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 |
|---|---|---|---|---|
3,508,730 | <p>I was trying to prove that <span class="math-container">$(N+\sqrt{N^2-1})^k$</span>, where k is a positive integer, differs from the integer nearest to it by less than <span class="math-container">$(2N-\frac{1}{2})^{-k}$</span>. Note: N is an integer greater than 1. </p>
<p>So, I tried to look for the answer of the... | Jean Marie | 305,862 | <p>Let <span class="math-container">$N$</span> be a fixed integer. </p>
<p>Using the expression you give, </p>
<p><span class="math-container">$$\underbrace{(N+\sqrt{N^2-1})^k}_{A_k} + \underbrace{(N-\sqrt{N^2-1})^k}_{B_k>0}=\underbrace{2(N^k + \binom{k}{2} N^{k-2}(N^2-1)+...}_{C_k, \ \text{an integer}},\tag{1}$$<... |
3,877,891 | <p>Suppose we consider two graphs to be the equivalent if they are isomorphic. The idea is that if we relabel the vertices of a graph, it is still the same graph. Using this definition of “being the same graph”, can you conclude that the set of trees over countably infinite vertices is countable?<br>
I know that for an... | Parcly Taxel | 357,390 | <p>Suppose the binary expansion of a real number <span class="math-container">$r\in[0,1]$</span> is given by <span class="math-container">$0.b_1b_2b_3\dots$</span>. Based on this real number, define a tree with countable-infinitely many vertices as follows:</p>
<ul>
<li>There are four special vertices <span class="math... |
3,379,893 | <p><a href="https://i.stack.imgur.com/pfWuG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pfWuG.png" alt="enter image description here"></a></p>
<p>I am completely lost, I have no idea where to even start. I am sure someone here would be able to do this easily which is why I'm posting this here. B... | Glorious Nathalie | 948,761 | <p><a href="https://i.stack.imgur.com/3UX1q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3UX1q.png" alt="enter image description here" /></a></p>
<p>Let the side length of the equilateral triangle be <span class="math-container">$x$</span>, and let the tilt angle <span class="math-container">$\the... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Alexandre Eremenko | 25,510 | <p>This is possible. I have at least two friends who studied mathematics (in the graduate school), did not defend their PhD, and found some jobs not related to mathematics. Still they do research, and publish papers from time to time.</p>
<p>Probably the most famous modern mathematician who never studied mathematics on... |
363,377 | <p>Is there a world where circle is square? (like when triangle can have sum of degrees more than 180 on sphere)
What is the mathematical or at least common-sense proof?</p>
| Jun Zhang | 72,848 | <p>As Arya mentioned, in topological sense this is possible. </p>
<p>You can also define the distance (or "norm") in other forms, say 1-norm $|x|+|y|$ or $\infty$-norm $\max \{x,y\}$, then the circle will "look like" a square.</p>
|
1,521,745 | <p>I'm really confused about this math problem. I'm currently taking Calculus, but this problem seems to be like something basic in Algebra that I should understand.... Unfortunately, I don't remember what equivalent expression was used for this. </p>
<p>The problem is:</p>
<p><a href="https://i.stack.imgur.com/cjudA... | heropup | 118,193 | <p>Assuming you have done the coordinate transformation correctly, then the basic idea is that you calculate the vertex and focus of the transformed parabola, then perform the inverse transformation on those coordinates to recover the vertex and focus in the untransformed (original) coordinates.</p>
<p>So for example,... |
2,492,071 | <p>There are $3$ boxes, each contains $n$ mangoes. A person takes a mango from one randomly chosen box. This procedure is repeated until one of the boxes becomes empty. Find the probability that two other boxes contain one mango each.</p>
<p>I am new in a probability theory. I know that its conditional probability pro... | JV.Stalker | 416,274 | <p>Using the following inequality: ${(1+x)}$$\le$${e^x}$</p>
<p>$\lim_{n\rightarrow\infty}\sum_{i=0}^n \ln \left( \left(1+\frac{i}{n}\right)^\frac{1}{n}\right)$$\lt$$\lim_{n\rightarrow\infty}{^\frac{1}{n}}\sum_{i=0}^n \ln \left(e^{{i\over n}}\right)$=</p>
<p>$\lim_{n\rightarrow\infty}{^\frac{1}{n^2}}\sum_{i=1}^n{({i}... |
271,088 | <p>I am studying the symmetries of a particular function,
$$
f: R^n \rightarrow R
$$
which leave $f(x)$ unchanged (i.e. so $f(Ax) = f(x)$ for some matrix $A \in R^{n \times n}$). I have found that my function is invariant under the action of a matrix group which satisfies the following equation:
$$
A X A^T = X
$$
where... | Igor Rivin | 11,142 | <p>I am a little confused by Paul's answer. Your relation can be written as
$Ax x^\perp A^\perp = x x^\perp,$ which is the same as saying that
$A$ is norm preserving, hence orthogonal.</p>
|
2,283,230 | <p>Let $f: \Bbb R^n \to R$ be a scalar field defined by</p>
<p>$$ f(x) = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j .$$</p>
<p>I want to calculate $\frac{\partial f}{\partial x_1}$. I found a brute force way of calculating $\frac{\partial f}{\partial x_1}$. It goes as follows:</p>
<blockquote>
<p>First, we eliminate... | M. Winter | 415,941 | <p>The following could be something that you might accept as a "general rule". We just compute the derivative of $\langle x,Ax\rangle$ explicitely, using our knowledge about inner products. Choose some direction $v$, i.e. $v$ is a vector with $\|v\|=1$. Then</p>
<p>$$\lim_{h\to 0} \frac{\langle x+hv,A(x+hv)\rangle-\co... |
1,377,595 | <p>Compute this integral:
$$\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$$</p>
<p>my solution is I used derivative of Cauchy integral formula, which is
$$f^{(n)}(z_0) = \frac{n!}{2\pi i}\int \frac{f(z_0)}{(z-z_0)^{n+1}}$$
Then, I got
$$\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$$
$$ = \int_{|z|=1} \frac{(z-2)^3... | Mark Viola | 218,419 | <p>You were close</p>
<p>$$f^{(2)}(1/2)=\frac{2!}{2\pi i}\oint_{|z|=1}\frac{(z-2)^3}{(2z-1)^3}dz=\frac{2!}{2\pi i}\oint_{|z|=1}\frac{\frac18(z-2)^3}{(z-1/2)^3}dz$$</p>
<p>where $f(z)=\frac18(z-2)^3$ and thus $f^{(2)}(1/2)=\frac18 (6)(-3/2)=-\frac98$</p>
<p>Finally, we have</p>
<p>$$\bbox[5px,border:2px solid #C0A00... |
494,932 | <p>Why can trigonometry as a geometrically defined concept be used to algebraic operations between complex numbers? What connects the two things together and how ?</p>
| Cameron Williams | 22,551 | <p>There are a couple connections. First of all: you can look at $\mathbb{C}$ as being isomorphic to $\mathbb{R}^2$ with regards to addition (meaning if $z_1 = x_1+iy_1$ and $z_2=x_2+iy_2$ and we associate $z = x + iy$ with the vector $(x,y)$ then with regards to addition, these behave the same - have the "same structu... |
4,058,947 | <p>Have got into a pretty heated debate with a friend, and looking online there's lacking proof
<a href="https://i.stack.imgur.com/S30uM.png" rel="nofollow noreferrer">Line contained in a plane</a></p>
<p>Is a line that is contained within a plane, considered parallel to it? By my understanding it is parallel , if at a... | Sam | 858,695 | <p>This is entirely a question of convention. I'd say yes, because I like "parallelness" to be an equivalence relation, but it frankly does not matter in the slightest, providing you're clear about how you are using the word.</p>
|
3,677,967 | <p>I've got a <span class="math-container">$\prod$</span> (product operator) function that I'm trying to make explicit. I've managed to convert everything else to explicit form, which we can call <span class="math-container">$g(x)$</span>, except for this one part, so overall I've got:</p>
<p><span class="math-contain... | timur | 2,473 | <p>We have
<span class="math-container">$$
h(x) = \prod_{1\leq n\leq x} n^k = ([x]!)^k,
$$</span>
for <span class="math-container">$x\geq1$</span>,
