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 |
|---|---|---|---|---|
3,869,945 | <p>I have a problem that is formulated as this:
<span class="math-container">$$\begin{matrix}\min\\x \in \mathbb{R}^2\end{matrix} f(\mathbf{x}) := (2 x_1^2 - x_2^2)^2 + 3x_1^2-x_2$$</span>
The task is: Perform <strong>one</strong> iteration using the steepest descent algorithm when <span class="math-container">$\mathbf... | Decaf Sux | 691,869 | <p><a href="https://math.stackexchange.com/users/139123/david-k">David K</a>
You were right about the <span class="math-container">$\alpha$</span> so I looked again and I managed to find my error.
The thing I forgot was that I need to minimize <span class="math-container">$\alpha$</span> using:
<span class="math-contai... |
3,626,839 | <p>I have a parking lot with dimensions <span class="math-container">$20 m \times 30 m$</span> which is illuminated by lights placed in different positions and heights as shown below:</p>
<p><a href="https://i.stack.imgur.com/27CaQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/27CaQ.png" alt="Park... | AUPlagiatcontrol | 773,630 | <p>Regarding the first question. You need to create a system of linear equations such that:</p>
<p><span class="math-container">$ X D = Y $</span></p>
<p>D is a <span class="math-container">$ 600 \times 30 $</span> matrix containing all the distances between the lamps and the squares, <span class="math-container">$ Y... |
2,575,149 | <p>I have been given the following problem: </p>
<p>A spherical balloon is expanding at the rate of 60 pie in^3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches? </p>
<p>I don't understand how the way I set the problem up is not giving me the correct answer. I hav... | Eddy | 110,654 | <p>You want to know $dA/dt$, so rearrange the equation you have for this quantity, substitute in your expression for $A$ as a function of $r$, and finally evaluate at the value of $r$ given. </p>
|
296,138 | <p>My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \right)^n \right\rfloor$, the fractional part of $\left( \frac{4}{3} \right)^n$, might help him improve his results. Not... | Jan-Christoph Schlage-Puchta | 37,555 | <p>A non-trivial lower bound can be obtained using linear forms in $p$-adic logarithms. Suppose that $\left\{\left(\frac{4}{3}\right)^n\right\}$ is small. Clearly it is a rational number with denominator $3^n$, so assume it is $\frac{a}{3^n}$. Then $3^n|4^n-a$, that is, $\nu_3(4^n-a)$ is exceptionally large.</p>
<p>In... |
847,719 | <p>How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.</p>
| Peter Woolfitt | 145,826 | <p>Hint:
By the <a href="http://en.wikipedia.org/wiki/Chain_rule" rel="nofollow">chain rule</a> we have that
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x).$$</p>
<p>In your question the answer would be $3(5x-2)^2$ if you were taking the derivative with respect to $5x-2$, but you are not, instead you are taking the derivative w... |
847,719 | <p>How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.</p>
| Cookie | 111,793 | <p>You simply forgot to remember using the chain rule:
\begin{align}
\frac{d}{dx}(5x-2)^3 = 3(5x-2)^2 \frac{d}{dx}(5x-2) =3(5x-2)^2 \cdot 5 = \boxed{15(5x-2)^2}
\end{align}</p>
|
61,438 | <p><strong>Bug introduced in 10.0 and fixed in 11.2</strong>
<br> Problem is due to a <code>BezierCurve</code> bug.</p>
<hr>
<p>I don't understand why the following does not work:</p>
<pre><code>Export["graph.pdf", Graph[{1 <-> 1, 1 <-> 2}, EdgeShapeFunction -> "Line",
EdgeStyle -> {Black}... | rhermans | 10,397 | <p>I can reproduce your problem (Win7 64, M10.0.1). Same problem if you try with <em>PDF</em> or <em>EPS</em> format.</p>
<p><img src="https://i.stack.imgur.com/jgyYXm.jpg" alt="Graph with bug"></p>
<p>When actually your graph should look like this</p>
<pre><code>Graph[{1 <-> 1, 1 <-> 2}, EdgeShapeFuncti... |
974,560 | <p>How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$?</p>
<p>I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, and my textbook doesn't offer a derivation.</p>
| Robert Israel | 8,508 | <p>As Travis noted, differentiation is the best way to do this.</p>
<p>But strangely enough, this very question was an important open question in the mid-17th century (before the invention of Calculus). See e.g. <a href="http://www.math.ubc.ca/~israel/m103/mercator/mercator.html">this web page of mine</a></p>
|
41,832 | <p>I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my own. </p>
<p>I know the formula for derivatives is: $$\lim_{\delta\rightarrow 0}\frac{f(x+\delta)
- f(x)}{\... | nucleic | 11,479 | <p>Here is another great way to calculate the derivative of $f(x)=(x+x)$, instead of using the <em>Limit Definition</em>, which is the method you are using above. </p>
<p>The other way to do this is to use a combination of methods called the <em>Power Rule</em>, and <em>Sum Rule</em>.</p>
<p>Here is a mathematical re... |
3,375,366 | <blockquote>
<p>How do we find the latus rectum of parabola when the equation is given in this polar form?
