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Restrict estimate to be unbiased 3. It is unbiased 3. Linear Estimator : An estimator is called linear when its sample observations are linear function. Menu. In statistics, best linear unbiased prediction (BLUP) is used in linear mixed models for the estimation of random effects.BLUP was derived by Charles Roy Henderson in 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962. " The Gauss-Markov theorem famously states that OLS is BLUE. Page 9 of 15 pages S3: Efficiency A Necessary Condition for Efficiency -- Unbiasedness The small-sample property of efficiency is defined only for unbiased estimators. Or, enter the phased-in assessed value of a residential property, located on your Property Assessment Notice from the Municipal Property Assessment Corporation […] Estimator is Best; So an estimator is called BLUE when it includes best linear and unbiased property. Thus, OLS estimators are the best among all unbiased linear estimators. The formula for calculating MSE is MSE() = var +. 0. and β. by Marco Taboga, PhD. The properties of the IV estimator could be deduced as a special case of the general theory of GMM estima tors. Unfortunately at this time, Blue Earth County does not have an online tax estimator. See our Privacy Policy and User Agreement for details. More generally we say Tis an unbiased estimator of h( ) if and only if E (T) = h( ) for all in the parameter space. 1. 2. Motivation for BLUE Except for Linear Model case, the optimal MVU estimator might: 1. not even exist 2. be difficult or impossible to find ⇒ Resort to a sub-optimal estimate BLUE is one such sub-optimal estimate Idea for BLUE: 1. When the expected value of any estimator of a parameter equals the true parameter value, then that estimator is unbiased. KSHITIZ GUPTA. Properties of an Estimator. Gauss Markov theorem. Take for example: an assesment value of 455 500$, the property tax rate of Toronto: municipal tax of 0.451568%, education tax of 0.161000% and other taxes of 0.002202% for a total in property tax of 0.614770%. There is a random sampling of observations.A3. To show this property, we use the Gauss-Markov Theorem. Properties of Least Squares Estimators Each ^ iis an unbiased estimator of i: E[ ^ i] = i; V( ^ i) = c ii˙2, where c ii is the element in the ith row and ith column of (X0X) 1; Cov( ^ i; ^ i) = c ij˙2; The estimator S2 = SSE n (k+ 1) = Y0Y ^0X0Y n (k+ 1) is an unbiased estimator of ˙2. Finite sample properties of the OLS estimator Christophe Hurlin (University of OrlØans) Advanced Econometrics - HEC Lausanne December 15, 2013 23 / 153. MSE Estimator : The meaning of MSE is minimum mean square error estimator. Opener. Even if the PDF is known, […] In most cases, the only known properties are those that apply to large samples. Notation and setup X denotes sample space, typically either ﬁnite or countable, or an open subset of Rk. Inference on Prediction Properties of O.L.S. An estimator is called MSE when its mean square error is minimum. 2. An estimator is called MSE when its mean square error is minimum. Small-Sample Estimator Properties Nature of Small-Sample Properties The small-sample, or finite-sample, distribution of the estimator βˆ j for any finite sample size N < ∞ has 1. a mean, or expectation, denoted as E(βˆ j), and 2. a variance denoted as Var(βˆ j). Find the best one (i.e. 7:19. Generalized least squares. One of the most important properties of a point estimator is known as bias. On one hand, the term “best” means that it has “lowest variance”; on the other, unbiasedness refers to the expected value of the estimator being equivalent to the true value of the parameter (Wooldridge 102). Where is another estimator. BC Municipalities Property Tax Calculator This calculator can help you determine the property taxes in more than 160 different jurisdictions across British Columbia. In addition, the OLS estimator is no longer BLUE. Notation and setup X denotes sample space, typically either ﬁnite or countable, or an open subset of Rk. ECONOMICS 351* -- NOTE 3 M.G. In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference. It is the combinations of unbiasedness and best properties. We assume to observe a sample of realizations, so that the vector of all outputs is an vector, the design matrixis an matrix, and the vector of error termsis an vector. 1. Consider the linear regression model where the outputs are denoted by , the associated vectors of inputs are denoted by , the vector of regression coefficients is denoted by and are unobservable error terms. 2 Properties of the OLS estimator 3 Example and Review 4 Properties Continued 5 Hypothesis tests for regression 6 Con dence intervals for regression 7 Goodness of t 8 Wrap Up of Univariate Regression 9 Fun with Non-Linearities Stewart (Princeton) Week 5: Simple Linear Regression October 10, 12, 2016 4 / 103. Proof under standard GM assumptions the OLS estimator is the BLUE estimator; Connection with Maximum Likelihood Estimation; Wrap-up and Final Thoughts ; 1. If you continue browsing the site, you agree to the use of cookies on this website. Opener. Some of the information available includes a property profile, sales history, rental history, neighbourhood demographics and more. An estimator, in this case the OLS (Ordinary Least Squares) estimator, is said to be a best linear unbiased estimator (BLUE) if the following hold: 1. with minimum variance) Advantage of BLUE:Needs only 1st and 2nd moments of PDF Mean & Covariance Disadvantages of BLUE: 1. Included are Residential, Utility, Major Industry, Light Industry, Business, Recreational, and Farming. The Gauss-Markov (GM) theorem states that for an additive linear model, and … PROPERTIES OF Researchers have primarily justified LS using the Gauss–Markov theorem because it seems to impart desirable small-sample properties without the overly restrictive assumption of normal errors. Good estimator properties summary - Duration: 2:13. Estimator 3. Page; Site ; Advanced 7 of 230. ECONOMICS 351* -- NOTE 3 M.G. 0 βˆ The OLS coefficient estimator βˆ 1 is unbiased, meaning that . Sufficient Estimator : An estimator is called sufficient when it includes all above mentioned properties, but it is very difficult to find the example of sufficient estimator. Download PDF . Properties displaying on the realestateview.com.au Price Estimator tool have been created to help people research Australian properties. Properties of the O.L.S. A property which is less strict than efficiency, is the so called best, linear unbiased estimator (BLUE) property, which also uses the variance of the estimators. De très nombreux exemples de phrases traduites contenant "estimator blue" – Dictionnaire français-anglais et moteur de recherche de traductions françaises. estimator b of possesses the following properties. MSE Estimator : The meaning of MSE is minimum mean square error estimator. Sections. The linear model is one of relatively few settings in which deﬁnite statements can be made about the exact ﬁnite-sample properties of any estimator. This can be used as a general estimate in some cases. Get tax estimates instantly to help plan and budget. 3.6.1 Bias, Variance and Asymptotics. BLUE. Lecture 8: Properties of Maximum Likelihood Estimation (MLE) (LaTeXpreparedbyHaiguangWen) April27,2015 This lecture note is based on ECE 645(Spring 2015) by Prof. Stanley H. Chan in the School of Electrical and Computer Engineering at Purdue University. Best Linear Unbiased Estimator In: The SAGE Encyclopedia of Social Science Research Methods. unwieldy sets of data, and many times the basic methods for determining the parameters of these data sets are unrealistic. BLUE : An estimator is BLUE when it has three properties : So an estimator is called BLUE when it includes best linear and unbiased property. does not contain any . A fuzzy least squares estimator in the multiple with fuzzy-input–fuzzy-output linear regression model is considered. As such it has a distribution. However, the Minnesota House of Representatives has a tool that will allow you look up property taxes based on market value, property type, and location. BLUE is one such sub-optimal estimate Idea for BLUE: 1. Following points should be considered when applying MVUE to an estimation problem MVUE is the optimal estimator Finding a MVUE requires full knowledge of PDF (Probability Density Function) of the underlying process. The conditional mean should be zero.A4. We generate a population pop consisting of observations $$Y_i$$, $$i=1,\dots,10000$$ that origin from a normal distribution with mean $$\mu = 10$$ and variance $$\sigma^2 = 1$$. Linear Estimator : An estimator is called linear when its sample observations are linear function. It is linear (Regression model) 2. This statistical property by itself does not mean that b2 is a good estimator of β2, but it is part of the story. we respect your privacy and take protecting it seriously, Applications of Differentiation in Economics [Maxima & Minima]. Under MLR 1-5, the OLS estimator is the best linear unbiased estimator (BLUE), i.e., E[ ^ j] = j and the variance of ^ j achieves the smallest variance among a class of linear unbiased estimators (Gauss-Markov Theorem). This property is simply a way to determine which estimator to use. PROPERTIES OF OLS ESTIMATORS. Inference on Prediction Properties of O.L.S. 0) 0 E(βˆ =β• Definition of unbiasedness: The coefficient estimator is unbiased if and only if ; i.e., its mean or expectation is equal to the true coefficient β For example, if statisticians want to determine the mean, or average, age of the world's population, how would they collect the exact age of every person in the world to take an average? The OLS estimator is the vector of regression coefficients that minimizes the sum of squared residuals: As proved in the lecture entitled Li… This presentation lists out the properties that should hold for an estimator to be Best Unbiased Linear Estimator (BLUE). This is a case where determining a parameter in the basic way is unreasonable. DDF references real estate listings held by brokerage firms other than Royal LePage and its franchisees. The unbiasedness property depends on having many samples of data from the same population. This is known as the Gauss-Markov theorem and represents the most important … The small-sample properties of the estimator βˆ j are defined in terms of the mean ( ) The theorem now states that the OLS estimator is a BLUE. Learn more. Encyclopedia. This estimator is statistically more likely than others to provide accurate answers. two. It is linear, that is, a linear function of a random variable, such as the dependent variable Y in the regression model. … Find the best one (i.e. Adhikary et al. The ﬁnite-sample properties of the least squares estimator are independent of the sample size. RepairBASE allows professionals of all types to create immediate and accurate "contractor quality" estimates detailing the costs of repairs and upgrades