qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,962,514 | <p>This is just curiosity / a personal exercise.</p>
<p><a href="https://what3words.com/" rel="nofollow noreferrer">What3Words</a> allocates every 3m x 3m square on the Earth a unique set of 3 words. I tried to work out how many words are required, but got a bit stuck.</p>
<p><span class="math-container">$$
Area
= ... | CrackedBauxite | 867,032 | <p>You need to choose an eigenvector that is also in the column space of the matrix <span class="math-container">$A-\lambda I$</span>. In this case, by looking at the matrix you have and at your eigenvector basis, one sees that an eigenvector that will work is <span class="math-container">$\begin{bmatrix}1\\0\\-2\end{b... |
184,699 | <p>First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$
Is there any "name" for the corresponding "homology" group that one can define
(Kernel mod image)? Has this "homo... | Donu Arapura | 4,144 | <p>Actually, I have seen something like this used before in the paper <em>Holomorphic vector fields and Kaehler manifolds</em>, Invent 1973, by Carrel and Lieberman. They prove that if a compact Kähler manifold admits a holomorphic vector field $v$ with nonempty zero locus $Z(v)$, then the Hodge numbers vanish for $|p... |
184,699 | <p>First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$
Is there any "name" for the corresponding "homology" group that one can define
(Kernel mod image)? Has this "homo... | Vladimir Dotsenko | 1,306 | <p>These constructions certainly are very recognisable and meaningful in the algebraic context, that is if you think of algebraic differential forms on an affine algebraic variety. Namely, the corresponding complexes both implement versions of the Koszul complex for a sequence of elements $f_1,\ldots,f_n$ in $R$, where... |
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | Lisa | 81,933 | <p>Yes.</p>
<p>In response to your edit: You squared both sides of the equation to correct your post. However, this doesn't make a difference, it's still valid. Since you squared both sides of the equation (applied the same operation), you didn't effectively change anything.</p>
<p>The solution is therefore:</p>
<ol... |
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | StasK | 97,144 | <p>If you are solving for $x$, it's OK as long as in the end you check whether the expression you divided by, $5x$, is non-zero.</p>
<p>In general, you would have to follow through with everything assuming all transformations are legal, and then check for weak spots like division by zero ($5x=0$ or $x^2-4=0$, or squar... |
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | Cameron Buie | 28,900 | <p>As others have noted, it is valid so long as $x\ne0,$ since otherwise you're dividing by $0$. Let me offer another approach, using the difference of squares formula $a^2-b^2=(a-b)(a+b)$, together with the fact that $(-c)^2=c^2.$ The following, then, are equivalent:</p>
<p>$$\left(\frac{x^2+6}{x^2-4}\right)^2=\left(... |
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | mfl | 148,513 | <p><strong>Hint</strong></p>
<p>If <span class="math-container">$x<-1$</span> then <span class="math-container">$$\frac1x+\frac {1}{x^2}<0.$$</span></p>
|
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | B. Goddard | 362,009 | <p>First, I think it's better to divide top and bottom by the maximum power <em>of the bottom</em>. In this case, by <span class="math-container">$x$</span>, to get</p>
<p><span class="math-container">$$\lim_{x\to -\infty} \frac{\frac{1}{x}(x-1)^2}{1+\frac{1}{x}} = \lim_{x\to -\infty} \frac{\frac{1}{x}(x^2-2x+1)}{1+\... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Ross Millikan | 1,827 | <p>Hint: take the number apart into digits. Each digit $d$ represents $d \cdot 10^n$ for some $n$. What is the remainder when you divide $10^n$ by $3$? (Think about $10^n-1$ what does it look like?)</p>
|
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Community | -1 | <p>$1$. First prove that $3 \mid 10^n - 1$ (By noting that, $10^n - 1 = (10-1)(10^{n-1} + 10^{n-2} + \cdots + 1)$).</p>
<p>$2$. Now any number can be written in decimal expansion as
$$a = a_n 10^n + a_{n-1} 10^{n-1} + \cdots + a_1 10^1 + a_0$$</p>
<p>$3$. Note that $a_k 10^k = a_k + a_k (10^k-1)$. Hence,
$$a = \overb... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | mathandy | 53,784 | <p>There's likely a more elegant proof, but you got me thinking about this, so here's what I came up with (this is for two digits, but it generalizes easily).</p>
<p>If some number 10a+b (for a,b integers) is divisible by three then there is some integer k such that
10a+b = 3k</p>
<p>This means that</p>
<p>10a+10b =... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | DeepSea | 101,504 | <p><strong>Divisibility by $3$ rule</strong>: $ 3\mid \overline {a_1a_2...a_n} \iff 3\mid (a_1+a_2+...+a_n)$, whereas $a_1,a_2,..a_n$ are digits in $\{0,1,2,...9\}$.</p>
<p><strong>Proof:</strong> $\overline{a_1a_2...a_n} = a_1\cdot 10^{n-1} + a_2\cdot 10^{n-2} + ... + a_{n-1}\cdot 10 + a_n = a_1\cdot (9+1)^{n-1} + a... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Bill Dubuque | 242 | <p>Since you tagged it <a href="/questions/tagged/abstract-algebra" class="post-tag" title="show questions tagged 'abstract-algebra'" rel="tag">abstract-algebra</a> let's use a little algebra to help clarify the essence of the matter.</p>
<p><span class="math-container">$\!\!\begin{align}\rm\ {\bf Hint}\ \ \ mo... |
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| Henry Lee | 541,220 | <p>you can try the most famous one which is:
<span class="math-container">$$\int_0^\infty\frac{\sin(x)}{x}dx$$</span>
good luck!</p>
|
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| Zacky | 515,527 | <p>Since this became quite popular I will mention here about an <a href="https://zackyzz.github.io/feynman.html" rel="nofollow noreferrer">introduction to Feynman's trick</a> that I wrote recently. It also contains some exercises that are solvable using this technique.</p>
<p>My goal there is to give some ideas on how ... |
1,131,970 | <p>Let $I$ be a proper ideal of a polynomial ring $A$ and $x \in A$ an irreducible element.</p>
<p>In a theorem of commutative algebra I will use the fact that, in this hypothesis, holds the following equality: $$\sqrt{(I,x^k)}=\sqrt{(\sqrt{I},x)}$$</p>
<p>The assert seems to be true, anyone has any counterexample/p... | orangeskid | 168,051 | <p>If you know a bit about the field $\mathbb{Q}_p$ and its subring $\mathbb{Z}_p$, things become easier, at least formally.</p>
<p>Say $x_0^2 = a \mod p^k$. Well, let's assume in fact that the exponent of $p$ in $x^2-a$ is larger than the exponent of $p$ in $a$. </p>
<p>$$x^2_0 = a + \delta'$$
with $e_p(\delta') >... |
1,810,055 | <p>I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial.</p>
<blockquote>
<p>$X^3-3X+1$</p>
</blockquote>
<p>Do we first need to find its roots, then construct a splitt... | msm | 340,064 | <p>For the second one, you should think about those polynomials separately. In that case, it's pretty easy to see that the splitting field of $(x^3 - 2)(x^2+3)$ is $\mathbb{Q}(\sqrt[3]{2} , \zeta_3, \sqrt{-3})$.</p>
<p>Since $\zeta_3 = \frac{-1 \pm \sqrt{-3}}{2}$, this is the same as $\mathbb{Q}(\sqrt[3]{2} , \sqrt{-... |
188,150 | <p>I'm reading the paper <em>Loop groups and twisted K-theory I</em> by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence. </p>
<p>I'm a bit confused with some of their computations, for instance $S^3$ (their example 1.4 in the first section). The... | Neil Strickland | 10,366 | <p>Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles on $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibl... |
2,372,698 | <p>If a function $f(x)$ is continuous on the closed interval $\left [ a,b \right ]$ then its bounded on this interval........the proof for this theorem i have is: </p>
<p>Since it's continuous on $\left [ a,b \right ]$ if we pick a random point on this interval let it be $c$ </p>
<p>$\implies$ $\forall$ $\epsilo... | Sahiba Arora | 266,110 | <p>Let $f(x)=x$ on $[0,1]$ and $c=\frac12$. Then $M=|f(c)|=\frac12$. But $|f(x)| \not <M$ for all $x \in [0,1]$ (for instance, $|f(1)|>M$)</p>
|
3,662,466 | <p>Given a sequence with the terms </p>
<p><span class="math-container">$$
a_{n}=\left\{\begin{array}{ll}
n, & \text { if } n \text { even } \\
\frac{1}{n}, & \text { if } n \text { odd }
\end{array}\right.
