Exercise:   Suppose that $f$ is continuous on $[a,b]$ and that $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .  Attempt of proof: Suppose that $f$ is continuous on $[a,b]$ and that $F(x) = \sup f([a,x])$. Then since $f$ is continuous on [a,b]. Then by definition  $f$ is continuous at a point $c$ in $(a,b)$ iff for every $ε &gt; 0$ there is a $δ &gt; 0$ such that $|c-x| &lt; δ$  $→$ $|f(c) - f(x)| &lt; ε$.  Then we need to show $F$ is continuous.   Then let $ε &gt; 0$, then there is a $δ&gt;0$ such that $|c-x| &lt; δ$.   Then, $|F(c) - F(x)| = |\sup f([a,c)] - \sup f([a,x)]|$.  Can someone please help me? I am stuck and don't know how to continue. Thank you in advance.
I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out.  Let $f$ be a continuous function on $[a,b] \subset \mathbb{R}$. Define $f^\star (x)$ as:  $$ f^\star(x) = \sup \{f(y)\mid a \leq y \leq x \} $$ Prove that $f^\star$ is a continuous increasing function on $[a,b]$.  Things I've figured out: (these are mostly trivial)   $f$ is uniformly continuous on $[a,b]$. Since $[a,b]$ is closed and bounded there exists some $c \in [a,b]$ such that $f(c) \geq f(x)\ \forall x \in [a,b]$. (In words: the supremum is actually attained.) If $c \in [a,b]$ is as above, then $f^\star$ is constant (and hence continuous) on $[c,b]$ (which is possibly a singleton).   This seems completely obvious when you actually draw a continuous function but translating that to a formal proof eludes me...
Intuition behind measurable random variables and $\sigma$-algebra I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture.  I'll try to explain the counter-intuition I get with the classical example of : the probability space $\Omega = \{ HH, HT, TH, TT \}$ and a r.v. $X(\omega)$ equal to the number of heads.  At times $0$, $1$ and $2$ the available information is represented using $\sigma$-algebras $\mathcal{F}_0=\{\emptyset,\Omega\}$, $\mathcal{F}_1=\{\emptyset, \Omega, \{HH,HT\},\{TH,TT\}\}$ and $\mathcal{F}_2=\{\emptyset, \Omega,\{HH,HT\},\{TH,TT\},\{HH\},\{HT\},\{TH\},\{TT\}\}$.  One can notice that $X(\omega)$ is not measurable with respect to $\mathcal{F}_0$ and $\mathcal{F}_1$, because $X^{-1}((\frac{3}{2}; +\infty))=\{HH\}$. To me it is quite surprising: intuitively $X$ makes perfect sense at all times. In particular it has an expected value at time $0$, which I interpret as that the probability and value of all outcomes $\{\omega\}$ can be computed. How do you think of a non-measurable function?  Here's another way of expressing the same confusion. The most natural choice of $\sigma$-algebra in a finite discrete case is $\mathcal{F}=2^\Omega$, and it is implicitly used in all elementary probability problems. However, this choice of $\mathcal{F}$ does not reflect the fact that some information is known or unknown, conditional probability does. Does it mean that the statement "$\sigma$-algebra is known information" make sense only in conditioning? Why is it convenient then? Can I sell starting gear in Adventurers League play? I'm starting a new AL game (as a player) and I have a question that's not addressed in the AL material.  According to Adventurers League Player's Guide (page 4)     When you create your D&amp;D Adventurers League character for the current season, take starting equipment as determined by your class and background. You cannot roll for your starting wealth.   Meaning I can't start with whatever I want. However, I am playing a ranged-focused fighter, so the second gear option is of no use to me:   (a) a martial weapon and a shield or (b) two  martial weapons   Can I choose any two martial weapons and sell them for half price to use for purchasing other gear before play? And is there a restriction on how much gold I can get?  For example, two hand crossbows (martial ranged weapons) can net me 75 gp when sold, or two greatswords nets me 50 gp when sold. I have a very large codebase (read: thousands of modules) that has code shared across numerous projects that all run on different operating systems with different C++ compilers. Needless to say, maintaining the build process can be quite a chore.   There are several places in the codebase where it would clean up the code substantially if only there were a way to make the pre-processor ignore certain #includes if the file didn't exist in the current folder. Does anyone know a way to achieve that?  Presently, we use an #ifdef around the #include in the shared file, with a second project-specific file that #defines whether or not the #include exists in the project. This works, but it's ugly. People often forget to properly update the definitions when they add or remove files from the project. I've contemplated writing a pre-build tool to keep this file up to date, but if there's a platform-independent way to do this with the preprocessor I'd much rather do it that way instead. Any ideas? Avoiding confusion over the term "Research Assistant" Downgrading from SQL Server 2008 to 2005 I want to show that  $$\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{a + \sin^2(\theta)} = \frac{\pi}{2\sqrt{a(a+1)}}$$  for $a &gt; 0$.  I try several methods   Substitutions to rationalize The famous $U = \tan(\frac{\theta}{2})$ Multiplying by conjugates   and other calculus techniques and still, I can not prove the equality.   Anyone can give me a hint of how should I begin to work this integral?  I'm very ashamed that I can not solve this problem. Finding the limit of $x_1=0 , x_{n+1}=\frac{1}{1+x_n}$ Why are new sites classified as "uncategorized"? Disabling module with config sync and removing with composer I have an issue disabling modules, I've searched but cannot find something useful:  - Developer A disable module and via composer remove it  - I pull his changes so i run composer install that remove the code, then drush cim to update configuration, but it complains that the module is missing while it's trying to delete its configurations.  