where <span class="math-container">$[x]$</span> is the integer part of <span class="math-container">$x$</span>.</p>
|
25,778 | <p>Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a> of a matrix $A(x)$, without approximations (except for the usual floating-point limitations)? The matrix $\frac{\mathrm{d}}{\m... | Eric O Lebigot | 3,810 | <p>The answer is known <a href="http://www.jstor.org/stable/2156365" rel="noreferrer">since at least 1973</a>: a formula for the derivative of the pseudo-inverse of a matrix $A(x)$ of constant rank can be found in </p>
<blockquote>
<p>The Differentiation of Pseudo-Inverses
and Nonlinear Least Squares Problems
Wh... |
2,906,314 | <p>If I have vectors a and b sharing a common point of intersection then I know how to calculate angle between them by using the formula for dot product. But whether b lies to the right or left of a if I am moving along a can not be gotten from this. </p>
<p>What would be the easiest way to find out whether b lies lef... | David K | 139,123 | <p>In order for "left" and "right" to be defined you need to have a sense of turning direction. Generally this requires not only to a sense of the "forward" direction given by the vector $\mathbf a,$ but also to some sense of which way is "up."</p>
<p>If the vectors are constrained to lie in a plane and we have a view... |
24,939 | <p>Sorry, I'm just starting to learn mathematica.</p>
<p>I have the following two-argument function:</p>
<pre><code>h[{x_, y_}] := x ^ y
</code></pre>
<p>When I do the following:</p>
<pre><code>Map[h, {{1, 2}, {2, 2}, {3, 2}}]
</code></pre>
<p>I get the expected output:</p>
<pre><code>{1, 4, 9}
</code></pre>
<p>... | Artes | 184 | <p>You should use <code>Apply</code> (at the level <code>1</code> (<code>@@@</code>)) rather than <code>Map</code>, in terms of a pure function as you looking for:</p>
<pre><code>#1^#2 & @@@ {{1, 2}, {2, 2}, {3, 2}}
</code></pre>
<blockquote>
<pre><code>{1, 4, 9}
</code></pre>
</blockquote>
<p>Instead of a pure... |
24,939 | <p>Sorry, I'm just starting to learn mathematica.</p>
<p>I have the following two-argument function:</p>
<pre><code>h[{x_, y_}] := x ^ y
</code></pre>
<p>When I do the following:</p>
<pre><code>Map[h, {{1, 2}, {2, 2}, {3, 2}}]
</code></pre>
<p>I get the expected output:</p>
<pre><code>{1, 4, 9}
</code></pre>
<p>... | Szabolcs | 12 | <p>First of all, let's clarify that if you define <code>h</code> as</p>
<pre><code>`h[{x_, y_}] := ...`
</code></pre>
<p>then it takes a <em>single</em> argument which is a list of two items. If you define it as</p>
<pre><code>`h[x_, y_] := ...`
</code></pre>
<p>then it takes two separate arguments.</p>
<p><code>... |
1,919,584 | <p>Given two non-colinear real unit vectors $v,w$, I believe the rank of $M=vv^\top + ww^\top$ is 2 and I'd like to prove it. $vv^\top$ and $ww^\top$ are obviously of rank one, $v$ is not in the kernel of $M$ because $vv^\top v=v$ and $\|ww^\top v\|<1$ (because $v$ and $w$ are not colinear), same with $w$. So the ra... | Ben Grossmann | 81,360 | <p>Yet another proof: note that
$$
vv^T + ww^T = \pmatrix{v&w} \pmatrix{v & w}^T
$$
and for any matrix $M$, $M$ has the same rank as $MM^T$.</p>
|
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Guy Fsone | 385,707 | <h2>Here is a nice and simple way. No special change of variable.</h2>
<p>Let $f: [0,1-\varepsilon]\to [0,f(1-\varepsilon)],~~\text{for}~~~~0<\varepsilon<1$ with $f(x)=\sqrt{\dfrac{x}{1-x}}~ $ <strong>is a bijection (even increasing) since</strong> $$f'(x) = \frac{1}{2(1-x)^2}\sqrt{\dfrac{1-x}{x}}>0~~~ \text... |
3,829,280 | <p>Let <span class="math-container">$H$</span> be a semisimple algebraic subgroup of <span class="math-container">$GL(V)$</span> without compact factors (I am not sure if this part is relevant) where <span class="math-container">$V$</span> is a finite dimensional vector space. From a paper I have read, it follows that ... | Tsemo Aristide | 280,301 | <p>I assume that <span class="math-container">$H$</span> is connected (and to simplify that <span class="math-container">$H$</span> is defined over <span class="math-container">$\mathbb{R}$</span>) <span class="math-container">$[Lie(H),Lie(H)]=Lie(H)$</span>, where <span class="math-container">$Lie(H)$</span> is the L... |
2,752,377 | <p>Let $p$ be a prime number. I could already show, that $\mathbb{Z}[\sqrt{-2}]$ is an euclidean domain and that $$p\text{ reducible }\Leftrightarrow\text{ }p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$$</p>
<p>Now I wonder how to prove</p>
<p>$$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\... | Rene Schipperus | 149,912 | <p>$p$ splits iff $$\left(\frac{-2}{p}\right)=1$$
That is either $$\left(\frac{-1}{p}\right)=1, \left(\frac{2}{p}\right)=1$$
in which case $p\equiv 1(\mod 8)$, or
$$\left(\frac{-1}{p}\right)=-1, \left(\frac{2}{p}\right)=-1$$
in which case $p\equiv 3(\mod 8)$</p>
|
2,993,166 | <p>Suppose I have an operator valued function, <span class="math-container">$\omega\mapsto A(\omega)$</span>; for each <span class="math-container">$\omega$</span>, <span class="math-container">$A(\omega):X\to Y$</span>, is a bounded linear operator with <span class="math-container">$X$</span> and <span class="math-con... | Community | -1 | <p>Guess the solution <span class="math-container">$x = 2$</span> by using the Rational Root Theorem. By synthetic division, we have</p>
<p><span class="math-container">$$x^{3} - x - 6 = (x - 2)(x^{2} + 2x + 3)$$</span></p>
<p>Using the quadratic equation on the second equation, we obtain the solution set</p>
<p><sp... |
2,993,166 | <p>Suppose I have an operator valued function, <span class="math-container">$\omega\mapsto A(\omega)$</span>; for each <span class="math-container">$\omega$</span>, <span class="math-container">$A(\omega):X\to Y$</span>, is a bounded linear operator with <span class="math-container">$X$</span> and <span class="math-con... | Oscar Lanzi | 248,217 | <p>Other answers give ways to get the roots, but here is where you went wrong.</p>
<p>You assumed <span class="math-container">$x=y-2$</span> to get one equation for <span class="math-container">$y$</span>, then <span class="math-container">$x=y+3$</span> to get another equation. Either is correct by itself, but you ... |
128,656 | <p><img src="https://i.stack.imgur.com/AyYxe.jpg" alt=""Put the alphabet in math..."" /></p>
<p><strong>variable</strong>: A symbol used to represent one or more numbers.</p>
<p>Or alternatively: A symbol used to represent any member of a given set.</p>
<p>High school students are justifiably confused by the... | Scott Carter | 722 | <p>There probably should be a strict distinction made between variables and constants. For example in a quadratic equation $f(x) = Ax^2+Bx+C,$ the letters $A,B$, and $C$ are constants in the sense that they are standing in for specific values. The variable in the equation (rather the independent variable) is $x$. The d... |
3,593,702 | <p>I've discovered and am trying to understand power sets, specifically how to calculate the power sets of a set. I found the <a href="http://www.ecst.csuchico.edu/~akeuneke/foo/csci356/notes/ch1/solutions/recursionSol.html" rel="nofollow noreferrer">algorithm's description</a>, which concluded with this:</p>
<p><span... | David K | 139,123 | <p>The short answer is <em>there is no reason</em> why you have to compute one probability according to the order of the objects for one problem and without regard to order for the other. You can do it either way for either problem if you count the orderings correctly.</p>
<hr>
<p>Here are some ways you can compute p... |
317,756 | <p>Let $L>1$. I am looking for the value, or the leading asymptotics for $L\to\infty$, of
$$\int_1^L\int_1^L\int_1^L\int_1^L \dfrac{\mathrm dx_1~\mathrm dx_2 ~ \mathrm dx_3 ~ \mathrm dx_4}{(x_1+x_2)(x_2+x_3)(x_3+x_4)(x_4+x_1)}$$