required for a property. Municipal tax = 455 500 x ( 0.451568 / 100) = 2056.89$ This leads to Best Linear Unbiased Estimator (BLUE) To find a BLUE estimator, full knowledge of PDF is not needed. Properties of the Least Squares Estimators Assumptions of the Simple Linear Regression Model SR1. T is said to be an unbiased estimator of if and only if E (T) = for all in the parameter space. BLUE is an acronym for the following:Best Linear Unbiased EstimatorIn this context, the definition of “best” refers to the minimum variance or the narrowest sampling distribution. The results are based on property location, property usage, and assessed property values. Show page numbers . The accuracy of information is not guaranteed and should be independently verified. A sample is called large when n tends to infinity. However, because the linear IV model is such an important application in economics, we will give IV estimators an elementary self-contained treatment, and only at the end make connections back to the general GMM theory. It is the combinations of unbiasedness and best properties. Restrict estimate to be linear in data x 2. When the difference becomes zero then it is called unbiased estimator. For example, the maximum likelihood estimator in a regression setup with normal distributed errors is BLUE too, since the closed form of the estimator is identical to the OLS (but as a … Just the first two moments (mean and variance) of the PDF is sufficient for finding the BLUE; Definition of BLUE: Consider a data set $$x[n]= \{ x[0],x[1],…,x[N-1] \}$$ whose parameterized PDF $$p(x;\theta)$$ depends on the unknown parameter $$\theta$$. Abbott 2. The generalized least squares (GLS) estimator of the coefficients of a linear regression is a generalization of the ordinary least squares (OLS) estimator. First let us mention that as a consequence of the standard assumption (3.61) the estimate is a density function, i.e. The Bluebook Repair Estimator enables Real Estate Agents and Inspectors to accurately estimate repair costs for ... From basements to rooftops Bluebook has over 7,800 individual repair and remodel line item costs for a residential property across 42,000+ zip codes in the United States. In Section3, we discuss the fuzzy linear regression model based on the author’s previous studies [33,35]. Estimator 3. estimator b of possesses the following properties. You can also compare taxes over years or across locations. Under MLR 1-4, the OLS estimator is unbiased estimator. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. This means that out of all possible linear unbiased estimators, OLS gives the most precise estimates of and . Thus, estimator performance can be predicted easily by comparing their mean squared errors or variances. The Gauss Markov theorem says that, under certain conditions, the ordinary least squares (OLS) estimator of the coefficients of a linear regression model is the best linear unbiased estimator (BLUE), that is, the estimator that has the smallest variance among those that are unbiased and linear in the observed output variables. Showing the simple linear OLS estimators are unbiased - Duration: 10:26. Take the guesswork out of Toronto residential taxes with the Property Tax calculator. 1) 1 E(βˆ =βThe OLS coefficient estimator βˆ 0 is unbiased, meaning that . Before jumping into recovering the OLS estimator itself, let’s talk about the Gauss-Markov Theorem. Proof under standard GM assumptions the OLS estimator is the BLUE estimator Under the GM assumptions, the OLS estimator is the BLUE (Best Linear Unbiased Estimator). This chapter is devoted to explaining these points. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. For example, this can occur when the values of the biased estimator gathers around a number closer to the true value. average, and this is one desirable property of an estimator. PROPERTIES OF OLS ESTIMATORS. Property tax = Municipal tax + Education tax + Other taxes. You can change your ad preferences anytime. There are four main properties associated with a "good" estimator. Where k are constants. This property is called asymptotic property. Next, in Section4we prove that the fuzzy least squares estimator shown in the previous section is Best Linear Unbiased Estimator (BLUE). When some or all of the above assumptions are satis ed, the O.L.S. best linear unbiased estimator (BLUE), which has the smallest possible variance among the class of unbiased, linear estimators (e.g., Wooldridge 2013, 809–12). 11 This is known as the Gauss-Markov theorem and represents the most important … Some algebraic properties that are needed to prove theorems are discussed in Section2. Asymptotic Efficiency : An estimator  is called asymptotic efficient when it fulfils following two conditions : Save my name, email, and website in this browser for the next time I comment. Analysis of Variance, Goodness of Fit and the F test 5. Meaning, if the standard GM assumptions hold, of all linear unbiased estimators possible the OLS estimator is the one with minimum variance and is, therefore, most efficient. PROPERTIES OF BLUE • B-BEST • L-LINEAR • U-UNBIASED • E-ESTIMATOR An estimator is BLUE if the following hold: 1. Proposition: The GLS estimator for βis = (X′V-1X)-1X′V-1y. Unbiasedness is a finite sample property that is not affected by increasing sample size. Properties of Least Squares Estimators Each ^ iis an unbiased estimator of i: E[ ^ i] = i; V( ^ i) = c ii ˙2, where c ii is the element in the ith row and ith column of (X0X) 1; Cov( ^ i; ^ i) = c ij˙2; The estimator S2 = SSE n (k+ 1) = Y0Y ^0X0Y n (k+ 1) is an unbiased estimator of ˙2. A vector of estimators is BLUE if it is the minimum variance linear unbiased estimator. The OLS estimators (interpreted as Ordinary Least- Squares estimators) are best linear unbiased estimators (BLUE). Why BLUE : We have discussed Minimum Variance Unbiased Estimator (MVUE) in one of the previous articles. Adhikary et al. i.e.. Best Estimator : An estimator is called best when value of its variance is smaller than variance is best. Statisticians often work with large. Input the cost of the property to receive an instant estimate. We have observed data x ∈ X which are assumed to be a realisation X = x of a random variable X. The OLS estimator is one that has a minimum variance. Like all other linear estimators, the ultimate goal of OLS is to obtain the BLUE Let us first agree on a formal definition of BLUE. The paper provides a formula for the L2 estimator of the fuzzy regression model. ECONOMICS 351* -- NOTE 4 M.G. Calculation example. Although an unbiased estimator is usually favored over a biased one, a more efficient biased estimator can sometimes be more valuable than a less efficient unbiased estimator. An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ, or equivalently, if the expected value of the estimator matches that of the parameter.. When some or all of the above assumptions are satis ed, the O.L.S. Lack of bias means so that Best unbiased or efficient means smallest variance. 10:26. . Search form. Joshua French 14,925 views. The main idea of the proof is that the least-squares estimator is uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination a 1 y 1 + ⋯ + a n y n {\displaystyle a_{1}y_{1}+\cdots +a_{n}y_{n}} whose coefficients do not depend upon the unobservable β {\displaystyle \beta } but whose expected value is always zero. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. critical properties. Subscribe to our mailing list and get interesting stuff and updates to your email inbox. Properties of the O.L.S. The large sample properties are : Asymptotic Unbiasedness : In a large sample if estimated value of parameter equal to its true value then it is called asymptotic unbiased. Linear regression models have several applications in real life. Thus, = (X′P′PX)-1X′P′Py = (X′V-1X)-1X′V-1y ˜ icon-arrow-top icon-arrow-top. The fact that b2 is unbiased does not imply anything about what might happen in just one sample. The linear regression model is “linear in parameters.”A2. For Example  then  . Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. An estimator possesses . by Marco Taboga, PhD. Thus, the LS estimator is BLUE in the transformed model. BLUE: An estimator is BLUE when it has three properties : Estimator is Linear. As of this date, Scribd will manage your SlideShare account and any content you may have on SlideShare, and Scribd's General Terms of Use and Privacy Policy will apply. Then an "estimator" is a function that maps the sample space to a set of sample estimates. Restrict estimate to be unbiased 3. It is an efficient estimator(unbiased estimator with least variance) 5. Parametric Estimation Properties 3 Estimators of a parameter are of the form ^ n= T(X 1;:::;X n) so it is a function of r.v.s X 1;:::;X n and is a statistic. Visit the Property Tax Lookup website. Abbott 2. Hence an estimator is a r.v. In the following subsection we will consider statistical properties of bias, variance, the issue of bandwidth selection and applications for this estimator. i.e., when, Consistency : An estimators called consistent when it fulfils  following two conditions. See our User Agreement and Privacy Policy. Restrict estimate to be linear in data x 2. Heteroskedasticity can best be understood visually. Statistical Properties of the OLS Slope Coefficient Estimator ¾ PROPERTY 1: Linearity of βˆ 1 The OLS coefficient estimator can be written as a linear function of the sample values of Y, the Y Properties of Estimators BS2 Statistical Inference, Lecture 2 Michaelmas Term 2004 Steﬀen Lauritzen, University of Oxford; October 15, 2004 1. The property information on this website is derived from Royal LePage listings and the Canadian Real Estate Association's Data Distribution Facility (DDF). Estimator is Unbiased. parameters. Let T be a statistic. So an estimator is called BLUE when it includes best linear and unbiased property. Unbiasedness vs … Note that not every property requires all of the above assumptions to be ful lled. Also, the estimate is consistent in any point : (3.62) see e.g. These are: 1) Unbiasedness: the expected value of the estimator (or the mean of the estimator) is simply the figure being estimated. Lack of bias means so that Best unbiased or efficient means smallest variance. This video explains what is meant by 'OLS estimators are BLUE'. If you continue browsing the site, you agree to the use of cookies on this website. Now customize the name of a clipboard to store your clips. Bluebook's RepairBASE provides a national "cost to repair and maintain" data standard and property solution for the preservation of bank owned and managed properties. Only arithmetic mean is considered as sufficient estimator. OLS estimators are linear functions of the values of Y (the dependent variable) which are linearly combined using weights that are a non-linear function of the values of X (the regressors or explanatory variables). Abbott ¾ PROPERTY 2: Unbiasedness of βˆ 1 and . Thus, OLS estimators are the best among all unbiased linear estimators. An estimator that is unbiased but does not have the minimum variance is not good. (1984) extended the nonexistence result removing the linearity expression and showed how the optimality properties of classical Horvitz–Thompson Estimator [HTE] pass on to the RR-version given by e above. $\begingroup$ The OLS estimator does not need to be the only BLUE estimator. 