$$</span></p>
<p>Prove <span class="math-container">$\limsup _{n \rightarrow \infty} a_{n} = \infty$</s... | QuantumSpace | 661,543 | <p>Recall that <span class="math-container">$\limsup a_n$</span> is the largest limit of a subsequence. But we have the sequence <span class="math-container">$(n)_{n=1}^\infty$</span> as subsequence of even terms and this tends to <span class="math-container">$+\infty$</span>. Thus we must have <span class="math-contai... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Timothy Chow | 3,106 | <p>I'm not sure I understand your question, since at first it sounds like you're thinking of $\in$ as a multi-valued function that sends a set $A$ to an element $x$ of $A$, but then I would expect you to be asking about the <i>range</i> of such a function rather than its <i>domain</i>. I'll assume that you are, loosel... |
2,763,974 | <blockquote>
<p>Find $\displaystyle \lim_{(x,y)\to(0,0)} x^2\sin(\frac{1}{xy}) $ if exists, and find $\displaystyle\lim_{x\to 0}(\lim_{y\to 0} x^2\sin(\frac{1}{xy}) ), \displaystyle\lim_{y\to 0}(\lim_{x\to 0} x^2\sin(\frac{1}{xy}) )$ if they exist.</p>
</blockquote>
<p>Hey everyone. I've tried using the squeeze theo... | The Phenotype | 514,183 | <p>$0 \le |x^2\sin(\frac{1}{xy})| \le |x^2| \xrightarrow{x\to0} 0$ means indeed that for any trajectory of $(x,y)$ going to $(0,0)$ we have that $x^2\sin(\frac{1}{xy})$ approaches $0$.</p>
<hr>
<p>Note that just by taking any $y\neq 0$ fixed we cannot just conclude that it would not approach $0$, since eventually you... |
285,591 | <p>If $$\lim_{x\to\infty} f(x) = \infty$$ what can be said about the rate at which $$\int_1^\infty f(x) \,dx$$ approaches infinity if $f(x) \geq 1$ for all values of $x$?</p>
| Benjamin Dickman | 37,122 | <p>Is this all the information given? If so, not much.</p>
<p>$f(x)$ could be $|x| + 1$ or $e^x + 1$. These are two pretty different functions, in terms of the rate at which their integrals grow. You would need more information to say much of anything at all.</p>
|
285,591 | <p>If $$\lim_{x\to\infty} f(x) = \infty$$ what can be said about the rate at which $$\int_1^\infty f(x) \,dx$$ approaches infinity if $f(x) \geq 1$ for all values of $x$?</p>
| Eric Naslund | 6,075 | <p>All that can be said is that $$\frac{\int_1^x f(y)dy}{x}\rightarrow \infty.$$ No better lower bound can be given, and nothing can be said about the rate at which this goes to infinity since nothing is given about $f$. Indeed, you can construct $f$ so that this ratio goes to infinity as slowly, or as quickly as desi... |
506,397 | <p>I would like to know the condition for a random variable <span class="math-container">$Y$</span> in order to make <span class="math-container">$\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$</span>, where <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> are ii... | karakfa | 14,900 | <p>start with $\max(x,y) = \frac12(x+y+|x-y|)$</p>
<p>$\mathbb{E}[\max\{X_1+Y,X_2\}]=\frac12\left(\mathbb{E}[X_1+Y+X_2] + \mathbb{E}\bigl[|X_1+Y-X_2|\bigr]\right) $
$\mathbb{E}[\max\{X_1,X_2\}]=\frac12\left(\mathbb{E}[X_1+X_2] + \mathbb{E}\bigl[|X_1-X_2|\bigr]\right) $</p>
<p>your condition will be satisfied when
$\m... |
985,103 | <p>The set $\{u_{1},u_{2}\cdots,u_{6}\}$
is a basis for a subspace $\mathcal{M}$ of $\mathbb{F}^{m}$ if and
only if $\{u_{1}+u_{2},u_{2}+u_{3}\cdots,u_{6}+u_{1}\}$
is also a basis for $\mathcal{M}$.
So far I have that the two basis are just rearranged sums of each other but don't know where else to go with it.</p>
| Kim Jong Un | 136,641 | <p>You don't have enough info to infer that $\{b_n\}$ converges.</p>
<p>Let $M>0$ be an upperbound of $\{b_n\}$. For a given $\epsilon>0$, there is $N>0$ such that for $n\geq N$, we have $|a_n/b_n|<\frac{\epsilon}{M}$. Then, for all such $n$,
$$
\frac{\epsilon}{M}>|a_n/b_n|=a_n/b_n\geq a_n/M\implies0<... |
3,572,967 | <p>Can I ask how to solve this type of equation:</p>
<blockquote>
<p><span class="math-container">$$\log_{yz} \left(\frac{x^2+4}{4\sqrt{yz}}\right)+\log_{zx}\left(\frac{y^2+4}{4\sqrt{zx}}\right)+\log_{xy}\left(\frac{z^2+4}{4\sqrt{xy}}\right)=0$$</span></p>
</blockquote>
<p>It is given that <span class="math-contain... | Michael Rozenberg | 190,319 | <p>By AM-GM
<span class="math-container">$$0=\sum_{cyc}\log_{yz}\frac{x^2+4}{4\sqrt{yz}}\geq\sum_{cyc}\log_{yz}\frac{2\sqrt{x^2\cdot4}}{4\sqrt{yz}}=\sum_{cyc}\left(\log_{yz}x-\frac{1}{2}\right)=$$</span>
<span class="math-container">$$=\sum_{cyc}\frac{2\ln{x}-\ln{y}-\ln{z}}{2(\ln{y}+\ln{z})}=\frac{1}{2}\sum_{cyc}\frac{... |
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | Brian Fitzpatrick | 56,960 | <p>Recall the following facts.</p>
<p><strong>Fact 1.</strong> $\sin t=0$ if and only if $t=k\pi$ for some $k\in\Bbb Z$</p>
<p><strong>Fact 2.</strong> $\cos t=0$ if and only if $t=k\pi+\frac{\pi}{2}$ for some $k\in\Bbb Z$</p>
<p>Also, recall the identity
$$
\sin u-\sin v=2\,\cos\left(\frac{u+v}{2}\right)\sin\left(\... |
3,527,004 | <p>As stated in the title, I want <span class="math-container">$f(x)=\frac{1}{x^2}$</span> to be expanded as a series with powers of <span class="math-container">$(x+2)$</span>. </p>
<p>Let <span class="math-container">$u=x+2$</span>. Then <span class="math-container">$f(x)=\frac{1}{x^2}=\frac{1}{(u-2)^2}$</span></p>
... | José Carlos Santos | 446,262 | <p>No, it is not, since what you got is <em>not</em> a power series (see what you get if you put <span class="math-container">$n=0$</span>).</p>
<p>Use the fact that<span class="math-container">\begin{align}\frac1{x^2}&=\frac14+\left(\frac1{x^2}-\frac14\right)\\&=\frac14+\int_4^x-\frac1x\,\mathrm dx\\&=\fr... |
1,694,159 | <p>I am prepping for my mid semester exam, and came across with the following question:</p>
<blockquote>
<p>Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that $k = 0, 1,...n$ and $n$ can be any natural number.</p>
</blockquote>
<p>So what I have done is to ... | Brian M. Scott | 12,042 | <p>The computational argument is easier and quicker, but for the record there is also a combinatorial argument.</p>
<p>As usual let $[n]=\{1,\ldots,n\}$; we’ll count the $k$-element subsets of $[n]$ that contain every element of $[n]$ in two ways.</p>
<p>First, it’s clear that there is such a set if and only if $k=n$... |
281,294 | <p>Let $G$ be a finite abelian group.</p>
<p>Is there a field $K$, and an elliptic curve $E$ over $K$ such that $E(K)_{tor} \cong G$?</p>
| Daniel Litt | 6,950 | <p>Even better, there exists an elliptic curve $E$ over a number field $K$, such that for any group of the form $\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/mn\mathbb{Z}$, there is a finite extension $K'/K$ such that $$E(K')_{\text{tors}}\simeq\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/mn\mathbb{Z}.$$
Indeed, take any $(E,K)$... |
3,682,900 | <p>I have a hard time solving this one.
