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 |
|---|---|---|---|---|
299,056 | <p>Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $K = \mathbb{Q}(\sqrt{d})$ and $\overline{K}$ defined to be the algebraic closure of $K$. Is it true that $\overline{K} \cong \overline{\mathbb{Q}}$?</p>
| Zach L. | 43,128 | <p>If $d \in \mathbb{Q}$, $K$ will be a subfield of $\overline{\mathbb{Q}}$, since $\sqrt{d} \in \overline{\mathbb{Q}}$. Since $\overline{\mathbb{Q}}$ is algebraically closed, it suffices to show that $\overline{\mathbb{Q}}$ is algebraic over $K$. But if $\alpha \in \overline{\mathbb{Q}}$, choose a polynomial in $\math... |
527,799 | <p>In Hatcher's Algebraic Topology section 1.3, Cayley complexes are explained. The book states that we get a Cayley complex out of a Cayley graph by attaching a 2-cell to each loop. There is an example showing the Cayley complex for $\mathbb{Z}\times\mathbb{Z}$ (the fundamental group of the torus). We attach one 2-cel... | Bombyx mori | 32,240 | <p>The Cayley complex of $\mathbb{Z}/n\mathbb{Z}$ contains $n$ vertices and $n$ edges connected in a circle. So if you want to attach a disk, you have $n$ choices because $(123),(231),(312)$ gives three different attaching disks. For example in the $\mathbb{R}\mathbb{P}^{2}$ case, you glue a disk along the relation $(1... |
2,226,150 | <blockquote>
<p>The following diophantine equation came up in the past paper of a Mathematics competition that I am doing soon: $$ 2(x+y)=xy+9.$$ </p>
</blockquote>
<p>Although I know that the solution is $(1,7)$, I am unsure as of how to reach this result. Clearly, the product $xy$ must be odd since $2(x+y)$ must b... | Sarvesh Ravichandran Iyer | 316,409 | <p>$2(x+y) = xy +9 \implies 2x - xy = 9 - 2y \implies x = \frac{9-2y}{2-y} = \frac{2y-9}{y-2} = 2 - \frac{5}{y-2}$.</p>
<p>If $x$ is an integer, then $\frac{5}{y-2}$ is also an integer. This will tell you what $y$ can be, then what $x$ can be. Trying these out will give you the solution $y=7, x=1$ and the solution $y=... |
4,249,789 | <p>My understanding is that the span of a set is a set of all vectors that can be obtained from the linear combination of all the vectors in the original set as shown in image #<a href="https://i.stack.imgur.com/3v6g4.png" rel="nofollow noreferrer">1</a>.</p>
<p><img src="https://i.stack.imgur.com/3v6g4.png" alt="Image... | Stanislav Bashkyrtsev | 24,684 | <p>Matrix rank shows how many linearly independent vectors there are in the matrix. And when determining the span you don't really need any other vectors (linearly dependent vectors are also in the span since they can be formed by linearly independent ones).</p>
<p>The dimension of span is the number of vectors which f... |
223,509 | <p>Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the <a href="https://en.wikipedia.org/wiki/G%C3%B6del_numbering" rel="nofollow">Godel number</a> of $\varphi$.
Let $T$ be a first-order arithmetic theory, e.g. PA.
Let $\Phi$ be a class of close... | Nik Weaver | 23,141 | <p>The law $${\rm Tr}(\ulcorner \phi\urcorner) \to \phi$$ is sometimes called the "release scheme". "T-Elim" and "T-Out" are also used, according to <a href="http://plato.stanford.edu/entries/liar-paradox/" rel="nofollow">this source</a>. The converse is "capture scheme" ("T-Intro", "T-In").</p>
|
2,554,380 | <p>With the metric space $(X,d) : X = \Bbb R$ and $d(x,y) = |x| + |y|$ for $x\neq y$, $d(x,x) = 0$ and $A = \{0\}$. Is interior of $A$ empty? </p>
<p>In usual metric it would be empty but with this metric I conclude that it is $A$ itself.</p>
<p>Def: $ a \in A $ is an interior point of $A$ if $\exists \epsilon > 0... | operatorerror | 210,391 | <p>You want, given a $\epsilon$, a $\delta$ which will work for any points in the domain of your function. </p>
<p>You have a direct control over the distance between values in the range by values in the domain however, $|f(x)-f(y)|< c|x-y|$. Hence, if
$$
|x-y|<\delta/c
$$
then
$$
|f(x)-f(y)|<\delta
$$
and... |
3,002,767 | <p><span class="math-container">$l^p$</span> appears frequently in undergrad real analysis courses, I wonder if there is any strong connection between <span class="math-container">$l^p$</span> and <span class="math-container">$L^p$</span> space? (Other than they look similar)</p>
<p>I give one definition of <span clas... | Nico | 592,433 | <p>Yes. The connection is, that both are the same kind of construction, but over different measure spaces.</p>
<p>The standard definition of <span class="math-container">$l^p$</span> spaces is: <span class="math-container">$l^p = \{x: \mathbb N \to \mathbb R|\quad ||x||_p<\infty\}$</span> where
<span class="math-c... |
878,914 | <p>I'm having a problem to solve this limit.</p>
<p>$$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$$</p>
<p>$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$ =
$\lim_{x \to \pi/4} \frac{\frac{\sin x}{\cos x}-1}{\sin x-\cos x}$=
$\lim_{x \to \pi/4} \frac{\frac{\sin x-\cos x}{\cos x}}{\sin x-\cos x}$=
$\lim_{x ... | lab bhattacharjee | 33,337 | <p>Safely cancel out $\sin x-\cos x$ as $\displaystyle x\to\frac\pi4,\sin x-\cos x\to0\implies\sin x-\cos x\ne0$</p>
<p>Things will be clearer if we write $$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}=\lim_{x\to\dfrac\pi4}\frac{\tan x-1}{\cos x(\tan x-1)}$$</p>
<p>As $\displaystyle x\to\frac\pi4,\tan x\to1\impl... |
3,771,972 | <blockquote>
<p>If gcd<span class="math-container">$(m,n)=1$</span>, then <span class="math-container">$\phi(mn)=\phi(m)\phi(n)$</span>.</p>
</blockquote>
<p>My text book write that <span class="math-container">$f:\Bbb Z/_{(mn)}\Bbb Z\to\Bbb Z/_m\Bbb Z \times \Bbb Z/_n\Bbb Z$</span> by <span class="math-container">$f(... | Dietrich Burde | 83,966 | <p>For a ring <span class="math-container">$R$</span>, the group of units is denoted either by <span class="math-container">$U(R)$</span> or by <span class="math-container">$R^{\times}$</span>. In your example the ring is <span class="math-container">$R=\Bbb Z/m\Bbb Z$</span> fo some integer <span class="math-container... |
50,023 | <p>two problems from Dugundji's book page $156$. (I don't know why the system deletes the word hi in this sentence)</p>
<p>$1$. Let $X$ be a Hausdorff space. Show that:</p>
<p>a) $\bigcap \{F: x \in F , F \ \textrm{closed}\} = \{x\}$.</p>
<p>b) $\bigcap \{U : x \in U, U \ \textrm{open}\} = \{x\}$.</p>
<p>c) The ab... | Stephen J. Herschkorn | 13,048 | <p>Your answer to 1(a) is too complicated. Note that $\{x\}$ is in the collection whereof you are taking the intersection.</p>
<p>Also, your answer to 1(c) is too complicated: The natural numbers under the cofinite topology provides a counterexample.</p>
<p>For 2, I haven't thought through the details, but I think ... |
4,605,772 | <p>I've been familiar with the formula of finding the <span class="math-container">$r$</span>-permutation of an <span class="math-container">$n$</span>-element set; but I've only been half-so familiar with deriving the formula. See, I know that to prove the truth of
<span class="math-container">$$
