Showing $\mathbb{Q}(\sqrt{a},\sqrt{b})= \mathbb{Q}(\sqrt{a}+\sqrt{b})$
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
Optimizing a "black box" function: Linear Regression or Bayesian Optimization... what's the difference? Goal: I have a function $f(x,y)=z$ (two variables for illustration only) which I know almost nothing about--it has a compact domain which I can determine, it is non-negative, and bounded above.  My goal is to find the maximum value $f(x,y)$ takes on its domain.  I can query the function and I know the values at a few points: $f(x_1, y_1)=z_1, f(x_2, y_2)=z_2,..., f(x_n, y_n)=z_n$.  Optimization via Linear Regression:  I have used linear regression (and related methods like the lasso) to model functions like this in the past; this is the method that I am very comfortable with.  Typically, I might try to find a polynomial (of degree $k$) which relatively accurately models the data. The end result would be to say that:   $$f(x,y)\approx \beta_1 + \beta_2 x + \beta_3 y + \beta_4 x^2 + \beta_5 xy + \beta_6 y^2 + ... + \beta_{m}y^{k}.$$  Determining the coefficients is done by solving the .  The "best degree" for the polynomial might be determined using $k$-fold cross validation.  Once the "best" polynomial has been determined, one can choose any of a number of methods to optimize it (Newton's method, gradient descent, hill climbing, etc).  Then, if all goes well, the optimal value on the polynomial corresponds to an optimal value of $f(x,y)$.  Bayesian Optimization:  Recently, I was describing this technique (linear regression) to a colleague who asked: "What is the difference between that and Bayesian Optimization?"  I've been attempting to answer this question for the last couple days, and so far I don't have anything really conclusive.  After reading , trying to read  and getting stuck reading  I think I understand the following about BO:   Start with a prior (a guess as to what family of functions the black box belongs to... or at least some family of functions which you believe might be able to describe $f$ well).  It looks using a  as a prior is popular. Specify whatever is needed to build your prior (for GP this is a covariance function). As you go along, you maintain two evolving functions, a "mean" function $m(x,y)$ and a "confidence interval" function $c(x,y)$.  Basically, you can predict the value at any point by evaluating $m(x,y)$ and you are confident that $f(x,y)\in [m(x,y)-c(x,y), m(x,y)+c(x,y)]$. Bayesian optimization provides a systematic way of updating/exploiting the model by alternatively selecting points where the confidence interval is wide (you are very uncertain about the value) or at optimal points on the model $m(x,y)$.   My Question: Is it fair/correct to say that the major difference between Bayesian Optimization and simply optimizing the linear regression model is that Bayesian Optimization provides the systematic way of sampling with the goal of optimizing the model and reducing uncertainty whereas the linear regression model ignores the idea of uncertainty and doesn't explicitly provide a means of updating the model?  Said another way, is the linear regression model an example of what might be a "mean" function $m(x,y)$ in Bayesian Optimization?  If I stick with linear regression, specify how to compute the uncertainty, and provide a means of updating/improving the model (exploit/explore) am I just doing Bayesian Optimization (without calling it that)?  Background: My background is in pure mathematics (graph theory) but my new job has me working with a lot of data (so I'm self learning statistics).  With that in mind, my statistics vocabulary is somewhat limited (so I appreciate answers that either avoid technical terminology or briefly explain any technical terminology used)--for example, Monday was the first time I heard/read the word "prior" used in a stats sense and I'm still not 100% sure I understand that correctly. Doubling the tails in two-sample permutation test Suppose we have two samples and we wish to determine if they are drawn from the same distribution, the samples A,B being composed of some integers say.  If we test this using a two-sample permutation test, specifically by looking at permutations where the differences in the means of the samples are as extreme as the difference observed: is there any reason to think that we can calculate the two-tailed p-value by looking at one tail and doubling the probability?  