More generally, I'd like to know the leading asymptotics of an expression like this wit... | joriki | 6,622 | <p>The leading term for $n=2$ is $\frac23\pi^2\log L$.</p>
<p>As has been mentioned in the comments, you can perform one half of the integrations explicitly:</p>
<p>$$
\begin{align}
&\int_1^L\int_1^L\int_1^L\int_1^L\frac{\mathrm dx_1\mathrm dx_2\mathrm dx_3\mathrm dx_4}{(x_1+x_2)(x_2+x_3)(x_3+x_4)(x_4+x_1)}
\\
=&... |
3,149,215 | <p>I know that my reasoning is incorrect, I just don't know where I went wrong. I did discuss this with my Maths teacher, and even she could not find what I did wrong.</p>
<p>Let us begin by assuming a function, <span class="math-container">$f(x)$</span> that is continuous and has an antiderivative in the interval <sp... | Community | -1 | <p><strong>Short answer:</strong></p>
<p><span class="math-container">$$\sin x=t$$</span> doesn't mean that</p>
<p><span class="math-container">$$x=\arcsin t.$$</span></p>
|
3,149,215 | <p>I know that my reasoning is incorrect, I just don't know where I went wrong. I did discuss this with my Maths teacher, and even she could not find what I did wrong.</p>
<p>Let us begin by assuming a function, <span class="math-container">$f(x)$</span> that is continuous and has an antiderivative in the interval <sp... | Sarvesh Ravichandran Iyer | 316,409 | <p>Longer Answer : </p>
<p>The change of variables formula(COV) does not apply in the given conditions.</p>
<p>Think of the change of variables formula like this : given a set whose area you cannot find, you change it in a manner such that its area does not change and it assumes some possibly nice shape (whose area y... |
3,775,647 | <p>The maths book I'm using shows:<br/>
<span class="math-container">$$ a^2b \div \frac13a^2b^3 $$</span>
Which would be something like:<br/>
<span class="math-container">$$ a^2 \cdot b ÷ \frac13 \cdot a^2 \cdot b^3 $$</span>
My understanding of order of operations it would equal:<br/>
<span class="math-container">$$ 3... | gnasher729 | 137,175 | <p>The usual interpretation is that writing two things side by side without a visible operator means they are multiplied with higher priority than normal multiplication or division using a visible operator. So we multiply 1/3, <span class="math-container">$a^2$</span> and <span class="math-container">$b^3$</span> first... |
2,978,793 | <p>I want to show that</p>
<p><span class="math-container">$$\sum_{i=1}^n i^{2-\alpha} \sim c n^{3-\alpha}$$</span></p>
<p>for some constant <span class="math-container">$c$</span>. Here, <span class="math-container">$\alpha$</span> is assumed <span class="math-container">$0 < \alpha < 1$</span>.</p>
<p>I know... | Kavi Rama Murthy | 142,385 | <p>Both sides are equal to <span class="math-container">$E(Y|X)$</span>. All you have to do is to use repeatedly the property <span class="math-container">$E(E(Y|\{X_i\}|\{Z_j\})=E(E(Y|\{Z_j\})$</span> whenever <span class="math-container">$\{Z_j\}$</span> is a subcollection of <span class="math-container">$\{X_i\}$</s... |
1,229,164 | <p>How to solve modular equations?
So for example $a \equiv i$ mod $x$, $a \equiv j$ mod $y$ for some given $i,j,x,y$ with $gcd(x,y)=1$, and I must find $a$ mod $x*y$.
Any tips on how to do this?
Specifically I want to calculate $a \equiv 1$ mod $16$, $a \equiv 3$ mod $17$, for example.</p>
| Shrey | 222,631 | <p>I recommend referring to this website – it is extremely comprehensive and gives many valuable examples: <a href="http://www.millersville.edu/~bikenaga/abstract-algebra-1/modular-arithmetic/modular-arithmetic.html" rel="nofollow">http://www.millersville.edu/~bikenaga/abstract-algebra-1/modular-arithmetic/modular-arit... |
4,303,589 | <p>To make mathematical annotations on the computer I use Notepad, but Notepad has neither a formula editor nor almost any editor, so it very quickly starts to get harder to read equations. The other pieces of software I have seen were robust, as Evernote. For Notepad, I press two buttons, type “notepad”, press enter, ... | Ethan Bolker | 72,858 | <p>If <span class="math-container">$\LaTeX$</span> will suffice and you are willing to learn it, use it. You can learn most of the math syntax right here on this site from a mathjax tutorial. Texlive is a good application to download. Overleaf does TeX in the cloud for you.</p>
|
4,303,589 | <p>To make mathematical annotations on the computer I use Notepad, but Notepad has neither a formula editor nor almost any editor, so it very quickly starts to get harder to read equations. The other pieces of software I have seen were robust, as Evernote. For Notepad, I press two buttons, type “notepad”, press enter, ... | marty cohen | 13,079 | <p>On the Mac,
I use Macdown:</p>
<p><a href="https://macdown.uranusjr.com/" rel="nofollow noreferrer">https://macdown.uranusjr.com/</a></p>
<p>Another one
that is available for Mac,
Windows, and Linux
is Haroopad:</p>
<p><a href="http://pad.haroopress.com/user.html" rel="nofollow noreferrer">http://pad.haroopress.com/... |
3,713,580 | <blockquote>
<p>Question: Prove that <span class="math-container">$$\lim_{n\to\infty}(n!)^\frac1n=+\infty.$$</span></p>
</blockquote>
<p>Solution 1: Let <span class="math-container">$\{a_n\}_{n\ge 1}$</span> be such that <span class="math-container">$$a_n:=(n!)^\frac{1}{n}, \forall n\in\Bbb N.$$</span> Now clearly <spa... | Z Ahmed | 671,540 | <p><span class="math-container">$$\log L=\lim_{n \infty} \frac{1}{n} \sum_{r=1}^{n} \log r=\lim_{n \infty} \frac{1}{n} \sum_{r=1}^{n} \log (r/n)+\log n= \int_{0}^{1} \log x dx+\infty= \infty$$</span>
<span class="math-container">$$\implies L= \infty .$$</span></p>
|
3,713,580 | <blockquote>
<p>Question: Prove that <span class="math-container">$$\lim_{n\to\infty}(n!)^\frac1n=+\infty.$$</span></p>
</blockquote>
<p>Solution 1: Let <span class="math-container">$\{a_n\}_{n\ge 1}$</span> be such that <span class="math-container">$$a_n:=(n!)^\frac{1}{n}, \forall n\in\Bbb N.$$</span> Now clearly <spa... | Mostafa Ayaz | 518,023 | <p><span class="math-container">$$
n!>n\times (n-1)\times \cdots \times ({n\over 2}+1)>({n\over 2})^{n\over 2}
$$</span>
hence
<span class="math-container">$$
(n!)^{1\over n}>\sqrt{n\over 2}
$$</span>
Done!</p>
|
1,399,921 | <p>Suppose that $A$ and $B$ are invertible, $p \times p$ matrices. If $A^n = B$ and I know all of the entries in $B$, can I find an $A$ for some or all integers $n \ge 0$? How many solutions for $A$ exist? If I'm thinking correctly, then $A = B * (A^{-1})^{n-1},$ but this is sort of self referential. Thanks!</p>
| parsiad | 64,601 | <p>Consider $A=\left(\begin{array}{cc} 1\\ & 1 \end{array}\right)$ and $A^{\prime}=\left(\begin{array}{cc} & 1\\ 1 \end{array}\right)$. Both matrices squared are the identity. Thus, if you know $A^2 = I$, you cannot determine (uniquely) $A$.</p>
|
1,744,832 | <p>Given $$A=\begin{bmatrix}
-4 & 3\\
-7 & 5
\end{bmatrix}$$ Find $A^{482}$ in terms of $A$</p>
<p>I tried using Characteristic equation of $A$ which is $$|\lambda I-A|=0$$ which gives</p>
<p>$$A^2=A-I$$ so $$A^4=A^2A^2=(A-I)^2=A^2-2A+I=-A$$ so</p>
<p>$$A^4=-A$$ but $482$ is neither multiple of $4$ nor Powe... | Oscar Lanzi | 248,217 | <p>There are in fact infinitely many solutions:</p>
<p>$k_1=1, k_2=4, k_3=15, k_n=4k_{n-1}-k_{n-2}$</p>
<p>$3\cdot15^2+1=676=26^2$, etc.</p>
|
27,233 | <p>A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gX)=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$ </p>
<p>A countable discrete group $G$ is inner amenable if there is ... | Andreas Thom | 8,176 | <p>The group $G:=SL_{\infty}(\mathbb Q) = \cup_n SL_n(\mathbb Q)$ is a concrete example. It is obviously simple and non-amenable. Let $g_n \in SL_{\infty}(\mathbb Q)$ be the matrix which is
$$g_n:= 1_n \oplus \left(\begin{matrix} 0 & 1 \newline -1 & 0 \end{matrix}\right) \oplus 1_{\infty}.$$
and let $$m_{n}(A)... |
3,589,178 | <p>I don't know how to do this limit.