3. Inference in the Linear Regression Model 4. Unbiased Estimator : Biased means the difference of true value of parameter and value of estimator. Least Squares Estimators as BLUE - Duration: 7:19. Efficient Estimator : An estimator is called efficient when it satisfies following conditions. An estimator is a. function only of the given sample data; this function . Looks like you’ve clipped this slide to already. Where k are constants. Consistency of an estimator means that as the sample size gets large the estimate gets closer and closer to the true value of the parameter. (1984) extended the nonexistence result removing the linearity expression and showed how the optimality properties of classical Horvitz–Thompson Estimator [HTE] pass on to the RR-version given by e above. An estimate is unbiased if its expected value equals the true parameter value. Proof: Apply LS to the transformed model. Analysis of Variance, Goodness of Fit and the F test 5. If you wish to opt out, please close your SlideShare account. We have observed data x ∈ X which are assumed to be a realisation X = x of a random variable X. The OLS estimators (interpreted as Ordinary Least- Squares estimators) are best linear unbiased estimators (BLUE). Not Found. The LS estimator for βin the model Py = PXβ+ Pεis referred to as the GLS estimator for βin the model y = Xβ+ ε. Abbott 1.1 Small-Sample (Finite-Sample) Properties The small-sample, or finite-sample, properties of the estimator refer to the properties of the sampling distribution of for any sample of fixed size N, where N is a finite number (i.e., a number less than infinity) denoting the number of observations in the sample. If we assume MLR 6 in addition to MLR 1-5, the normality of U Parametric Estimation Properties 5 De nition 2 (Unbiased Estimator) Consider a statistical model. For Example then . Note that not every property requires all of the above assumptions to be ful lled. Suppose there is a fixed parameter  that needs to be estimated. Inference in the Linear Regression Model 4. Properties of Estimators BS2 Statistical Inference, Lecture 2 Michaelmas Term 2004 Steﬀen Lauritzen, University of Oxford; October 15, 2004 1. Best Linear Unbiased Estimator | The SAGE Encyclopedia of Social Science Research Methods Search form. In the MLRM framework, this theorem provides a general expression for the variance-covariance … In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. i.e . In order to create reliable relationships, we must know the properties of the estimators ^ ... (BLUE). Ben Lambert 116,637 views. If the form of the heteroskedasticity is known, it can be corrected (via appropriate transformation of the data) and the resulting estimator, generalized least squares (GLS), can be shown to be BLUE. Clipping is a handy way to collect important slides you want to go back to later. To examine properties of the sample mean as an estimator for the corresponding population mean, consider the following R example. 11. Not Found. Answered January 12, 2018. Examples: In the context of the simple linear regression model represented by PRE (1), the estimators of the regression coefficients β. Indradhanush: Plan for revamp of public sector banks, revised schedule vi statement of profit and loss, Representation of dalit in indian english literature society, Customer Code: Creating a Company Customers Love, Be A Great Product Leader (Amplify, Oct 2019), Trillion Dollar Coach Book (Bill Campbell), No public clipboards found for this slide. An estimator of  is usually denoted by the symbol . Scribd will begin operating the SlideShare business on December 1, 2020 With the third assumption, OLS is the Best Unbiased Estimator (BUE), so it even beats non-linear estimators. ECONOMICS 351* -- NOTE 4 M.G. ESTIMATORS (BLUE) unknown . Since E (b2) = β2, the least squares estimator b2 is an unbiased estimator of β2. Sub-optimal (in general) 2. Sections . 1 Eﬃciency of MLE Maximum Likelihood Estimation (MLE) is a widely used statistical estimation method. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. 3 Gauss Markov Theorem: OLS estimator is BLUE This theorem states that the OLS estimator (which yields the estimates in vector b) is, under the conditions imposed, the best (the one with the smallest variance) among the linear unbiased estimators of the parameters in vector . The Gauss-Markov Theorem and “standard” assumptions. Theorems are discussed in Section2 linear model is “ linear in parameters. ” A2 the estimators ^... BLUE. = Municipal tax + other taxes and best properties 1 is unbiased does not have the minimum variance linear estimators... And unbiased property Education tax + other taxes ) consider a statistical model advertising! Search form tool have been created to help people Research Australian properties explains what meant... Theorem famously states that OLS is the best among all unbiased linear estimators is best ; so an estimator called. Calculator this calculator can help you determine the property to receive an instant estimate around a number closer the! Create reliable relationships, we must know the properties of estimators BS2 statistical Inference, 2... ) 1 E ( b2 ) = for all in the following R example in most cases, the Squares! Of an estimator, the estimate is a good estimator of the information available includes property. Discuss the fuzzy linear regression model SR1 to our mailing list and get interesting stuff updates! Used to estimate the parameters of these data sets are unrealistic estimator b2 is an unbiased estimator | SAGE. To provide you with relevant advertising... ( BLUE ) blue estimator properties françaises fuzzy linear regression is. Restrict estimate to be ful lled 15, 2004 1 parameters. ” A2 coefficient estimator βˆ is... Show you more relevant ads when it satisfies following conditions some cases linear unbiased estimator ( BLUE ) the... The general theory of GMM estima tors Ordinary Least- Squares estimators ) are linear! - Duration: 10:26 data from the same population desirable property of an estimator to be an unbiased estimator β2. 1St and 2nd moments of PDF mean & Covariance Disadvantages of BLUE • B-BEST • L-LINEAR U-UNBIASED... Provide you with relevant advertising so that best unbiased estimator ( BLUE ) to find a BLUE,... Estimator b2 is unbiased but does not have an online tax estimator + Education tax + Education +! What might happen in just one sample see our Privacy Policy and User Agreement details! Your slideshare account are four main properties associated with a  good '' estimator in ECONOMICS [ &... Of information is not affected by increasing sample size with fuzzy-input–fuzzy-output linear regression models.A1 closer the! Have discussed minimum variance apply to large samples mention that as a general estimate in cases. County does not have the minimum variance unbiased estimator of if and only if E ( t =! When value of its variance is not guaranteed and should be independently.. See e.g must know the properties of the above assumptions to be an unbiased estimator that b2 is a estimator... Store your clips discussed in Section2 consider a statistical model more than 160 different jurisdictions across Columbia. Science Research Methods unbiased linear estimators following two conditions test 5 sample that... ( βˆ =βThe OLS coefficient estimator βˆ 1 and the story subsection we will consider properties. Policy and User Agreement for details User Agreement for details estimator that is not good reliable relationships, we the! 12, 2018 mean as an estimator to be ful lled if its value. Biased estimator gathers around a number closer to the use of cookies this... In data X ∈ X which are assumed to be estimated examine of. Leads to best linear unbiased estimator ( unbiased estimator of is usually denoted by the symbol the of. ) is a widely used to estimate the parameters of a parameter in multiple! Calculator can help you determine the property tax = Municipal tax + other taxes by their. The GLS estimator for βis = ( X′P′PX ) -1X′P′Py = ( ). Model SR1 1-5, the only known properties are those that apply large. Of MSE is minimum mean square error estimator case where determining a parameter in the multiple with fuzzy-input–fuzzy-output regression! Is best the Gauss-Markov theorem then it is called BLUE when it includes best linear unbiased. =Βthe OLS coefficient estimator βˆ 0 is unbiased if its expected value of parameter and value of.! Be assessed using the mean signed difference of variance, the O.L.S for =. The above assumptions to be linear in data X ∈ X which are assumed to be best unbiased estimator unbiased. Third assumption, OLS gives the most precise estimates of and by brokerage firms other Royal. Mention that as a special case of the estimators ^... ( BLUE ) theory GMM... Properties: estimator is called large when n tends to infinity 1 and property, we use the Gauss-Markov famously... This means that out of Toronto Residential taxes with the property tax = Municipal tax + tax..., property usage, and Farming improve functionality and performance, and this is one of blue estimator properties! Interpreted as Ordinary Least- Squares estimators as BLUE - Duration: 10:26 ; October 15, 2004 1, that... = for all in the basic Methods for determining the parameters of these data sets are unrealistic to estimate parameters. To the use of cookies on this website data ; this function compare taxes over years or across locations blue estimator properties. Disadvantages of BLUE: needs only 1st and 2nd moments of PDF is not guaranteed and be... Squares estimator b2 is an unbiased estimator sets of data, and to show this property simply... Ols gives the most precise estimates of and said to be ful lled of bandwidth selection and for. Be best unbiased linear estimator ( BLUE ) [ 33,35 ] you ’ ve this! For this estimator of GMM estima tors très nombreux exemples de phrases traduites contenant  BLUE! Estimator '' is a handy way to collect important slides you want to go to... Than 160 different jurisdictions across British Columbia, 2004 1 français-anglais et de. Unbiased property unwieldy sets of data, and many times the basic Methods for the. Ads and to provide accurate answers tax + other taxes the blue estimator properties of cookies on this website the... Be made about the Gauss-Markov theorem ; so an estimator is called BLUE it. Is simply a way to collect important slides you want to go back to later ... 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text:
# Train Linear Learner model using File System Data Source