P(n,r) = \frac{n!}{(... | Ben Grossmann | 81,360 | <p>In case it's helpful (e.g. perhaps it can be modified to work using pseudoinverses), here's an approach that works if <span class="math-container">$\Gamma_j$</span> for <span class="math-container">$j = 1,\dots,n$</span> are invertible.</p>
<p>First of all, note that for the block matrix
<span class="math-container"... |
4,605,772 | <p>I've been familiar with the formula of finding the <span class="math-container">$r$</span>-permutation of an <span class="math-container">$n$</span>-element set; but I've only been half-so familiar with deriving the formula. See, I know that to prove the truth of
<span class="math-container">$$
P(n,r) = \frac{n!}{(... | tch | 352,534 | <p>If it is not possible to store a matrix of the dimensions of <span class="math-container">$\mathbf{C}_n$</span>, one possibility is to use a randomized estimator for the diagonal in conjunction with a linear-system solver.</p>
<p>In particular, note that for a matrix <span class="math-container">$\mathbf{A}$</span>,... |
112,263 | <p>When Mathematica outputs nested lists, it just gloms one record right after the next record and it wraps around in the space available, for example:</p>
<pre><code>In[10]:= FactorInteger[FromDigits["1000000000000101",Range[2,10]]]
Out[10]= {{{13,1},{2521,1}},{{11,1},{251,1},{5197,1}},{{3,2},{229,1},{520981,1}},{{47... | mgamer | 19,726 | <p>Maybe you are looking for something like this:</p>
<pre><code>Print[#] & /@ Flatten[
FactorInteger[FromDigits["1000000000000101", Range[2, 10]]], 1];
</code></pre>
|
3,848,517 | <p>I have a conjecture in my mind regarding Arithmetic Progressions, but I can't seem to prove it. I am quite sure that the conjecture is true though.</p>
<p>The conjecture is this: suppose you have an AP (arithmetic progression):
<span class="math-container">$$a[n] = a[1] + (n-1)d$$</span>
Now, suppose our AP satisfie... | Kosh | 270,689 | <p>The condition you are looking for is uniform convergence. If you want a counterexample just think, on the real line, about a sequence of functions with compact support obtained translating by <span class="math-container">$+n$</span> a given nonnegative and non identically zero function with compact support.</p>
|
766,694 | <p>I am stuck trying to get the values for x, y, and z. I keep moving variables around but I end up getting answers like x = x or z = z and I do not think that is what I want. It's really just algebra but I seem to have forgotten that. </p>
<p>(x/2) + (2z/3) = x</p>
<p>(x/2) + (y/3) = y</p>
<p>(2y/3) + (z/3) = z</p>... | Amzoti | 38,839 | <p>From the first equation, we have:</p>
<p>$$z = \dfrac{3x}{4}$$</p>
<p>From the second equation, we have:</p>
<p>$$y = \dfrac{3x}{4}$$</p>
<p>Substituting these into the third equation gives an identity. So, we have:</p>
<p>$$z = y = \dfrac{3x}{4},~ x ~~\mbox{is a free variable}$$</p>
<p>Note, you could have re... |
2,282,359 | <p>I am trying to calculate this limit:
$$\lim_{x \to \infty} x^2(\ln x-\ln (x-1))-x$$
The answer is $1/2$ but I am trying to verify this through proper means. I have tried L'Hospital's Rule by factoring out an $x$ and putting that as $\frac{1}{x}$ in the denominator (indeterminate form) but it becomes hopeless afterwa... | Community | -1 | <p>$$\ln x - \ln(x-1)=\ln\frac{1}{1-\frac{1}{x}}=\frac{1}{x}+\frac{1}{2}\frac{1}{x^2}+\frac{1}{3}\frac{1}{x^3}+\cdots, $$
so $$x^2(\ln x - \ln(x-1))-x=\frac{1}{2}+\frac{1}{3}\frac{1}{x}+\cdots$$
and thus $$\lim_{x\rightarrow\infty}[x^2(\ln x - \ln(x-1))-x]=\frac{1}{2}.$$</p>
|
729,415 | <blockquote>
<p>If $f$ has a relative extremum at $c$, then either $f'(c) = 0$ or $f'(c)$ does not exist. Can one prove the opposite?</p>
</blockquote>
<p>The statement is due to Fermat. It shows that one can find the relative extrema of $f$ by finding the points $c$ such that $f'(c)=0$ or $f'(c)$ doesn't exist.</p>... | Nitish | 61,574 | <p>It's <em>not</em> true.</p>
<p>\begin{align}
f(x)&=x^3\\
f'(0)&=0
\end{align}</p>
<p>However, $0$ is neither a local minima or maxima. It is in fact a <a href="http://en.wikipedia.org/wiki/Saddle_point" rel="nofollow"><em>saddle point</em></a>.</p>
<p>You can have points like these which are stationary (i... |
729,415 | <blockquote>
<p>If $f$ has a relative extremum at $c$, then either $f'(c) = 0$ or $f'(c)$ does not exist. Can one prove the opposite?</p>
</blockquote>
<p>The statement is due to Fermat. It shows that one can find the relative extrema of $f$ by finding the points $c$ such that $f'(c)=0$ or $f'(c)$ doesn't exist.</p>... | Jared | 138,018 | <p>No, because the other way isn't true. Here's an example:</p>
<p>$$
f(x) = x^3 \\
f'(x) = 3x^2 \rightarrow f'(0) = 0
$$</p>
<p>But $x = 0$ isn't an extremum. The only definition for an extremum is that if the derivative's sign changes then you definitely have an extremum. If the function is differentiable (i.e. ... |
11,315 | <p>I just answered this question:</p>
<p><a href="https://math.stackexchange.com/questions/528880/boolean-formula">Boolean formula over 64 Boolean variables X</a></p>
<p>By the time I had posted my answer, another user had edited the question so as to remove all the mathematical interest.</p>
<p>What is going on her... | Dennis Meng | 35,665 | <p>Anyone is able to edit a question or answer; this is common across <em>all</em> stackexchange sites, not just Mathematics. Those with less than 2000 reputation must first have their edit approved.</p>
<p>It's unfortunate when a question is edited to either change the question asked or otherwise make the question lo... |
950,937 | <p><strong>Specifications and Data in question</strong></p>
<p>1.We have a skew symmetric matrix
$ A(t)_{3\times 3}= \begin{bmatrix}\,0&\!-a_3(t)&\,\,a_2(t)\\ \,\,a_3(t)&0&\!-a_1(t)\\-a_2(t)&\,\,a_1(t)&\,0\end{bmatrix} \tag 1$ and</p>
<p>$B(t)_{3 \times 3}= -\int_{0}^{t} A(s)\ ds \tag 2$ ... | Slade | 33,433 | <p>Suppose I have $n$ light switches, numbered $1$ through $n$. I'd like at least one light on, but other than that I can select any configuration. How many configurations are there?</p>
<p>Now, for some $i$ with $1\leq i\leq n$, how many configurations are there with the property that switch $i$ is on, but all swit... |
950,937 | <p><strong>Specifications and Data in question</strong></p>
<p>1.We have a skew symmetric matrix
$ A(t)_{3\times 3}= \begin{bmatrix}\,0&\!-a_3(t)&\,\,a_2(t)\\ \,\,a_3(t)&0&\!-a_1(t)\\-a_2(t)&\,\,a_1(t)&\,0\end{bmatrix} \tag 1$ and</p>
<p>$B(t)_{3 \times 3}= -\int_{0}^{t} A(s)\ ds \tag 2$ ... | miniparser | 99,402 | <p>I have my own combinatorial proof for this.</p>
<p>$\sum_{i=0}^{n-1}2^i=2^n-1$</p>
<p>Interpreted as $2^0+2^1+2^2+...+2^{n-1}$</p>
<p>Each increase of $n\ge 1$ by $1$ LHS increases the total by the current total plus one, making the total $(2^{n-1}-1)+2^{n-1}=2(2^{n-1})-1=2^n-1$.</p>
|
446,948 | <blockquote>
<p>Suppose that <span class="math-container">$X$</span> is a geometric random variable with parameter (probability of success) <span class="math-container">$p$</span>.</p>
<p>Show that <span class="math-container">$\Pr(X > a+b \mid X>a) = \Pr(X>b)$</span></p>
</blockquote>
<p>First I thought I'd s... | André Nicolas | 6,312 | <p>There are two ways to get $\Pr(X\gt n)$, the easy way and the hard way.