<span class="math-container">$$1/r = 1 + \cos t$$</span></p>
</blockquote>
<p>This curve cuts the <span class="math-container">$x$</span> axis on <span class="math-container">$1/2$</span> and <span class="mat... | José Carlos Santos | 446,262 | <p>Use the fact that<span class="math-container">$$a^3-b^3=(a-b)(a^2+ab+b^2).$$</span></p>
|
445,069 | <p>I am reading a textbook "Representation theory" by Fulton and Harris and I have a question.</p>
<p>They proved the following theorem on page 16.</p>
<p>With an Hermitian inner product on a set of class function, the characters of the irreducible representation of a finite group $G$ are orthonormal.</p>
<p>For a c... | Javier | 445,081 | <p>To prove that the characters of the irreducible representations span the centrum of the associated group algebra it is enough to prove that every central function can be spanned by the characters of the irreducible representations (by their orthonormality they are linearly independent). We already know from the Pete... |
387,542 | <p>e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?</p>
| RHS | 36,878 | <p>Here is a summary of my comments on Harald Hanche-Olsen's ans.</p>
<p>To proof g is the inverse of f iff they reflect each other along $y=x$.</p>
<p>First, we need $y′\equiv g(x′)$, g is the inverse of f. W/ the original y values of f we could get back the x's w/ g. So, $x′=y$ & $y′=x$.</p>
<p>Second, we need... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| PlatoManiac | 240 | <p>One More way! The means of each data is the blue dot. bars are color coded according to the standard deviation within each sub list.</p>
<pre><code>ListLinePlot[Mean /@ data,
Prolog ->
MapThread[{Thickness[.04], ColorData["SandyTerrain"][#3],
Line[{{#2, First@#1}, {#2, Last@#1}}], Opacity[0.7], White,... |
163,934 | <p>Suppose $F : [0,1]^n \to [0,1]^n$ is continuously differentiable and $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ for all $i =1,\dots,n$. Conjecture: there exists unique $x^* = F(x^*)$, and moreover, $x_1^* \leq \dots \leq x_n^*$.</p>
<p>Proof of the... | Włodzimierz Holsztyński | 8,385 | <p>Here is the question by @TomH (exact quote from the above):</p>
<p><code>Suppose</code> $F : [0,1]^n \to [0,1]^n$ <code>is continuously differentiable and</code> $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ <code>for all</code> $i =1,\dots,n$. <code>... |
272,193 | <p>It is well known that the Petersen Graph is not Hamiltonian. I can show it by case distinction, which is not too long - but it is not very elegant either.</p>
<p>Is there a simple (short) argument that the Petersen Graph does not contain a Hamiltonian cycle? </p>
| kerpoo | 93,480 | <p>if it had a hamiltonian circle we will find a complete matching...
so if we check all 6 matching of petersen graph we can see after removing a complete matching from petersen the second graph isn't one circle.
this is proving that petersen graph dosn't have a hamiltonian circle.</p>
|
2,793,380 | <blockquote>
<p>How many ways are there to seat n people in 4 benches so that no bench
is left empty with order?</p>
</blockquote>
<p>Hints from the teacher</p>
<ul>
<li><p>Each bench should have at least 1 person</p></li>
<li><p>This question is similar to distributing different object among n children</p></li>
... | Narek Bojikian | 494,145 | <p>We have $n!$ permutations of students.