<span class="math-container">$$\lim_{x\to\infty}\left(\frac1{x^2\sin^2\frac 1x}\right)^\frac 1{x\sin\frac 1x-1}$$</span>
And here it is as an image, with bigger font:
<a href="https://i.stack.imgur.com/uKwCN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uKwCN.p... | Jean-Claude Colette | 742,526 | <p>Considering <span class="math-container">$\frac{1}{x\sin\frac1x -1}\ln\left(\frac{1}{x^2\sin^2\frac1x}\right)$</span>,</p>
<p>We substitute <span class="math-container">$t=1/x$</span></p>
<p><span class="math-container">$\frac{1}{\frac{\sin t}{t}-1}\ln\left(\frac{1}{\frac{1}{t^2}\sin^2 t}\right)=\frac{t}{\sin t -t... |
3,589,178 | <p>I don't know how to do this limit.
<span class="math-container">$$\lim_{x\to\infty}\left(\frac1{x^2\sin^2\frac 1x}\right)^\frac 1{x\sin\frac 1x-1}$$</span>
And here it is as an image, with bigger font:
<a href="https://i.stack.imgur.com/uKwCN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uKwCN.p... | Andrei | 331,661 | <p>You need to use a couple of tricks. We are going to use <span class="math-container">$$\lim_{x\to\infty}\frac{1}{x\sin\frac 1x}=1$$</span> and <span class="math-container">$$\lim_{y\to\infty}\left(1+\frac 1y\right)^y=e$$</span>
So we rewrite your original expression as:<span class="math-container">$$\begin{align}L=\... |
3,926,603 | <p>So im learning about limits from tutorials and at the same time im solving the examples myself. Im getting solutions that sometimes differ from the explainer's.</p>
<p>Lets take this limit and the way it is solved on the video:</p>
<p><span class="math-container">$$\lim_{x \to -2} \dfrac { (x^3 - x^2 - 6x)}{(x^2+2x)... | Hagen von Eitzen | 39,174 | <p>By squaring<span class="math-container">$$\sqrt{2+\sqrt{2-\sqrt{2+x}}} = x,$$</span>
you arrive at
<span class="math-container">$${2+\sqrt{2\color{red}-\sqrt{2+x}}} = x^2,$$</span>
then
<span class="math-container">$$ 2\color{red}-\sqrt{2+x}=(x^2-2)^2$$</span>
and
<span class="math-container">$$ 2+x=((x^2-2)^2-2)^2.... |
3,640,587 | <blockquote>
<p>How do I find the <span class="math-container">$\angle BCA$</span>?</p>
<p><a href="https://i.stack.imgur.com/zOux6m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zOux6m.png" alt="enter image description here" /></a></p>
</blockquote>
<p>I tried to find the formula of the parabola, ... | Community | -1 | <p>The interior of a set is the largest open set contained in it. Or, it is the union of all open sets contained in it.</p>
|
1,447,110 | <p>This is from an MCQ contest.</p>
<blockquote>
<p>Consider the two functions: $f(x)=\ln(1+x^2)+x+2$ et $g(x)=ch(x)+sh(x)$.</p>
<p>The real number $c$ such that: $(f^{-1})'(2)=g(c)$</p>
<ul>
<li>$1]$ $c=-1$</li>
<li>$2]$ $c=0$</li>
<li>$3]$ $c=1$</li>
<li>$4]$ None of the above statements is corre... | Mihail | 201,204 | <p>$$f^{-1}(2)=t$$
$$f(t)=2$$
$$\ln(1+t^2)+t+2=2$$
Should I continue?</p>
|
2,050,426 | <p>My question is really simple. How can I show intuitively to my complex analysis students that the sine function is unbounded? What kind of behavior makes the complex sine function different from the real one in this sense?</p>
| Paramanand Singh | 72,031 | <p>This is one approach you can provide as far as boundedness of sine function is concerned. We know that $$\sin^{2}z + \cos^{2}z = 1\tag{1}$$ for all $z\in\mathbb{C}$ and if $z \in \mathbb{R}$ then $\cos z, \sin z$ are also real and then from the above equation it follows that $\sin z, \cos z$ are bounded.</p>
<p>If ... |
201,370 | <p>I would like to plot entropy on a 2-simplex, i.e., I want to plot a function for x,y,z s.t. x+y+z=1. My strategy is to take bounds {x,0,1}, {y,0,1-x}, and compute z=1-x-y. However, there is some problem with logarithm in this approach. The following yields an empty plot:</p>
<pre><code>Plot3D[x + Log[y], {x, 0, 1},... | Michael E2 | 4,999 | <p>It works for me:</p>
<pre><code>Plot3D[x + Log[y], {x, 0, 1}, {y, 0, 1 - x}]
</code></pre>
<p><a href="https://i.stack.imgur.com/pBMkI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pBMkI.png" alt="enter image description here"></a></p>
|
3,587,851 | <p>I know that for any two sets <span class="math-container">$A,B$</span>, it holds that <span class="math-container">$A\subseteq B$</span> iff every element of <span class="math-container">$A$</span> is in <span class="math-container">$B$</span>, intuitively. I also know that it is reflexive, antisymmetric and transit... | QuantumSpace | 661,543 | <p>You can still define order relations on a class that is not a set, but it is then not a partially ordered <strong>set</strong>.</p>
<p>See: <a href="https://en.wikipedia.org/wiki/Preordered_class" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Preordered_class</a></p>
<p>Given a set <span class="math-cont... |
748,013 | <p>Definition: $f(X)=${$f(x)|x\in X$}, "$|A|$"represents the number of elements in the set A.
<br/>In the title, $f:S\to T$, "$iff$" means "if and only if".$S$and $T$ are finite sets.
Two definitions for being onto:
<br/>1.If for every element $ t$ in $T$, there exists some $s\in S$, such that $f(s)=t$, then it's onto... | Christian Blatter | 1,303 | <p>An $n$-tuple is just a list of $n$ numbers $x_k$ $\>(1\leq k\leq n)$, written as $$(x_1,x_2,\ldots, x_n)\ .\tag{1}$$ Unless you want to do linear algebra with it you can leave it at that. </p>
<p>In linear algebra tuples are used for various purposes and become part of a certain algebraic technique called <em>ma... |
1,872,234 | <p>If we have</p>
<p>$$f(x) = \sum_{n=0}^\infty a_n x^n$$</p>
<p>The $k$th derivative is </p>
<p>$$f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} x^n$$</p>
<p>Which also means that</p>
<p>$$f^{(k)}(0) = k! a_k$$</p>
<p>Implying</p>
<p>$$a_k = \frac{f^{(k)}(0)}{k!}$$</p>
<p>So we can substitute this ... | Doug M | 317,162 | <p>Same logic still applies</p>
<p>$f(x) = \sum_{n=0}^{\infty} a_{n} (x-a)^n\\
f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} (x-a)^n\\
f^{(k)}(a) = a_{k} k!\\
a_k = \frac {f^{(k)}(a)}{k!}$</p>
|
4,325,440 | <p>I need to compute <span class="math-container">$$I:=\int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}dx$$</span>
(<span class="math-container">$a>0$</span>)</p>
<p>Probably there is a way to compute it with residue theorem.</p>
<p>My thoughts:</p>
<ul>
<li>The singularity at <span class="math-container">$x... | Sam Ginrich | 1,001,541 | <h3>Not a solution</h3>
<p>Partial Fraction Decomposition of the denominator is straight forward as</p>
<p><span class="math-container">$$ \frac{A}{x} + \frac{B}{\pi-ax} + \frac{C}{\pi+ax} $$</span></p>
<p>Remains the <em>Sine Integral</em> type. You might consider their asymptotic behavior or win with Residue Calculu... |
4,325,440 | <p>I need to compute <span class="math-container">$$I:=\int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}dx$$</span>
(<span class="math-container">$a>0$</span>)</p>
<p>Probably there is a way to compute it with residue theorem.</p>
<p>My thoughts:</p>
<ul>
<li>The singularity at <span class="math-container">$x... | robjohn | 13,854 | <p>As the singularities are removable, I like to translate the contour so that it misses the singularities when we apply <span class="math-container">$\sin(x)=\frac1{2i}\left(e^{ix}-e^{-ix}\right)$</span>.