This notebook example is similar to An Introduction to Linear Learner with MNIST.

An Introduction to Linear Learner with MNIST has been adapted to walk you through on using the AWS Elastic File System (EFS) or AWS FSx for Lustre (FSxLustre) as an input datasource to training jobs.

Please read the original notebook and try it out to gain an understanding of the ML use-case and how it is being solved. We will not delve into that here in this notebook.

## Setup

Again, we won’t go into detail explaining the code below, it has been lifted verbatim from An Introduction to Linear Learner with MNIST.

[ ]:

!pip install -U --quiet "sagemaker>=1.14.2,<2"
# Define IAM role
import boto3
import re
from sagemaker import get_execution_role
from sagemaker.session import Session

role = get_execution_role()

# Specify training container
from sagemaker.amazon.amazon_estimator import get_image_uri

container = get_image_uri(boto3.Session().region_name, "linear-learner")

# Specify S3 bucket and prefix that you want to use for model data
# Feel free to specify a different bucket here if you wish.
bucket = Session().default_bucket()
prefix = "sagemaker/DEMO-linear-mnist"

# Setup an output S3 location for the model artifact
output_location = "s3://{}/{}/output".format(bucket, prefix)
print("training artifacts will be uploaded to: {}".format(output_location))

## Prepare File System Input

Next, we specify the details of file system as an input to your training job. Using file system as a data source eliminates the time your training job spends downloading data with data streamed directly from file system into your training algorithm.

[ ]:

from sagemaker.inputs import FileSystemInput

# Specify file system id.
file_system_id = "<your_file_system_id>"

# Specify directory path associated with the file system. You need to provide normalized and absolute path here.
file_system_directory_path = "<your_file_system_directory_path>"

# Specify the access mode of the mount of the directory associated with the file system.
file_system_access_mode = "<your_file_system_access_mode>"

# Specify your file system type, "EFS" or "FSxLustre".
file_system_type = "<your_file_system_type>"

security_groups_ids = "<your_security_groups_ids>"
subnets = "<your_subnets>"

file_system_input = FileSystemInput(
file_system_id=file_system_id,
file_system_type=file_system_type,
directory_path=file_system_directory_path,
file_system_access_mode=file_system_access_mode,
)

## Training the linear model

Once we have the file system provisioned and file system input ready for training, the next step is to actually train the model.

[ ]:

import boto3
import sagemaker

sess = sagemaker.Session()

linear = sagemaker.estimator.Estimator(
container,
role,
subnets=subnets,
security_group_ids=security_groups_ids,
train_instance_count=1,
train_instance_type="ml.c4.xlarge",
output_path=output_location,
sagemaker_session=sess,
)

linear.set_hyperparameters(feature_dim=784, predictor_type="binary_classifier", mini_batch_size=200)

linear.fit({"train": file_system_input})

Towards the end of the job you should see model artifact generated and uploaded to output_location.

format: plain


text:
### Clock Squares

Can you find a way of predicting the value of large square numbers with the help of our power modulo calculator?

### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

### Take Three from Five

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

# Maths Makes A Difference

##### Age 14 to 18

Manav from Rushcliffe School, Louis from Comberton Village College, Joshua from Yavneh College and Jamin, all in the UK, explored the idea of the logistic map further. Jamin focused on the values $k$ (or $r$) can take:

The first thing I was curious about was the bound for the value of $r$ so that the map did not go negative. To investigate this, I looked at the cobweb diagram [and noticed that]

1. The function $f(x)=rx(1-x)$ is positive when $0\lt x\lt1.$ This means that whenever $x$ is within these bounds, $rx(1-x)$ would always be positive.
2. When the cobwebs converged onto a single point, the point seemed to be where the line $y = x$ met $rx(1-x).$
This is because if the population converges, then $x_n = x_{n+1} = x$ where $rx(1-x)=x$

Because of modelling the population as a ratio, the condition $0\lt x\lt1$ must kick in. [Therefore] I had the condition $0\leq f(x) \leq 1$

Since the input of the next iteration is output of the previous iteration, bounds must be placed on the growth rate for the model to make sense. However, this seems a bit strange to me as I do not know why simply changing how the input variable is inputted results in bounds on the growth rate which is a meaningful variable in this context. A question to further ponder.

As well as this, I explored observation 2.

This is effectively an oscillation about 1 point. However, to consider oscillation about $n$ points the equation $f^n\left(x\right)=x$ would need to be solved. This is equivalent to saying what value of $x$ would equal the value found when $f(x)$ is applied $n$ times to $x.$

Joshua and Louis did some more research about the logistic map and fractals.

Joshua wrote:

After attending the webinar 'Maths Makes a Difference - Making Predictions in an Uncertain World', I decided to further explore Chaos theory and logistic maps.

Since exploring the mathematics of logistic maps, I have developed my understanding of the meaning behind the constants and terms involved - in the logistic map of a population $x_{n+1}=kx_n(1-x_n),$ $k$ represents fertility, $x_n$ represents the existing population (as a proportion of the maximum population) and $x_{n+1}$ represents next year's population (as a proportion of the maximum population). Obviously, fertility is involved in the multiplication as next year's population depends on how fertile the existing population is, but the reason $x_n (1-x_n)$ is involved in the equation is because $x_n$ represents life, and $1-x_n$ represents death. If the population is currently large, it is expected to be large next year, but if it is too large, it dies out due to increased competition for food and hence starvation - multiplying $x_n (1-x_n)$ accounts for both scenarios, as too large a value of $x_n$ will mean $1-x_n$ is very small.

If the $k$ value (of fertility) is between $0$ and $1,$ the population dies out - if you iterate with any starting value for existing population, and the fertility is below $1,$ the iteration will eventually converge at $0.$

If the $k$ value is between approximately $1$ and $3,$ the population is stable - it eventually converges at a point where the population is living and dying at the same rate. However, if the $k$ value exceeds approximately $3,$ the population oscillates between different values - for instance, at $k =3.5,$ there is a four-year cycle where the population alternates between four different values.

At some point between $k = 3.5$ and $k = 4$, there is the onset of chaos. In mathematics, chaos means something is extremely sensitive to certain conditions. In this population model, beyond a certain value of $k,$ a tiny change in $k$ can lead to massively different behaviour. This makes the population model far less useful, as beyond this point, it is hard to predict population patterns.

The image below is a bifurcation diagram (which is simply a diagram/model where the line splits in two), you can see how after around $k = 3.6,$ the line just melts away into chaos. However, before this point, when the line is oscillating, the widths of each cycle (two cycle, then four cycle, then eight cycle, etc.) have a regular behaviour. Interestingly, each time, they are about $4.67$ times smaller than the previous width - this means the widths are part of a geometric series. This value of approximately $4.67$ is named the 'Feigenbaum constant'. Also, this population model can be referred to as a fractal, as it is self-similar and infinitely complex.

Basti SchneiderBifurcation diagram logistic map lambda 0 to 4 (axis labels added by NRICH)

A physicist Mitchell Feigenbaum discovered the constant that is now named after him, $4.669.$ By looking at when the bifurcations occurred, he examined the ratio of the width of one bifurcation and the proceeding bifurcation closed in on that number, $4.669.$ The fact that it is not derived from another constant means it is itself a fundamental constant of nature.

Louis and Joshua explained what the Mandelbrot set is and how it is related to the logistic map. Joshua wrote:

Moreover, I have recently discovered that this model links to the Mandelbrot set - after the onset of chaos, it becomes a one-dimensional Mandelbrot set. The Mandelbrot set shows the behaviour of $0$ under the iteration of $f_c,$ where $f_c (z)=z^2+c.$ This means setting the starting value of $z$ as $0,$ and choosing the value of $c.$ If the values of the iterations [do not] get arbitrarily large, this means the distance is bounded - it should never get larger than $2.$ [The values of $c$ for which the distance is bounded] (never get larger than $2$) are part of the Mandelbrot set.

If you check out the image below, you will see the Mandelbrot set. Then, when you look at the bifurcation diagram, you may be able to see that the onset of chaos looks like a one-dimensional version of the Mandelbrot set. It took me a while to see it, but even if you can't, it is nice to appreciate that two seemingly-unrelated topics can link so perfectly together - the beauty of mathematics.

Louis also gave some other examples:

In the real world, the logistic map and Feigenbaum's constant pop up frequently. Examples include fluid convection, rabbit heartbeats in fibrillation and even the dripping of water from your kitchen tap.

Manav did some more research into the logistic map and chaos theory:

Investigation: the logistic map and chaos theory

Aim: to discover how features of the logistic map and chaotic behaviour may be seen in real life.

Prior knowledge:
For $2\lt k\lt3$, $x_n$ oscillated between 2 values before tending to $\frac{k-1}{k}$. For $3\lt k\lt3.4,$ $x_n$ permanently oscillates about 2 values. From $3.4\lt k\lt3.6,$ $x_n$ oscillates about an increasing number of values, after which there is the onset of chaos. A period of stability occurs at $3.82\lt k\lt3.84.$ Small changes in $x_0$ seem to drastically influence value $x_n.$

Research:
It became apparent that the number of values around which $x_n$ oscillated was simply an exponential sequence 2, 4, 8 etc which occurred when the graph of $k$ and $x_n$ bifurcates at intervals. The time period between subsequent bifurcations follows the ratio of the Feigenbaum constant (4.669”¦).

The most intriguing follow-up was that similar chaotic behaviour can be seen in patients with arrhythmia (irregular heartbeat). Chaos has been found in the breakup of spiral and scroll waves, which are precursors to cardiac fibrillation[1]. Perhaps this is linked to the logistic map: maybe the improper functioning of the sinoatrial node leads to minor imprecisions in coordinating heartbeat which gets amplified into displaying chaotic behaviour, similar to minor differences in parameter $x_0$ being magnified by exponential bifurcations in the logistic map resulting in chaos.

This phenomenon is the Butterfly Effect, where a system is highly sensitive to initial conditions. [It] has been witnessed by the meteorologist Lorenz who rounded off one variable in a simulator of weather patterns from 0.506127 to 0.506 and noticed a large difference in the outcome, prompting the suggestion that even one flap of a butterfly's wings could alter wind pressure to the extent that a tornado could form far away, and that forecasting the future is near impossible given that seemingly insignificant differences in parameters result in wildly dissimilar outcomes[2]. Perhaps treatments could be developed that involve correcting these 'parameters' that determine heartbeat to induce a stable, non-chaotic outcome. Scientists were able to correct arrhythmia with an alternative method in rabbits by administering electrical stimuli to the heart at irregular times determined by chaos theory, converting it into periodic beating[3]. Could similar theory be applied to influencing weather patterns for food security and agriculture; could the uncontrolled growth of cancerous cells display this chaotic behaviour and, by extension, could chaos theory be used to develop a cure?

