Constructing an orthogonal basis with a choice of inner product
Orthogonal basis for infinite-dimensional vector spaces
Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you ! Footnote to a footnote How can I make a footnote to a footnote? E.g.,  Playfair's axiom\footnote{so called in honor of mathematician John Playfair, etc, etc., but was actually discovered by Proclus\footnote{Greek mathematician, etc, etc.}} PHP class: Global variable as property in class I am learning basics of the category theory (CT).  I do understand that CT is a modern powerful framework to describe various branches of mathematics in a unified way.  I do admit that category's definition tells a lot about the whole thing and focuses on the most important properties, shadowing unnecessary details. However, the more I go forward, the more it feels like it should explicitly require a notion of morphisms equality (like it does with morphism composition). It feels so because equality or inequality of morphisms arises everywhere, since the very beginning: any commutative diagram eventually reduces to an equality of two different paths and many other things involve equality as well. Even composition is defined grounding upon an (implicit) equality!  I'll try to give an example of my concern. Consider $\mathbb{SET}$. Its objects are sets and morphisms are functions. Now, let's recall that functions are sets as well - to be more precise, a function $f : A \mapsto B$ is a subset of a $A \times B$ obeying certain properties, which I omit. Thus, one could argue that $f$ is both a morphism between objects $A, B \in Obj(\mathbb{SET})$ and an object defined as a set of pairs $\{ (a_1 \in A, b_1 \in B), ..., (a_n \in A, b_n \in B) \}$ and those are equal because both are essentially exactly the same thing.  I hope my concern is clear and I need some kind of "conceptual insight" from knowledgeable people to move forward. Please give me the right perspective. What are point objects? Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ Also why $$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}.$$ I understand it, but I want a mathematical approach as a math student proves it. Also I want to know the mathematics topic of this question (number theory, logic, etc). A full answer is not necessary. Just a reference. Change object pivot point orientation For some reason the pivot point on my object has become rotated (other objects show normally oriented pivot points). I have to set the "transformation orientation" (see the bottom of my screenshot) to be able to see it though. Does anyone know how to change it back? That is, change it to be aligned with the "world orientation". Same question listed twice in reputation list This might not be a bug but I see the same question list twice in my reputation display, I know it is grouping things by a six hour difference, but is this really expected?    The first list says 2 events but shows all 5, similarly the second list says 3 events and shows all five. error while loading shared libraries: libssl.so.1.1 (and libcrypto.so.1.1) As you can see I'm missing libcrypto and libssl files. I locked out myself from my server after running apt-update and apt-upgrade, I'm running this from web console, because ssh nor any other program won't work because of missing SSL files.  I tried removing and installing apt install openssl so did libssl-dev and/or compiling own openssl from a distro. None of these create the needed .so files, I ran locate, those files are not present anywhere on the server. There's also libssl1.0.0 which cannot be reinstalled and removing it would cause removal of 700MB of packages that would literally kill server and delete all data. How to prevent book class from vertically justifying the contents? If we use the book class, the output looks like the following screenshot.    But if we use the article class, the output looks better as follows.    The MWE is given as follows. Note: please enable -shell-escape!  \documentclass[12pt]{article}% please change to article and compare the output.  \usepackage{filecontents} \begin{filecontents*}{"article sample.tex"} \documentclass{article} \usepackage{lipsum} \begin{document} \lipsum[1-20] \end{document} \end{filecontents*}  \usepackage{showexpl}  \lstset {     breaklines=true,     breakindent=0pt,     tabsize=2,     columns=flexible,     language={[LaTeX]TeX},     basicstyle=\small\ttfamily\hbox{}, }  \newcount\x  \newcommand\MyLTXinputExample[2][2mm]{% % #1: length of gap, % #2: filename with .tex,      \lstinputlisting[caption=\url{#2.tex}]{"#2.tex"}     \bgroup         \parskip=#1\relax         \fboxsep=0pt         \pdfximage{#2.pdf}%         \x=0         \loop             \ifnum\x&lt;\pdflastximagepages                 \advance\x by 1                 \noindent\hfill                 \fbox{\includegraphics[width=0.5\dimexpr\linewidth-4\fboxrule-#1\relax,page=\x]{"#2"}}%                                  \ifnum\x&lt;\pdflastximagepages                 \advance\x by 1                 \hfill                 \fbox{\includegraphics[width=0.5\dimexpr\linewidth-4\fboxrule-#1\relax,page=\x]{"#2"}}%             \fi                 \hfill\null\par         \repeat     \egroup }  \PassOptionsToPackage{obeyspaces}{url} \usepackage{hyperref}  \begin{document} \immediate\write18{pdflatex "article sample"} \MyLTXinputExample{article sample} \end{document}   How to prevent book class from vertically justifying the contents? In other words, I want to use the book class but its output (only the justification is under consideration) must look like the output that article class produces. How to properly handle collision in a component-based game? Trying to wrap my head around the ways to properly handle collision in a game designed around components.  