Since there is something to be learned from both, we do it both ways.</p>
<p><strong>The hard way:</strong> The probability that $X=n+1$ is the probability of $n$ failures in a row, then success. This is $(1-p)^np$.</p>
<p>Similarly, the prob... |
48,237 | <p>I have found by a numerical experiment that first such primes are:
$2,5,13,17,29,37,41$. But I cannot work out the general formula for it.<br>
Please share any your ideas on the subject.</p>
| Zarrax | 3,035 | <p>If you know a little group theory it's not too hard. It holds for $p = 2$ so assume $p > 2$. Under multiplication ${\mathbb Z}_p$ is a cyclic group of order $p - 1$. To say an $x$ in ${\mathbb Z}_p$ satisfies $x^2 = -1$ is the same as saying that $x$ is an element of order $4$ in this group. To see why, first not... |
48,237 | <p>I have found by a numerical experiment that first such primes are:
$2,5,13,17,29,37,41$. But I cannot work out the general formula for it.<br>
Please share any your ideas on the subject.</p>
| André Nicolas | 6,312 | <p>The following argument depends on knowing (or separately proving) <strong>Wilson's Theorem</strong>.</p>
<p><strong>Theorem</strong> Let $p$ be prime. Then $(p-1)! \equiv -1 \pmod{p}$.</p>
<p>Now we use Wilson's Theorem to prove the result. For the sake of concreteness, I will use $p=17$, but the same idea exactl... |
3,275,966 | <p>During the drawing lottery falls six balls with numbers from <span class="math-container">$1$</span> to <span class="math-container">$36$</span>. The player buys the ticket and writes in it the numbers of six balls, which in his opinion will fall out during the drawing lottery. The player wants to buy several lotter... | Ross Millikan | 1,827 | <p><span class="math-container">$10$</span> tickets are enough. Start by buying <span class="math-container">$(1,2,3,4,5,6)$</span> and the other five blocks of six numbers. If you have not won yet there must be one number in each block of six. Now buy <span class="math-container">$(1,2,3,7,8,9),(1,2,3,10,11,12),(4,... |
1,042,285 | <p>How can i prove boundedness of the sequence
$$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.</p>
| Bumblebee | 156,886 | <p>$$\Big|\frac{sin (n)}{8+\sqrt{n}}\Big|\le\Big|\frac{1}{8+\sqrt{n}}\Big|\le\dfrac{1}{9}, \forall n\in\mathbb{N}. $$</p>
|
4,415,037 | <p>I would like to prove upper and lower bounds on <span class="math-container">$|\cos(x) - \cos(y)|$</span> in terms of <span class="math-container">$|x-y|$</span>. I was able to show that <span class="math-container">$|\cos(x) - \cos(y)| \leq |x - y|$</span>. I'm stuck on the lower bound. Does anyone know how to appr... | Community | -1 | <p>You might consider this a "satisfactory" answer instead of the obvious <span class="math-container">$0$</span> and <span class="math-container">$-|x-y|$</span>.</p>
<p>What we want to achieve first here is find some <span class="math-container">$f\left|t\right|$</span> so <span class="math-container">$\lef... |
3,104,210 | <p>Is there a commonly accepted notation for a vector whose first <span class="math-container">$k$</span> entries are <span class="math-container">$1$</span>'s, with <span class="math-container">$0$</span>'s afterwards? I have seen <span class="math-container">$\mathbf{e}_i$</span> for a vector with a <span class="mat... | neofelis | 641,909 | <p>As noted in comments, your <span class="math-container">$\phi$</span> is not a property of sets -- it's a class. You're correct when you say that Comp says that given a set, there is a subset of that set whose elements all satisfy some property. What's the property we want? "Being identical to x or being identical t... |
651,174 | <p>I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I let a computer calculate it, it just seems like it finds the form and then multiplies by $i$ and negative $i$. Have a ... | Claude Leibovici | 82,404 | <p><strong>Hint</strong> </p>
<p>Click the "Approximate Forms" button and think about what the numbers could be !</p>
|
2,623,544 | <p>I have to diagonolize a matrix A
\begin{bmatrix}0&-3&-1&1\\2&5&1&-1\\-2&-3&1&1\\2&3&1&1\end{bmatrix}</p>
<hr>
<p>I do $det(A-λ)=0$ and I get $λ_{1}=1$, $λ_{2}=2$, $λ_{3}=2$, $λ_{4}=2$</p>
<p>So possible diagonal matrix looks like:
\begin{bmatrix}1&0&0&0\... | Konstantin | 509,087 | <p>Your computations are totally correct, but you made a tiny mistake in the order of the decomposition of $A$.
It has to be $A = PDP^{-1}$.</p>
<p>Just remind yourself, what if A was e.g. symmetric, since then one could find orthogonal transformations and there you immediately see, that the most right matrix has to c... |
2,249,929 | <p>Is there a more general form for the answer to this <a href="https://math.stackexchange.com/q/1314460/438622">question</a> where a random number within any range can be generated from a source with any range, while preserving uniform distribution? </p>
<p><a href="https://stackoverflow.com/q/137783/866502">This</a>... | hardmath | 3,111 | <p>A general strategy for obtaining a uniform discrete distribution on $1$ to $N$ using an $M$-sided die is to find the least power $r$ such that $M^r\ge N$, then roll the die $r$ times. Assign radix $M$ place-values $0$ to $M-1$ to each face of the die, and compute the radix $M$ value:</p>
<p>$$ m_r m_{r-1} \ldots m... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | user246336 | 246,336 | <p>Here is another approach. Let us define the function </p>
<p>$$\phi(x):=\sum_{n=0}^\infty (n+1)(n+2) x^n.$$</p>
<p>If we differentiate we obtain</p>
<p>$$\phi'(x):=\sum_{n=0}^\infty n(n+1)(n+2) x^{n-1}=\sum_{n=0}^\infty(n+1)(n+2)(n+3)x^n.$$</p>
<p>Now, </p>
<p>$$\phi'(x)-3\phi(x)=\sum_{n=0}^\infty n(n+1)(n+2)x^... |
98,932 | <p>Suppose I would like to use a method for data prediction, and that I have some empirical data (i.e., sequence of samples of the form [time, value]). Would it be possible to know in advance, based on the data only, if it makes sense to use a model for prediction (based on the samples, used as a training set). </p>
... | Community | -1 | <p>Why don't you omit a few data points, when working out the parameters of your model? Then see if the model correctly predicts the data points that you omitted. You'll need to decide ahead of time how accurately the model needs to predict the missing data point, in other words, what constitutes "success" of the mod... |
3,409,342 | <p>Let <span class="math-container">$X$</span> be a set containing <span class="math-container">$A$</span>.</p>
<p>Proof:
<span class="math-container">$y\in A \cup (X \setminus A) \Rightarrow y\in A$</span> or <span class="math-container">$y \in (X \setminus A)$</span></p>
<p>If <span class="math-container">$y \in ... | PrincessEev | 597,568 | <p>The reverse implication is simple enough.</p>
<p>If <span class="math-container">$y \in X$</span>, then either <span class="math-container">$y \in A$</span> or <span class="math-container">$y \not \in A$</span>, since <span class="math-container">$A$</span> is a subset of <span class="math-container">$X$</span>.</p... |
1,361,948 | <p>$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$</p>
<p>$\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$</p>
<p>So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$.</p>
<p>$x(1-x^2-y^2) = y(1+x^2+y^2)$</p>
<p>$x-x^3-xy^2 = y + x^2y + y^3$</p>
<p>Is the guessing the values of the variables the only way of so... | Community | -1 | <p>For $t/\delta=\pi$ and ignoring the inessential $\sigma\delta$,</p>
<p>$$z=\frac{1+i}{1+e^{-\pi}}$$</p>
<p>and</p>
<p>$$\Re(z)=\frac1{1+e^{-\pi}}.$$</p>
<p>This doesn't match</p>
<p>$$\frac1{1-e^{-\pi}}.$$</p>
<p>So, no.</p>
|
