Is an entire function upper bounded by a polinomial a polynomial?
Property of Entire Functions
XmlInclude : List and array The proof I have starts of with $\;xy=0\;$ in a field.   Then $x^{-1}$ exists because it is a field.  Then $x^{-1} xy=x^{-1} 0$.  Therefore $y=0$.  But surely if an integral domain can not have any zero divisors, how can we end the proof by saying $y=0$? Surely then y is a zero divisor and hence the field is not an integral domain? The characteristic and minimal polynomial of a companion matrix Find asymptotic of recurrence sequence Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$.  The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$.  I tried to find explicit formula for $x_n$, but did not succeed. I stucked at that point: if $a_1=(\frac{1}{4})^2$, $a_n=(a_{n-1}+\frac{1}{4})^2$, then $x_n=-a_{n-2}+\frac{1}{4}$ for $n\ge3$.   Thanks for any ideas and help. Proving the sum of two independent Cauchy Random Variables is Cauchy Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt Characteristic Functions. I do not know anything about Complex Analysis, Residue Theorem, etc.   I would want to prove the statement only using Real Calculus. Feel free to use Double Integrals if you please.   On searching, I found . However, I was wondering if I could get some help directly on the convolution formula:  $$f_Z(z)=\int_{-\infty}^\infty f_X(x)f_Y(z-x)\,dx=\int_{-\infty}^\infty\frac{1}{\pi^2}.\frac{1}{1+x^2}.\frac{1}{1+(z-x)^2}dx\tag{1}$$  Here I have supposed that $X,Y$ are Independent Standard Cauchy. But I think the general formula can be derived easily after some substitutions. I need some help on how to proceed from $(1)$.  EDIT: Just as what the hint in the hyperlink said, I got the answer using that hint. However, I am not quite sure that the hint is algebraically correct. Maybe there has been some typing mistake in the book. How did the word "Bollywood" originate? As we know, American film industry is referred as Hollywood. The American film industry is called so because the film industry emerged at the place called (Correct me if I am wrong). The place adopted the name Hollywood which is informally known as Tinseltown.  But my question is not about Hollywood but the . This is widely referred as Bollywood.   From the  page of Bollywood:      Bollywood is the sobriquet for India's Hindi language film industry, based in the city of Mumbai, Maharashtra. It is more formally referred to as Hindi cinema. The term "Bollywood" is often used by non-Indians as a synecdoche to refer to the whole of Indian cinema; however, Bollywood proper is only a part of the larger Indian film industry, which includes other production centres producing films in many other Indian languages. There is no place in India called Bollywood like the Hollywood.    Many people prefer calling it as Hindi film industry and against naming film industries with wood. There are other film industries in India too in other languages. Eg: Telugu film industry is named as Tollywood, Tamil industry is named Kollywood etc.,   is the list of Hollywood inspired nicknames.  My question:   Who is the creator of the word Bollywood? Since when did the word become popular? How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$ Clear named (and lored) item from players inventory 1.13 In 1.12, the command:  clear @p paper 0 1 {display:{Name:"[Winning Ticket] Musical Minecarts",Lore:[" Take this ticket to the reward "," collections at the entrance of the ", " Games area to claim your prize! "]}}   would remove a piece of paper with that specific name and lore from the player's inventory. I assumed the 1.13 conversion would be:   execute at @p as @p run clear @s paper{display:{Name:"[Winning Ticket] Musical Minecarts",Lore:[" Take this ticket to the reward "," collections at the entrance of the ", " Games area to claim your prize! "]}} 1   but this doesn't work. As well as this, the command:  give @p minecraft:paper{display:{Name:"[Winning Ticket] Musical Minecarts",Lore:[" Take this ticket to the reward "," collections at the entrance of the ", " Games area to claim your prize! "]}} 1   only gives the item, without the custom display name, despite the lore being correct. What is the correct syntax for these two commands? iPhone, reproduce the magnifier effect Why is Euclid's proof on the infinitude of primes considered a proof? I've expressed Euclid's proof on the infinitude of primes on Mathematica:  f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]   Which results in:  $\begin{array}{ll}  3 &amp; \text{True} \\  7 &amp; \text{True} \\  31 &amp; \text{True} \\  211 &amp; \text{True} \\  2311 &amp; \text{True} \\  30031 &amp; \text{False} \\  510511 &amp; \text{False} \\  9699691 &amp; \text{False} \\  223092871 &amp; \text{False} \\  