I'm sure there is trick that should be used but if so, I can't spot it.</p>
<p><span class="math-container">$$(3\cdot4^{-x+2}-48)\cdot(2^x-16)\leqslant0$$</span></p>
<p>Here is what I get but I'm anything but confident about this:</p>
<p><span class="math-container">$$3\cdot(2... | hamam_Abdallah | 369,188 | <p>If we put <span class="math-container">$t=2^x>0$</span>, the inequation becomes</p>
<p><span class="math-container">$$48(\frac{1}{t^2}-1)(t-16)\le 0$$</span></p>
<p>which equivalent to
<span class="math-container">$$(1-t)(1+t)(t-16)\le 0$$</span></p>
<p>or
<span class="math-container">$$(1-t)(t-16)\le 0$$</spa... |
1,400,399 | <p>Here is an indefinite integral that is similar to an integral I wanna propose for a contest. Apart from
using CAS, do you see any very easy way of calculating it?</p>
<p>$$\int \frac{1+2x +3 x^2}{\left(2+x+x^2+x^3\right) \sqrt{1+\sqrt{2+x+x^2+x^3}}} \, dx$$</p>
<p><strong>EDIT:</strong> It's a part from the gener... | lab bhattacharjee | 33,337 | <p>HINT: </p>
<p>As $\dfrac{d(2+x+x^2+x^3)}{dx}=1+2x+3x^2,$</p>
<p>let $\sqrt{1+\sqrt{2+x+x^2+x^3}}=u\implies 2+x+x^2+x^3=(u^2-1)^2$</p>
<p>and $(1+2x+3x^2)dx=(u^2-1)2u\ du$</p>
<p>Now use <a href="http://mathworld.wolfram.com/PartialFractionDecomposition.html">Partial Fraction Decomposition</a>, </p>
<p>$\dfrac1... |
3,417,227 | <p><strong>Problem</strong>:</p>
<p>Let <span class="math-container">$f : \Bbb R \to \Bbb R$</span> be a differentiable function such as <span class="math-container">$f(0) = 0$</span>, compute </p>
<p><span class="math-container">$$\lim_{r\to 0^{+}} \iint_{x^2 + y^2 \leq r^2} {3 \over 2\pi r^3} f(\sqrt{x^2+y^2}) dx$$... | Gabriel Palau | 619,715 | <p><span class="math-container">$\lim_{R\to 0^{+}} \iint_{x^2 + y^2 \leq R^2} {3 \over 2\pi r^{3}} f(\sqrt{x^2+y^2}) dxdy
=\lim_{R\to 0^{+}} \int_{0}^{2\pi}\int_{0}^{R} {3 \over 2\pi r^3}r^{2} f(r) drd\theta
=\lim_{R\to 0^{+}} \int_{0}^{2\pi}\int_{0}^{R} {3 \over 2\pi r} f(r) drd\theta$</span></p>
<p>integrating in <s... |
770,430 | <p>How to find the value of $X$?</p>
<p>If $X$= $\frac {1}{1001}$+$\frac {1}{1002}$+$
\frac {1}{1003}$. . . . $\frac {1}{3001}$</p>
| Lucian | 93,448 | <blockquote>
<p><em>How to find the value of X ?</em></p>
</blockquote>
<p>You don't. All you can do is to approximate it with $\ln3000-\ln1000=\ln\dfrac{3000}{1000}=\ln3$.</p>
|
770,430 | <p>How to find the value of $X$?</p>
<p>If $X$= $\frac {1}{1001}$+$\frac {1}{1002}$+$
\frac {1}{1003}$. . . . $\frac {1}{3001}$</p>
| pi37 | 46,271 | <p>According to Wolfram Alpha, the exact answer is $\frac{p}{q}$, where
p=75328031318485390324661526737425033504382783697307257420576924210199013865675494588453357597732611836759615279262897804228616138318504937188371116285762401242413015264220245625479603089419098916457166245209822607501071955143499155953588750190801... |
49,544 | <p>In reading section 2.2, page 14 of <a href="http://www.gaussianprocess.org/gpml/chapters/" rel="nofollow noreferrer">this book</a>, I came across the term "singular distribution".</p>
<p>Apparently, a multivariate Gaussian distribution is singular if and only if its covariance matrix is singular. One way (... | Shai Covo | 2,810 | <p>It turns out that there are two common definitions for singular distribution; see <a href="https://encyclopediaofmath.org/wiki/Singular_distribution" rel="nofollow noreferrer">this article</a> (Singular distribution - Springer Online Reference Works).</p>
<p>According to one definition, it is a probability distribut... |
396,088 | <p>Let <span class="math-container">$K$</span> be a field and let <span class="math-container">$\Lambda_{1}$</span> and <span class="math-container">$\Lambda_{2}$</span> be two finite-dimensional <span class="math-container">$K$</span>-algebras with Jacobson radicals <span class="math-container">$J_{1}$</span> and <spa... | Benjamin Steinberg | 15,934 | <p>Here is an alternate version of @Mare's answer. First recall that a <span class="math-container">$K$</span>-algebra <span class="math-container">$A$</span> is separable if it is semisimple under all base extensions; its enough to check over an algebraic closure of <span class="math-container">$K$</span>.</p>
<p>Le... |
1,829,342 | <p>So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone point me to a formula with proof for these two sums? My searches thus far have only turned up those first two sums wi... | Brian M. Scott | 12,042 | <p>Here’s a solution not using generating functions. Let </p>
<p>$$a_n=\sum_kk\binom{n}{2k}\;,$$</p>
<p>the first of your two sums. Suppose that you have a pool of players numbered $1$ through $n$; then $k\binom{n}{2k}$ is the number of ways to choose $2k$ players from the pool to form a team and then designate one o... |
2,198,454 | <p>My professor's solution to this is as follows: "Create a 2 x 2 matrix with the first row (corresponding to $X=0$) summing to $P(X=0)=1-p$, the second row summing to $P(X=1)=p$, the first column ($Y=0$) summing to $P(Y=0)=1-q$ and the second column summing to $P(Y=1)=q$. We want to maximize the sum of the diagonal wh... | Misha Lavrov | 383,078 | <p>The first diagonal entry is the probability that $X=Y=0$, and we know that both of the following statements are true:</p>
<ul>
<li>Since $\Pr[X=Y=0] \le \Pr[X=0]$, it is at most $1-p$.</li>
<li>Since $\Pr[X=Y=0] \le \Pr[Y=0]$, it is at most $1-q$. </li>
</ul>
<p>However, $p>q$, so the first constraint is strong... |
2,539,888 | <p>I have an polynomial $x^4+x+1 \in \mathbb{Z}\left\{ x\right\}$ and I want to construct an extension field of $\mathbb{Z}_2$ that include the roots of that polynomial. So is this the right approach?</p>
<p>Let E be the extension field.