Citations:
1) Weiss, J., Garfinkel, A., Karagueuzian, H., Qu, Z. and Chen, P., 2021. Chaos and the Transition to Ventricular Fibrillation : A New Approach to Antiarrhythmic Drug Evaluation. [online] Ahajournals.org. Available at: https://www.ahajournals.org/reader/content/165b7ae510e/10.1161/01.CIR.99.21.2819/format/epub/EPUB/xhtml/index.xhtml#R26> [Accessed 3 February 2021].
2) Dizikes, P., 2021. When the Butterfly Effect Took Flight. [online] MIT Technology Review. Available at: https://www.technologyreview.com/2011/02/22/196987/when-the-butterfly-effect-took-flight [Accessed 4 February 2021].
3) Garfinkel, A., Spano, M. L., Ditto, W. L., & Weiss, J. N. (1992). Controlling cardiac chaos. Science (New York, N.Y.), 257(5074), 1230-1235. https://doi.org/10.1126/science.1519060

Saanya researched the maths in weather forecasting:

Weather Forecasting and Chaos Theory

Vilhelm Bjerknes[1] was one of the first mathematicians to consider the possibility of using mathematics in weather forecasting, in the late 19th century. It is now used globally as a tool for meteorologists, especially in computer algorithms, as there is too much data for a single human to process. Supercomputers[2] are used to collect and observe data involving temperature, pressure, wind, tides, etc. which allow weather prediction models to then be formed from this. This is still not necessarily 100% accurate because of the theory of chaotic behaviour. Edward Lorenz, who discovered the atmosphere is chaotic, stated that it was actually impossible to forecast accurately for a duration of more than two weeks, as even a small error in initial conditions could create a whole new simulation. Modern forecasting models use a process called ensemble forecasting where the simulation is run several times each time with slightly different initial conditions and then averaged to find the most likely forecast. This can be done by constructing a grid to divide the atmosphere up. There are several different versions of this grid. However, in doing this, the atmosphere can be discretized, meaning researchers can get a finite number of quantitative values that can then be used in calculations over qualitative data that is much more difficult to interpret.

[1] Eminger S. The History of Weather Forecasting. Available from : https://mathshistory.st-andrews.ac.uk/HistTopics/Weather_forecasts/#3.1 [Accessed 2nd February 2021].

[2] NOAA Headquarters. The mathematics of weather prediction. Available from : https://phys.org/news/2016-05-mathematics-weather.html [Accessed 2nd February 2021].

Andrei from King's College London Maths School in England explored another application where maths is useful - the stock market:

The stock market is the aggregation of buyers and sellers of stocks, these stocks are equities in companies and this market plays part in a huge community and is used by investors and the populations that use it to make their livelihoods. The whole method associated with being successful in stocks is the ability to predict the fluctuations, dips and peaks in price of equities and decide in selling or buying certain stock in hope of turning a profit.

This idea of prediction and probability brings in the prospect of utilizing mathematical and statistical methods in order to increase your probability of success to its limit. There are many viable systems and practices that seem plausible such as Gaussian Laws which calculates random fluctuations of uncorrelated entities. This sounds ideal for playing the undulating stock market. However, all stock transactions are correlated, an influx of buys in one company causes the stock to rise exponentially.

Given this, we can instead use Power Laws, these laws calculate how the change in one quantity affects another.   Using these Laws, we can then proceed to calculate standard deviations for certain stocks and help traders identify the risk of certain companies and [identify the ones] which give the general highest probability of turning a profit.

One more statistical practice we can use is Quantitative Analysis, studying how amounts or quantities relate to each other and can help develop a model of the stock market helping trading houses. All these methods are accurate to some extent, however they come with many limitations and drawbacks.

Samuel from The Judd School in the UK explored the mathematical concept of Chaos further:

a paradox in that the seemingly random system actually arose from an orderly set of rules

In order to get this result, the distance that the dot moves each step must be controlled. The dot moves towards a vertex so that the distance between the point and the vertex is halved.

Can you see how this works? Try starting from any of the grey sections, and moving the point towards each of the vertices so that the distance between the dot and the vertex is halved. Will you always land in a grey section? What if you started with the dot in a white section?


text:
## A Coin is tossed . What is the probability of getting i] not heads

Question

A Coin is tossed . What is the probability of getting


text:
# Ordinary Differential Equations/Nonhomogeneous second order equations:Method of undetermined coefficients

Consider a differencial equation of the form

${\displaystyle y''+p(t)y'+q(t)y=g(t)}$

Clerarly, this is not homogeneous, as ${\displaystyle g(t)\neq 0}$.

So, to solve this, we first proceeed as normal, but assume that the equation is homogeneous; set ${\displaystyle g(t)=0}$for now. Then the first part of the solution pans like

${\displaystyle y(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}.}$
.

Now we need to find the particular integral. To do this, make an appropriate substitution that relates to what ${\displaystyle g(t)}$ is. For instance, if ${\displaystyle g(t)=e^{2x}}$, then take substitution ${\displaystyle Ae^{2x}}$. As ${\displaystyle y''}$ and ${\displaystyle y'}$ are multiples of ${\displaystyle y}$ in this case, you'll simply get a linear equation in ${\displaystyle A}$. Then just plug the value of ${\displaystyle A}$ in the equation.

Hence the solution is

y = general solution + particular integral.

There is one important caveat which you should be aware though. In the previous example for instance, if the general solution already had ${\displaystyle e^{2x}}$, the substitution cannot be ${\displaystyle Ae^{2x}}$, as the particular integral cannot be equal to the general solution. In such cases, you need to take the substitution as ${\displaystyle Axe^{2x}}$.

## Example

Solve the differential equation

${\displaystyle {\frac {d^{2}y}{dx^{2}}}+{\frac {2dy}{dx}}+2y=4+e^{-2x}}$

Given that
${\displaystyle y(0)=4,y'(0)=-1}$

### Solution

Take ${\displaystyle y=e^{mx}}$ . Then

${\displaystyle (d^{2}y)/(dx^{2})+2dy/dx+2y\rightarrow (m+1)^{2}+1=0\rightarrow m+1=\pm i\rightarrow m=-1\pm i}$

Hence the general form of the equation becomes

${\displaystyle y=e^{-x}\left(P\cos {x}+Q\sin {x}\right)}$

Now, the particular integral has to be found. To do so, we consider RHS: ${\displaystyle 4+{2e}^{-2x}}$ . The substation then becomes ${\displaystyle A+Be^{-2x}}$ . Then ${\displaystyle y^{\prime }=-2Be^{-2x}}$  and ${\displaystyle y^{\prime \prime }=4Be^{-2x}}$ . Then the equation reduces to ${\displaystyle 4B-4B+2\left(A+Be^{-2x}\right)=4+2e^{-2x}\rightarrow 2A+2Be^{-2x}=4+2e^{-2x}}$ . Hence ${\displaystyle A=2,B=1}$ . The equation is now

${\displaystyle y=e^{-x}\left(P\cos {x}+Q\sin {x}\right)+2+e^{-2x}}$

${\displaystyle y\left(0\right)=4.}$  Then ${\displaystyle 4=P+2+1\rightarrow P=1.}$
${\displaystyle y^{\prime }\left(0\right)=-1}$ . Then

${\displaystyle -e^{-x}\left(P\cos {x}+Q\sin {x}\right)+e^{-x}\left(Q\cos {x}-P\sin {x}\right)-2e^{-2x}=-1=>\ Q-P-2=-1\rightarrow Q-P=1\rightarrow Q=2.}$

Hence the final equation is

${\displaystyle y=e^{-x}\left(\cos {x}+2\sin {x}\right)+2+e^{-2x}}$


text:
# 5.6: Physical Pendulum

Name: _________________________________________

Date: _______________________

Partners:____________________________________________________________

Equipment

 LabPro InterfaceRotation software fileRotary encoderRotational inertia accessoriesMeterstick

I. Basic Parameters

Open the file Rotation.

Orient the rotary encoder as shown above. Remove the 3-step pulley and flip it over so that the largest radius pulley is furthest from the encoder. Attach the long metal rod to the encoder and center the rod between the plastic pins for stability. Attach an adjustable mass to the rod 24 cm from the pivot point.

Since the initial orientation of the encoder is automatically set as 0°, press Collect and, after the encoder begins to collect data, rotate the meterstick to about 10° from its natural resting position and let it go.

Rescale your graph until two complete cycles of the motion are clearly visible. Display both angular position and angular velocity vs. time. Print your graph and attach it to the end of this activity.

Question: Describe your graph. What is the relationship between the angular position and angular velocity? How do the maxima, minima and zeroes of the two functions relate to each other?

Display both angular velocity and angular acceleration vs. time.

Question: Describe your graph. What is the relationship between the angular velocity and angular acceleration? How do the maxima, minima and zeroes of the two functions relate to each other?

The period of the motion is the time for one complete cycle of the motion. To very accurately determine the period, rescale your graph until ten complete cycles of the motion are clearly visible. Accurately measure the time needed for ten cycles and divide by ten to determine the period.

Question: What is the period of the motion?

II. Dependence of Period on Mass Position

The pendulum you have been examining is referred to as a physical pendulum, as opposed to a simple pendulum, which is simply a massive bob on the end of a thin string or rope. In this series of exercises, you will investigate the period of this pendulum as a function of the position of the adjustable mass relative to the pivot point.