What's the right way to do this? I can't import config before running composer install because along the uninstalled module there could be also new modules needed for the configuration import. How to install the Pantheon desktop environment? Is there any way to tell the compiler, inside the main file body, how to find the class? I would like to use a class that belongs not found in my MiKTeX database. I know I might as well pick up this class and put it in some directory inside the base and update MiKTeX, but I do not want to do that. What I want is to organize it into a subdirectory within the same directory as my main tex file. However, when compiling the tex file, the class is not found. Of course, if I put the direct class in the same directory as the main file I will solve the problem, but that's not what I want. So, I ask: Is there any way to tell the compiler, inside the main file body, how to find the class?  Note: I'm using MiKTeX as an example, but I'm really having the problem in overleaf. There was a revtex upgrade from version 4.1 to 4.2. However, overleaf does not find revtex4-2.cls, which is, in fact, the class I'd like to use.  Below, I show an example of how I would like to organize the files. Note that the class is in a subdirectory. I am flying from Orange County, California to San Francisco, California and then on to Seoul, South Korea. If I check a bag will I have to pick it up in San Francisco and re-check it and go through security or customs for the international flight?  Everything I am reading leads me to believe that it's only necessary to re-check baggage and go through security and/or customs when you are first arriving into a country. However, since my first leg is staying within the United States I assume the airline will transfer my bag to the next flight and I can just go straight to the gate for my international flight?  Both legs of the flight are through the same airline (United Airlines) and there are only the two legs: OC -> SF, then SF -> Seoul. how to prove that this positive sequence converges to 1 I have a positive infinite sequence $a_n$ and I know that $a_n$$a_{n+1}$ converges to 1 and also $a_n$$a_{n+2}$ converges to 1.  Why does it mean that $a_n$ also converges to 1? How can I prove it?  Thanks. RF Energy Harvesting Why can energy invested in RF radiation not be harvested fully? What are the main factors that affect this? Is there any significant research nowadays that fully describes this process? I stand on the conclusion: "Energy moves from one form to another.", but in this case why can it not be harvested in that quantity to use it for powering low-power devices or store it in significant quantities in some batteries for later use? Why is it not profitable nowadays and why is it not used today like energy from water, wind and other energy sources? I have requested for a French long term (Type D) student visa from a consulate in India. It's a national Type D visa and not a Schengen visa. My question is that can we still go to other Schengen countries on this visa? If not then can we get a Schengen visa from a consulate in France? Why can't a single photon produce an electron-positron pair? In reading through old course material, I found the assignment (my translation):     Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.   My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.  How to solve this task? Can't install jdk on Fedora with yum nor with rpm How to remove indentation in footnote? Categorical Principal Component Analysis - using Count, Continuous, Ordinal variables together I have some variables and I want to reduce their number for further analysis. I initially thought of combining them using factor analysis. But since the variables are of all kinds (rating, count, ordinal, continuous dollar amount) I am thinking about using CATPCA for it. However, I have some questions about the technique -   Is CATPCA the right approach or Latent Class Factor Analysis would be the better one? By using the different kinds of variables together - how will the object scores be interpreted? For example - The rating question is on a 10 point scale but the dollar amount ranges from 0 to 1000s, so what will the object score obtained after CATPCA represent? And is it comparable to the Factor Score obtained from traditional PCA? For ordinal variables - Do all of them need to be in the same form, e.g. 1 meaning lowest and 10 meaning highest? Or can they be used in whatever form they are? Is it good to categorize the count and continuous variables based on their distribution and then treat then as ordinal/categorical variables?   I would highly appreciate any help you can provide! Proximal Operator Fixed Point Property for Matrices $\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ $\newcommand{\dom}{\operatorname{dom}}$  Recall again that the proximal operator for vectors $\prox_{f}: R^n \rightarrow R^n$ of $f$ is defined as:  $\prox_f(v) := \argmin_{x} \left(f(x) +(1/2)\|x-v\|_2^2 \right) $,  where $f: R^n \rightarrow R \cup \infty$ is a closed proper convex function and $\|\cdot \|_2$ is the Euclidean norm. $\prox$ is strongly convex and not everywhere infinite, so there it will have a unique minimizer for every $v \in R^n$  The crucial property of proximal operator is that $x^*$ minimizes $f(x)$ iff $x^* = \prox_f(x^*)$, i.e. $x^*$ is a fixed point of $\prox_f$.  Let us consider the proof of this property. The first part, namely, that if $x^*$ minimizes $f$ then $\prox_f(x^*)=x^*$, is trivial:  $f(x) +(1/2)\|x-x^*\|_2^2 \le f(x^*) = f(x^*) +(1/2)\|x^*-x^*\|_2^2$  The second part uses the notion of subdifferential. In the proof authors say that $\tilde{x}$ minimizes  $f(x) +(1/2)\|x-v\|_2^2$  i.e. $\tilde{x} = \prox_f(v)$ iff $0 \in \partial f(\tilde{x}) + (\tilde{x}-v)$,   where the sum is of a set and a point. Recall the definition of the subdifferential:  $\partial f(x) = \{ y$ |  $ f(z)\le f(x) + y^T(z-x)$ $\forall z \in \dom{f}\}$  Take $\tilde{x}=v=x^*$, it follows that $0 \in \partial f(x^*)$ so $x^*$ minimizes $f$.  Question 1:  Consider proximal operator defined for matrices now:  $\prox_f(Y) := \argmin_{X} \left(f(X) +(1/2)\|X-Y\|^2 \right)$,  where $X$ is some real $m$ by $n$ matrix. What norm function should be taken in the definition of the $\prox$ in this case? Frobenius? Or spectral norm (induced 2nd norm) which is the largest eigenvalue (is spectral norm even differentiable?)? Or something else?  Question 2:   Can the norm for the definition of $\prox$ in case of matrices be chosen in different ways? What conditions does the proof above impose on the norm function?