recursion and while loop
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Prove $\mathbb{Z}$ is not a vector space over a field This is an exercise from Chapter 3 of Golan's linear algebra book.  Problem: Show $\mathbb{Z}$ is not a vector space over a field.  Solution attempt:  Suppose there is a such a field and proceed by contradiction. I will write multiplication $FV$, where $F$ is in the field and $V$ is an element of $\mathbb{Z}$.   First we rule out the case where the field has characteristic 2. We would have   $$0=(1_F+1_F)1=1_F1+1_F1=2$$  a contradiction.  Now, consider the case where the field does not have characteristic 2. Then there is an element $2^{-1}_F$ in the field, and   $1=2_F(2^{-1}_F1)=2^{-1}_F1+2^{-1}_F1$  Now $2^{-1}_F1\in\mathbb{Z}$ as it is an element of the vector space, but there is no element $a\in\mathbb{Z}$ with $2a=1$, so we have a contradiction.  Is this correct? Executing Java programs through Python How do I do this? What's the best way of checking if an object property in JavaScript is undefined? I'm looking for the name of a sci-fi series, intended for children. Robots have control over the humans. There is a squad of humans who attack them. Every member in this squad has a badge on their suit, and when they touch this badge the suit becomes armed. This series was not cartoon series, it was performed by real people. The main robot character is very similar in style to Darth Vader in Star Wars. The series aired in the 1990s. I am sorry but I can't recall many details. What to do about a 2-year-old who won't sleep without someone else in the room? My 2-year-old son won't fall asleep without having someone in his room with him. He doesn't want to be touched or held. He just wants you to sit in the chair. And after he does go to sleep he wakes up about 4 to 5 hours later, crying and screaming and is out of his bed and by the door. I go to him and give him a hug and put him right back in his bed. If I walk out he screams and won't go back to sleep for the rest of the night. If I stay he goes right back to sleep. But he wakes right back up when I walk out of his room.   I'm at a loss as to what to do. I've tried lots of stuff and nothing seems to work. It's almost as if he is having separation anxiety. I dont know what to do. Please help. Windows 10 high memory usage (unknown reason) Can misused Schema.org hurt SEO? I've inherited a site where I noticed that    there is an excessive use of Schema.org in all navigation elements pages describing products are marked as itemtype="http://schema.org/CreativeWork"   So I'm wondering if these tags are sending the wrong message and not truly optimized. I realize that the nav (1) is done in some themes, but I'm more worried about (2). Can it be a hit SEO-wise to mislabel pages instead of products? I'm working from the URL I found here:    My ssh client is Ubuntu 64 bit 11.10 desktop and my server is Centos 6.2 64 bit. I have followed the directions. I still get a password prompt on ssh.  I'm not sure what to do next. How can I calculate magnification ratio I get by reversing a lens? I am looking for a way to mathematically determine the maximum magnification ratio that is possible/practical to achieve by reverse mounting a non-macro lens, prior to purchasing the lens.     See related questions: Possible to create new Application Handler? Blender's application handlers are listed , but is it possible to create a new one?  For example, I'd like an application handler which is a callback for every time I add a strip to the Video Sequence Editor. Prepositions used with "inquire" Query with BETWEEN returns Zero or ONE These questions are really bothering me. Any help would be much appreciated.  Let $p$ be prime  a) If $(G, \oplus)$ has order $2p$, prove that every proper subgroup of $(G, \oplus)$ is cyclic  b) If $(G, \oplus)$ has order $p^2$, prove that $(G, \oplus)$ has a subgroup of order $p$ can't get placeholder attribute to work in IE using iframe What are the prerequisites to encounter the legendaries in ORAS? There are a lot of legendaries available in ORAS, however a lot of them will only show up/be available if certain prerequisites are met.  I know Groudon/Kyogre are available through the Story, and      Rayquaza &amp; Deoxys   through the postgame Delta episode, but what about the other available legendaries? What do I need to do to catch them? "Repository doesn't support architecture" when trying to update on Landscape client Conjectures Involving floor(x)/piecewise continuity with regards to Integration and Differential Equations [This answer has been heavily edited in response to a long chain of comments on Eric Stucky's answer.]  I came up with these few theorems and I am curious whether or not my hypothesis are true. I don't really have a method for proving these, mind you. I'm more considering these as things I've noticed to be true. I just don't know whether they are always true.  Some of these are definitions to which I will append "def." to the end of the title. Think of these as a given. It's like how the integral is defined abstractly as area. I'm defining terms.  Jump Series Theorem def.  A function $f$ [consisting of floor] is said to have a jump series if there is a series $\Sigma=\sum_{n=1}^\infty a_nH_n$ of scaled step functions such that $f+\Sigma$ no longer has any discontinuity attributable to floor. The sequence of scaling factors $a_n$ is called the jump sequence for the function.  Definition of Implied Derivative def.  We define the evaluation of the implied derivative of a function $y$ at the point $x$ as the set of values $y^{\to}(x) = \{y | y = \lim_{h \to 0^+} \frac {y(x+h) - f(h)}{h} \lor y = \lim_{h \to 0^-} \frac {y(x+h) - f(h)}{h}\}$. It is in this sense that the implied derivative has the potential of being a multi-valued operator.  Jump Series Resolution Theorem  [The original wording is preserved; I have a formalism but it assumes that having a jump function is equivalent to being piecewise continuous, which I'm not sure is true.]   The integral of a floor-based function is floor treated as a constant minus the appropriate jump series of the integral.  Integration Continuity Theorem  The (ordinary) indefinite integral of any function $f$ having a jump series must be continuous (or fixably discontinuous) on the domain of $f$. However, this can vary for the implied indefinite integral.  Calculus Jump Theorem def.  [Again, the original wording is preserved; the meaning of this phrase has not been properly illuminated]  Any form of jumping within a function representable as a floor function can be considered as a portion of the constant of the implied integration for floor.  Jump Location Theorem  For any function $a(x)$ with a jump series, the function $\lfloor a(x)\rfloor$, has points of discontinuity precisely at those $x$ such that $a(x) - [a(x)] = 0$.  Composite Floor Function Jump Sequence Theorem One  If $a$ is a function such that $a(x) - \lfloor a(x) \rfloor = 0$ when $x - \lfloor x \rfloor = 0$, then the jump sequence of the composite function $f(x - \lfloor a\rfloor)$ is a constant value $f(0) - f(1)$.  Implied Differential Equations Conjecture  Any continuous solution to an equation constructed with implied derivatives of various orders is the same as the solution to the similarly constructed differential equation. I want to find all files which contain a specific string of text. The grep command works, but I don't know how to use it for every directory (I can only do it for my current directory). I tried reading man grep, but it didn't yield any help. How do I create a D&D character correctly as a total newbie? Redefining \listoffigures and \listoftables as tables