3,689,051 | <p>Let <span class="math-container">$(X,d_{X})$</span> be a metric space, and let <span class="math-container">$(Y,d_{Y})$</span> be another metric space. Let <span class="math-container">$f:X\to Y$</span> be a function. Then the following two statements are logically equivalent:</p>
<p>(a) <span class="math-container... | Steven Lu | 789,699 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$X$</span>. If <span class="math-container">$V$</span> is open in <span class="math-container">$Y$</span>, <span class="math-container">$f^{-1}(V)$</span> is in X. Given a point <span class="math-container">$x \in f^{-1}(V)... |
492,392 | <p>I found this math test very interesting. I would like to know how the answer is being calculate?</p>
| Brian M. Scott | 12,042 | <p>Since $49a+49b=6272$, we can divide by $49$ to find that $a+b=128$. By definition the average of $a$ and $b$ is $\frac12(a+b)$, which is $\frac12\cdot128=64$. (Of course you can do this all at once by dividing by $2\cdot49$, but the logic is probably a little clearer the way I did it.)</p>
|
4,095,715 | <p>I know how to do these in a very tedious way using a binomial distribution, but is there a clever way to solve this without doing 31 binomial coefficients (with some equivalents)?</p>
| epi163sqrt | 132,007 | <p>Your approach is also fine. In the following it is convenient to denote with <span class="math-container">$[x^k]$</span> the coefficient of <span class="math-container">$x^k$</span> in a series.</p>
<blockquote>
<p>We obtain
<span class="math-container">\begin{align*}
\color{blue}{[x^{12}]}&\color{blue}{\left(\f... |
1,930,722 | <p>$$\lim_{x\to 0}\frac{x}{x+x^2e^{i\frac{1}{x}}} = 1$$</p>
<p>I've trying seeing that:</p>
<p>$$e^{\frac{1}{x^2}i} = 1+\frac{i}{x^2}-\frac{1}{x^42!}-i\frac{1}{x^63!}+\frac{1}{x^84!}+\cdots\implies$$</p>
<p>$$x^2e^{\frac{1}{x^2}i} = x^2+\frac{i}{1}-\frac{1}{x^22!}-i\frac{1}{x^43!}+\frac{1}{x^64!}+\cdots$$</p>
<p>bu... | DonAntonio | 31,254 | <p>First some simplification and then assuming $\;x\in\Bbb R\;$:</p>
<p>$$\frac x{x+x^2e^{i/x}}=\frac1{1+xe^{i/x}}=\frac1{1+x\left(\cos\frac1x+i\sin\frac1x\right)}\xrightarrow[x\to0]{}\frac1{1+0}=1$$</p>
<p>since </p>
<p>$$\left|e^{i/x}\right|=\left|\cos\frac1x+i\sin\frac1x\right|\le1\;\;\;\text{and this is then a b... |
33,311 | <p>I'm (very!) new to <em>Mathematica</em>, and trying to use it plot a set in $\mathbb{R}^3$. In particular, I want to plot the set</p>
<p>$$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{u^2+v^2+w^2}(vw,uw,uv) \text{ for some } (u,v,w) \in \mathbb{R}^3\setminus\{0\} \big\}. $$
This is just the image of function from $\mathbb{... | Michael E2 | 4,999 | <p>Here's a fairly simple way to fix your <code>Manipulate</code> by applying <code>Dynamic</code> to <code>ListPlot</code>.</p>
<pre><code>Manipulate[
(* Beep[]; *)
data = function @ Range[-Pi*10., Pi*10, Pi/1000];
Dynamic @ ListPlot[data, PlotRange -> {{start, stop}, Automatic}],
{function, {Sin, Cos, Tan}},
... |
313,298 | <p>For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?</p>
<p>I'm using the Buchanan textbook, but I'm not understanding their proofs at all.</p>
| Alex Becker | 8,173 | <p>Well we want to show that for any $\epsilon>0$, there is some $N\in\mathbb N$ such that for all $n>N$ we have $|c_n-1/2|<\epsilon$ (this is the definition of a limit). In this case we are looking for a natural number $N$ such that if $n>N$ then
$$\left|\frac{n^2+10}{2n^2}-\frac{1}{2}\right|=\frac{5}{n^2}... |
730,253 | <p>let $x,y$ such
$$2^x+5^y=2^y+5^x=\dfrac{7}{10}$$</p>
<p>prove or disprove $x=y=-1$ is the only solution for the system.</p>
<p>My try: since
$$2^x-2^y=5^x-5^y$$</p>
<p>But How can prove or disprove $x=y$?</p>
| barak manos | 131,263 | <p><strong>Here is my idea of proving it:</strong></p>
<ol>
<li><p>For $x=y=-1$, you can verify that $2^{-1}+5^{-1}=0.7$</p></li>
<li><p>In order to refute any other possible solution where $x=y$:</p>
<ul>
<li>You can prove that $f(x)=2^x+5^x$ is monotonously increasing</li>
<li>Do it by showing that $f'(x)=2^x\ln2+5... |
730,253 | <p>let $x,y$ such
$$2^x+5^y=2^y+5^x=\dfrac{7}{10}$$</p>
<p>prove or disprove $x=y=-1$ is the only solution for the system.</p>
<p>My try: since
$$2^x-2^y=5^x-5^y$$</p>
<p>But How can prove or disprove $x=y$?</p>
| Christian Blatter | 1,303 | <p>Write $x=u-v$, $\>y=u+v$ and put $r:={5\over2}$. Then we have to solve the equations
$$2^{u-v}+5^{u+v}=5^{u-v}+2^{u+v}={7\over10}\ .$$
Dividing the first equation by $2^u$ we obtain
$$(s:=)\qquad 2^{-v} +r^u 5^v = r^u 5^{-v}+ 2^v\ ,$$
or
$$r^u(5^v-5^{-v})=2^v-2^{-v}\ .\tag{1}$$
Equation $(1)$ is obviously fulfill... |
3,561,860 | <p>Given <span class="math-container">$x_i, x_j \in [0,1]$</span>, and a payoff function <span class="math-container">$u_i(x_i, x_j) = (\theta_i + 3x_j - 4x_i)x_i$</span> if <span class="math-container">$x_j < 2/3$</span>, and <span class="math-container">$= (3x_j-2)x_i$</span> if <span class="math-container">$x_j \... | marty cohen | 13,079 | <p>Since they are each alternating series and eventually the terms are decreasing you can use the alternating series rule on each one. If you make your error criteria on each series to be half the desired error, then the overall error when you combine the two will be what you want.</p>
|
351,534 | <p>It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over <span class="math-container">$\mathbb{F}_p$</span> would allow for a possible solution to the Weil conjectures. </p>
<p>It was also understood that such a cohomology theory could not take values in vector spac... | R. van Dobben de Bruyn | 82,179 | <p>I think an important motivation for the <span class="math-container">$\ell$</span>-adic theory comes from the Riemann existence theorem/the Grauert–Remmert theorem. This says that a finite (topological) covering space <span class="math-container">$Y \to X$</span> of a normal complex variety can again be equipped wit... |
3,565,727 | <blockquote>
<p>Show that the antiderivatives of <span class="math-container">$x \mapsto e^{-x^2}$</span> are uniformly continuous in <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>So we know that for a function to be uniformly continuous there has to exists <span class="math-container">$\var... | the_candyman | 51,370 | <p>Let <span class="math-container">$F(x) = \int_{-\infty}^{x} e^{-t^2}dt$</span> be the antiderivative of <span class="math-container">$e^{-x^2}.$</span> </p>
<p>Without loss of generalities (indeed, the problem is symmetric if we exchange <span class="math-container">$x$</span> and <span class="math-container">$y$</... |
85,117 | <p>Let $\mathcal{A}$ and $\mathcal{B}$ be two $2$-categories and $F : \mathcal{A} \to \mathcal{B}$ be a lax $2$-functor. Given $1$-cells $(f_{i})_{0 \leq i \leq n}$ of $\mathcal{A}$ such that the composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}$ makes sense, this data together with the structural $2$-cells of ... | Buschi Sergio | 6,262 | <p>For composable couple of morphisms $g\circ f$ let </p>
<p>$T(g, f): F(g)\circ F(f) \Rightarrow F(g\circ f)$ the canonical cell. </p>
<p>For a triple of composable $h\circ g\circ f$ let
$T_r(h, g, f):=T(T(h, g),f)$ (A short way to write $T(hg, f)\ast T(h, g)f$) </p>
<p>and $T_l(h,g,f):= T(h,T(g, f))$ for the ... |
3,770,391 | <p>According to <span class="math-container">${\tt Mathematica}$</span>, the following integral converges if
<span class="math-container">$\beta < 1$</span>.</p>
<p><span class="math-container">$$
\int_{0}^{1 - \beta}\mathrm{d}x_{1}
\int_{1 -x_{\large 1}}^{1 - \beta}\mathrm{d}x_{2}\, \frac{x_{1}^{2} + x_{2}^{2}}{\l... | Claude Leibovici | 82,404 | <p>Changing a little notations (and trying to show the steps), considering
<span class="math-container">$$I=\int^{1-\beta}_{0}dx \int^{1-\beta}_{1-x} \frac{x^2+y^2}{(1-x)(1-y)}\,dy$$</span>