This is what it appears to say in my lecture notes, but I don't understand why we could assume that the tails are symmetrical (or why it doesn't entail that assumption). Explanations were not forthcoming. Is there a graceful way to handle passing a list of ids as a parameter to a stored procedure?   For instance, I want departments 1, 2, 5, 7, 20 returned by my stored procedure. In the past, I have passed in a comma delimited list of ids, like the below code, but feel really dirty doing it.   SQL Server 2005 is my only applicable limitation I think.  create procedure getDepartments   @DepartmentIds varchar(max) as   declare @Sql varchar(max)        select @Sql = 'select [Name] from Department where DepartmentId in (' + @DepartmentIds + ')'   exec(@Sql) About big parenthesis larger than Bigg How do you make the shapefile a Read-only so the client can not make any edits using 10.2.2 ArcGIS.  It is not within a geodatabase, just a stand alone shapefile.  I am using Windows 8 Search All Fields In All Tables For A Specific Value (Oracle) Which Standard Visitor visa should I apply for? Sometimes when reading through user comments on a particular answer, some usernames that appear like user[digit] e.g user12345 are not links. Is this a known behaviour or any other thing I am not aware of? calculating mean of a random variable raised to a power File issues after importing Drupal site I recently tried to move my Drupal site to a new server using this page: . But now the style doesn't seem be loaded and more importantly I get this error on every page in the site (even on the frontpage):  Warning: file_put_contents(temporary://filef76T9k): failed to open stream: &quot;DrupalTemporaryStreamWrapper::stream_open&quot; call failed in file_unmanaged_save_data() (line 1900 of /Library/Server/Web/Data/Sites/Default/drupal/includes/file.inc). The file could not be created.  Any ideas what this could be about? I'd like to call a function in python using a dictionary.  Here is some code:  d = dict(param='test')  def f(param):     print(param)  f(d)   This prints {'param': 'test'} but I'd like it to just print test.  I'd like it to work similarly for more parameters:  d = dict(p1=1, p2=2) def f2(p1, p2):     print(p1, p2) f2(d)   Is this possible? How to add coloured line before caption? Selenium + FF : How to fix error, Unable to connect to host 127.0.0.1 on port 7055 after 45000 ms? Recover harddrive data Why a backslash displays as "W"+strikethrough in Netbeans? I'm facing a strange problem with Netbeans. In the editor, when I type a backslash(), I see a W with a strikethrough (as shown in the attached image).  So the c:\\abc is displaying as:    But, when I copy and paste that string into notepad, it gets pasted normally (i.e. c:\\abc).  Has anyone faced this problem before? How to fix it? How to use a string column with pgfplotstable to label datapoints in a pgfplot? I have a comma-separated list of data points. They should be plotted as a scatter plot, and should be labeled using data from the .csv file. Having specified the first column of my file as string type, I still keep running into errors saying Could not parse input [...] as a floating point number, sorry.  Given a CSV file of the following format, how can I read Name and Z as labels?  Name,X,Y,Z ODIN,250,45,2001   Here is my MWE:  \documentclass{article}  \usepackage{tikz,pgfplots,pgfplotstable}     \pgfplotsset{compat=1.5.1}  \begin{document}  \pgfplotstabletypeset[columns/Name/.style={string type}]{table.csv};  \begin{tikzpicture}     \begin{axis}[     visualization depends on=\thisrow{Name} \as \labela,     visualization depends on=\thisrow{Z} \as \labelb,     every node near coord/.append style={font={\tiny}},     nodes near coords=\labela/\labelb,     nodes near coords align={horizontal},]      \addplot[only marks] table [x=X,y=Y,col sep=comma] {table.csv};      \end{axis} \end{tikzpicture}  \end{document}   Note that if the data in column Name is just floats it should work fine. Database schema documentation for the public data dump and SEDE How can I stand on the ground? EM or/and Pauli? Use of *args and **kwargs Prove that $\sum_{n=2}^\infty (\ln n)^{- \ln n}$ converges As the title suggests, I'd like to prove that the sum $$ \sum_{n=2}^\infty (\ln n)^{- \ln n} $$ is finite.  The root and ratio test both fail here, but  suggests that there is a comparison that can be used to show convergence.  The only thought I have is that it may help to write the terms as $e^{-\ln(n)\ln(\ln(n))}$, but this has not led me to any particular insight.  Any ideas are appreciated.