<span class="math-container">$$\newcommand{\Res}{\operatorname*{Res}}
\begin{align}
&\int_{-\infty}^\infty\fra... |
31,562 | <p>How to evaluate the number of ordered partitions of the positive integer <span class="math-container">$ 5 $</span>?</p>
<p>Thanks!</p>
| HEKTO | 92,112 | <p>I'm so late to this discussion, however I think I can add something useful. The method with pennies and grains of rice, described here by @AndreNicolas, can be made more clear. Let's have <span class="math-container">$N$</span> pennies in a row with <span class="math-container">$N-1$</span> gaps between them. Let's ... |
3,080,402 | <p>I need to show that the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^{-n}}{n!}$</span> is convergent.</p>
<p>I invoked the limit comparison with the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^n}{n!}$</span> which is absolutely convergent (and hence convergent).</p>
<p>I got ... | Dr. Sonnhard Graubner | 175,066 | <p>Use that we get from <span class="math-container">$$a+b=c+d$$</span> so <span class="math-container">$$a^3+b^3+3ab(a+b)=c^3+d^3+3cd(c+d)$$</span> and <span class="math-container">$$x^3+y^3=(x+y)(x^2-xy+y^2)$$</span></p>
|
917,849 | <p>I tried following but then I got stuck</p>
<p>$676 = 26*26$ </p>
<p>$12^{39} + 14^{39}$ is divisible $26$ for sure since $a^n + b^n$ is divisible by $(a+b)$ when $n$ is odd. But what to do next?</p>
| lab bhattacharjee | 33,337 | <p>Clearly, $$12^{39}+14^{39}$$ is divisible by $4$</p>
<p>Again, $$12^{39}=(13-1)^{39}\equiv-1+\binom{39}113\pmod{169}\equiv-1$$</p>
<p>$$14^{39}=(13+1)^{39}\equiv1+\binom{39}113\pmod{169}\equiv1$$ </p>
<p>More generally if $\displaystyle a\equiv b\pmod p, a^p\equiv b^p\pmod{p^2}$</p>
<p>Here $\displaystyle p=13,... |
3,520,313 | <p>Let <span class="math-container">$I\subset\mathbb R$</span> be an interval and <span class="math-container">$\left(f_n\right)_{n\in\mathbb N}:I\to\mathbb R$</span> be a sequence that is uniformly convergent on <span class="math-container">$I$</span>. Is the sequence <span class="math-container">$g_n=\left\{\frac{f_n... | William M. | 396,761 | <p>Let <span class="math-container">$\varphi(t) = \dfrac{t}{1 + t^2}.$</span> Then <span class="math-container">$\varphi'(t) = \dfrac{1 + t^2 - 2t^2}{(1 + t^2)^2}$</span> is bounded on <span class="math-container">$\mathbf{R};$</span> this shows <span class="math-container">$\varphi$</span> is <em>uniformly continuous.... |
1,452,903 | <p>Has the aforementioned constant ever been used in any major proofs? Can it be expressed in terms of $e$, $\pi$, or both? Does it appear in any sort of geometric sense like $\pi$ does? Or is it just a kind of pretty number somebody thought would be nice to experiment with?</p>
| quid | 85,306 | <blockquote>
<p>Or is it just a kind of pretty number somebody thought would be nice to experiment with?</p>
</blockquote>
<p>This. </p>
<hr>
<p>More specifically, it is constructed in such a way that its (decimal) digits are easy to investigate. This allows to establish fairly easily that it is normal in its base... |
2,405,709 | <p><strong>Problem</strong></p>
<p>A math book wants me to prove that given two natural numbers $m, \ n$ are not divisible by $5$, then the difference $m^4 - n^4$ <em>is</em> divisible by $5$.</p>
<p><strong>Thoughts</strong></p>
<p>The only method I can think of now, is to go through all the possible ways of writin... | Tsemo Aristide | 280,301 | <p>This is an appplication of <a href="http://mathworld.wolfram.com/FermatsLittleTheorem.html" rel="nofollow noreferrer">Fermat’s little theorem</a>: since $5$ is a prime, $m^4 \equiv 1 \pmod{5}$.</p>
|
234,845 | <p>How do we prove that a function $f$ is measurable if and only if $\arctan(f)$ is measurable?</p>
<p>If I use the definition of measurable functions, that is, a function is measurable if and only if its inverse is measurable?</p>
| Amr | 29,267 | <p>Since f is measurable and arctan is continuous therefore arctan f is Lebesgue measurable. If arctan f is measurable, then tan arctan f is Lebesgue measurable (because tan is continuous) therefore f is Lebesgue measurable.</p>
|
10,948 | <p>This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.</p>
<p>Some quick facts:</p>
<ul>
<li>Pretty much by c... | j.c. | 353 | <p>If you consider the two-dimensional version of your sum you are quickly led to the <a href="http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions" rel="nofollow">Weierstrass elliptic functions</a>:</p>
<p>Naively, one is led to consider $$\sum_{m,n\in \mathbb{Z}}\frac{1}{(z+m+\omega n)^2}$$ (where $\omega... |
4,365,318 | <p>I recently came across the following:</p>
<p><span class="math-container">$\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...=1$</span></p>
<p>The factorials in the denominators reminded me of a Taylor Series. In particular, I found that it fit the Maclaurin series, evaluated at 1:</p>
<p><span class="math-cont... | b00n heT | 119,285 | <p>Notice that your sum is of the form
<span class="math-container">$$\sum_{n=2}^{\infty}\frac{n-1}{n!}$$</span>
and we know that
<span class="math-container">$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$</span>
So that
<span class="math-container">$$\frac{e^x-1}{x}=\sum_{n=1}^{\infty}\frac{x^{n-1}}{n!}$$</span>
and now di... |
4,365,318 | <p>I recently came across the following:</p>
<p><span class="math-container">$\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...=1$</span></p>
<p>The factorials in the denominators reminded me of a Taylor Series. In particular, I found that it fit the Maclaurin series, evaluated at 1:</p>
<p><span class="math-cont... | PierreCarre | 639,238 | <p>Similar to @b00nheT but without differentiation,
<span class="math-container">$$
\sum_{n=2}^{\infty} \frac{n-1}{n!} = \sum_{n\ge 2}\frac{1}{(n-1)!}-\sum_{n\ge 2}\frac{1}{n!} = \sum_{n\ge 1}\frac{1}{n!}-\sum_{n \ge 2}\frac{1}{n!} = (e-1)-(e-1-1)=1.