<span class="math-container">$$\int \frac{x^2+y^2}{(1-x)(1-y)}\,dy=\frac{\left(x^2+1\right) \log (y-1)+\frac{1}{2} (y-1)^2+2 (... |
3,273,830 | <p>Okay so I've started to study derivatives and there is this idea of continuity. The book says <em>"a real valued function is considered continuous at a point iff the graph of a function has no break at the point of consideration, which is so iff the values of the function at the neighbouring points are close enough ... | Community | -1 | <p>Intuitively, the idea is that the closer and closer you get to <span class="math-container">$x$</span>, the closer and closer <span class="math-container">$y$</span> gets to <span class="math-container">$f(x)$</span>. A small neighborhood of <span class="math-container">$x$</span> must correspond to a small neighbor... |
937,443 | <blockquote>
<p>Evaluate $\displaystyle \int \tan^2x\sec^2x\,dx$</p>
</blockquote>
<p>I tried several methods: </p>
<ul>
<li>First method was I changed $\tan^2x = \sec^2x-1$, and then substitute $\sec x$ to $t$, but it doesn't work.</li>
<li>Second method was to use substitute $\tan^2x = v$, $\sec x = u$. And, it ... | RE60K | 67,609 | <p>$$I=\int\sec x\tan^2xdx\\=\int\sec x(\sec^2x-1)dx\\=\int\underbrace{\sec x}_{\mathtt u}.\underbrace{\sec^2x}_{\mathtt v}dx-\int\sec xdx\\=\left(\sec x\int\sec^2 xdx-\int (\sec x\tan x)\tan xdx\right)-\int\sec x$$
$$I=\sec x \tan x-\ln|\sec x+\tan x|-I+C$$
$$I=\frac12\left(\sec x\tan x-\ln\left|\sec x+\tan x\right|\r... |
4,131,208 | <p>Does the following converge to <span class="math-container">$0$</span> for <span class="math-container">$\theta>-1/2$</span>?</p>
<p><span class="math-container">$$\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^ni^{3\theta}}{\left(\sum_{i=1}^ni^{2\theta}\right)^{3/2}}$$</span></p>
<p>I'd like to use the comparison tes... | Ninad Munshi | 698,724 | <p>Rearranging terms we have that</p>
<p><span class="math-container">$$\frac{\sum_{i=1}^ni^{3\theta}}{\left(\sum_{i=1}^ni^{2\theta}\right)^{\frac{3}{2}}}\cdot\frac{\frac{1}{n^{3\theta}}}{\frac{1}{n^{3\theta}}}\cdot\frac{\frac{1}{n^{\frac{3}{2}}}}{\frac{1}{n^{\frac{3}{2}}}} = \frac{\sum_{i=1}^n\left(\frac{i}{n}\right)^... |
2,107,990 | <p>I'm looking for suggestions for textbooks covering multivariable calculus. I am looking for two textbooks, one which covers the theory and the other which covers the computational aspects. I have already taken a (not so taught well) first course in multivariable calculus, but I'd ideally like to to keep a computatio... | IQ WANTER | 511,144 | <p>I'm also a Bangladeshi, reading in class IX in DRMC. Probably the problem was:</p>
<blockquote>
<p>How many solution triads <span class="math-container">$(a,b,c)$</span> are there for the equation <span class="math-container">$a^2 + b^2 + c^2 - \frac{a^3 + b^3 + c^3 - 3abc}{a+b+c} = 2 + abc?$</span> Where <span c... |
673,928 | <p>How could I solve, apart from brute force, the congruence</p>
<p>$$ \frac{x(x-1)}{2} \equiv 0\ ( \text{mod} \ 2) \ ? $$</p>
<p>I mean, for what values of $y$ has the polynomial $ x(x-1)=2y $ only integer solutions and even for example $ x=f(y) $ is an EVEN number ?
Thanks.</p>
| Peter | 111,826 | <p>One also should add that $|e^{j\omega t}|=1$ and therefore $\lim_{t\to\infty} e^{-(a - j\omega) t} = 0 $ provided $a>0$ and $\omega\in \mathbb R$.</p>
|
484,281 | <p>I recently got interested in mathematics after having slacked through it in highschool. Therefore I picked up the book "Algebra" by I.M. Gelfand and A.Shen</p>
<p>At problem 113, the reader is asked to factor $a^3-b^3.$</p>
<p>The given solution is:
$$a^3-b^3 = a^3 - a^2b + a^2b -ab^2 + ab^2 -b^3 = a^2(a-b) + ab(... | Tyler Holden | 68,577 | <p>The second equality is the typical trick of adding/subtracting things until you get something you want. It is usually not terribly insightful but is nonetheless standard. In general, one can factor $a^n-b^n$ as
$$ a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b^1 + a^{n-3}b^2 + \cdots + a^2 b^{n-3} + a b^{n-2} + b^{n-1}).$$
One ... |
4,106,575 | <p>Consider a number drawn from <span class="math-container">$U(1,100)$</span>. When we make an incorrect guess, we are told whether the target number is smaller or larger. So we employ a binary search approach, where the first guess is 50. If that is not the target, then that means our next guess is going to be 25 or ... | roddur | 915,875 | <p>The probability that it ends in 1 guess is 1 in 100.
For ending in 2 guesses, you must first miss the first then make the second.</p>
<p>Missing the first is probability 99/100.
But now, your solution set is cut down to either 49 or 50 numbers (depending on whether you were above or below).
If you are above, your pr... |
4,106,575 | <p>Consider a number drawn from <span class="math-container">$U(1,100)$</span>. When we make an incorrect guess, we are told whether the target number is smaller or larger. So we employ a binary search approach, where the first guess is 50. If that is not the target, then that means our next guess is going to be 25 or ... | saulspatz | 235,128 | <p>The second calculation is correct.</p>
<p>The mistake in the first calculation is that <span class="math-container">$\Pr(B|A)$</span>is not <span class="math-container">$\frac2{100}$</span> but <span class="math-container">$\frac2{99}$</span> Once we know that <span class="math-container">$50$</span> is not the num... |
222,701 | <p>Below in code are two versions. I believe plot1 might be correct but not sure why its shifted to the left and how to fix it. plot2 is copied directly from the wolfman online documentation but using my own data. It doesn't look normalized and the continuous curve is almost flat. I know something is likely wrong with ... | prog9910 | 59,860 | <p>I get this plot: followed @JimB advice. Time to learn some statistics, again.</p>
<p><a href="https://i.stack.imgur.com/RXT0p.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RXT0p.jpg" alt="Histogram Plot"></a></p>
|
1,037,621 | <p>There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (and who has the black ones). </p>
<p>I thought it might be $$\large2^{\frac{20(20-1)}2}$... | Graham Kemp | 135,106 | <p>A set of twenty distinct items can be ordered in $20!$ ways.</p>
<p>A set of $10$ distinct pairs can be ordered in $10!$ ways.</p>
<p>A pair of distinct items can be ordered in $2!$ ways.</p>
<p>Since we don't care about the order <em>of</em> the pairs, and only care about the the order within <em>each</em> pair,... |
1,037,621 | <p>There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (and who has the black ones). </p>
<p>I thought it might be $$\large2^{\frac{20(20-1)}2}$... | pmunoz | 693,872 | <p>I would like to share how I ended up thinking about this.</p>
<p>To come up with a single instance of pairings first choose who gets to play white. There are <span class="math-container">$20\choose{10}$</span> ways to pick who gets to play white. Then permute the remaining 10 players, and pair them up with the whit... |
140,459 | <p>There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of r... | Carlo Beenakker | 11,260 | <p>Do applications to physics count?</p>
<p><A HREF="http://arxiv.org/abs/hep-th/9403014">Supersymmetry Breakings and Fermat's Last Theorem</A>, Hitoshi Nishino, Mod.Phys.Lett. A <strong>10</strong> (1995) 149-158.</p>
<blockquote>
<p>In this paper, we give the first application of Fermat's last theorem
(FLT) to ... |
140,459 | <p>There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of r... | Chandan Singh Dalawat | 2,821 | <p>It is perhaps an indication of the average age of today's MOers that nobody remembers the work of Hellegouarch who introduced around 1970 what is nowadays called the Frey curve precisely in order to deduce information about elliptic curves from (the then) known results about Fermat's Last Theorem.</p>