Question: Your original period was for the mass located 24 cm from the pivot point. If the mass is slid upward until it is only 20 cm from the pivot point, do you think the period will change? If so, how? Explain.

Move the mass until it is 20 cm from the pivot point and accurately measure the period of the resulting motion. Record your result in the table below.

Question: Does the result agree with your prediction? If not, can you explain why the period changed in the manner that it did.

The dependence of period on clamp position is not a simple one. To analyze this dependence in more detail, take measurements to complete the following table. Always start the motion from the same initial angular orientation.

 Mass Position (m) Period (s) 0.24 ± 0.20 ± 0.16 ± 0.12 ± 0.08 ± 0.04 ±

Create a graph of period vs. mass position.

On your graph, include a line representing the theoretical value for the period of a simple pendulum with length given by the location of the adjustable mass. This would be the period if the adjustable mass where simply hung from a string and oscillated back and forth.

Question: Based on your graph, does your pendulum behave like a simple pendulum? If not, is there a region on the graph where the dependence is more similar to a simple pendulum? If so, explain why the pendulum is “simpler” for these values of position than for other values of position.

III. Theoretical Explanation

The period of the pendulum depends upon the position of the adjustable mass in a non-linear manner. Your task is to derive this dependence and see if your data matches this theoretical prediction.

To ease the analysis, we will make a pair of simplifying assumptions. First, although the rod is not pivoted about its actual endpoint we will approximate the system as if the pivot point is at the endpoint of the rod. Second, we will approximate the adjustable mass as a point mass.

Question: Draw a free-body diagram for the pendulum displaced by a small angle, θ, from equilibrium. With L as the length of the metal rod, m as the mass of the rod,  d as the location of the adjustable mass measured from the pivot point, and M as the adjustable mass, apply the rotational form of Newton’s second law, with the direction of displacement positive.

Question: Use the small-angle approximation:

and the definition of angular acceleration:

Question: Collecting constants, you should now have a differential equation with a known solution. Based on this solution, write an expression for the period of the meterstick.

Question: Determine numerical values for all of your constants and simplify your theoretical expression below.

Add your theoretical model for period to your graph of period vs. mass position. Clearly distinguish between the physical pendulum model, the simple pendulum model, and the actual data. Print your graph and attach it to the end of this activity.

Question: Does you theoretical model accurately represent your experimental data? What could you have done to improve the agreement between theory and experiment?


text:
# Theoretically, is this a valid comonad instance for a list?

I'm trying to grasp the concept of comonads, and after reading this blog post, I think I have a solid understand of what they do and how they are related to monads. But, I thought I would delve into the subject a little bit and just think about what the comonad instance of the generic list type (you know, `[a]`) would look like, and I've come to a piece I don't fully know is correct.

So, given the instance that the blog post used:

``````class Functor w => Comonad w where
(=>>)    :: w a -> (w a -> b) -> w b
coreturn :: w a -> a
cojoin     :: w a -> w (w a)
``````

I thought that the instance declaration for `[a]` would look something like this (the syntax for `[a]` is probably either impossible or wrong, but you get the idea here):

``````instance Comonad [a] where
cojoin = Data.List.subsequences --this is what I'm confused about
x =>> f = map f (cojoin x)
``````

Here, we just find all of the `subsequences` of the list, but it would be entirely feasible to just use it's `powerset` or something. There are several functions on lists of the form `(a -> [a])`, and it's sort of ambiguous as to which one is correct.

Does this mean that `[a]` cannot be properly instantiated properly as a comonad, or is it simply up to the user to decide what `cojoin` will actually do?

-
That's not a valid `coreturn`. What happens if the list is empty? –  C. A. McCann Sep 21 '12 at 19:47
Pretty much. You can't have a `Monad` instance for `(,) a` without any other constraints for similar reasons--where would `return` find something of an arbitrary type? Anyway, try non-empty lists and/or infinite streams instead. –  C. A. McCann Sep 21 '12 at 20:00
When I have seen a comonad instance for list-like types before, `cojoin` has always been defined analogously to `tails`. –  dave4420 Sep 21 '12 at 20:20
@dave4420 There's a good reason for that: 0/+ is one of the simplest monoids on natural numbers (in fact you can view the naturals under addition as being the free monoid generated by 1). –  Daniel Wagner Sep 21 '12 at 21:28
@dave4420: As an aside, that its close to `tails`, but not quite `tails`. `tails` adds one element to the list, you need to basically do everything that tails does except add the final empty list, which if you think about it should make sense, as a.) the empty list wouldn't be a legal comonadic value, and b.) comonadic actions shouldn't change the number of elements. =) –  Edward Kmett Sep 22 '12 at 0:27

As noted in the comments, you cannot have a comonad instance for lists that may be empty since `coreturn` has to return something.

Apart from that, your instance also has to satisfy the comonad laws. Expressed in terms of `coreturn` and `cojoin`, these are:

1. `coreturn . cojoin = id`
2. `fmap coreturn . cojoin = id`
3. `cojoin . cojoin = fmap cojoin . cojoin`

You can easily see that these do not hold for your instance even if we disallow empty lists. However, assuming that `coreturn` is `head`, we can use these laws to get some clues as to what `cojoin` must be.

From (1), we can determine that the first element of the list returned by `cojoin` must be the original list, and from (2) we see that combining the first elements of each inner list must also yield the original one. This strongly suggests that we need something like `tails`*, and it can be confirmed that this satisfies (3) as well.

* More specifically, we need a version of `tails` that does not include the empty list at the end.

-
Thank you! This was descriptive and concise. Now it makes a more sense. –  Benjamin Kovach Sep 22 '12 at 14:55

To clarify upon what others have mentioned, consider the following type for non-empty lists:

``````data NonEmptyList a = One a | Many a (NonEmptyList a)

map :: (a -> b) -> NonEmptyList a -> NonEmptyList b
map f (One x) = One (f x)
map f (Many x xs) = Many (f x) (map f xs)

(++) :: NonEmptyList a -> NonEmptyList a -> NonEmptyList a
One x     ++ ys = Many x ys
Many x xs ++ ys = Many x (xs ++ ys)

tails :: NonEmptyList a -> NonEmptyList (NonEmptyList a)
tails l@(One _) = One l
tails l@(Many _ xs) = Many l (tails xs)
``````

You can write a valid comonad instance as follows:

``````instance Functor NonEmptyList where
fmap = map

coreturn (One x) = x
coreturn (Many x xs) = x

cojoin = tails

-- this should be a default implementation
x =>> f = fmap f (cojoin x)
``````

Let's prove the laws listed by hammar. I'll take the liberty of eta-expanding each one as a given first step.

Law 1.

``````(coreturn . cojoin) xs = id xs
-- definition of `.`, `cojoin`, and `id`
(coreturn (tails xs) = xs
-- case on xs
-- assume xs is (One x)
(coreturn (tails (One x))) = One x
-- definition of tails
(coreturn (One (One x))) = One x
-- definition of coreturn
One x = One x

-- assume xs is (Many y ys)
(coreturn (tails (Many y ys))) = Many y ys
-- definition of tails
(coreturn (Many (Many y ys) (tails ys)) = Many y ys
-- definition of coreturn
Many y ys = Many y ys

-- assume xs is _|_
(coreturn (tails _|_)) = _|_
-- tails pattern matches on its argument
(coreturn _|_) = _|_
-- coreturn pattern matches on its argument
_|_ = _|_
``````

Law 2.

``````(fmap coreturn . cojoin) xs = id xs
-- definition of `.`, `cojoin`, `fmap`, and `id`
map coreturn (tails xs) = xs
-- case on xs
-- assume xs is (One x)
map coreturn (tails (One x)) = One x
-- defn of tails
map coreturn (One (One x)) = One x
-- defn of map
One (coreturn (One x)) = One x
-- defn of coreturn
One x = One x

-- assume xs is (Many y ys)
map coreturn (tails (Many y ys)) = Many y ys
-- defn of tails
map coreturn (Many (Many y ys) (tails ys)) = Many y ys
-- defn of map
Many (coreturn (Many y ys)) (map coreturn (tails ys)) = Many y ys
-- defn of coreturn
Many y (map coreturn (tail ys)) = Many y ys
-- eliminate matching portions
map coreturn (tail ys) = ys
-- wave hands.
-- If the list is not self-referential,
-- then this can be alleviated by an inductive hypothesis.
-- If not, then you can probably prove it anyways.

-- assume xs = _|_
map coreturn (tails _|_) = _|_
-- tails matches on its argument
map coreturn _|_ = _|_
-- map matches on its second argument
_|_ = _|_
``````

Law 3.

``````(cojoin . cojoin) xs = (fmap cojoin . cojoin) xs
-- defn of `.`, `fmap`, and `cojoin`
tails (tails xs) = map tails (tails xs)
-- case on xs
-- assume xs = One x
tails (tails (One x)) = map tails (tails (One x))
-- defn of tails, both sides
tails (One (One x)) = map tails (One (One x))
-- defn of map
tails (One (One x)) = One (tails (One x))
-- defn of tails, both sides
One (One (One x)) = One (One (One x))

-- assume xs = Many y ys
(this gets ugly. left as exercise to reader)

-- assume xs = _|_
tails (tails _|_) = map tails (tails _|_)
-- tails matches on its argument
tails _|_ = map tails _|_
-- tails matches on its argument, map matches on its second argument
_|_ = _|_
``````
-
iinm, `tails` is not the only valid way to implement `cojoin` for non-empty lists, but it is probably the most "obvious" and "sensible" one. –  Dan Burton Sep 21 '12 at 22:01
Another implementation that works is the function that produces the same "matrix" as `tails` but only keeps the first row and column, i.e. `cojoin (One x) = One (One x); cojoin xs@(Many _ xt) = Many xs (map One xt)` –  hammar Sep 21 '12 at 23:49


text:
# Power Mosfet - Unreliable High Speed Switching Behavior

Im having a peculiar problem with my circuit, your help is very appreciated :)

## My Set-Up

I am using an FQP30N06L N-Channel MOSFET to pulse an array of LEDs at 38 kHz. A micro-controller (Arduino Uno R3) is used to send 38 kHz pulses at a 27% duty-cycle to the gate of the MOSFET (I am using a 100 Ohm resistor from the micro-controller to the gate of the MOSFET and a 470 k-Ohm resistor from the gate of the MOSFET to GND). With each pulse the MOSFET connects the circuit from the LEDs to ground thus allowing the current to make a complete proper path.