$$</span></p>
<p>This is sufficient to prove the equality.</p>
|
1,350,147 | <p>I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined?</p>
<p>Afais for Riemannian manifolds, there exists always exactly one Levi-Civita connection, but the calculation is rather cumbersome.</p>
<p>Now, if we consider manifolds without a metric, i... | Jeffrey Rolland | 329,781 | <p>I'm not sure how on-point this answer is, but it may give an answer in some interesting cases.</p>
<p>If one has a (complete?) differential equation <span class="math-container">$\xi$</span> on an manifold <span class="math-container">$M$</span>, the differential equation has a flow <span class="math-container">$\Ph... |
2,310,413 | <p>A coin is flipped twice. $A$ is the result of the first coin flip and $B$ the result of the second one. I know that $A$ and $B$ are independent. Following $$P(A\cap B)=P(A)P(B),$$ when it comes to write down the event of the intersection of $A$ and $B$ I am stuck because the event $A$ excludes the event $B$, and yet... | drhab | 75,923 | <p>You call $A$ a "result". </p>
<p>That is okay but then it can take several values and e.g. you can look at it as function $A:\Omega\to \{H,T\}$. </p>
<p>Likewise $B$ is such a function. </p>
<p>Behind all this we have a probability space $\langle\Omega,\mathcal A,P\rangle$.</p>
<p>The independence tells us thing... |
126,251 | <p>Suppose one has a finite number of distances $d_1,\ldots,d_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.</p>
<p>Define for each pair of points $x$ and $y$ in the plane
$$d(x,y) = \inf\left\lbrace d_{i_1}(x_0,x_1) + \cdots d_{i_l}(x_{l-1},x_l) \right\rbrace$$
where the infimum is tak... | Sergei Ivanov | 4,354 | <p>This is possible even on the real line.</p>
<p>There is a strictly increasing continuous function $f:[0,1]\to\mathbb R$ whose derivative is zero almost everywhere. It is a suitable sum of a series of Cantor functions. See, for example, Gelbaum and Omsted, "Counterexamples in analysis" (1964), Chapter 8, Example 30.... |
119,981 | <p>Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence</p>
<p>$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$</p>
<p>gives a homomorphism from $\operatorname... | Igor Rivin | 11,142 | <p>For a mathematical model see <a href="https://dl.dropbox.com/u/5188175/fiatmoney.pdf" rel="nofollow">Hayashi and Matsui, 1994.</a> For an in-depth discussion without too many (actually, any) equations, see many books by Murray Rothbard (all available on Amazon.com).</p>
|
4,076,584 | <blockquote>
<p>Let <span class="math-container">$G$</span> be the relation in <span class="math-container">$\Bbb {R}^2$</span> given by <span class="math-container">$$ G = \{((a, b), (c, d)) \in \Bbb {R}^2 \times \Bbb {R}^2: a ^ 2 + b ^ 2 = c ^ 2 + d ^ 2 \}. $$</span> Prove that <span class="math-container">$ G $</spa... | Jens Renders | 131,972 | <p>This notation is often formally used, notably in the definition of the <a href="https://en.wikipedia.org/wiki/Laplace_operator" rel="nofollow noreferrer">Laplace operator</a>. We can make it explicit though.</p>
<p>For convenience, lets only work with functions that are <span class="math-container">$C^\infty$</span>... |
3,986,554 | <p>I'm trying to work out if the following two curves are birationally equivalent:</p>
<p><span class="math-container">$$Y^2 = 2X^4 + 17X^2 + 12$$</span>
<span class="math-container">$$2Y^2 = X^4 - 17$$</span></p>
<p>(I'm considering the above as the affine shorthands for the projective curves they represent)</p>
<p>I ... | José Carlos Santos | 446,262 | <p>Yes, the solutions of <span class="math-container">$x^2-2x+2=0$</span> are <span class="math-container">$1+i$</span> and <span class="math-container">$1-i$</span>. So, the solutions of <span class="math-container">$e^{2z}-2e^z+2=0$</span> are all those numbers <span class="math-container">$z$</span> such that <span ... |
3,986,554 | <p>I'm trying to work out if the following two curves are birationally equivalent:</p>
<p><span class="math-container">$$Y^2 = 2X^4 + 17X^2 + 12$$</span>
<span class="math-container">$$2Y^2 = X^4 - 17$$</span></p>
<p>(I'm considering the above as the affine shorthands for the projective curves they represent)</p>
<p>I ... | fleablood | 280,126 | <p>Yes.....</p>
<p>Once you get <span class="math-container">$e^z = 1 +i, 1-i$</span> you can take the following as a formula (assuming <span class="math-container">$a,b$</span> are real):</p>
<p><span class="math-container">$a + bi= \sqrt{a^2 + b^2}e^{\arctan \frac ab i}= e^{\ln(a^2 + b^2)}e^{\arctan \frac ab i}= e^{\... |
4,203,235 | <p>Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$ a \in G$</span>. If <span class="math-container">$A = \{a \}$</span>, prove <span class="math-container">$C_G(a) = C_G(a^{-1})$</span></p>
<p>Proof:
Since <span class="math-container">$C_G(a) \le G$</span>, it is given that... | user1729 | 10,513 | <p><strong>Claim.</strong> Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$ a \in G$</span>. If <span class="math-container">$A = \{a \}$</span>, prove <span class="math-container">$C_G(a) = C_G(a^{-1})$</span>.</p>
<p>Quick comment: Centralisers are typically defined for se... |
1,649,076 | <p>Could anyone guide along with this question?</p>
<p>I was trying $(A−I)(A−I)^{−1} = I$ and was figuring if there's a way out to expand $(A−I)^{−1}$. I also tried $(A−I)x=0$ but to no avail.</p>
| Community | -1 | <p>See from the relation you have given.
$f (x)=x^3-2$ is a polynomial that annihilates the operator.hence the minimal polynomial must divide this polynomial .and hence as you can see that the eigenvalues are.cube root of 2....i hope this hepls...proceed on these lines</p>
|
1,649,076 | <p>Could anyone guide along with this question?</p>
<p>I was trying $(A−I)(A−I)^{−1} = I$ and was figuring if there's a way out to expand $(A−I)^{−1}$. I also tried $(A−I)x=0$ but to no avail.</p>
| MooS | 211,913 | <p>Of course, in this case, we can just immediately see the inverse after using the factorization of $x^3-1$.</p>
<p>But I think it is good to know the general method:</p>
<p>Let $A$ be a matrix with $f(A)=0$ for some polynomial $f$ and $g$ some other polynomial, which is co-prime to $f$. The euclidean algorithm (so ... |
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Chappers | 221,811 | <p>Tom Körner's got a couple:</p>
<ul>
<li><em>Calculus for the Ambitious</em></li>
<li><em>A Companion to Analysis: A second first and first second course in analysis.</em></li>
</ul>
<p>All of his stuff is extremely readable, whether it's formatted like a textbook or otherwise. (His book on Fourier analysis is easi... |
2,847,359 | <p>So say you had $5^x=25$ where $x$ is obviously $2$, how would you work $x$ out if the question wasn't obvious?</p>
<p><strong>edit:</strong> What if the question was something like $a^x=-1$ (where $a$ is any number).</p>
<p>PS to all the maths elitists out there: Feel free to down vote, I just want to know how to ... | 0x.dummyVar | 575,408 | <p>You would want to use a logarithmic function.</p>
<p>$a^x = b$, $x = \log_a(b)$</p>
<p>So in your case:</p>
<p>$5^x = 25$, $x = \log_5(25)=2$</p>
|
599,467 | <p>I would like to summarize my formula.
$p$ and $y$ are constant value, $10000$ and $0.65$.</p>
<p>When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to summarize or other formula for that equivalent?</p>
<p>$$x_n=(p+x_{n-1})y$$</p>
<p><strong>Update :</strong></p>... | Tobias | 114,571 | <p>Already answered at <a href="http://www.stackoverflow.com/questions/20465190/summarize-my-formula">http://www.stackoverflow.com/questions/20465190/summarize-my-formula</a>.
