress it everytime (what is send to the server should also be zlib compressed).

![Alt shacolla01](./images/shacolla01.png?raw=true)

We are required to send two different strings with same sha1 value. It is easy to recall the two pdfs with same sha1 that Google release several weeks ago. First 320 byte of the two files are different.

Experiments show that, if we use the different first 320 bytes and a same suffix to construct a string, their sha1 will also be the same. The sha1 should end with certain 5 bytes, but this is easily worked out with brute force. So again write a script (kinda ugly though)

import zlib
import hashlib
from pwn import *
m=hashlib.sha1()
fp1=open('shattered-1.pdf','rb')
fp2=open('shattered-2.pdf','rb')
p=remote('66.172.27.77',52317)
r=p.recv()
print zlib.decompress(r)
p.send(zlib.compress('Y'))
r=p.recv()
print zlib.decompress(r)
r=p.recv()
string=zlib.decompress(r)
required=(string.split('\n'))[0][-5:]
for i in xrange(10000000):
#for i in xrange(100):
suffix=str(hex(i))[2:]
if len(suffix)%2==1:
suffix='0'+suffix
m=hashlib.sha1()


text:
# Derivative of an operator valued function

1. Feb 11, 2012

### giova7_89

If I have a function
$$f:R\rightarrow L$$
where L is the space of linear operators from an hilbert space to itself, how can i define the derivative of f at a particular point of R? I mean, it is "obvious" that one should try:

$$f'(s_0)=lim_{\Delta s\rightarrow0}\frac{|f(s_0 + \Delta s) -f(s_0)|}{\Delta s}$$

but i'm concerned with the fact that when in the numerator i use ||, i mean the norm on L, which as i said is the space of linear operators from an hilbert space to itself. Since in physics one often deals with operators that have infinite operator norm, I wanted to know if the norm of that difference of operators is finite or not, even if we take unbounded operators.

It is a mathematical question, but since in QM one encounters these things all the time, I wanted some clarifications..

thanks!

2. Feb 11, 2012

### Staff: Mentor

Yea I was going to suggest to define the norm on the space of operators by the following inner product of operators (A,B) = Trace (Abar B) - here Abar means the adjoint which reduces to Trace (AB) for the usual Hermtian operators in QM.

The above is called the Frobenius norm (older texts refer to it as the Hilbert–Schmidt norm or the Schur norm).

But ouch - even the identity operator has an infinite norm.

Its probably better to use the operator norm:
http://en.wikipedia.org/wiki/Operator_norm

But even then I don't know how to guarantee it being finite.

Thanks
Bill

Last edited: Feb 12, 2012
3. Feb 12, 2012

### Bill_K

I would think the "obvious" thing to do is to take the ||'s off and just let f'(s) be an operator.

4. Feb 12, 2012

### giova7_89

Ok, one doesn't need to put || in THAT definition, and I know that f'(s_0) is an operator, what I meant was that when i use the usual $$\epsilon,\delta$$

definition of limit, the norm on L will surely pop out: I mean that f'(s_0) is the operator that does this thing:

$$\forall\epsilon>0\exists\delta>0:if\,\,0<|s-s_0|<\delta\Rightarrow\bigg|\frac{f(s) - f(s_0) - (s-s_0)f'(s_0)}{s-s_0}\bigg|_L<\epsilon$$

where the $$||_L$$ is the norm on L. Here I wanted to know if there are any problems if i'm dealing with unbounded operators. I admit I made a mistake, and in my first post the || were not needed, but they surely are in this definition (which is that of the differential of a function from a vector space to another, i think)

5. Feb 12, 2012

### giova7_89

In the original post, things should look like this:

$$f'(s_0)$$

is an operator such that:

$$lim_{\Delta s\rightarrow 0}\frac{|f(s_0+\Delta s) - f(s_0) - \Delta s\,f'(s_0)|_L}{\Delta s} = 0$$

The oroginal post was wrong since it compared a number to an operator

6. Feb 12, 2012

### strangerep

For bounded operators, one can use the supremum norm (aka operator norm), which bhobba mentioned already: http://en.wikipedia.org/wiki/Operator_norm

For unbounded operators, one can talk in terms of densely-defined operators on an ordinary Hilbert space, or else pass to a rigged Hilbert space and use the theory of generalized functions (distributions). But for that, one must first understand how operators on (say) the Schwartz space are extended to operators on its dual space of tempered distributions via the dual pairing. (If that all sounds like gobble-de-gook, I could probably post some introductory references. It depends what you're really trying to do.)

7. Feb 13, 2012

### giova7_89

My knowledge about this is very limited.. So i'd be glad if you could post some references. I just started a course on quantum field theory and since (but it also happens in the mathematical formalism of quantum mechanics) we're dealing with operator valued functions, I wanted to know more about the mathematics behind that..