|
2,399,216 | <p>How to find the coefficient of $x^4$ in the expansion of $(1+x+x^2)^{20}$?</p>
| Franklin Pezzuti Dyer | 438,055 | <p>Let us treat this trinomial
$$1+x+x^2$$
instead as a binomial:
$$=(1+x)+x^2$$
Then, by the binomial theorem, we have
$$(1+x+x^2)^{20}=\sum_{n=0}^{20} \binom{20}{n}x^{2n}(1+x)^{20-n}$$
Now notice that it in this sum, only the $n=0,1,2$ terms can have an $x^4$ term, because otherwise the $x^{2n}$ would produce some po... |
314,236 | <p>The integral $\int_{-\infty}^{\infty} e^{ix} dx$ diverges. I have read (in a wikipedia article) that the principal value of this integral vanishes: $P \int_{-\infty}^{\infty} e^{ix} dx = 0$. How can one see that?</p>
<p>Thank you for your effort!</p>
| Antonio Vargas | 5,531 | <p>The integral can be regularized using the <a href="http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation#Ces.C3.A0ro_summability_of_an_integral" rel="nofollow">integral analogue of Cesàro summation</a>. By definition we have</p>
<p>$$
\begin{align}
\textrm{C} \int_{-\infty}^{\infty} e^{ix}\,dx &= \lim_{a \to \inf... |
266,981 | <p>Wikipedia presents a recursive definition of the Pfaffian of a skew-symmetric matrix as $$ \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_{ij}\operatorname{pf}(A_{\hat{\imath}\hat{\jmath}}),$$ where $\theta$ is the Heaviside step function. I have failed to find a proper reference for t... | Todd Trimble | 2,926 | <p>To answer the question raised in the most recent edit: abelian sheaves are tensored and powered over abelian groups. </p>
<p>First of all, abelian presheaves have tensors and powers that are computed <em>pointwise</em>. By "abelian presheaves" I mean the $\textbf{Ab}$-category of functors $F: C \to \textbf{Ab}$ whe... |
72,876 | <p>A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he might have felt graph or digraph was not a good choice of terminology than why he thought quiver is a good name. (I r... | Dag Oskar Madsen | 18,756 | <p>Gabriel actually gave a short explanation himself in [Gabriel, Peter. Unzerlegbare Darstellungen. I. (German) Manuscripta Math. 6 (1972), 71--103]:</p>
<blockquote>
<p>Für einen solchen 4-Tupel schlagen wir die Bezeichnung <em>Köcher</em> vor, und nicht etwa Graph, weil letzerem Wort schon zu viele verwandte Begr... |
4,528,823 | <p>I'm trying to understand the proof that every bounded sequence has a convergent subsequence. The proof goes as follows:</p>
<blockquote>
<p>Let <span class="math-container">$\{a_{n}\}$</span> be a bounded sequence of real numbers. Choose <span class="math-container">$M\ge 0$</span>
such that <span class="math-contai... | DonAntonio | 31,254 | <p>For <span class="math-container">$\;k=1\;$</span> :</p>
<p><span class="math-container">$$E_1=\overline{\{a_j\;|\; j\ge1\}}$$</span></p>
<p>For <span class="math-container">$\;k=2\;$</span></p>
<p><span class="math-container">$$E_2=\overline{\{a_j\;|\; j\ge2\}}\subset E_1$$</span></p>
<p>and etc.</p>
|
184,487 | <p>For any domain $A$ let $A^\times$ be its group of units.</p>
<p>Let $A$ be a noetherian domain with only finitely many prime ideals, and field of fractions $K$. </p>
<p>Is the group $K^\times/A^\times$ finitely generated?</p>
| t.b. | 5,363 | <p>I think this (good) exercise in the use of the calculus of fractions is made a bit more difficult than necessary because Gel'fand–Manin formulate the axioms in a self-dual way. Since we're dealing with (what I call) <em>right fractions</em>, it seems like a good idea to isolate the necessary parts of the axiom... |
224,670 | <p>If I have a data that I fit with <code>NonlinearModelfit</code> that fits a data based on two fitting parameters, <code>c1</code> and <code>c2</code>.</p>
<p>When I used <code>nlm["ParameterTable"] // Quiet</code> I get the following table:</p>
<p><a href="https://i.stack.imgur.com/ZmQQO.png" rel="nofollow... | JimB | 19,758 | <p>If you're wanting an estimate of the standard error for <code>eq</code>, one approach is to use the Delta Method (a.k.a Propagation of Error if you're in the physical sciences).</p>
<pre><code>(* Data from first example in `NonlinearModelFit` documentation *)
data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}, {6, 4}, {7, 5}};
... |
3,288,651 | <p>I have a problem understanding a proof about ideals, which states that every ideal in the integers can be generated by a single integer. And with that I realized that I also don't really understand ideals in general and the intuition behind them. </p>
<p>So let me start by the definition of an ideal. For <span clas... | Bill Dubuque | 242 | <p>It seems you may be missing some arithmetical intuition on these ideas so we emphasize that below.</p>
<blockquote>
<p>For <span class="math-container">$a, b \in \mathbb{Z}$</span>, the ideal generated by <span class="math-container">$a$</span> is the set <span class="math-container">$ (a) := \{ua : u \in \mathbb... |
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | J. M. ain't a mathematician | 498 | <blockquote>
<p>Is there a general approach to find a function to approximate a constant?</p>
</blockquote>
<p>For the square root case, sure. Bahman Kalantari has found <a href="http://dx.doi.org/10.1016/S0377-0427%2897%2900014-9" rel="nofollow">a "basic family" of iteration functions</a> for finding $\sqrt c$. The... |
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | NoChance | 15,180 | <p>The link below shows to some extent the rationale behind the formula you have provided as well as others - See in particular the "Babylonian method"</p>
<p><a href="http://en.wikipedia.org/wiki/Methods_of_computing_square_roots" rel="nofollow">Approximating square roots</a></p>
|
1,008,610 | <blockquote>
<p>If <span class="math-container">$a$</span> is a group element, prove that <span class="math-container">$a$</span> and <span class="math-container">$a^{-1}$</span> have the same order.</p>
</blockquote>
<p>I tried doing this by contradiction.</p>
<p>Assume <span class="math-container">$|a|\neq|a^{-1}|$</... | Asinomás | 33,907 | <p>Let $a^n$ be $e$, then $e=(aa^{-1})^n=a^n(a^{-1})^n=e(a^{-1})^n=(a^{-1})^n$.</p>
<p>Let $(a^{-1})^n=e$, then $e=(aa^{-1})^n=a^n(a^{-1})^n=a^ne=a^n$.</p>
<p>So, $a^n=e \iff (a^{-1})^n=e$.</p>
|
4,242,133 | <p>I was having trouble with the question:</p>
<blockquote>
<p>Prove that
<span class="math-container">$$I:=\int_0^{\infty}\frac{\ln(1+x^2)}{x^2(1+x^2)}dx=\pi\ln\big(\frac e 2\big)$$</span></p>
</blockquote>
<p><strong>My Attempt</strong></p>
<p>Perform partial fractions
<span class="math-container">$$I=\int_0^{\infty}... | Ilovemath | 956,199 | <p>Let <span class="math-container">$$\int_0^{\infty} \frac{\log(1+x^2)}{1+x^2}dx= J$$</span>.</p>
<p>Substituting <span class="math-container">$x=tan(\theta)$</span>,</p>
<p><span class="math-container">$$J=-2\int_0^{\frac{π}{2}} \log{\cos{\theta}}d{\theta}$$</span>.</p>
<p>Hence, <span class="math-container">$$J=π\lo... |
777,691 | <p>I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$</p>
<p>I'm stuck.<br>
$n = a \cdot k_1$<br>
$n = b \cdot k_2$<br>
$\therefore a \cdot k_1 = b \cdot k_2$</p>
<p>EDIT: so for <a href="http://en.wikipedia.org/wiki/Fizz_buzz" rel="... | Eric Towers | 123,905 | <p>Updated: The edited version is still not true. At least two counterexamples in the comments below.</p>
<p>Prior counterexample: This is not true. $12|36$ and $9|36$, but $12\cdot9 = 108 \not | 36$.</p>
|
777,691 | <p>I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$</p>
<p>I'm stuck.<br>
$n = a \cdot k_1$<br>
$n = b \cdot k_2$<br>
$\therefore a \cdot k_1 = b \cdot k_2$</p>
<p>EDIT: so for <a href="http://en.wikipedia.org/wiki/Fizz_buzz" rel="... | Xelvonar | 147,242 | <p>This is false. For example, 3 | 30 and 6 | 30, but their product, 18, does not divide 30 even though $3 \times 6 < 30$.</p>