$$E= \mathbb{Z}_2 / <x^4+x+1> $$?</p>
<p>If so, how do I find the root of ... | Dr. Sonnhard Graubner | 175,066 | <p>i have your term factorized in the form $$-(c-b)^2 (2 b+c) (b+2 c)$$</p>
|
2,924,380 | <p><span class="math-container">$\sum_{k=0}^{n}{k\binom{n}{k}}=n2^{n-1}$</span></p>
<p><span class="math-container">$n2^{n-1} = \frac{n}{2}2^{n} = \frac{n}{2}(1+1)^n = \frac{n}{2}\sum_{k=0}^{n}{\binom{n}{k}}$</span></p>
<p>That's all I got so far, I don't know how to proceed</p>
| Sri-Amirthan Theivendran | 302,692 | <p><strong>Two approaches:</strong></p>
<p><strong>First Approach:</strong> Consider <span class="math-container">$(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k$</span>, differentiate both sides with respect to <span class="math-container">$x$</span> and substitute <span class="math-container">$x=1$</span>.</p>
<p><strong>Sec... |
2,924,380 | <p><span class="math-container">$\sum_{k=0}^{n}{k\binom{n}{k}}=n2^{n-1}$</span></p>
<p><span class="math-container">$n2^{n-1} = \frac{n}{2}2^{n} = \frac{n}{2}(1+1)^n = \frac{n}{2}\sum_{k=0}^{n}{\binom{n}{k}}$</span></p>
<p>That's all I got so far, I don't know how to proceed</p>
| Quiver | 564,698 | <p>I want to give another way of doing it, but all the other answers are very good (I really like the differentiation one). Mine is going to be more of simple arithmetic and change of indices.</p>
<p>Note that the binomial coefficient can be written as <span class="math-container">$${n\choose k} = \frac{n!}{k!(n-k)!}$... |
2,425,337 | <p>What would be an example of a real valued sequence $\{a_{n}\}_{n=1}^{\infty}$ such that $$\frac{a_{n}}{a_{n+1}} = 1 + \frac{1}{n} + \frac{p}{n \ln n} + O\left(\frac{1}{n \ln^{2}n}\right)\ ?$$</p>
| robjohn | 13,854 | <p>$$
f_n(x)=\frac{\alpha\,}{1+\left(\frac{2x}{1-2x}\right)^n}
$$
For $\alpha=1$ and $n=1,2,3,4,5$, we get
<a href="https://i.stack.imgur.com/B5WoM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B5WoM.png" alt="enter image description here"></a></p>
|
676,171 | <blockquote>
<p>Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal.</p>
</blockquote>
<p>Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any suggestions on this would be appreciated. </p>
| ricardio | 124,869 | <p>If $F$ is a field, $F[x]$ is a Euclidean Domain (just the normal division of polynomials you learn in high school). Thus $F[x]$ is a PID and a UFD. In a PID or UFD, proper prime ideals are maximal.</p>
|
3,294,402 | <p>Prove that it is impossible for three consecutive squares to sum to another perfect square.
I have tried for the three numbers <span class="math-container">$x-1$</span>, <span class="math-container">$x$</span>, and <span class="math-container">$x+1$</span>.</p>
| John Omielan | 602,049 | <p>As you stated, you let the <span class="math-container">$3$</span> consecutive numbers be <span class="math-container">$x-1$</span>, <span class="math-container">$x$</span>, and <span class="math-container">$x+1$</span>. This will give you a sum of their squares to be <span class="math-container">$3x^2 + 2$</span>. ... |
3,294,402 | <p>Prove that it is impossible for three consecutive squares to sum to another perfect square.
I have tried for the three numbers <span class="math-container">$x-1$</span>, <span class="math-container">$x$</span>, and <span class="math-container">$x+1$</span>.</p>
| James Arathoon | 448,397 | <p>I will attempt a proof by contradiction.</p>
<p>Assume that the given sum of three consecutive squares sums to the square <span class="math-container">$(x+n)^2$</span>, <span class="math-container">$n$</span> being a positive integer like <span class="math-container">$x$</span>. Thus the statement or proposition we... |
2,782,726 | <p>I'm reading through some lecture notes to prepare myself for analysis next semester and stumbled along the following exercises: </p>
<p>a) Prove that $\lim_{x\to0} f(x)=b$ is equivalent to the statement $\lim_{x\to0} f(x^3)=b$.</p>
<p>b) Give an example of a map where $\lim_{x\to0} f(x^2)$ exists, but $\lim_{x\to0... | user | 505,767 | <p>Yes your example for point b) is a good example.</p>
<p>For a) the property is true for continuity of the function $x^3$ and since $x^3 \to 0$ as $x\to 0$. Yes we can prove that by the $\epsilon-\delta$ definition.</p>
<p>Refer also to <a href="https://math.stackexchange.com/questions/167926/formal-basis-for-varia... |
2,782,726 | <p>I'm reading through some lecture notes to prepare myself for analysis next semester and stumbled along the following exercises: </p>
<p>a) Prove that $\lim_{x\to0} f(x)=b$ is equivalent to the statement $\lim_{x\to0} f(x^3)=b$.</p>
<p>b) Give an example of a map where $\lim_{x\to0} f(x^2)$ exists, but $\lim_{x\to0... | Community | -1 | <p>Suppose, for every $\epsilon>0$, whenever $|x-0|< \delta$, it holds true that $|f(x)-b|<\epsilon$. With this in mind we can make the case for $x^3$,</p>
<p>By using the fact that $x^3$ is bijective and </p>
<p>for any $x \in \mathbb{R}$, $p(x)$ is true$\iff$ for any $x^3 \in \mathbb{R}$, $p(x^3)$ is true ... |
180,839 | <p>Is there any software which can be used for computing Thurston's unit ball (for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy?</p>
<p>PS: even a table for Thurston's ball of two component links would be helpful for me.</p>
| Igor Rivin | 11,142 | <p>I am not aware of any implementation. The best known algorithm is <a href="http://arxiv.org/abs/0706.0673" rel="noreferrer">Cooper and Tillmann's</a>, the closest (which is not very) to a table is in <a href="http://www.math.harvard.edu/~ctm/papers/home/text/papers/alex/alex.pdf" rel="noreferrer">Curt McMullen's cla... |
408,344 | <p>I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly theoretical unfortunately).</p>
<p>Right now I'm interested in the cohomology of quiver and character varieties and their li... | Libli | 37,214 | <p>M. Haiman "Notes on Macdonald polynomials and the geometry of the Hilbert scheme of points on <span class="math-container">$\mathbb{P}^2$</span>". By one of the greatest specialists of interactions between combinatorics and algebraic geometry.</p>
|
408,344 | <p>I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly theoretical unfortunately).</p>
<p>Right now I'm interested in the cohomology of quiver and character varieties and their li... | Per Alexandersson | 1,056 | <p>I am not that strong on the representation-theory side, but know more about the combinatorics side. If you want to get an overview of the symmetric functions and the associated combinatorics (crystal bases, RSK etc), then one starting point (with references!) is <a href="https://www.symmetricfunctions.com" rel="nore... |