I see many examples have some sort of PhysicsComponent that gets added to the entity's list of components but the actual implementation is confusing me.  For this to work, the PhysicsComponent would need access to the world around it. This doesn't make intuitive sense to me. Shouldn't a component be unaware of not only its container (the entity), but its container's container (the world)?  To me, it sounds like the level or scene should maintain a list of these entities and every game update, loop through the entities to determine which collide.  My question is firstly, whether or not this is good design, and secondly, how to determine which entities can collide. I suppose solid entities could implement an empty IRigidBody interface so the level could determine which entities in the list support collision. But is this breaking the component design?  Instead, should they contain an empty RigidBody component? This may actually be better because it may not always be empty and this approach is more future-proof. The only problem with this is the complexity. The scene would have to loop through not only every entity, but also every entity's components to determine if it had this RigidBody component.  Thirdly, when they do collide both entities should be informed somehow and I am unsure on how to accomplish this either.  Let's say that both entities contained a HealthComponent and when they collided both their healths would be decreased by some arbitrary value, 5. I suppose it would be the scene's responsibility to handle this when it detects a collision between two entities?  But then is the scene responsible for too much? I could see this possibly getting out of hand and becoming unwieldy when the scene is responsible for many things entities should not (?) have access to.  Edit: Question updated with more details. In which way does quantum mechanics disprove determinism? I've heard this pop up in a discussion with my physicist/engineer roommates, but didn't care to ask at the time.  Now I'm mighty curious about it.   doesn't really seem to say much on this issue.  From what I understand about the Uncertainty Principle, it says that there are certain properties of electrons and stuff that cannot be measured, and are therefore uncertain.  Then Wikipedia (under ) states that Sir Arthur Eddington says that the Uncertainty Principle isn't really so because we can't measure these properties, but because turns out nature is indeterministic.  At least that's what I took from those paragraphs.  Even without my biased wording, it sounds more like an assertion than evidence.  I've also read a few things about how other scientific conventions perceive the issue, like how a ball on the peak of a perfect mound might randomly roll down in any direction, and I'm still unconvinced.  My belief of determinism is generally that if you knew every single variable that existed as a factor at the very beginning and birth of the universe, you could correctly determine all properties of any individual particle at any point in time.  Could anyone provide some more background about this?  Especially regarding quantum mechanics? Is LED properly implemented in the constant current circuit? Formatting Ticks and FrameTicks labels with a function How do I search a specific user's answers? Can "whatever" be split into two words? I tend to write, "say whatever they want", but I'm always tempted to write "say what ever they want". Is it acceptable to split the word in this context? How can I get useful error messages in PHP? What's the difference between dot syntax and square bracket syntax? I'm trying to find a story for my father, from the 70s or earlier  Earth is at war with aliens (possibly Martians) and was unable to effectively fight back as the aliens had planetary (or at least city-protecting) force fields through which no bombardment was possible.  Earth eventually wins through invention of a device that allows them to teleport weapons (nukes?) under the force fields.  originally believed to be one of Harry Harrison's 'One step from earth' shorts, but this is not the case.  Thanks in advance for any help here! Why doesn't the "zig-zag" comb deformation retract onto a point, even though it's contractible? I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6(c) in Chapter $0$. Unfortunately a picture is involved so it doesn't quite make sense for me to repeat it here - but I'll do it anyways (maybe it helps)     Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by the heavier line in the picture:       Show there is a deformation retraction in the weak sense of $Y$ onto $Z$, but no true deformation retraction."   Now the definition for deformation retraction and weak deformation retraction as as follows:  We say that $f: X \to X$ is a deformation retraction of a space $X$ onto a subspace $A \subset X$ if there exists a family of maps $f_t : X \to X$ with $t \in [0,1]$ such that $f_0 = \mathbb{1}$ (the identity) and $f_1 (X) = A$ and also $f_t$ restricts to the identity on $A$ for each $t$.   A weak deformation retraction is almost the same, only that we now relax the conditions $f_1(X) = A$ to $f_1 \subset A$ and, for each $t \in [0,1]$ we require that $f_t(A) \subset A$.  Hence, as far as I understand the question and the concepts involved, I need to show that basically no map can be constructed such that the bold zigzag - line stays put while the thin lines retract to the bold line over a finite time interval $0 \leq t \leq 1$. However, I should be able to show that I can pull the thin lines continously onto the thick line, provided I am allowed to move points on the zigzag line .. is that correct ?  Now, the problem is I can't see why I shouldn't be able to do the former, given that I could find a weak deformation retract. Any help would be great!