|
1,067,762 | <p>Is what I am doing below correct, please assist. </p>
<p>$$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ </p>
<p>Is this the rule on how to "invert" the limits, and does it matter if there are imaginary numbers in the sum; or not or is it all the same with both pure real an... | Ahaan S. Rungta | 85,039 | <p>Note that the statement is true if and only if $ \angle B + \angle C = 90^\circ $. Also, note that $$ \angle B + \angle C = \arctan \left( \sqrt{2} - 1 \right) + \arctan \left( \sqrt{2} + 1 \right). $$Now, we <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities" ... |
929,243 | <p>Hey guys I just need help solving this solution here. Sorry if I didn't type the symbols correctly.</p>
<p>My solution so far:
$$
(¬p \vee ¬(p\wedge¬q)) \wedge (¬p \vee ¬q)≡
(¬p \vee (¬p \vee q)) \wedge (¬p \vee ¬q)≡
$$
at this point I'm stuck. Is there any way I can take care of the not-$p$ $\vee$ not-$p$?</p>
<p... | zeta | 88,778 | <p>\begin{align*}
&(¬p \vee (¬p \vee q)) \wedge (¬p \vee ¬q) \\
&\equiv((\lnot p \lor q) \land (\lnot p \lor \lnot q)) \\
&\equiv\lnot p \lor(q \land \lnot q)) \\
&\equiv\lnot p
\end{align*}</p>
|
1,387,454 | <p>What is the sum of all <strong>non-real</strong>, <strong>complex roots</strong> of this equation -</p>
<p>$$x^5 = 1024$$</p>
<p>Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. Is there any way to determine number of real and non-real roots of an ... | juantheron | 14,311 | <p>$\bf{My\; Solution::}$ Given $$x^5 = 1024 = 2^{10} = 4^5\Rightarrow x^5-4^5 = 0$$</p>
<p>So $$(x^5-4^5) = (x-5)\cdot (x^4+x^3\cdot 4+x^2\cdot 4^2+x\cdot 4^3+4^4) = 0$$</p>
<p>Above we use the formula $\displaystyle x^n-a^n = (x-a)\cdot (x^{n-1}+x^{n-2}\cdot a+x^{n-3}\cdot a^2+.......+a^{n-1})$</p>
<p>So We Get $x... |
1,591,278 | <p>$$f(x) =\frac{ (x - 1) }{(x^2 - 1)}$$
$$g(x) = \frac{1}{(x + 1)}$$</p>
<p>Is $f = g$? why?</p>
<p>Solution: answer is no.
I don't get it. why can't it be same when it is the alt form?? Can anyone explain this further...</p>
<p>additional questions for further clarification:
$$f(x) = \lim_{x\to 1}\frac{ (x^2 - 1) ... | pancini | 252,495 | <p>What they don't tell you until later in life is that a function is TWO things, a rule and a domain (plus the codomain).</p>
<p>That said, we often don't specify the domain since it is usually obvious. In the first function, for example, the domain can by any real number EXCEPT when the denominator is zero. Thus, th... |
3,921,252 | <p>Let <span class="math-container">$M$</span> be a connected topological <span class="math-container">$n$</span>-manifold. Note this implies <span class="math-container">$M$</span> is path-connected.</p>
<p>Let <span class="math-container">$a,b \in M$</span>. Must there always exist a continuous <span class="math-cont... | user3482749 | 226,174 | <p>For manifolds with boundary: no, you can't take boundary points to non-boundary points.</p>
<p>For manifolds without boundary: yes. For 1-manifolds, this is easy. For <span class="math-container">$n > 1$</span>, take a neighbourhood of a (shortest) path between your points. After removing problems (by taking a su... |
1,063,599 | <p>There is a deck made of $81$ different card. On each card there are $4$ seeds and each seeds can have $3$ different colors, hence generating the $ 3\cdot3\cdot3\cdot3 = 81 $ card in the deck.
A tern is a winning one if,for every seed, the correspondent colors on the three card are or all the same or all different.</... | Brian M. Scott | 12,042 | <p>HINT: The problem is equivalent to the following one. Let $D=\{0,1,2\}^4$, the set of $4$-tuples of numbers from the set $\{0,1,2\}$. Each member of $D$ corresponds to a card; the four components of the $4$-tuple are the seeds; and $0,1$, and $2$ are the three ‘colors’. A tern is a set of three members of $D$.</p>
... |
1,521,739 | <p>The following is the Meyers-Serrin theorem and its proof in Evans's <em>Partial Differential Equations</em>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/XnzXY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XnzXY.png" alt="enter image description here"></a>
<a href="https://i.stack... | Brevan Ellefsen | 269,764 | <p>The transform equations to use for rotating any curve are
$$ x' = (x \cos θ - y \sin θ)$$
$$y' =(x \sin θ + y \cos θ)$$
(These basically rotate the axes, but when we view them with static axes the graph rotates) </p>
<hr>
<p>In this case, we get
$$x' = \frac{x+y}{\sqrt{2}}$$
$$y' = \frac{y-x}{\sqrt{2}}$$
$$$$$... |
832,715 | <blockquote>
<p>Suppose that the distribution of a random variable
$X$ is symmetric with respect to the point $x = 0$. If $\mathbb{E}(X^4)>0$ then $Var(X)$ and $Var(X^2)$ are both positive.</p>
</blockquote>
<p>How is that true? I am getting $Var(X)=\mathbb{E}(X^2)$ and $Var(X^2)=\mathbb{E}(X^4)-(\mathbb{E}(X... | M. Vinay | 152,030 | <p>We know that the variance of a random variable is $0$ if and only if it is constant (so $X = \text{E}(X)$).</p>
<blockquote>
<p>Proof:<br>
If $X$ has constant value $c$, then $\text{E}(X) = c$ and $\text{E}(X^2) = \text{E}(c^2) = c^2$, $\Rightarrow \text{V}(X) = \text{E}(X^2) - \text{E}(X)^2 = 0$.<br>
Convers... |
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | fretty | 25,381 | <p>I am in the same boat at the moment, I am partway through my first year as a postgrad and I feel I am forgetting many results that I used to know through lack of use/practice or full understanding.</p>
<p>However, I think at this level maths is more about understanding the situation rather than studying the results... |
1,232,036 | <p>I have a midterm tomorrow, and while studying for that I saw this question, however don't have any idea how to solve it. (I could not come up with a legitimate proof. All I could do was, by putting some functions, approving what the problem claims.) I will appreciate if you can help.</p>
<p>Suppose that $ h(t)$ is ... | RTJ | 223,807 | <p>For $V=y^2$ we have
$$\frac{dV}{dt}=2y(t)\frac{dy}{dt}(t)=-2y^2(t)+2y(t)h(t)\leq -y^2(t)+h^2(t)=-V(t)+h^2(t)$$
Multiplying by $e^t$ we equivalently have
$$\frac{d}{dt}\left[V(t)e^t-\int_0^t{h^2(s)e^s ds}\right]\leq 0$$
If we now integrate over $[0,t]$ we obtain
$$y^2(t)\leq y^2(0)e^{-t}+e^{-t}\int_0^t{e^sM^2ds}=y^2(... |
1,453,330 | <p>I want to prove that this function is one-to-one</p>
<p>$ f: \mathbb R \rightarrow \mathbb R$ ,
$$f(x) = \left\{\begin{array}{ll}
2x+1 & : x \ge 0\\
x & : x \lt 0
\end{array}
\right.$$
And for this function</p>
<p>$ f: \mathbb R \rightarrow \mathbb R$,
$$f(x) = \left\{\begin{array}{ll}
x^2 & : x \gt... | Maciej | 241,823 | <p><strong>First function</strong></p>
<p>Notice, that $f[\mathbb{R}_{\ge 0}]\cap f[\mathbb{R}_{< 0}] = \emptyset$, so you need to show, that $f|_{\mathbb{R}_{\ge 0}}$ and $f|_{\mathbb{R}_{< 0}}$ are one-to-one.</p>
<p><strong>Second function</strong></p>
<p>Notice, that if $m \ge 0$, than $f$ is not one-to-on... |
3,444,463 | <p>Everyone knows that the shortest distance between two points is a straight line. How can we prove it mathematically? </p>
| Matthew Leingang | 2,785 | <p>If <span class="math-container">$y$</span> is a <span class="math-container">$C^2$</span> function of <span class="math-container">$x$</span> and traces a curve between points <span class="math-container">$(a,c)$</span> and <span class="math-container">$(b,d)$</span>, then the length of the curve is
<span class="ma... |
1,047,489 | <p>let $f(x)$ be a function such that </p>
<p>$$f(0) = 0$$</p>
<p>$$f(1) =1$$ $$f(2) = 2$$ $$f(3) = 4$$ $$f'(x) \text{is differentiable on } \mathbb{R}$$ Prove that there is a number in the interval $(0,3)$ such
that $0 < f''(x)<1$</p>
<p>I'm really stuck.
