Question: Let $B^n \subset \mathbb{R^n}$ be open ball in the Euclidean metric. Prove that the complement of $B^n$ in $\mathbb{R^n}$ has exactly one unbounded component (components of a set are class partitions defining largest connected sets...).   This is an exercise of the book 'C. Adam - Topology'. Obviously, it does not hold for $n=1$, since $\mathbb{R}-(-a,a)$ is disconnected and is made of TWO unbounded components. But for the case of $n=2,3$ it is easy to prove that the conjecture holds (each 'side' of the open balls is homeomorphic to $\mathbb{R^n}$ and has non-empty intersection with its neighbour-side...)  How to elevate the conjecture for the case of $n\ge 4$, since it is not possible by intuition?    Thank you.  EDIT - This is not dublicate, since my question is about complement of an open ball not a bounded set in general. I read  before I wrote my question; the answer doesn't prove $\mathbb{R^n}−B^n$ is connected, which I need to prove.
Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected?  Idea One could then perhaps make use of the facts that every $2$-dimensional Banach space is homeomorphic to the Euclidean plane and that - for the Euclidean plane - my question has an obvious &quot;YES&quot; answer (since every open ball is the interior of a circle).
Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ideas  like $$S(x)+S'(x)+S''(x)=e^x-1\,.$$Maybe if I get the third equation it will allow me to construct a kind of differential equation. Then, by solving it, obtain $S(x)$. Assume there is a public-key encryption scheme $(KeyGen, Enc, Dec)$ with perfect correctness (i.e., for all messages M and valid key-pairs (PK,SK), we have $Dec_{SK}(Enc_{PK}(M))=M$).  Will there always be a function $A$ such that for all messages M and valid public-keys PK, $A_{PK}(Enc_{PK}(M))=M$?  In other words, can a computationally unbounded adversary decrypt the message, having only the public key? Is it safe to delete from C:\Windows\Installer? Pages: How to put a word between horizontal rules How can I make something that looks like this?  --------   Text   ---------  where the horizontal rules extend to the margin on either side? Why is the content of /run/resolvconf/interface/NetworkManager changed every time I reboot? I have my Git repository which, at the root, has two sub directories:  /finisht /static   When this was in , /finisht was checked out in one place, while /static was checked out elsewhere, like so:  svn co svn+ssh://admin@domain.com/home/admin/repos/finisht/static static   Is there a way to do this with Git? Is it legal to add your publications to ResearchGate? I recently created a profile on . And to add your publications list it requires you to upload the paper. Is it legal to upload published journal articles on public servers, where everyone can download it for free? Taking the * of these two large shapefiles   locally using ArcMap takes appr. 10 min.   Over the last few weeks, I've tried to obtain the same results using other, open-source approaches without success. I am curious why only ArcMap seems to generate results in a reasonable amount of time without errors.   The approaches I've tried so far:  QGIS  The most straight-forward alternative. There are two "union" functions available in QGIS. One default (part of vector overlay), one based on SAGA. Both functions create the desired results on smaller samples but fail to run (or take forever) on my shapefiles. I tried running union for 30 minutes after which the progress bar was at appr. 4%    PostGIS  I've ingested the shapefiles on an AWS RDS instance running postGreSQL with PostGIS enabled. The approach is not so straight forward and requires a hideous long SQL query to obtain the same result. Most importantly, the query did not create a meaningful result on the full datasets. It did run properly on a sample though.   