3,195,618 | <p>Prove that topological space <span class="math-container">$ \mathbb{R^2} $</span> with dictionary order topology is first countable, but not second countable.</p>
<p>I am a bit stuck. Some hints would help. For first countability I am having trouble finding a local base for each <span class="math-container">$ (x,y)... | JJacquelin | 108,514 | <p><span class="math-container">$$\frac{d^2y}{dx^2} + \frac{2}{4x} \frac{dy}{dx} + \frac{9}{4x} y = 0 \text{ with transformation } t = \sqrt{x}$$</span>
<span class="math-container">$\frac{dt}{dx}=\frac{1}{2\sqrt{x}}=\frac{1}{2t}$</span></p>
<p><span class="math-container">$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\fr... |
237,446 | <p>I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$
what gives $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \geqslant \lim_{n\rightarrow\inft... | André Nicolas | 6,312 | <p><strong>Hint:</strong> We look at the behaviour of
$$x\left(1-\sqrt[x]{\log x}\right)$$
for large $x$. Rewrite the expression as
$$\frac{1-e^{\frac{\log\log x}{x}} }{\frac{1}{x}}.$$
Top and bottom both approach $0$ as $x\to\infty$, so the conditions for using L'Hospital's Rule hold. The rest is a calculation. </p>... |
3,255,644 | <p>I need help understanding the definitions and context for a homework question:</p>
<blockquote>
<p>Consider a 3 by 7 matrix A over GF(2) containing distinct columns. The row space C of A is the
subspace over GF(2) generated by the 3 rows. (Extra note: This is a “simplex” code [7,3] with generator
matrix A. It... | ArsenBerk | 505,611 | <p>Row space is a set of vectors generated by row vectors of <span class="math-container">$A$</span>, i.e. any linear combination of <span class="math-container">$0000111, 0011001, 0101010$</span>.</p>
<p>Distance between words <span class="math-container">$u$</span> and <span class="math-container">$v$</span>, genera... |
3,547,995 | <p>I've been trying to figure out the way to solve this for a while now, and I'm hoping someone could point me in the right direction to find the answer (or show me how to solve this).</p>
<p>The problem I'm having is with this equation: <span class="math-container">$(2i-2)^{38}$</span> and I need to solve it using de... | Pythagoras | 701,578 | <p><strong>Hint</strong>: <span class="math-container">$(2i-2)^{38}=2^{38}(i-1)^{38}$</span> and <span class="math-container">$(i-1)^2=-2i$</span>. So de Moivre’s formula is not required.</p>
|
1,902,878 | <p>If $a^x=bc$, $b^y=ca$ and $c^z=ab$, prove that: $xyz=x+y+z+2$.</p>
<p>My Approach;
Here,</p>
<p>$$a^x=bc$$
$$a={bc}^{\frac {1}{x}}$$</p>
<p>and,</p>
<p>$$b={ca}^{\frac {1}{y}}$$
$$c={ab}^{\frac {1}{z}}$$</p>
<p>I got stopped from here. Please help me to continue </p>
| Aryabhatta | 406,086 | <p>Notice here, quite simple approach... </p>
<p>$a^x.b^y.c^z=(bc).(ca).(ab)$
$a^x.b^y.c^z=a^2.b^2.c^2$</p>
<p>Comparing the corresponding exponents...
$x=y=z=2$.</p>
<p>Now,
Ĺ.H.S$=xyz =2\times 2\times 2=8$</p>
<p>R.H.S$=x+y+z+2=2+2+2+2=8$
Proved. </p>
<p>.....))</p>
|
1,336,869 | <p>Does every mod p have at least one element with a non-identical inverse?</p>
<p>I very much suspect this is true, but how can I prove it? For example, in mod 5, some elements have inverses that are not themselves ${2,3}$ and some have themselves as inverses ${1,4}$. Am I assured that every prime $p\gt 2$ will have ... | egreg | 62,967 | <p>An element which is its own inverse modulo $p$ is represented by an integer $x$ such that $x^2\equiv 1\pmod{p}$, that is, $p\mid (x^2-1)$ which means
$$
p\mid x-1 \quad\text{or}\quad p\mid x+1
$$
In other words, either $x\equiv 1\pmod{p}$ or $x\equiv p-1\pmod{p}$.</p>
<p>So, as soon as $p>3$, there are elements ... |
2,886,675 | <p>I suspect the following is exactly true ( for positive $\alpha$ )</p>
<p>\begin{equation}
\sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} }
\end{equation}</p>
<p>If the above is exactly true, then I would like to know a proof of it.
I accept showing a particular limit is true, may b... | Jack D'Aurizio | 44,121 | <p>It is not an exact equality. By the <a href="https://en.wikipedia.org/wiki/Poisson_summation_formula" rel="nofollow noreferrer">Poisson summation formula</a>, assuming $\alpha>0$,</p>
<p>$$ \sum_{n\in\mathbb{Z}}e^{-\alpha n^2} = \sqrt{\frac{\pi}{\alpha}}\sum_{n\in\mathbb{Z}}e^{-\pi^2 n^2 / \alpha} \tag{1}$$
henc... |
417,064 | <p>Let T be a totally ordered set that is <strong>finite</strong>. Does it follow that minimum and maximum of T exist?
Since T is finite, I believe there exists a minimal of T. From that it maybe able to be shown that the minimal is the minimum but not quite sure whether it is the right approach. </p>
| Cameron Buie | 28,900 | <p>Yes, so long as $T$ is nonempty. Since $T$ is totally ordered, then minimal is equivalent to minimum (one direction is easy, the other follows by totality/comparability). Similarly for maximal and maximum.</p>
|
379,669 | <p>So I was exploring some math the other day... and I came across the following neat identity:</p>
<p>Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1... | Sidharth Ghoshal | 58,294 | <p>Having been quite a bit of time I thought about this some more!</p>
<p>If we cut our original radical at a finite time we have (as pointed out by Aryabhata) we have</p>
<p>$$y = n!\frac{d^n}{dx^n} $$</p>
<p>Solutions to this include $$ y= C_1 e^{x\sqrt[n]{n!}_1} +C_2 e^{x\sqrt[n]{n!}_2}... C_n e^{x\sqrt[n]{n!}_n}... |
2,021,217 | <p>I tried to solve this limit
$$\lim_{x\to 1} \left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$$
and, without thinking, I thought the result was 1. But, using wolfram to verify, I noticed that the limit is $1/2$. </p>
<p>How can I solve it without Hopital/series/integration, just with known limits (link in comments) /sque... | marco2013 | 79,890 | <p>Let $E=(\mathbb{R}/\mathbb{Z})^6$. Let $x=(\overline{a},\overline{b},\overline{c},\overline{d},\overline{e},\overline{f})\in E$. Let $F=\mathbb{N}x$.</p>
<p>$\overline{F}$ is a subgroup of $E$.</p>
<p>$a,b,c,d,e,f$ are irrationnal, so it exists $(u_n)\in\mathbb{N}^{\mathbb{N}}$ and $(\delta_a,\delta_b,\delta_c,\de... |
3,991,351 | <p>As stated in the title.</p>
<p>Any arbitrary function can be expressed as
<span class="math-container">$$f(x)=\frac{a_0}{2}+\sum^{\infty}_{n=1}(a_n\cos(nx)+b_n\sin(nx)) \tag{1}$$</span>
where
<span class="math-container">$$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx \tag{2}$$</span>
<span class="math-container"... | GEdgar | 442 | <p>Let's try this definition: <span class="math-container">$f(x) = e^{ix}$</span> is the unique differentiable function <span class="math-container">$f : \mathbb R \to \mathbb C$</span> that satisfies
<span class="math-container">$$
f'(x) = if(x),\qquad f(0)=1.
