I have the next problem. Prove that any base $a&gt;1$, each positive integer $m$ has a unique expression as the form $$a^nr_n+a^{n-1}r_{n-1}+...+ar_1+r_0$$ where the integers $r_k$ satisfy $$0\le r_k \lt a,  r_n\neq0.$$  I have tried this.   Proof: By induction above n $$m_1=ar_1+r_0$$  We suppose that it is true for n=k  $$m_k=a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$ Prove for n=k+1. By hypothesis $$m_k=a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$ We add $a^{k+1}r_{k+1}$.  $$m_k+a^{k+1}r_{k+1}=a^{k+1}r_{k+1}+a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$  As $a^{k+1}r_{k+1}$ and $m_k$ are positive integers then be  $m_{k+1}=m_k+a^{k+1}r_{k+1}$ a positive integer, hence      $$m_{k+1}=a^{k+1}r_{k+1}+a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$  but I am not sure and still I need to prove the uniqueness.Thank you for your advices.
Base-b Representation of an Integer: Why can I make the assumption about number of terms in expansions in uniqueness part?   Reading the book of Koshy I have come across the below theorem. It states that I can write any integer $N$ in a base $b\ge2$ representation $N=a_kb^k+...+a_1b+a_0$ such that $0\le a_i&lt;b$, $a_k \neq 0$ for some $k\ge0$.  In the uniqueness part, why can I assume the expansions contains the same number of terms ? I must prove that all expansions of $N$ equal the one in the theorem, but if I add terms to the expansion by zero coefficients, well, then the expansion is not in the proper form anymore ? (the greatest term in the expansion is $0$ if I add zero coefficients) .  Can someone help clear this out ?     Why can I assume the expansions contain the same number of terms ? If I add terms to the expansion, well then I've violated the form of the expansion ?
How do I prevent object distortion when applying a curve modifier? How to boot from non-Ubuntu live ISO images like Fedora or CentOS? Was Daenerys actually trying to hatch the eggs? I was reading @Aegon' s response   and I realized that I've never been clear on whether Dani was intentional in her pyre building to do blood magic and wake the dragons, or if she was just grieving Drogo and killing Mirri and her statement about "only death can pay for life" was a slap in the Maegi's face, since in the end she sacrificed her unborn child for the husk of Drogo.    I know she had tried baking the eggs previously and had felt like they could hatch then, but was clearly disappointed and chastised herself for foolishness when they did not. I don't recall inner monologue or statements after that where she said, "ya know what I should have done..."   I guess what I'm asking is whether there is evidence outside her statement to the Maegi that she was intentional in her blood magic rite or if the miracle was a happy accident? Examples of the Pigeonhole Principle As most of you might know, the Pigeonhole Principle basically  that      If $n$ items are put into $m$ containers, with $n&gt;m$, then at least one container must contain more than one item   It always surprises me how this trivial - and at the same time powerful - idea might be the key in order to solve extremely complicated math olympiad-problems...  Quick and beautiful solutions are characteristic of pigeonhole problems, which are often a three-part process   Recognize that the problem requires the Pigeonhole Principle Figure out what the pigeons and what the pigeonholes might be After applying the pigeonhole principle, there is often more work to be done   I'll illustrate this with an example I've always liked...     (Example-)Problem: Given a $n\times n$ square, prove that if $5$ points are placed randomly inside the square, then two of them are at most $\frac{n}{\sqrt2}$ units apart.   Step 1: This problem can be solved with the Pigeonhole Principle  Step 2: We divide the $n\times n$ square into four $\frac{n}{2}\times\frac{n}{2}$ squares (pigeonholes). Consequently, at least two points (pigeons) are inside the same $\frac{n}{2}\times\frac{n}{2}$ square.  Step 3: The maximal distance between two points in an $\frac{n}{2}\times\frac{n}{2}$ square is the diagonal, which has the length $\frac{n}{\sqrt2}\qquad\square$  Another problem which can be solved with the Pigeonprinciple is the following:     IMO $1972/1$           Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.      At this point, you might have noticed how useful the Pigeonhole Principle can be, if you know how to recognize and use it.     Question: I would like to work on this amazing principle with my students for a week and was, therefore, gathering problems related to the Pigeonhole Principle with beautiful solutions.   Could you suggest some more? I stopped dual booting and I don't want to have to choose the OS any more. I want to boot into Ubuntu 12.04 directly without seeing the GRUB menu.  How can I achieve that? Why are some muggles allowed to know of the magical world? If $f(x)=x-\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx$ find the explicit function for $f(x)$ in its simplest form When has Wonder Woman died? I know Wonder Woman has died in canon before, but in which issue did this happen? Can I force a link to open in a specific browser? What's the difference between correlation and simple linear regression? In particular, I am referring to the Pearson product-moment correlation coefficient. Reopened questions get a "modified" time stamp in the questions list just showed up on 's front page like this:    ... which makes no sense, because the question hasn't been modified since May 25th, and the person who edited it wasn't CatLord anyway:    Turns out what actually happened was that CatLord cast a fifth reopen vote to reopen the question.    So that stamp in the questions list should probably say this instead:     reopened 4m ago CatLord 10.7k   Or just:     reopened 4m ago PHP __PHP_Incomplete_Class Object with my $_SESSION data Android/Java - Date Difference in days In MTG, what happens when a player loses in multiplayer? Immutability of Strings in Java Allow embedded HTML5 YouTube video How to calculate the signal-to-noise ratio (SNR) in an image? I am working with an image X, I apply the "adaptive median filter" in it and I get the image Y. I'd like to measure the SNR in both in order to evaluate the quantity of noise deleted.   I know the formula to calculate the SNR is:   SNR = Psignal / Pnoise  but I don't know how to get these values from both images I have. I was thinking in substract the image X from Y and get the noise value. But I am not sure about the method. A slick proof that a field which is finitely generated as a ring is finite It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,\dots,x_n]\to k$ for some $n\in \mathbb{N}$, then $k$ must be finite. Since finite generation as a ring implies finite generation over the prime field ($\mathbb Q$ or $\mathbb F_p$), by Noether normalization it follows that $k$ must be a finite extension of its prime field. In positive characteristic this finishes the job and in zero characteristic, should lead quickly to a contradiction, though I don't see immediately how. Is there an elementary/slick proof of this fact? How can I produce an encrypted Swap Partition for 14.04 ? All the advices I saw here are for versions &lt;14.04 and do not work for me.  I tried a lot but nothing lasted longer than a reboot.  On installation I decided to encrypt my home and 9 GiB partition for swap but this swap was never used by Kubuntu. Gparted finds an "unknown filesystem".  I many times formated it as swap, edited fstab and crypttab and used all the tricks mentioned in different answers here. The often mentioned PunyGeek letter does not work for me probably because /etc/initramfs-tools/conf.d/resume does not even exist on my system. Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then (UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)        $f$ maps a bounded sequence to a bounded sequence.   $f$ maps a Cauchy sequence to a Cauchy sequence.   $f$ maps a convergent sequence to a convergent sequence.   $f$ is uniformly continuous.      I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)  i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $ \forall x,y$.  Hence all other options are bound to be true.  Am I correct?