6469693231 &amp; \text{False} \\  200560490131 &amp; \text{True} \\  7420738134811 &amp; \text{False} \\  304250263527211 &amp; \text{False} \\  13082761331670031 &amp; \text{False} \\  614889782588491411 &amp; \text{False} \\  32589158477190044731 &amp; \text{False} \\  1922760350154212639071 &amp; \text{False} \\  117288381359406970983271 &amp; \text{False} \\  7858321551080267055879091 &amp; \text{False} \\  557940830126698960967415391 &amp; \text{False} \\ \end{array}$  The proof flaws for all those values, how is it considered a proof then? I guess that there might be infinite prime numbers according to the proof, but what is the guarantee that at some point it won't fail indefinitely? Can't do reviews after failing 1 audit Good keyboard layouts for typing (La)TeX When typing (La)TeX some keys are used a lot more often then in plain text, especially \, {, }, [, ], $, ^ and _. On most keyboard layouts these keys are rather cumbersome to type. The English QWERTY keyboard is a lot better than the German QWERTZ, but is still far from optimal.  Given that I mostly type mathematical texts in English, what keyboard layout would you recommend?  Bonus points for the following:   Ability to type non-English Latin characters, especially German umlauts. Some similarity to QWERTZ or QWERTY as I'm most used to these layouts (I currently use the US-International layout from Linux) Ability to type math symbols directly (Greek letters, \times, etc.) How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like? If we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G_f \subseteq A \times B$ is the graph of the function. Now my question is: what is $f$ if   $\bullet \ A=\emptyset, \ B\neq\emptyset$,?  $ \bullet \ B=\emptyset, \ A\neq\emptyset$ ?  $ \bullet \ B=\emptyset, \ A=\emptyset$ ?  (Another way to formulate this, I think, would be: How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like? Are they all $\emptyset$ ?) I'm trying to connect to another computer in my house via ssh. Both computers have ubuntu and openssh installed. When I run the command "ssh username@ip" on terminal I'm asked for the user's password I'm trying to connect to. I type it but I get this error "Permission denied, please try again." I'm sure the password is correct, then why can't I access? Explicitly finding the sum of $\arctan(1/(n^2+n+1))$ This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$ Finding the fallacy in this broken proof Today, a friend gave me a "proof" of $1=2$ and challenged me to find the fallacy.     $1 = 1$      $1 = 1 + 0 + 0 + 0 ...$      $1 = 1 + 1 - 1 + 1 - 1 + 1 - 1 ...$      $1 = 2 - 1 + 1 - 1 + 1 - 1 ...$      $1 = 2 + 0 + 0 ...$      $1 = 2$   My answer was that once you turn the initial $1 + 1$ into a 2, everything is offset so a $-1$ is always left at the end no matter how many times it is repeated. This negative one balances out the $2$ at the beginning so $1=1$ still holds true. I.e.  $$1 = 1 + 0 + 0 + 0 ... = 1 + (1 - 1) + (1 - 1) + (1 - 1) = 2 + (-1 + 1) + (-1 + 1) - 1$$  However, my friend claimed that my answer only applies if the $+ 1 - 1$ repeats for a finite number of times. He argues that because the sequence repeats infinitely and things work differently when working with infinity, my answer is not valid.  Can anyone enlighten me to the true fallacy in this proof? Any class of a nonempty $X$ generates a base for a topology on $X$. Theorem Any class $\mathcal{A}$ of subsets of a nonempty set $X$ is the subbase for a unique topology $\tau$ on $X$. That is to say, the finite intersections of members of $\mathcal{A}$ form a base for the topology $\tau$ on $X$. Example Consider $X=\{a,b,c,d\}$ Let $\mathcal{A}=\{\{a,b\},\{b,c\},\{d\}\}$ The finite intersection of members of $\mathcal{A}$ give us: $$\mathcal{B}=\{\{a,b\},\{b,c\},\{d\},\{b\},\varnothing,X\}$$  How is it true that $X \in \mathcal{B}$ ? By definition of subbase, the finite intersection of members of $\mathcal{A}$ are the members of some base $\mathcal{B}$. But for $a_i \in \mathcal{A}, \displaystyle\bigcap a_i \neq X$ for any finite intersection of $a_i$'s. Look: $\{a,b\}\cap\{b,c\}\cap \{d\} \neq \{a,b,c,d\}$ nor do any other intersection of elements of $\mathcal{B}$ result in $X$. Anybody know what's going on here? Why is $X \in \mathcal{B}$? Negative Mass and gravitation Since Newtonian gravity is analogous to electrostatics shouldn't there be something called negative mass? Also, a moving charge generates electric field, but why doesn't a moving mass generate some other field? What differentiates the scientific method from other methods From the controversy in , I would like to know what differentiates "the scientific method" from other method.  I'm asking for differentiation, rather than a definition, because definitions call for an all encompassing answer, which is often hard to find.  