\tag4$$</span>
Then let's try to evaluate the required int... |
3,354,684 | <p>I was trying to prove following inequality:</p>
<p><span class="math-container">$$|\sin n\theta| \leq n\sin \theta \
\text{for all n=1,2,3... and } \
0<\theta<π $$</span></p>
<p>I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?</p>
| Conrad | 298,272 | <p>Not sure that this is what you want, but a neat way to do it is noticing that if <span class="math-container">$0 < \theta < \pi$</span>:</p>
<p><span class="math-container">$|1+e^{2i\theta}+...+e^{2i(n-1)\theta}|=\frac{|\sin (n\theta)|}{\sin (\theta)}$</span> and then use the triangle inequality on LHS</p>
|
88,319 | <p>I've had the same effect in Mathematica 9 and 10.</p>
<p>I'm trying to color a 3D Plot with another function, let's call it colorFun ( it should highlight the areas where the colorFun is above a certain threshold), but ColorFunction seems to use the wrong coordinates.</p>
<p>Horribly colored minimal example</p>
<... | GeckoGeorge | 26,921 | <p>And immediately after submitting, I find the answer <a href="https://mathematica.stackexchange.com/questions/58951/set-the-scale-in-colorfunction">here</a>:</p>
<p>Using ColorFunctionScaling->False</p>
<pre><code>Plot3D[Evaluate[x^2+y^2],{x,0,1},{y,0,2},ColorFunction->colorFun,ColorFunctionScaling->False]
</... |
552,395 | <p>I have $f(x)$=$(2x,e^x)$
what does this notation mean? Notation: $Df(\frac{∂}{∂x})$</p>
<p>Certainly $Df(x)$=$(2,e^x)$
but how can I replace $x$ with $\frac{∂}{∂x}$?</p>
<p>Particularly, how can I make sense of $e^{\frac{∂}{∂x}}$</p>
| dezign | 32,937 | <p>In your case $f$ is a map from $\mathbb{R}^1$ to $\mathbb{R}^2$, so at any point $x\in \mathbb{R}^1$ $Df$ is a linear map from $T\mathbb{R}^1_x$ to $T\mathbb{R}^2_{f(x)}$, and is given by the $1\times 2$ matrix $(2,e^x)^T$. To clarify the notation, $\frac{\partial}{\partial x}$ is notation for the unit tangent vecto... |
4,364,421 | <p>Are all solutions of the equation <span class="math-container">$x^2-4My^2=K^2$</span>, multiples of <span class="math-container">$K$</span>? I am considering <span class="math-container">$M$</span> not perfect square. Any tests in Python show be true, but...</p>
<p>My code:</p>
<pre><code>for x in range (1,8000):
... | Thomas Andrews | 7,933 | <p>For the case <span class="math-container">$M=5, K=6,$</span> all solutions have <span class="math-container">$x$</span> divisible by <span class="math-container">$6.$</span></p>
<p>Modulo <span class="math-container">$2,$</span> you'd get <span class="math-container">$x^2\equiv 0\pmod{2},$</span> so <span class="mat... |
4,027,889 | <p>I'm trying to graph <span class="math-container">$f(x,y)=\ln(x)-y$</span>, however, I am not sure how as all of my tools are refusing to graph it.</p>
<p>Can you please help me?</p>
<p>Thanks</p>
| VIVID | 752,069 | <p>You cannot draw it on the <span class="math-container">$Oxy$</span>-plane. You have two input values <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and so you need one more axis to represent the value of <span class="math-container">$f$</span>. Below is the graph drawn via GeoGeb... |
121,897 | <p>I want to check if a user input the function with all the specified variables or not. For that I choose the replace variables with some values and check for if the result is a number or not via a doloop. I am thinking there might be more elegant way of doing it such as <a href="http://reference.wolfram.com/language... | kglr | 125 | <p>Another <em>undocumented</em> function: <code>Internal`LiterallyOccurringQ</code>:</p>
<pre><code>And @@ (Internal`LiterallyOccurringQ[u, #] & /@ vas)
</code></pre>
<blockquote>
<p>True</p>
</blockquote>
<pre><code>And @@ (Internal`LiterallyOccurringQ[u, #] & /@ {x, y, t})
</code></pre>
<blockquote>
... |
381,036 | <p>I must show that $f(x)=p{\sqrt{x}}$ , $p>0$ is continuous on the interval [0,1). </p>
<p>I'm not sure how I show that a function is continuous on an interval, as opposed to at a particular point. </p>
| Ben | 33,679 | <p>Firstly consider the case where $x_0 \in (0,1)$ then: $$ |f(x)-f(x_0)| =|p{(\sqrt{x}-\sqrt{x_0})}|= |p|\lvert{x-x_0 \over \sqrt{x} +\sqrt{x_0}}\rvert = |p|{|x-x_0| \over \sqrt{x} +\sqrt{x_0}} \tag{1}$$</p>
<p>We want to show that this gets small as we move toward $x_0$. Take the distance of $x$ from $x_0$ to be le... |
2,546,497 | <p>987x ≡ 610 (mod 1597)</p>
<p>Is this correct way of applying little Fermat's theorem for linear congruences? Does it make any sense? If not could someone advice a bit.</p>
<p>Since gcd(987,1597)=1</p>
<p>-> 987ˆ1597-1 ≡ 1 (mod 1597)</p>
<p>-> 987ˆ1596 ≡ 1 (mod 1597)</p>
<pre><code>610 ≡ 610 (1597)
987ˆ1596 * 6... | Green | 357,732 | <p>It's because all the 1's are indistinguishable. For example, if you have 1001 and the 1's and 0's were distinguishable, you have $4!$ options, but since they aren't you have $4!$ divided by $2! * 2!$ (which is equivalent to 4 choose 2). </p>
<p>As a result, to find out the total number of ways to have r 1's in a st... |
3,693,735 | <p><span class="math-container">$X \not= \emptyset$</span>,<span class="math-container">$Y \not= \emptyset$</span>,<span class="math-container">$(X,T)$</span> and <span class="math-container">$(Y,V)$</span> are topological space. Let <span class="math-container">$f:X \rightarrow Y$</span> function is a homeomorphizm an... | Community | -1 | <p>I think that a solution based solely on the Class Equation can be given, as follows.</p>
<p>For your group, the Class Equation reads (see <em>e.g.</em> <a href="https://math.stackexchange.com/q/4185478/943729">here</a>):</p>
<p><span class="math-container">$$pq=1+k_pp+k_qq \tag 1$$</span></p>
<p>where <span class="m... |
1,771,920 | <p>Okay so here's the question </p>
<blockquote>
<p>Seventy percent of the light aircraft that disappear while in flight
in a certain country are subsequently discovered. Of the aircraft that
are discovered, 60% have an emergency locator, whereas 90% of the
aircraft not discovered do not have such a locator. S... | Luis Vera | 178,730 | <p>Intuitively, you can think of $P(E \mid D)$ in the numerator as "given that you are already inside the $70 \%$ that has been discovered, what is the probability that inside that $70 \% $ it does have an emergency locator?" so you don't take $.7$ into account in the calculation, you only take $.6$.</p>
<p>You can al... |
1,771,920 | <p>Okay so here's the question </p>
<blockquote>
<p>Seventy percent of the light aircraft that disappear while in flight
in a certain country are subsequently discovered. Of the aircraft that
are discovered, 60% have an emergency locator, whereas 90% of the
aircraft not discovered do not have such a locator. S... | heropup | 118,193 | <p>Let's use a frequency table of a deterministic cohort of light aircraft that disappear. Suppose there are $N = 100$ such aircraft. As $70\%$ of these are discovered, this means $70$ aircraft belong in the group $D$, indicating that they are subsequently discovered, and $30$ aircraft belong in the group $\bar D$, i... |
1,919,880 | <p>Let $B=\begin{bmatrix}
0 & 0 & 0 & -8 \\
1&0&0 & 16\\
0 &1&0& -14\\
0&0&1&6
\end{bmatrix}$</p>
<p>Consider B a real matrix. Find its Jordan Form. So, the characteristic polynomial for B is $(x^2-2x+2)(x-2)^2$. Suppose $B$ represents $T$ in the standard basis. B... | Sarvesh Ravichandran Iyer | 316,409 | <p>You are getting something wrong. The point is, that the eigenvalues of this expression are $2,1+i$ and $1-i$, of which the eigenvalue $2$ has multiplicity $2$, hence the Jordan canonical form $J$ will look like:$$
\begin{pmatrix}
2 \qquad 1\qquad 0\qquad 0 \\
0\qquad 2 \qquad 0 \qquad 0 \\
0\qquad 0 \ \quad 1-i \ ... |
2,339,521 | <p>In a game, each trial consists of two possible outcomes, success or failure. Two trials $H$ and $K$ are carried out. The probability of success for trial $H$ is $x$, and the probability of success for trial $K$ is $2/5$ if trial $H$ is a success, and $x/2$ if trial $H$ is a failure. Given that the probability of tri... | G Tony Jacobs | 92,129 | <p>There are two ways to obtain exactly one success: Success on $H$ and failure on $K$, or failure on $H$ and success on $K$. The probability of the first option is $(x)(3/5)$, and the probability of the second option is $(1-x)(x/2)$. The sum of those two probabilities should be $1/5$.</p>
<p>Does that help?</p>
|
202,379 | <p>Suppose for some constants $\alpha,\beta,\gamma$ that we're given the following ODE: $$\alpha y''+\beta xy'+\gamma y=0.$$ Now, I know how to find the general solution for $y(x)$ if any of $\alpha,\beta,\gamma$ should turn out to be $0$, but I've just ended up with the ODE $$2y''+xy'+y=0.$$ Can anybody give me the fi... | Ayman Hourieh | 4,583 | <p>Write $xy'+y$ as $(xy)'$ and integrate to get:
$$
y' + \frac{x}{2}y = c_1
$$</p>
<p>Which can be solved using the integrating factor $\exp\left(\int \frac{x}{2} \, dx\right) = \exp\left(\frac{x^2}{4}\right)$. The solution cannot be written in terms of elementary functions though:</p>
<p>\begin{align*}
\exp\left(\f... |
701,241 | <p><code>¬(p∨q)∧(p∨r)</code> Does this mean the negation of both <code>(p∨q)</code> and <code>(p∨r)</code> or just <code>(p∨q)</code>?