See separate question   CREATE TABLE test.sqlislelijk AS  -- input data with polys1 AS (   SELECT      pfaf_id as df1,     geom as g   FROM hybas06_v04 ), polys2 AS (   SELECT      aqid as df2,     geom as g   FROM y2018m11d14_rh_whymap_to_rds_v01_v01 ), -- intersections intersections AS (   SELECT df1, df2, ST_INTERSECTION(a.g, b.g) i, a.g AS g1, b.g AS g2    FROM polys1 a, polys2 b WHERE ST_INTERSECTS(a.g, b.g) ), -- per-row union of intersections with this row diff1 AS (   SELECT df1, ST_UNION(i) i FROM intersections GROUP BY df1 ), diff2 AS (   SELECT df2, ST_UNION(i) i FROM intersections GROUP BY df2 ), -- various combinations of intersections pairs AS (   SELECT df1, df2, i AS g FROM intersections   UNION ALL   SELECT      p.df1,     NULL,     CASE       WHEN i IS NULL THEN g        ELSE ST_DIFFERENCE(g, i)     END   FROM polys1 p LEFT JOIN diff1 d ON p.df1 = d.df1   UNION ALL   SELECT     NULL,     p.df2,     CASE       WHEN i IS NULL THEN g       ELSE ST_DIFFERENCE(g, i)     END   FROM polys2 p LEFT JOIN diff2 d ON p.df2 = d.df2   ) SELECT    df1 as pfaf_id,   df2 as aqid,   g as geom FROM pairs WHERE NOT ST_IsEmpty(g);   Google BigQuery  Google's GIS extension for BQ looks promising but I experienced weird behavior and internal errors, probably due to mixing of GEOMETRY and GEOGRAPHY objects. Also, the syntax is similar to PostGIS and a command of 1 line in geopandas takes >20 lines in SQL.   GeoPandas  Probably my favorite approach. The out-of the box approach does't give any results after running it for several hours. My latest approach includes tiling the shapefiles, use multiprocessing on each tile and merge them. Not very straight forward and one 10x10 degree tile till takes 40+ minutes to calculate.   See   It is the first time in my career that I am impressed by the performance of ArcMap. It seems that ArcMap handles this use-case much, much better than anything else. The issue seems persistent over all solutions I've tried so far. Is there something wrong in the underlying libraries of the open-source/alternative solutions?    Note that Union in ArcMap means something different than in the Shapely / PostGIS world. We have to evaluate the following integral:  $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$  I tried this:  I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$.  But after that I got stuck.  The book where this is taken from gives the following as the answer: $$\ln(1+t)-\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$ BeanFactory vs ApplicationContext Reducing voltage with resistors I have been looking around for an easy way to convert 12V to 5V. I have seen some people saying that a simple resistor is all that is needed.  $$ Volts = Ohms \cdot Amps $$ $$ Amps  = \frac{Volts}{Ohms} $$ $$Ohms  = \frac{Volts}{Amps}$$  So applying a resistor will diminish the voltage of the circuit. That should mean that an appropriately sized resistor could simply be placed in the path of a 12V circuit, converting it to 5v.   If this is the case how would one reduce amps? Would series vs parallel make a difference in this area?   I have seen designs that include a regulator IC and some capacitors, but if a simple resistor/fuse/diode setup will do the trick, I would really prefer that. I'm having a really hard time trying to understand what the differences between a desktop environment and a window manager are?  EDIT: From this :  There are basically three layers that can be included in the Linux desktop: X Windows – This is the foundation that allows for graphic elements to be drawn on the display.  builds the primitive framework that allows moving of windows, interactions with keyboard and mouse, and draws windows. This is required for any graphical desktop. Window Manager – The Window Manager is the piece of the puzzle that controls the placement and appearance of windows. Window Managers include: , , , , , etc. Requires X Windows but not a desktop environment. Desktop Environment – This is where it begins to get a little fuzzy for some. A Desktop Environment includes a Window Manager but builds upon it. The Desktop Environment typically is a far more fully integrated system than a Window Manager. Requires both X Windows and a Window Manager.  Examples of desktop environments are , , ,  among others) How do I create a villager with custom trade offers? Possible Duplicate:          In  the following claim was mentioned:  $R/I \otimes_R M \cong M / IM$  So I tried to prove it. Can you help me finish my proof? Thanks!    We recall that $M \otimes_R -$ is a covariant right-exact functor that it is exact if $M$ is flat and we observe that the following is an exact sequence: $$ 0 \to IM \xrightarrow{i} M \xrightarrow{\pi} M / IM \to 0$$  $R/I$ is an $R$ module since $R/I$ is a subring of $R$ closed with respect to multiplication from $R$ but it is not necessarily flat hence we only get exactness on one side:  $$  (R/I) \otimes_R IM \xrightarrow{id \otimes i} (R/I) \otimes_R M \xrightarrow{id \otimes \pi} (R/I) \otimes_R M / IM \to 0$$  Then $$ \mathrm{Im(id \otimes \pi)} = (R/I) \otimes_R M / IM  \cong (R/I) \otimes_R M / \mathrm{Ker} (id \otimes i) $$  so we want to show two things:   (i) that $\mathrm{Ker} (id \otimes i) = \{0\}$. To this end let $r + I \otimes im \in (R/I) \otimes_R  IM $ and assume $ id \otimes i(r + I \otimes im ) = r + I \otimes im = 0 + I \otimes 0$. I'm not sure how to proceed from here.   (ii) and that $M /IM \cong (R/I) \otimes_R M / IM $  Can you help me? Thanks. If we set $A$ and $B$. Assuming that there exist an injective map from $A$ to $B$ and an injective map from $B$ to $A$. How to prove that there is a bijection between $A$ and $B$. Redirecting stdout to a file you don't have write permission on I'm using MySQL's AUTO_INCREMENT field and InnoDB to support transactions. I noticed when I rollback the transaction, the AUTO_INCREMENT field is not rollbacked? I found out that it was designed this way but are there any workarounds to this? At  I've read that being $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using  we get $\sin'(x)=\sum_{n=0}^{\infty}(2n+1) \frac{(-1)^n}{(2n+1)!}x^{2n}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=\cos(x)$ but the page  says that if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$?  Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\sin'(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified?   (I mean, I think we can't just say "oh, there's a missing term!" after having applied the theorem and magically shift the index of the series to obtain the desired result) How can I implement a comprehensive apa-style citation and bibliography I am really new to LaTeX and I need a comprehensive solution for my Thesis  I have this problem since two days and I couldn't find any solution in any other post.  I work with BibDesk on Mac  And I used Packages apacite and natbib  @article{Ha89, Author = {Peter M. Haas}, Date-Added = {2017-02-06 21:15:57 +0000}, Date-Modified = {2017-02-06 21:45:06 +0000}, Journal = {International Organization}, Lastchecked = {03/02/2010 12:54}, Number = {3}, Pages = {377-403}, Publisher = {Peter Haas}, Read = {0}, Title = {Do regimes matter? Epistemic communities and Mediterranean pollution control}, Urldate = { http://www.jstor.org/stable/2706652}, Volume = {43}, Year = {1989}}   this is a stripped version of my document:  \documentclass[11pt, oneside]{article}  \usepackage{lipsum} \usepackage[margin=1in, left=1.5in, includefoot]{geometry}   \usepackage{ragged2e} % REFERENCE PREAMBLE %\usepackage{url} \usepackage{apacite} \usepackage{natbib} %\usepackage{biblatex-apa}                                                   % Bullet preamble \renewcommand{\labelitemi}{$\bullet$} \renewcommand{\labelitemii}{$\diamond$} \renewcommand{\labelitemiii}{$\circ$} % graphics preamble \usepackage{graphicx}   %Allows you to import images \usepackage{float}      %Allows for control of float positions %HEADER AND FOOTER STUFF \usepackage{fancyhdr}  \pagestyle{fancy} \usepackage{graphicx}        \usepackage{amssymb} %Zeilenabstand 1,5 wenn aktiviert  \usepackage{setspace}\makeatletter\newcommand{\MSonehalfspacing}{\setstretch{1.44}\ifcase \@ptsize \relax\setstretch {1.448}\or\setstretch {1.399}\or\setstretch {1.433}\fi}\newcommand{\MSdoublespacing}{\setstretch {1.92}\ifcase \@ptsize \relax\setstretch {1.936}\or\setstretch {1.866}\or\setstretch{1.902}\fi}\makeatother\MSonehalfspacing %Zeilenabstand Code fertig \usepackage[hidelinks]{hyperref}   %allows for clickable reference %                       BEGIN OF DOCUMENT \citeA{source} %                       REFERENCES  \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}} \fancyhf{} \rhead{\fancyplain{}{Identifying the Regime}} \lhead{\fancyplain{}{List of References}} \cfoot{\fancyplain{}{\thepage}}  \setlength\bibitemsep{1.9\itemsep} \bibliography{/User/Desktop/Thesis/references/mybib.bib}                   \bibliographystyle{apa} \addcontentsline{toc}{section {\numberline{}References}                                                       \end{document}   I would like to have the bibliography entry include a retrieved from    I tried to deactivate hyperref – as suggested in another post – but it didn't work. I also tried to activate and deactivate natbib and apacite as well.   I would really appreciate any help... Any Set with Associativity, Left Identity, Left Inverse is a Group