The general tool to deal with difference equations is Z-Transformation. The solution for the wanted initial condition $x_0=0$ is
\begin{align*}
... |
657,162 | <p>Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$</p>
<p>I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$</p>
<p>Also a tighter upper bound is appreciated.</p>
| Przemo | 99,778 | <p>Let us denote
\begin{equation}
{\mathcal I}_{n,k} := \prod\limits_{i=3}^k (n-i)
\end{equation}
Then the quantity ${\mathcal I}_{n,k}$ is a polynomial of order $k-2$ in $n$.We have:
\begin{eqnarray}
{\mathcal I}_{n,k} &:=& \sum_{l=0}^{k-2} n^{k-2-l}
\cdot (-1)^l \sum\limits_{3 \le i_1 < i_2 < \dots &l... |
2,192,314 | <p>Once upon a time, I was looking at interesting properties of prime numbers. One thing I noticed was that if we take the <strong>absolute values of the differences</strong> between each prime, and repeat this process on the differences recursively, the first column turns out to always be $1$ (With the exception of th... | Brian Storie | 638,362 | <pre><code>2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101
1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4
1 1 2 2 2 2 2 4 4 4 2 2 2 2 0 4 4 2 2 4 2 2 2 4
... |
3,283,101 | <p>I was working on a problem:</p>
<blockquote>
<p>"Find the number of those <span class="math-container">$5$</span>-digit numbers whose digits are non-decreasing." </p>
</blockquote>
<p>I was able to calculate the number of decreasing <span class="math-container">$5$</span>-digit numbers, which I found to be <span... | N. F. Taussig | 173,070 | <p>The flaw in your attempt is that not every number whose digits are not strictly decreasing has nondecreasing digits. For instance, the number <span class="math-container">$32856$</span> is a five-digit number with digits that are not strictly decreasing. However, it is also a number in which the digits are not non... |
134,848 | <p>In short, I look for a <strong>concise</strong> definition of a function in Mathematica that calculates the following:</p>
<p>$f\left(\begin{pmatrix}a_{11}&\dots&a_{1m}\\\vdots&\ddots&\vdots\\a_{n1}&\dots&a_{nm}\\\end{pmatrix},\left(b_1,\dots,b_p\right)\right)=\begin{pmatrix}a_{11}&\dots... | kglr | 125 | <pre><code>aa = Array[a, {4, 4}];
bb = Array[b, {2}];
ArrayPad[#, {{0}, {0, Length@#2}}, #2] &[aa, bb]
(* or *) Distribute[{#, {#2}}, List, List, List, Join] &[aa, bb]
</code></pre>
<blockquote>
<p><img src="https://i.stack.imgur.com/05D3c.png" alt="Mathematica graphics"></p>
</blockquote>
|
1,308,045 | <p>consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$.</p>
<ol>
<li>How would I transform the first term by hand to the second. What are the exact steps?</li>
<li>Does this transformation always lead to the same resul... | gt6989b | 16,192 | <p>When $a$ is the average of the $x_i$, you have
$$
\begin{split}
\sum_{i=1}^n (x_i - a)^2
&= \sum (x_i^2 -2ax_i + a^2) \\
&= \sum x_i^2 - 2a \sum x_i + a^2 n \\
&= \sum x_i^2 - 2a (na)+ a^2 n \\
&= \sum x_i^2 - 2n a^2 + a^2 n \\
&= \sum x_i^2 - a^2 n
\end{split}
$$</p>
|
1,308,045 | <p>consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$.</p>
<ol>
<li>How would I transform the first term by hand to the second. What are the exact steps?</li>
<li>Does this transformation always lead to the same resul... | Mark Viola | 218,419 | <p>If $x_6=70$, then the sum $\sum_{i=1}^{6}x_i=270=6a$. So, $a$ is the average of the $x_i$'s.</p>
<p>Then noting that $(x_i-a)^2=x_i^2-2ax_i+a^2$, we have</p>
<p>$$\sum_{i=1}^{6}(x_i-a)^2=\sum_{i=1}^{6}x_i^2-2a\sum_{i=1}^{6}x_i+6a^2.$$</p>
<p>But, recalling that $a$ is the average of the $x_i$'s, we have $-2a\sum... |
2,751,831 | <p>Are there any results along the following lines: </p>
<p>Let $\Gamma_1$ and $\Gamma_2$ be groups with respective finite index subgroups $\Gamma_0^i$ for $i=1,2$. If $\Gamma_1 \cap \Gamma_2 \leq \Gamma_0^i$ for $i=1,2$ can we conclude that $[\Gamma_1 \ast_{\Gamma_1 \cap \Gamma_2} \Gamma_2: \Gamma_0^1 \ast_{\Gamma_1 ... | Robert Bell | 4,009 | <p>The following example shows that this cannot hold: let $\Gamma = <a_1, a_2 ; >$, the free group on $\{a_1,a_2\}$. Let $\Gamma_i = gp<a_i>$ for $i=1,2$, where $gp<S>$ means the subgroup of $\Gamma$ generated by $S \subset \Gamma$. Let $\Gamma_0^i = gp<a_i^2>$ for $i = 1, 2$. Then $\Gamma_1 ... |
1,227,759 | <p>$$x \mod 1000 \mod 5$$</p>
<p>I would have thought that it was $x \mod 5000$ except that it doesn't hold true for $x = 5005$ since you'll get zero, but $5005 \mod 5000 = 5$.</p>
| MonkeyKing | 225,981 | <p>The Chinese Remainder Theorem provides solution to the following
$$\begin{cases}
a_1 & \mod n_1\\
a_2 & \mod n_2\\
\dots
\end{cases}$$
if $n_1, n_2, \dots$ are relative primes. If not, you can simply factorize $n_x = p_1 p_2 \dots$ and express $a_x$ with $\mod p_1, p_2, \dots$ and in your question, $1000 = 5... |
1,573,141 | <p>Consider the following equality:</p>
<p>$$x=(1-2)(1+2+4)+(2-3)(4+6+9)+(3-4)(9+12+16)+....+(49-50)(2401+2450+2500)$$</p>
<p>Solve for $x$.</p>
<p>The only thing I noticed is the first part like $(1-2)$,$(3-4)$ gives us $-1$ but then I just don't see what the trick behind this problem is.</p>
| Gottfried Helms | 1,714 | <p>Hint: have you ever tried to look at a formula like this: $$ {a^3 - b^3 \over a-b} = (?)(?)$$ ? Just expand and observe the two occuring factors and look like the astronomers over long distances ... </p>
<p><strong><em>[update]</em></strong> </p>
<p>As $ {a^3-b^3 \over a - b }= a^2+ab+b^2$... |
2,216,161 | <p>How to prove with identities:
(A - B) ∩ (C - D) = (A ∩ C) - (B ∪ D) </p>
| DanielWainfleet | 254,665 | <p>Easily seen to be false . Suppose $f(p)=f(q)$ and there is an open set $U$ in $X$ with $p\in U$ and $q\not \in U.$ If $V\subset Y$ and $p\in f^{-1}V$ then $f(p)\in V$, so $$f^{-1}V\supset f^{-1}\{f(p)\}=f^{-1}\{f(q)\}\supset \{q\}$$ so $f^{-1}V\ne U.$</p>
<p>Even if $f$ is a continuous injection it can fail. For ex... |
55,509 | <p>I have a measured experimental dataset which is well approximated by the sum of several basis functions in linear combinations. Linear least squares of course gives me the optimal weight for each basis function. These basis functions are all unrelated and may or may not be correlated (or even repeated). That still ... | guy | 12,806 | <p>There is a lot of literature on model selection of this nature; I'm somewhat surprised you haven't found anything on google. The Stepwise method you came up with yourself is among them. Obviously, though, there is probably too much on this topic to contain in this post, so a reference should be more useful. There is... |
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| Kanwaljit Singh | 401,635 | <p>$140^{67}+153^{51}\equiv 4^{67} + 0^{51}$</p>
<p>Now $4^{67} \equiv 4^3 \equiv 13 \mod 17$</p>
|
3,643,773 | <p>I have a circle with center (0,0) and radius 1. I have calculated a random point inside a circle by generating a random angle <span class="math-container">$a=random()\cdot 2\pi $</span> and a random distance smaller than or equal to the radius <span class="math-container">$b=random()\cdot r$</span>. The center of th... | Community | -1 | <p>In polar coordinates,</p>
<p><span class="math-container">$$\rho\cos(\theta-a)=b,$$</span></p>
<p>hence
<span class="math-container">$$x\cos a+y\sin a=b.$$</span></p>
|
1,208,269 | <p>let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Find the minimum of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$</p>
<p>Is the answer $3\sqrt{3}$ by any chance?</p>
| John Hughes | 114,036 | <p>You can read the first equation as saying that the vector $(x, y)$ is perpendicular to the vector $(a, 2)$; the second says that it's perpendicular to $(1, -b)$. Under what conditions on $a$ and $b$ is there a nonzero vector perpendicular to both of these? </p>
|
4,582,099 | <p>Find limit of the given function:</p>
<p><span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})}
$$</span></p>
<p>I tried putting 0 instead of x
<span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)... | Ninad Munshi | 698,724 | <p>The limit of a product is the product of the limits (provided they exist). Multiply and divide by terms so that the resulting limmand is a product of known limits:</p>
<p><span class="math-container">$$\frac{\left(4^{\arcsin(x^2)}-1\right)\left(\sqrt[10]{1-\arctan(3x^2)}-1\right)}{(1- \cos \tan 6x)\log(1-\sqrt{\sin ... |
1,108,019 | <p>Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show that $T(Y)=\frac{1}{n}\sum_iY_i$ is complete:
$$
h\text{ being a function s.t. }E_\theta(h(T))=0 \forall \theta\in\Th... | Michael Hardy | 11,667 | <p>You want a function $h$ such that for all $\theta\in\mathbb R$, the following integral is zero:
$$\int_{-\infty}^\infty \exp\left(-\frac{n}{2}(t-\theta)^2\right)h(t) \, dt.$$
This is
$$
\int_{-\infty}^\infty \exp\left(-\frac n 2 \theta^2\right) \exp(nt\theta)\exp\left(-\frac n 2 t^2\right)h(t) \, dt.