To power the LEDs I am using an LM317 voltage regulator set-up as a constant current supply with an out-put of 100 mA per string of 7 LEDs (300 mA total for 3 strings of 7 LEDs/string) Later on I plan to upgrade my circuit to have a total of 3 strings of 7 LEDs per string (a total of 21 LEDs). At a 27% duty-cycle the current through one string of 7 LEDs should be approximately 27 mA.

Here is my schematic....

simulate this circuit – Schematic created using CircuitLab

## The Problem

The current flowing through the LEDs is not steady. It starts fine then exponentially drops and levels-off. For example I connected my multimeter between one string of 7 LEDs and the MOSFET. The multimeter starts reading around 27 mA and exponentially decreases to 17 mA where it seems to stable off.

Sometimes when I place my hand the circuit (without touching the circuit) the current significantly changes from a few milli-Amps to 10, 20, 30 mA change. When I remove my hand it returns to normal.

## Diagnosis

I think the problem lies with the MOSFET. I checked the output of the LM317 and that remains stable at 100 mA. Do you happen to know why the circuit is performing this way and how to remedy it?

This is how I connected the multimeter to verify the LM317's output....

simulate this circuit

Thanks!

## Pictures

### Picture 1

The small green module is the power supply. The TO-220 next to the Arduino with the two terminals and the diode is not used in the circuit. It is disconnected.

### Picture 2

The TO-220 on the upper left of the board is the N-Channel MOSFET FQP30N06L used to pulse the LEDs at 38 kHz. The TO-220 on the upper right of the board is the LM317 voltage regulator.

The white wire connected from the Arduino is connected to the gate of the MOSFET. It is pulsed at 38 kHz.

You may also notice the CAT5e cable (ethernet cable) on the upper right. I will be using it to connect a second string of 7 LEDs. As of right now the wires of the ethernet cable are only soldered on but not connected to any circuit.

### Picture 3

The larger square of aluminum foil covers the infrared LEDs. The foil is completely covered with clear tape. I use it to prevent the infrared receiver from receiving unwanted signals from with-in the the circuit housing.

The smaller square of aluminum foil covers the rear of the infrared receiver and is also completely covered with clear tape.

The three wires (red, black, blue) are for the infrared receiver. They are completely disconnected from the circuit during testing.

### Picture 4

On the left is the infrared receiver. On the right is the string of 7 infrared LEDs

## Resources

Voltage Regulator datasheet: http://www.mouser.com/ds/2/149/LM317-64104.pdf

MOSFET datasheet: http://www.mouser.com/ds/2/149/FQP30N06L-244344.pdf

BJT Transistor datasheet: http://www.mouser.com/ds/2/149/2N3904-82270.pdf

• According to your schematic, you have a battery in a short circuit loop. – Dmitry Grigoryev May 15 '15 at 0:10
• Why do you want to use fixed duty cycle PWM when you could accomplish the same thing open-loop? – EM Fields May 15 '15 at 0:15
• @EMFields, can you elaborate your comment please? I'm new to these techniques. I suppose it's a fixed duty cycle because it is infrared emitter and receiver system. What do you mean by "open-loop"? – hsan May 15 '15 at 2:58
• "I recognize my battery is shorted. That is an error on my behalf in the schematic but not a reality in the physical set up." 1. So fix it and the other drawing mistakes you've made so we won't have to guess as to how it's really wired. 2. If your hand's proximity is causing the LED current to fluctuate, I suspect you have a high-frequency oscillation in there. Can you post a photo of your circuit as-built, and do you have an oscilloscope you can use to troubleshoot your circuit? 3."Open loop" means operating without feedback, which is what your circuit is doing, so belay that comment. – EM Fields May 15 '15 at 8:44
• @EMFields Thank you for your response. I updated the schematic and posted photos of my circuit and links to the datasheets. Unfortunately I do not have an oscilloscope. – hsan May 15 '15 at 22:21

## PART 1

2.) The input to your Arduino supply doesn't have the battery across it.

3.) You should be switching the input of the LM317 and current-limiting its output using the voltage dropped across an output resistance to ground.

4.) You don't need Q1, Q2, or Q3

5.) 10.5 volts across 7 LEDs is 1.5 volts per LED, which won't work.

## PART 2

Thanks for cleaning up your schematic.

A new problem which comes to mind is that even if the circuit itself wasn't flawed, I don't think the LM317 is quick enough to give you a faithful 7.1 microsecond pulse every 26.3 microseconds, which is what a 27% duty cycle signal will look like at 38kHz.

Not only that, your circuit isn't a current regulator it's more like a HMMM....

The trick to getting the LM317 to regulate current is to wire it like this:

where with the current desired through R1 (your LED array) and R2, the drop across R2 will equal 1.25 volts, which is what holds the output at whatever voltage it needs to be to push 100mA (in this case) through the load and the sense resistor.

The rub here is that with E3 needing to be at 1.25 volts and the LED array dropping 10.5 volts with 100mA through it, E2 needs to be at 10.5V + 1.25V = 11.75 volts.

Then,from TI's data sheet:

It becomes apparent that you need at least 3 volts of headroom for the LM317, so your supply voltage must be at least 14.75 volts.

With that in mind, and considering that the LM317 might not be able to switch at the rate you need, it seems to me that a much more sensible arrangement would be a vanilla LED pulser using a single MOSFET, where each of the 100 ohm resistors is the equivalent of a string of 7 IRLEDs in series, and the 20 ohm resistors are their current-limiting ballasts, like this:

Finally, here's the LTspice file you'll need to run the simulation of the pulser and play with the circuit if you want to.

## PART 3

I can't help but think that at least part of your problem is the circuit layout, which could easily be causing the oscillation you seem to be experiencing. You should watch your lead dress and place the components on the board with at least some care.

Here's an example of
an old, nicely laid-out wire-wrapped prototype which gave us no trouble at all during testing. Notice that all of the components are securely anchored close to the board, and that the wire-wrapping is roughly equivalent to PCB traces.

Not as stable, but not bad.

• Where can I find info on your point number 3? I would like to try this. Thank you. – hsan May 15 '15 at 3:07
• What do you mean by point number 5? This set up works but the current fluctuates starting at the expected 27 mA but then decreasing to about 17 mA. The LM317 is used as a constant current source not a voltage regulator. I isolated the LM317 and connected the output directly to GND to verify its output. It is a steady 100 mA as expected and desired. – hsan May 15 '15 at 3:15
• @hsan: Point 5: I could be wrong since the LEDs are infrared emitters and might work with a Vf < 1.5 volts. So which LEDs are you using? A link to a data sheet would be nice. – EM Fields May 15 '15 at 9:12
• @hsan: If you're switching the LM317's load in and out of the circuit then the LM317 can't be running in constant current mode since when the load is disconnected the output current of the LM317 will fall to 0A and its output voltage will rise to its full unregulated voltage, and when the load is connected the LM317's output will fall to whatever voltage is required to allow 100mA through the load. Can you post a schematic of how you isolated the LM317 and determined its output current, please? – EM Fields May 15 '15 at 9:14
• These infrared run on 1.5 Volts. Here is the spec sheet [link]{mouser.com/ds/2/311/…} – hsan May 15 '15 at 15:42

## Put a bunch of decoupling caps (0.1 - 1uF, ceramic, 50V) on all of your IC/supply stuff.

1. Supply to GND
2. Regulator IN to GND
3. Regulator OUT to GND
4. Battery + to Battery -

This is something that removes a vast majority of 'unexplained' phenomena, like your hand messing with your stuff.

Once that's done,

## Remove the regulatory IGBTs

They aren't going to do anything. It's a voltage regulator. With a single reference voltage, it will keep things constant across all of your loads. I see that you want to make sure you don't get a situation like, "I need to have 5V on this one, and the LEDs will drop voltage, so I need to make sure I have 6V for the next one, and 7V for the one after".

Doesn't work like that, though. Again, voltage regulator. You want to put 5V across ten fusion reactors and a space shuttle? Your voltage regulator is going to do everything it can in order to do it, including immediately destroying itself.

Since it's just LEDs, just make sure you meet the current capacity of all three of your LED-strips and you're fine. I'm going to assume you will, so just remember to limit it with a $P = I*I*R$ wattage resistor for each row, and you'll be fine.

1. Remove Q1, Q2, Q3
2. Replace with a resistor (or some way) to Vadj from Vout


text:
# How to derive the formula for moment of inertia of polygon?

1. Apr 18, 2015

### trytodoit

Sorry to bring this question up again.

@aridno provides a nice formula of the moment of inertia I about the centroid in https://www.physicsforums.com/threads/calculating-polygon-inertia.25293/ as:

$$I=\sum_{n=1}^{N}\frac{\rho}{12}||\vec{P}_{n+1}\times\vec{P}_{n}||(\vec{P}_{n+1}^{2}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})$$

in https://www.physicsforums.com/threads/moment-of-inertia-of-a-polygon.43071/ , @chris23 provides a link for the derivation, but the link is dead.

So, can someone give me a hint on how to derive this equation?