If it was just <code>p∨q</code> it would make more sense to me being inside the brackets like <code>(¬p∨q)</code> but maybe that's just the programmer in me. I have also seen <code>(¬p∨... | homegrown | 125,659 | <p>Firstly, the $\neg$ is applied only to $(p\lor q).$ If it was intended for the whole statement to be negated, then it would look like $\neg[(p\lor q)\land (p\lor r)].$ Now, for your second question, $\neg(p\lor q)$ is <strong>not</strong> logically equivalent to $(\neg p\lor \neg q).$ According to De Morgan's Law... |
578,961 | <p>Let $\mathbf{T}=[\mathbf{t}_1,\dots,\mathbf{t}_d]$ be a $m\times d$ matrix with $\mathbf{t}_i$ as its linearly independent columns. Also I assume $d<\min(m,n)$. Let $\mathbf{H}$ be a $n\times m$ matrix. Let $\mathbf{W}$ be a $n \times n$ positive definite matrix. For $i=1,\dots,d$, let me define the matrices
\beg... | user1551 | 1,551 | <p>Let $u_i=W^{-1/2}Ht_i$ and $U=\pmatrix{u_1&u_2&\cdots&u_d}\in M_{n\times d}(\mathbb{C})$. Then $\alpha_i=\|C_i^{-1/2}Ht_i\|^2=u_i^H(W+UU^H - u_iu_i^H)^{-1}u_i$ and $T^HH^HW^{-1}HT=U^HU$. In this formulation, since $U^HU$ depends solely on $U$ but $\alpha_i$ depends on both $U$ and $W$, there is no reason... |
3,956,828 | <p>I can find the nth integral of <span class="math-container">$\ln(z)$</span> as follows:
<span class="math-container">\begin{aligned}
\left(\frac d{dz}\right)^{-n}\ln(z)&=\frac1{\Gamma(n)}\int\limits_0^z(z-t)^{n-1}\ln(t)dt\\
&=\frac1{n!}\left[\int\limits_0^z\frac1t(z-t)^ndt-z^n\ln(0)\right]\\
&=\frac1{n!}... | Claude Leibovici | 82,404 | <p><em>This is not a proof</em></p>
<p>Consider
<span class="math-container">$$f_1=\int_0^z \log \left(1+\frac{z}{k}\right)\,dz\qquad \text{and} \qquad f_{n}=\int_0^z f_{n-1}\,dz$$</span>
Define
<span class="math-container">$$a_n=n\, a_{n-1} \qquad \text{with} \qquad a_1=1$$</span> and
<span class="math-container">$$g_... |
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Shaurya Gupta | 102,986 | <p>Let x be rational and y be irrational. So let us assume that x + y is rational.<br>
(x+y) - x will also be a rational since rational - rational is always rational.<br>
therefore y is also rational. but that is a contradiction. Hence proved!</p>
|
2,530,788 | <p>$x + y + z = 0$;</p>
<p>$x^2 + y^2 + z^2 = 1$;</p>
<p>$x^3 + y^3 + z^3 = 0$;</p>
<p>I understand that there are multiple solutions which are the permutations of $(\sqrt{ 2 }/2, 0, -\sqrt{2}/2).$</p>
<p>How do i go about solving for it? I have tried the normal brute force gaussian elimination method, Cramer's rul... | Michael Rozenberg | 190,319 | <p>We can use the Viete's theorem.</p>
<p>Indeed, $$0=(x+y+z)^2=1+2(xy+xz+yz),$$ which gives
$$xy+xz+yz=-\frac{1}{2}.$$
Also, since $$(x+y+z)^3=x^3+y^3+z^3+3(xy+xz+yz)(x+y+z)-3xyz,$$ we obtain $$xyz=0,$$
which gives that $x$, $y$ and $z$ are roots of the equation:
$$t^3-\frac{1}{2}t=0.$$</p>
|
1,416,275 | <p>I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very important, but I'm having trouble understanding it. Here is the text:</p>
<blockquote>
<p>Let $\sigma$ be an element of t... | uniquesolution | 265,735 | <p>The textbook chooses to define the action of the dual space as multiplication of row and column vectors. In this approach, an element in $V$ is a column vector, i.e, a matrix of order $n\times 1$, whereas the elements of the dual space $V^{*}$ are row vectors, i.e., matrices of order $1\times n$. So the action of th... |
3,751,780 | <p>Given positive real numbers <span class="math-container">$a, b, c$</span> with <span class="math-container">$ab + bc + ca = 1.$</span> Prove that <span class="math-container">$$ \sqrt{a^{2} + 1} + \sqrt{b^{2} + 1} + \sqrt{c^{2} + 1}\leq 2(a+b+c).$$</span></p>
<p>I have no idea to prove this inequality.</p>
| Z Ahmed | 671,540 | <p>Note that <span class="math-container">$$\sqrt{a^2+1}= \sqrt{a^2+ab+bc+ca}=\sqrt{(a+b)(a+c)}\le \frac{(2a+b+c)}{2}~~\text{AM-GM}\,.$$</span>
Addling three similar results we prove that
<span class="math-container">$$\sqrt{a^2+1}+\sqrt{b^2+1}+\sqrt{c^2+1}\le \frac{4(a+b+c)}{2}=2(a+b+c)\,.$$</span></p>
|
1,072,524 | <p>Suppose $X$ is an integral scheme. I would like to show that the restriction maps
$res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could give me some assistance with this. This is exercise 5.2 I on Ravi Vakil's notes. Thank you!</p>
| Ayman Hourieh | 4,583 | <p>Let $X$ be an integral scheme. Let $\xi$ be its generic point. If we show that the canonical map $\mathcal O_X(U) \to \mathcal O_\xi$ is injective, we're done. Since $U$ can be covered by affine open subsets, we can assume that $U = \operatorname{Spec} A$ is affine. Now the map $\mathcal O_X(U) \to \mathcal O_\xi$ c... |
228,651 | <p>When testing to determine the convergence or divergence of series with positive terms, there's a common way by comparing them with other series which we already know converge or diverge.</p>
<p>My question is, how do we choose the proper to-be-compared series? I hope to get some detailed <strong>methodology</strong... | Scorpio19891119 | 40,071 | <p>Usually, you can choose the highest order of the denominator and numerator, respectively. So in your problem, for the numerator choose $1$, whose highest order is $0$, and for denominator choose $n$. Then, the sequence you should choose is $1/n$. </p>
|