So, if you know an introductory book about the mathematics of QFT, feel free to tell me about that, too

8. Feb 13, 2012

### Fredrik

Staff Emeritus
If you want to learn this stuff, you have to start with topology and functional analysis. Kreyszig's book on functional analysis covers the topology you need as well, so it's much easier to read than many of the others. (I haven't read it myself, but I know how crazy hard some other books are).

9. Feb 13, 2012

### lugita15

I second the recommendation of Kreyszig. It's probably the only functional analysis book that assumes so little background and yet covers so much ground.

10. Feb 13, 2012

### Staff: Mentor

I also recommend Kreyszig and I am thinking of actually purchasing a copy.

I also recommend the following book on Distribution Theory that I have in my library. It helped me a lot with this sort of stuff:
https://www.amazon.com/Theory-Distr...=sr_1_1?s=books&ie=UTF8&qid=1329180440&sr=1-1

You can usually define what you want on a test space then use distribution theory extend it. And if you are doing QFT then the above IMHO is a must - they are really distributions rather than functions.

Thanks
Bill

11. Feb 14, 2012

### strangerep

Yeah, I was going to say something about how basic stuff on distributions and generalized functions is probably more important to understand clearly (for QM/QFT purposes) before tackling Functional Analysis books (which usually don't cover generalized functions anyway). I didn't know a good basic book on distributions and generalized functions, so I'll take at look at the one you suggested (Richards & Youn) and see what it's like.

Cheers.

12. Feb 14, 2012

### Tarantinism

The space of BOUNDED operators on a Hilbert space is itself a Banach space, so it is easy to define the usual Fréchet derivative:

http://en.wikipedia.org/wiki/Fréchet_derivative

I wouldn't know how to define this derivative, on the other hand, if you map into some unbounded operators, typical in physics.

13. Feb 14, 2012

### strangerep

Gel'fand triples, i.e., using topological duals of a suitable nuclear space.

14. Feb 15, 2012

### lugita15

Can you spell this out in more detail? Are you taking the Frechet derivative of an unbounded operator on the dense nuclear space, and then doing ... something with the dual pairing between the nuclear space and the space of continuous antilinear functionals?

15. Feb 15, 2012

### strangerep

Well, it can't be the standard Frechet derivative, since that's for Banach spaces, but the small nuclear space $\Omega$ in a Gel'fand triple
$$\Omega ~\subset~ \bar{\Omega} ~\subset~ \Omega' ~,~~~~ \Big(\mbox{where}~ \Omega' ~\mbox{is the topological dual of}~ \Omega\Big) ~.$$
is not complete, hence not Banach.

But here's my (quick) understanding of what's going on. (Warning: I could be wrong about parts of this, in which case I hope someone more knowledgeable will say so.)

Let $\phi$, $\Psi$ be arbitrary vectors in $\Omega, \Omega'$ respectively. Let A be an operator defined everywhere on $\Omega$. Then by standard theorems (cf. Gelfand & Vilenkin vol 4) it can be extended uniquely to an operator A' on $\Omega'$ via the dual pairing
$$(\phi, A'\Psi) ~:=~ (A\phi, \Psi)$$
such that A' coincides with A when acting on elements of $\Omega$.

Now let A(t) be an operator-valued function of a real parameter t such that A(t) is defined everywhere on $\Omega$. Then, by the above construction it can be extended to $\Omega'$.

If, in addition, there is an operator B such that
$$\frac{A(t+h) - A(t) - B}{h} ~\rightarrow~ 0 ~~~~~~~~ (1)$$
in weak topology, we can call B a derivative of A at t. By duality, we can extend B to all of $\Omega'$.

The subtleties lie in what it means for the LHS of (1) to approach 0 weakly. The standard meaning for weak operator topology is that a sequence of operators $\{T_n\}$ converges weakly to $T$ iff $T_n \phi \to T \phi$ for all $\phi$.

(Once again: caveat emptor.)

16. Feb 15, 2012

### Staff: Mentor

He is talking about Rigged Hilbert Spaces which is an extension of distribution theory - probably best to get the book I recommended on distribution theory first but do check out:
http://www.abhidg.net/RHSclassreport.pdf [Broken]

Thanks
Bill

Last edited by a moderator: May 5, 2017


text:
Greater Equal Threshold - MapleSim Help

Greater or Equal Threshold

True if input is greater than or equal to threshold

 Description The Greater or Equal Threshold (or Greater Equal Threshold) component has a boolean output that is true when the Real input $u$ is greater than or equal to parameter $\mathrm{threshold}$, otherwise it is false.
 Equations $y=\left(u\ge \mathrm{threshold}\right)$

Connections

 Name Description Modelica ID $u$ Real input signal u $y$ Boolean output signal y

Parameters

 Name Default Units Description Modelica ID $\mathrm{threshold}$ $0$ $1$ Threshold level threshold

 Modelica Standard Library The component described in this topic is from the Modelica Standard Library. To view the original documentation, which includes author and copyright information, click here.