$$
The first fa... |
1,711,266 | <p>The circle inscribed in the triangle $ABC$ touches the sides $BC$ , $CA$ , and $AB$ in the points $A_1,B_1,C_1$ respectively. Similarly the circle inscribed in the triangle $A_1B_1C_1$ touches the sides in $A_2,B_2,C_2$ respectively, and so on. If $A_nB_nC_n$ be the $n^{th}$ $\triangle$ so formed, Prove its angles ... | mathlove | 78,967 | <p>Quang Hoang has already provided a good hint.</p>
<p>Let us prove that by induction on $n$.</p>
<p>Let $O$ be the incenter of $\triangle{ABC}$. </p>
<p>Then, noting that $OB\perp A_1C_1,OC\perp A_1B_1$, we have
$$\begin{align}\angle{B_1A_1C_1}&=\pi -\angle{B_1A_1C}-\angle{C_1A_1B}\\&=\pi-\left(\pi-\frac{\... |
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| noname1014 | 27,743 | <p>$f$ is a subset of $[0,1] \times R$, while $f(x)$ is an element of $\mathbb{R}$ .</p>
<p>They are not the same.</p>
<p>In your example, $f=\{\left<x,x^2\right>|x \in [0,1]\}=\{\left<x,f(x)\right>|x \in [0,1]\} $.</p>
|
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| Gold | 39,106 | <p>I think what confuses you a little is the lack of the formal notion of what a function is: given sets $A$ and $B$ we can think of them just as collections of some types of objects. You can have a set $A$ whose elements are potatoes and a set $B$ whose elements are apples and so on. Given two sets, we can <em>relate<... |
2,783,922 | <p>I will start by the usual definition of a path connected set:</p>
<blockquote>
<p>A subset $A$ of a topological space $X$ is path connected when, for every pair of points $a,b\in A$, there exists a continuous function $f:[0,1]\rightarrow A$ such that $f(0)=a$ and $f(1)=b$.</p>
</blockquote>
<p>This question may ... | mallan | 557,176 | <p>With $a=-2$, find $f^{-1}(a)$ by the following</p>
<p>$$f(b)=a$$</p>
<p>$$b^3+b^2+2b+2=0$$</p>
<p>$$b^2(b+1)+2(b+1)=(b^2+2)(b+1)=0$$</p>
<p>Real solutions give that $b=-1$</p>
<p>$$\therefore-1=f^{-1}(a)$$</p>
<p>Now</p>
<p>$$f^{-1}{'}(a)=\frac{1}{f'(f^{-1}(a))}$$</p>
<p>$$f'(-1)=\left.\left(3x^2+2x+2\right)... |
2,280,696 | <p>For (a)$\sum{\frac{1}{(\ln(n))^n}}$</p>
<p>$\ln(n)>2$ $ \forall n > e^2$</p>
<p>$(\ln(n))^n>2^n$ $ \forall n > e^2$</p>
<p>$\frac{1}{(\ln(n))^{n}}<\frac{1}{2^n}$ $ \forall n > e^2$</p>
<p>$\sum{\frac{1}{2^n}}$ is convergent by geometric series test</p>
<p>$\sum{\frac{1}{(\ln(n))^n}}$ i... | Olivier Oloa | 118,798 | <p>Your answer to $(a)$ is correct.</p>
<p>Concerning the series $(b)$ one may consider the partial sums, for $N\ge1$,
$$
\begin{align}
S_{2N}:=\sum_{n=0}^{2N}\left({\frac{2}{(-1)^n-3}}\right)^n=&\sum_{p=0}^N\left({\frac{2}{1-3}}\right)^{2p}+\sum_{p=1}^N\left({\frac{2}{-1-3}}\right)^{2p-1}
\\\\=&\sum_{p=0}^N1-... |
1,449,845 | <p>So the biquadratic equation is $x^4+(2-\sqrt3)x^2+2+\sqrt3=0$. Let $a_1,a_2,a_3,a_4$ be its roots. So we have to find the value of $(1-a_1)(1-a_2)(1-a_3)(1-a_4)$ . <br>
<strong>My attempt:</strong> <br>
So of we put $x^2=t$, and let the roots of the new quadratic equation be $a_1,a_2$. So we get that $a_1=-a_3;a_2=-... | lab bhattacharjee | 33,337 | <p>Let $x=1-y$</p>
<p>So, we have $(1-y)^4+(2-\sqrt3)(1-y)^2+2+\sqrt3=0\iff y^4+\cdots+1+(2-\sqrt3)+(2+\sqrt3)=0$</p>
<p>$\implies\prod_{r=1}^4(1-a_r)=(-1)^4\dfrac{1+(2-\sqrt3)+(2+\sqrt3)}1$</p>
|
1,449,845 | <p>So the biquadratic equation is $x^4+(2-\sqrt3)x^2+2+\sqrt3=0$. Let $a_1,a_2,a_3,a_4$ be its roots. So we have to find the value of $(1-a_1)(1-a_2)(1-a_3)(1-a_4)$ . <br>
<strong>My attempt:</strong> <br>
So of we put $x^2=t$, and let the roots of the new quadratic equation be $a_1,a_2$. So we get that $a_1=-a_3;a_2=-... | Bob Hsu | 866,558 | <p><span class="math-container">\begin{align}
f(x) &= (x-a_1)(x-a_2)(x-a_3)(x-a_4) \\
&= x^4 +(2−√3) x^2 +2 + √3 \\
f(1)& = 1 + 2 - √3 + 2 - √3 = 5
\end{align}</span></p>
|
2,931,936 | <blockquote>
<p>Is the set of vectors of the form <span class="math-container">$[ a, b, a+2b ]$</span> a subspace of <span class="math-container">$\mathbb R^3$</span>?</p>
</blockquote>
<p>How to show this? The definition of a subspace <span class="math-container">$S$</span> requires showing that if <span class="mat... | KM101 | 596,598 | <p>You can convert it to a quadratic equation, where it can be solved rather simply.<br>
<span class="math-container">$$12g = 12(\frac{2}{3g} -1)+11$$</span>
<span class="math-container">$$12g = \frac{24}{3g}-12+11 \implies 12g = \frac{8}{g}-1$$</span>
<span class="math-container">$$\implies 12g^2 = 8-g\text{ (Multipl... |
4,244,708 | <p>Let <span class="math-container">$S$</span> denote the set of all solutions of the following differential equation defined on <span class="math-container">$C^3[0,\infty)$</span>;</p>
<p><span class="math-container">$$
\begin{align}
\frac{d^3x}{dt^3} + b \frac{d^2x}{dt^2} + c \frac{dx}{dt} + dx = 0
\end{align}
$$</... | River Li | 584,414 | <p>Using the identities
<span class="math-container">\begin{align*}
\frac{1}{x} &= \int_0^\infty \mathrm{e}^{-x t}\, \mathrm{d} t, \\
\ln x &= \int_0^\infty \frac{\mathrm{e}^{-s} - \mathrm{e}^{-xs}}{s}\,\mathrm{d} s,
\end{align*}</span>
we have
<span class="math-container">\begin{align*}
I &= \int_0^\inf... |
2,068,643 | <p>I'm solving a problem about a <a href="https://en.wikipedia.org/wiki/Chemical_reactor#PFR_.28Plug_Flow_Reactor.29" rel="nofollow noreferrer">plug flow reactor</a> and I have this limit to compute. Just to control my result I asked <a href="http://www.wolframalpha.com/input/?i=lim_%7BR%20%5Cto%20%2B%5Cinfty%7D%20(1-e... | Marko Gulin | 400,710 | <p>You should use L'Hopital theorem. That gives us:</p>
<p>$$\lim_{R\rightarrow+\infty} = \frac{\frac{d}{dR}\bigl( 1 - \exp(\frac{x}{R+1}) \bigr)}{\frac{d}{dR}\bigl( \frac{R}{R+1} - \exp(\frac{x}{R+1} \bigr)}$$</p>
<p>This is equal to:</p>
<p>$$\lim_{R\rightarrow+\infty} \frac{\frac{x}{(R+1)^2}}{ \frac{R+1-R}{(R+1)^... |
2,068,643 | <p>I'm solving a problem about a <a href="https://en.wikipedia.org/wiki/Chemical_reactor#PFR_.28Plug_Flow_Reactor.29" rel="nofollow noreferrer">plug flow reactor</a> and I have this limit to compute. Just to control my result I asked <a href="http://www.wolframalpha.com/input/?i=lim_%7BR%20%5Cto%20%2B%5Cinfty%7D%20(1-e... | Arnaldo | 391,612 | <p>Use the fundamental limit:</p>
<p>$$\lim_{x\rightarrow \infty}x(e^{1/x}-1)=1 \Rightarrow \lim_{x\rightarrow \infty}x(e^{a/x}-1)=a$$</p>
<p>and write your limit like:</p>
<p>$$\lim_{R \rightarrow \infty}\left(\frac{(R+1)(1-e^{x/(R+1)})}{-1+(R+1)(1-e^{x/(R+1)})}\right)=\frac{x}{1+x}$$</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.