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Will Jagy | 3,324 | <p>I have a story in the middle. I hurt my right shoulder over time, by 1994 it was simply too much to write on a blackboard, at least overhead. So, pre-Beamer, I wrote up these slides on transparencies with colored pens. These were unusually well-prepared lessons for me, I had everything worked out, it was all clearly... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Christopher Perez | 4,566 | <p>Most of the non-mathematics courses I've taken in college were done with lecture slides, and I have to say that there are a number of advantages and disadvantages to them that actually amount to more disadvantages if you were to do the same in math. The one obvious advantage is that the slides can be posted online, ... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Brian M. Scott | 12,042 | <p>In topology the letter $F$ is commonly used to denote a closed set, from French <a href="http://fr.wikipedia.org/wiki/Ferm%C3%A9_%28topologie%29" rel="nofollow"><em>fermé</em></a> 'closed [set]'. The common use of $K$ to denote a compact set is probably from German <em>kompakt</em>, as in <a href="http://de.wikipedi... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MartianInvader | 8,309 | <p>The etymology of the $\sin$ function has a colorful history - it comes from <em>sinus</em>, the latin word for... well, bosom. This was due to a mistranslation from Arabic text in the 12th century: The word <em>jaib</em> means bosom, and since Arabic is written without short vowels, it was written essentially as <... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MJD | 25,554 | <p>The center of a group $G$ is denoted $Z(G)$. The $Z$ is for “Zentrum”, which is the German word for ‘center’.</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| charlotte | 195,474 | <p>Most of the good examples have been mentioned already.</p>
<ul>
<li>$+$ probably comes from Latin <em>et</em>, "and"</li>
<li>$\tau$ is sometimes used for the golden ratio, from Greek <em>tomos</em>, "section" (same root as <em>atom</em>).</li>
<li>$\sinh$ from Latin <em>sinus hyperbolus</em> (and so on)</li>
<li>$... |
2,432,817 | <p>Let $X$ and $Y$ be topological spaces and let $A \subseteq X$ be a subspace of $X$. Suppose $A$ is homeomorphic to some subspace $B \subseteq Y$ of $Y$. Let $f$ explicitly denote this homeomorphism.</p>
<p>If $f : A \to B$ is a homeomorphism, does $f$ extend to a homeomorphism between $\text{Cl}_X(A)$ and $\text{Cl... | Henno Brandsma | 4,280 | <p>$A=S^1\setminus \{p\} \subseteq X= S^1$ (for any $p \in S^1$) is homeomorphic to $B=(0,1) \subseteq Y = [0,1]$. But their respective closures $X$ and $Y$ are not.</p>
<p>More trivially: $A = (0,1) \subseteq X=\mathbb{R}$ and $B = Y = \mathbb{R}$, where $\overline{B} = B$ but $\overline{A}$ becomes compact.</p>
|
2,573,572 | <p>Here is the expression to take the derivative of.
$$C = \frac{1}{2}\sum_j (y_j - a_j^L)^2$$</p>
<p>Here is the result.
$$\frac{\partial C}{\partial a_j^L} = 2(a_j^L-y_j)$$</p>
<p>Multiplying by 2, then again by the derivative of the inside (-1) seems reasonable, but what happened to the summation?</p>
| user | 505,767 | <p>The derivative is with respect to a component $a_j^L$ thus the others cancel out.</p>
|
2,864,992 | <p>It starts by someone asking an exercise question that whether negation of</p>
<pre><code>2 is a rational number
</code></pre>
<p>is</p>
<pre><code>2 is an irrational number
</code></pre>
<p>Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irra... | Somos | 438,089 | <p>Let $\ f(x) := \log_2(1-x) + \sum_{n=0}^\infty x^{2^n}, \ $
$\ g(x) := f(e^{-x}) = \log_2(1-e^{-x}) + \sum_{n=0}^\infty e^{-x2^n}, \ $
and $\ a_k := g(2^{-k}) = b_k +
\sum_{n=0}^\infty e^{-2^{n-k}} \ $ where
$\ b_k := \log_2(1-e^{-2^{-k}}) \approx -k - 2^{-1-k}/\log(2). \ $
Now $\ \sum_{n=0}^\infty e^{-2^{n-... |
90,673 | <p>let $M$ be a closed ortientable irreducible 3-mfd, let $T$ be a non-separating torus in $M$, we cut $M$ along $T$ and glue two solid tori along the two boundary tori, we get a new closed 3-mfd $M_1$ (we need $M_1$ also is irreducible). Now If there exists nonseperable torus $T_1$ in $M_1$, we go on the above process... | Bruno Martelli | 6,205 | <p>The answer is "no", although it seems that a homological argument is not enough as Kevin's and Bin's examples show. I describe an argument which uses geometrization.</p>
<p>There is a quantity which decreases strictly at each operation. It is crucial to suppose that both $M$ and $M_1$ are irreducible. The quantity ... |
1,170,088 | <blockquote>
<p>In a group of $10$ people, $60\%$ have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will
have brown eyes?</p>
</blockquote>
<p>How do I do this problem? $6$ people have brown eyes and $4$ people don't. </p>
<p>The possibil... | Stupid Man | 219,196 | <p>EDIT: I was overcounting in my previous solution.</p>
<p>Probability of an event = Number of favourable outcomes / Total number of outcomes</p>
<p>Here total outcomes = Number of ways to choose 2 people out of 10, which is $10C2$</p>
<p>Number of favourable outcomes = Number of ways to choose one non brown eyed p... |
272,114 | <p>Yesterday, my uncle asked me this question:</p>
<blockquote>
<p>Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.</p>
</blockquote>
<p>How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you are thinking about Fermat's Last Theorem.<... | jim | 44,551 | <p>$$f(x) = \left(\dfrac{3}{5}\right)^x + \left(\dfrac{4}{5}\right)^x -1$$</p>
<p>$$f^ \prime(x) < 0\;\forall x \in \mathbb R\tag{1}$$</p>
<p>$f(2) =0$. If there are two zeros of $f(x)$, then by Rolle's theorem $f^\prime(x)$ will have a zero which is a contradiction to $(1)$.</p>
|
272,114 | <p>Yesterday, my uncle asked me this question:</p>
<blockquote>
<p>Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.</p>
</blockquote>
<p>How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you are thinking about Fermat's Last Theorem.<... | Community | -1 | <p>Let $f(x)=5^x-4^x-3^x$. Then $f(2)=0$.</p>
<p>If $k>0$, then</p>
<p>$f(2+k)=f(2+k)-f(2)$</p>
<p>$=25(5^k-1)-16(4^k-1)-9(3^k-1)$</p>
<p>$>25(5^k-1)-16(5^k-1)-9(5^k-1)=0$.</p>
<p>If $k<0$, then</p>
<p>$f(2+k)=f(2+k)-f(2)$</p>
<p>$=25(5^k-1)-16(4^k-1)-9(3^k-1)$</p>
<p>$<25(5^k-1)-16(5^k-1)-9(5^k-1)=... |
2,203,066 | <p>The definition I have is the following:</p>
<blockquote>
<p>A vector space V is said to be <strong>finite-dimensional</strong> if there is a finite set of vectors in V that spans V and is said to be <strong>infinite-dimensional</strong> if no such set exists.</p>
</blockquote>
<p>However, with this definition I ... | Christopher Pompetzki | 429,330 | <p>A vector space is called finite-dimensional, if some list of vectors in it spans the space. By definition, every list has finite length. For every positive integer $n,$ $\mathbb{F^n}$ is a finite-dimensional vector space. </p>
|
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