thanks</p>
| FundThmCalculus | 153,550 | <p>I would suggest as a first attempt, curve fitting a polynomial. Let's fit to a cubic, since the general cubic equation has four parameters (and you have four known points).
$$f(x) = ax^3+bx^2+cx+d$$
$$f(0) = a*0^3+b*0^2+c*0+d = 0 \rightarrow d = 0$$
$$f(1) = a*1^3+b*1^2+c*1+d = 1 \rightarrow a+b+c=1$$
$$f(2) = a*2^3... |
25,413 | <p>Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.</p>
<p>We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant ... | Steven Gubkin | 117 | <p>The very limit you have chosen as an example shows the necessity of the squeeze theorem.</p>
<p>When deriving the fact that the derivative of sine is cosine, you first use the angle sum formula for sine, which then reduces the computation to the derivative of sine at 0. This is the limit you have chosen as an examp... |
4,294,714 | <p>The problem is as follows:</p>
<p>You have an unlimited number of marbles and each marble is one of 16 different colours. You have to choose 6 marbles and order is irrelevant. How many different combinations of 6 marbles are there?</p>
| gnasher729 | 137,175 | <p>To meet the condition that A, B and C leave at different stations: A can leave at 7 stations, B at 6 and C at 5.</p>
<p>There are 6 possible orderings, but only one is allowed. Therefore (7<em>6</em>5)/7^3/6 = 5/49.</p>
|
4,031,633 | <p>Let <span class="math-container">$R$</span> be a commutative ring and <span class="math-container">$M$</span> is an <span class="math-container">$R$</span>-module. The following statement is well-known.</p>
<p>If <span class="math-container">$M$</span> is finitely presented flat module, <span class="math-container">... | Dolors | 1,078,427 | <p>The result can be extended to the setting of non-commutative rings:</p>
<p><strong>Prop:</strong> Let M be a finitely generated flat right module over a ring R, and let P be a projectivemodule. If M/MJ(R)is isomorphic to P/PJ(R), then M is isomorphic to P</p>
<p>For the proof you can check the Proposition 7.3 in the... |
2,916,887 | <p>The formula for Shannon entropy is as follows,</p>
<p>$$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i
$$</p>
<p>Thus, a fair six sided dice should have the entropy,</p>
<p>$$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1}{6} = \log_2 (6) = 2.5849...$$</p>
<p>However, the entropy should also correspond to the average ... | metamorphy | 543,769 | <p>In your setting, the Shannon entropy is "just" a lower bound for an entropy of any decision tree (including optimal ones). These don't have to coincide. To get closer to what the Shannon entropy is, imagine an optimal decision tree identifying outcomes of throwing a dice $N$ times with some large $N$ (assuming indep... |
2,916,887 | <p>The formula for Shannon entropy is as follows,</p>
<p>$$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i
$$</p>
<p>Thus, a fair six sided dice should have the entropy,</p>
<p>$$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1}{6} = \log_2 (6) = 2.5849...$$</p>
<p>However, the entropy should also correspond to the average ... | celtschk | 34,930 | <p>To recover entropy, you have to consider a <em>sequence of</em> dice throws, and ask how many questions <em>per roll</em> you need in an optimal strategy, in the limit that the number of rolls goes to infinity. Note that each question can cover all the rolls, for example for two rolls, you could ask at some point: “... |
2,916,887 | <p>The formula for Shannon entropy is as follows,</p>
<p>$$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i
$$</p>
<p>Thus, a fair six sided dice should have the entropy,</p>
<p>$$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1}{6} = \log_2 (6) = 2.5849...$$</p>
<p>However, the entropy should also correspond to the average ... | A. Webb | 59,557 | <p>If you have $1$ die, there are $6$ possible outcomes. Label them 0 through 5 and express as a binary number. This takes $\lceil\log_2{6}\rceil = 3$ bits. You can always determine the 1 die with 3 questions, just ask about each bit in turn. </p>
<p>If you have $10$ dice, then there are $6^{10}$ possible outcomes. La... |
309,234 | <p>I am looking for good, detailed references for "mod $p$ lower central series".</p>
<p>So far I only find papers such as (<a href="https://core.ac.uk/download/pdf/81193793.pdf" rel="nofollow noreferrer">https://core.ac.uk/download/pdf/81193793.pdf</a>, <a href="https://www.sciencedirect.com/science/article/pii/00409... | Andy Putman | 317 | <p>This is a fundamental in the theory of pro-$p$ groups. A good reference is</p>
<p>J. D. Dixon, M. Du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, second edition, Cambridge Studies in Advanced Mathematics, 61, Cambridge University Press, Cambridge, 1999.</p>
|
201,358 | <p>I would like to be able to extract elements from a list where indices are specified by a binary mask.</p>
<p>To provide an example, I would like to have a function <code>BinaryIndex</code> doing the following:</p>
<pre><code>foo = {a, b, c}
mask = {True, False, True}
BinaryIndex[mask, foo]
(* expected output *) {a... | Sjoerd Smit | 43,522 | <p>Since this question is about performance, I'd like to add that often it's better to work with lists of integers than lists of booleans since lists of integers can be packed. For example, if you want to pick the positive elements from an array using <code>Pick</code>, you can either use <code>Positive</code> to make ... |
711,168 | <p>Let $V$ be an $n$-dimensional real inner product space and let $a=\lbrace v_1,v_2,\dots v_n \rbrace$ be an orthonormal basis for $V$. Let $W$ be a subspace of $V$ with orthonormal basis $B = \lbrace w_1, w_2,\dots w_k\rbrace$. Let $A = \lbrace [w_1]a, [w_2]a,\dots [w_k]a\rbrace$ and let $P_w$ be the orthogonal proje... | amWhy | 9,003 | <p>It is true that the group axioms can be "weakened" to include: </p>
<ul>
<li>associativity</li>
<li>the existence of a <strong>left</strong> identity for all elements in the group</li>
<li>for each element in the group, there exists a <strong>left</strong> inverse.</li>
</ul>
<p>Note that in this weakened definiti... |
1,790,311 | <p>Show the following equalities $5 \mathbb{Z} +8= 5\mathbb{Z} +3= 5\mathbb{Z} +(-2)$.</p>
<p>$5 \mathbb{Z} +8=\{5z_{1}+8: z_{1} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +3=\{5z_{2}+3: z_{2} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +(-2)=\{5z_{3}+(-2): z_{3} \in \mathbb{Z}\}$.</p>
<p>So, how can prove to use these de... | Stefan Hante | 42,060 | <p>Let $a \in 5\mathbb Z + 8$, then there is $b\in\mathbb Z$ such that $a = 5b + 8$. Then you can do
\begin{align*}
a = 5b + 8 = 5(b-1) + 3,
\end{align*}
thus $a\in5\mathbb Z + 3$. This proves
$$
5\mathbb Z + 8 \subseteq 5\mathbb Z + 3,
$$
because every element of $5\mathbb Z + 8$ is an element of $5\mathbb Z + 3$.</p>... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| SChepurin | 53,883 | <p>When you apply transformations to the systems/objects represented by matrices, and you need some characteristics of these matrices you have to calculate eigenvectors (eigenvalues). </p>
<blockquote>
<p>"Having an eigenvalue is an accidental property of a real matrix (since it may fail to have an eigenvalue), but ... |
2,325,421 | <blockquote>
<p>If $f$ is a linear function such that $f(1, 2) = 0$ and $f(2, 3) = 1$, then what is $f(x, y)$?</p>
</blockquote>
<p>Any help is well received.</p>
| kvicente | 452,277 | <p>I suppose you refer to a function <em>f</em> from the real plane to the real line, then note that <code>(1,2);(2,3)</code> is a base for the real pane vector space. Then any element of the plane can be represented as a linear combination of this elements. The applying linearity you get form for the required function... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.