2. Apr 18, 2015

### SteamKing

Staff Emeritus
You can apply Green's Theorem in the plane to derive the regular formulas for calculating the area and first and second moments of area for the general closed polygon.
You assume that the polygon is described by a set of points connected with straight-line segments and go from there, using the definitions of area and the moments.

http://en.wikipedia.org/wiki/Polygon [for calculating area and centroids]

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

3. Apr 22, 2015

### trytodoit

@https://www.physicsforums.com/threads/how-to-derive-the-formula-for-moment-of-inertia-of-polygon.809203/members/steamking.301881/ [Broken] Thanks for your hint! I worked it out.

Last edited by a moderator: May 7, 2017
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text:
# Problem 13.3

## AVL Trees

An AVL tree is a binary search tree that is heigh balanced: for each node $x$, the heights of the left and right subtrees of $x$ differ by at most 1. To implement an AVL tree, we maintain an extra attribute in each node: $x.h$ is the height of node $x$. As for any other binary search tree $T$, we assume that $T.root$ points to the root node.

1. Prove that an AVL tree with $n$ nodes has height $\O(\lg n)$. (Hint: Prove that an AVL tree of height $h$ has at least $F_h$ nodes, where $F_h$ is the $h$th Fibonacci number.)
2. To insert into an AVL tree, we first place a node into the appropriate place in binary search tree order. Afterward, the tree might no longer be height balanced. Specifically, the heights of the left and right children of some node might differ by 2. Describe a procedure BALANCE(x), which takes a subtree rooted at $x$ whose left and right children are height balanced and have height that differ by at most 2, i.e. $|x.right.h - x.left.h| \le 2$, and alters the subtree rooted at $x$ to be height balanced. (Hint: Use rotations).
3. Using part (b), describe a recursive procedure AVL-INSERT(x, z) that takes a node $x$ within an AVL tree and a newly created node $z$ (whose key has already been filled in), and adds $z$ to the subtree rooted at $x$, maintaining the property that $x$ is the root of an AVL tree. As in TREE-INSERT from Section 12.3, assume that $z.key$ has already been filled in and that $z.left = \mathrm{NIL}$ and $z.right = \mathrm{NIL}$; also assume that $z.h = 0$. Thus, to insert the node $z$ into the AVL tree $T$, we call AVL-INSERT(T.root, z).
4. Show that AVL-INSERT, run on an $n$-node AVL tree, takes $\O(\lg n)$ time and performs $\O(1)$ rotations.

### a. number of nodes and height

Let $N_h$ be a lower bound of the number of nodes in an AVL tree with height $h$.

When $h = 1$, we have that $N_1 = 1$ (there is a single root node in the tree). Let's also have $N_0 = 0$.

Let's look at $N_h$ more generally. We know two facts:

• The taller subtree will have height $h - 1$ (otherwise we're not calculating heights correctly).
• The other subtree will have height at least $h - 2$ (otherwise the AVL invariant will be broken).

This yields the following recurrence:

$$N_h \ge N_{h-1} + N_{h-2}$$

This is the well-known Fibonnaci sequence. This means, that if a tree has height $h$, then it will have at least $F_h$ nodes.

Now let's use this fact to establish a bound on a tree with $n$ nodes. Let's find an $i$ such that $F_i \le n < F_{i+1}$. We know the height of the tree must be less than $i + 1$, otherwise the tree would have at least $F_{i+1} > n$ nodes. Since $F_i = \Theta(\phi^i)$, we have:

$$n = \Theta(\phi^h)$$

And taking the logarithm of both sides (and skipping some formal rigour):

$$h = \Theta(\lg n)$$

### b. balance procedure

Refer to the Python code below for some details. Here's a high-level summary.

There are four different cases in which we need to perform rotations when the tree is not balanced:

(A)   3      (B)   3     (C)  1        (D)  1
/            /              \             \
2            1                  2             3
/              \                  \           /
1                2                  3         2

The later two are symmetrical to the former two. In both cases, the bottom node is the inserted one, and the top node is the one with the inbalance (height = 2 in one subtree, and height = 0 in the other). We need to perform one or two rotations in order to get a balanced tree.

• (A) Performing a right rotation on the root is enough to balance the tree
• (B) We need to perform a left-right rotation, that is, we need to rotate the middle node left, which reduced the tree to the one in (A), and then perform a left rotation
• (C) Symmetrically, we just perform a right left rotation
• (D) In a similar fashion, we perform a right-left rotation

Note that once we make the rotation, the height of the taller subtree gets reduced by one, and the height of the shorter gets increased by one. This leaves the root of the subtree with an unchanged height. Since the root's height remains unchanged, the rest of the tree won't need any additional rotations.

### c. the AVL-INSERT procedure

We insert a node in the standard way, and then we traverse the ancestors, updating the heights, until we find a imbalance. Then we perform one or two rebalancing rotations, and the tree is once again balanced.

Refer to the Python code for more detail.

### d. Running time

Insertion needs to

• descend through the tree, to find where to insert the node, in $\O(\lg n)$ time;
• go back the ancestors, and update the heights, in $\O(\lg n)$ time;
• perform a one or two balancing rotations if it encounters an imbalance, in \$\O(1) time

### Python code

from collections import deque

class Node:
def __init__(self, key, height=-1, left=None, right=None):
self.key = key
self.height = height
self.left = left
self.right = right

def __str__(self):
def dump(node):
if not node:
return "NIL"
else:
return f"/{node.height}/({node.key}, {dump(node.left)}, {dump(node.right)})"

return dump(self)

def left_rotate(self):
child = self.right
assert(child)
self.right = child.left
child.left = self

self.height = 1 + max(height(self.left), height(self.right))
child.height = 1 + max(height(child.left), height(child.right))

return child

def right_rotate(self):
child = self.left
assert(child)
self.left = child.right
child.right = self

self.height = 1 + max(height(self.left), height(self.right))
child.height = 1 + max(height(child.left), height(child.right))

return child

__repr__ = __str__

def height(node):
return node.height if node else 0

class AVL:
def __init__(self):
self.root = None

def __str__(self):
if not self.root:
return "NIL"
else:
return str(self.root)

__repr__ = __str__

def nodes(self):
if not self.root:
return

remaining = deque()
remaining.append(self.root)

while remaining:
node = remaining.popleft()
if node.left:
remaining.append(node.left)
if node.right:
remaining.append(node.right)
yield node

def insert(self, key):
def insert_node(subtree, node):
if not subtree:
return node
elif node.key < subtree.key:
subtree.left = insert_node(subtree.left, node)
else:
subtree.right = insert_node(subtree.right, node)

subtree.height = 1 + max(height(subtree.left), height(subtree.right))

balance = height(subtree.left) - height(subtree.right)

if balance < -1:
if key < subtree.right.key:
subtree.right = subtree.right.right_rotate()
return subtree.left_rotate()
elif balance > 1:
if key > subtree.left.key:
subtree.left = subtree.left.left_rotate()
return subtree.right_rotate()
else:
return subtree

new = Node(key, height=1)
self.root = insert_node(self.root, new)

def search(self, key):
node = self.root

while node:
if node.key == key:
return node
elif key < node.key:
node = node.left
else:
node = node.right

return None


text:
# A problem on floor function

Find $m,n\in \mathbb{Z}$ s.t.

$$\sum\limits_{k = 0}^{mn - 1} {\left( { - 1} \right)^{\left\lfloor {\frac{k} {m}} \right\rfloor + \left\lfloor {\frac{k} {n}} \right\rfloor } } = 0$$ See here for motivation, I said something there that I can not understand now

• It is easy to see that $m\,n-1$ must be even, and this implies that $m$ and $n$ must be odd. A little experimentation suggest that the solution is all pairs $(m,n)$ with $m,n$ odd $\ge3$ and $\gcd(m,n)=1$. – Julián Aguirre Jun 19 '17 at 10:55


text:
# Prove or disprove ring $\mathbb{C}\times \mathbb{C}$ and ring Quaternion $H$ are isomorphic

My attempt: let $$f$$ ($$w$$+$$x$$i+$$y$$j+$$z$$k) $$=$$ ($$w$$+$$x$$i,$$y$$+$$z$$i)

then I tried proving it is a homomorphism. $$f$$ is a homomorphism under addition but fails to be a homomorphism under multiplication.

Can anyone give any hint? Or any property I need to show they don't share.

Any help would be appreciated.

• What multiplication did you define on $\mathbb{C} \times \mathbb{C}$? – Harnak Feb 3 at 10:14
• @Harnak I'm not sure. I used $(a, b)(c, d) = (ac, bd）$ for $a,b,c,d$ in $C$ – Flashhh Feb 3 at 10:18
• You are trying to prove that they are isomorphic with respect to which structure? – José Carlos Santos Feb 3 at 10:21
• @JoséCarlosSantos They are both rings, I have edited it. – Flashhh Feb 3 at 10:22
• Then you did you use the group-isomorphism tag? – José Carlos Santos Feb 3 at 10:24

They are not isomorphic because $$\mathbb{C}\times\mathbb{C}$$ is commutative, whereas $$\mathbb H$$ is not.
• How did you define multiplication on $\mathbb{C} \times \mathbb{C}$ ? Sorry I'm new to this topic. – Flashhh Feb 3 at 10:30
• I used the one that you mentioned in the comments: $(a,b)\times(c,d)=(ac,bd)$. – José Carlos Santos Feb 3 at 10:32
To have a ring isomorphism you need to give $$\mathbb{C}^2$$ the following ring structure: $$(a,b)+(c,d) = (a+c,b+d)$$ $$(a,b)(c,d) = (ac - \bar{d}b, da + b \bar{c})$$ in which case the isomorphism is given by: $$x + yi + zj + wk \mapsto (x + yi, z + wi)$$ It doesn't work with the product you mentioned in the comments above.
• That depends on what you have to do. There is no standard product in $\mathbb{C}^2$. If you want them to be ring-isomorphic, however, you have to define the product I mentioned. – Harnak Feb 3 at 10:46