If someone asks "what is an alternative to the scientific method," I would like to be able to start by being able to reliably say whether a particular method is the scientific method or not.  Of course, being a linguistic issue, the definitions of words themselves are flexible.  For instance, it is not uncommon to hear one in science talk of "truth" in a sense which is incompatable with the mathematical meaning of "truth."  Accordingly, I would like to stick to the following ideals in wording whenever possible:   Prefer mathematically accurate terminology.  Ideally the word "proof" should be used in the form found in proof theory, and "true" and "false" should be used in the form of Boolean algebra.   No method should claim a hypothesis which predicts the state of the future is "true" unless they claim soothsaying abilities (most scientific theories include a random variable to avoid needing to claim soothsaying abilities... this is legitimate)  Words which imply direction should be clarified carefully.  "Good" and "bad" are notoriously difficult, but phrases such as "advancing society" can also be very tricky as well. The physical world and the world of information theory should be treated differently.  In particular, the concept of "testing a hypothesis" will likely need to be broken down into testing and observing the physical world, taking measurements (which should have errors associated with them), and only then is the hypothesis compared against the results of the measurements. Mathematically extending arguments into infinity (such as claiming some specific cellular level learning process qualifies as the scientific method) are valid, but may rapidly make it harder to distinguish non-scientific methods from scientific ones.  Such claims should be made carefully, but they are not inherently invalid.   If these ideals are too strict, you may bend them, but you should qualify such choices with an explanation of why my recommended wording choices are too strict.  Their goal is to put different methodologies on the levelest playing fields that I can manage in this medium.  Being stack exchange, bonus points for references to existing theories, though you may need to paraphrase their results to fit with the ideals I'm recommending for these answers.  Answers may be tested two ways:   Positive: If we think something is the scientific method, does it meet the requirements set forth? Negative: If we think something is not the scientific method, does the answer explain why it is not? I created this style that shifts the labels and put them on the bottom of the ybars.  However, I have a problem with the bars that have too short length (for example, see the 8.7 and 5.1 in the example below). I want to check if the values are less than some threshold, or the length of the bar is shorter than the label (number) then change its color.  How can I overlay another label on top of the one in white when the bar is shorter? or maybe change the color of the label to a gradient color? or just overlay the part that is outside of the bar?    \documentclass{standalone}  \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable} \usetikzlibrary{calc}  \begin{document} \makeatletter \begin{tikzpicture} \begin{axis}[ ybar, every node near coord/.style={   check for zero/.code={ % If meta=0, make the node a coordinate (which doesn't have text)     \pgfmathfloatifflags{\pgfplotspointmeta}{0}{         \pgfkeys{/tikz/coordinate}     }{}   },   check for zero, color=white, text opacity=1, font=\footnotesize, inner ysep=0.5pt, },% % Bottom nodes  calculate full offset/.code={     \pgfkeys{/pgf/fpu=true,/pgf/fpu/output format=fixed}     \pgfmathsetmacro\testmacro{((\pgfplotspointmeta-\pgfkeysvalueof{/pgfplots/ymin})*10^\pgfplots@data@scale@trafo@EXPONENT@y)*\pgfplots@y@veclength}     \pgfkeys{/pgf/fpu=false} },% nodes near coords bottom/.style={     every node near coord/.append style={         /pgfplots/calculate full offset,         yshift=-\testmacro,         rotate=90, anchor=west,%I need the rotate text here and not in the          %general style, as the ybar redefines the style     } },% ymin=0, nodes near coords={\pgfmathprintnumber[fixed zerofill,precision=1]{\pgfplotspointmeta}}, nodes near coords bottom, ] \pgfplotstableread{ % set real imag mag 1 8.7 2 5.1 3 91.8367  }\mydata \addplot table[x index=0,y index=1,header=false] {\mydata}; \end{axis} \end{tikzpicture} \makeatother \end{document}   * EDIT *  I need to create some special gradient on each label that is below the bar height. For example, I need to adapt something like  to each label that is not showing. However, I cannot find a way to put the macro in the style section of the codes. Or is there any equivalent solution?