Now lets take a permutation $i_1, i_2, .., i_n$ we want to distribute it in 4 slots. That means we have n-1 positions at which we can cut the set and we need to cut it at three of them, so the answer is</p>
<p>$n! * {n-1 \choose 3}$</p>
<p>P.s. no partition will be empty beca... |
2,091,761 | <p>If we define a set with $2$ elements in it $S=\{a,b\}$ and a variable "density" $d = 1$ here.</p>
<p>Then if we continue to expand the set with more elements relative to variable $d$ arithmetically, in such a way that:</p>
<p>$$(d=2) \to S= \{a, \frac{a+b}{2},b\}$$
$$(d=3) \to S= \{a, \frac{2a+b}{3}, \frac{a+2b}{3... | quid | 85,306 | <p>First, your definition of $S$ is not quite precise. Let us sidestep this and take for the final set the union of all the sets you write down. That is, </p>
<p>$$(d=2) \to S_2= \{a, \frac{a+b}{2},b\}$$
$$(d=3) \to S_3= \{a, \frac{2a+b}{3}, \frac{a+2b}{3}, b\}$$
$$(d=4)\to S_4= \{a, \frac{3a+b}{4}, \frac{a+b}{2}, \fr... |
3,276,699 | <p>If in a game of chance you have a certain probability <span class="math-container">$p$</span> to receive <span class="math-container">$x$</span> dollars else you receive <span class="math-container">$y$</span> dollars. How would you calculate the average money you would make per game? To clarify, for each game you e... | Parcly Taxel | 357,390 | <p>You are correct – the expected payout is <span class="math-container">$px+(1-p)y$</span>, which follows from the definition of expected value and the set-up of the problem.</p>
<p>That you can simply do this calculation is known as the linearity of expectation.</p>
|
1,786,306 | <p>Let $\sum\limits_{n=1}^\infty$ $a_n$ be a convergent series of positive terms. What can be said about the convergence of the following series: </p>
<p>$\sum\limits_{n=1}^\infty$ $\frac{a_1 + a_2 + ... + a_n}{n}$</p>
<p>The series above diverges. We know that $\sum\limits_{n=1}^\infty$ $\frac{a_1 + a_2 + ... + a_n}... | zhw. | 228,045 | <p>No, your proof has problems: $\sum a_n/n$ need not diverge, and $a_n(1+1/2 +\cdots +1/n)$ need not diverge. Your first assertion</p>
<p>$$\sum_{n=1}^{\infty}\frac{a_1 + \cdots + a_n}{n} \ge \sum_{n=1}^{\infty}\frac{a_1}{n}$$</p>
<p>gives you all you need, because the series on the right diverges.</p>
|
1,074,740 | <p>We know that torsion-free plus finitely generated <span class="math-container">$\rightarrow$</span> free and that <span class="math-container">$\mathbf{Q}$</span> is torsion-free is easy. </p>
<blockquote>
<p>But how to show <span class="math-container">$\mathbf{Q}$</span> is not finitely generated and not free?<... | Lee Mosher | 26,501 | <p>Here is a more fundamental approach to showing that $\mathbb{Q}$ is not finitely generated, based on a general method.</p>
<p>Here's the general method. In a group $G$ with a finite generating set $\{g_1,…,g_K\}$, for any increasing sequence of subgroups $H_1 \subset H_2 \subset \cdots$ whose union $\cup_i H_i$ equ... |
2,589,965 | <p>I'm working through multivariable functions and derivatives of multivariable functions. Since I am not very familiar yet with multivariable functions I wondered about the following: </p>
<p>In a function like $f(x,y)=x^2+y$, are x and y independent of each other and are we allowed to pick values for each deliberate... | user | 505,767 | <p>In general $$f(x,y): D \subseteq \mathbb{R^2}\to\mathbb{R}$$is defined in a subset $D$ of $\mathbb{R^2}$ that is the real plane.</p>
<p>In your case $$f(x,y)=x^2+y$$</p>
<p>is defined for each value of (x,y) thus $D\equiv \mathbb{R^2}$.</p>
|
2,085,664 | <p>Let $1$ be the multiplicative identity, so that $1\cdot a = a$ (where $a\in \mathbb{F})$. Let $0$ be the additive identity, so that $a+0=a$. Prove that $0\ne 1$. (Here we don't yet know that $0$ and $1$ must be unique, nor do we know that $0\cdot a = 0$).</p>
<p>My approach:</p>
<p>Suppose that $1=0$, then $a+1 = ... | Xam | 133,781 | <p>If $1=0$, then $a=a\cdot 1=a\cdot 0=0$, i.e. every element is the zero element, which means that the ring is the <em>zero</em> ring. About why $a.0=0$, just note that $a\cdot 0=a\cdot (0+0)=a\cdot 0+a\cdot 0$, so $a\cdot 0=0$.</p>
<p>Actually the fact that $1\neq 0$ comes from the definition of integral domain. As ... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Joachim König | 127,660 | <p>In a finite Frobenius group, the set of all fixed point free elements together with the identity forms a subgroup.</p>
<p>This might not have such a shocking effect to us, since usually when we first hear about Frobenius groups, it's precisely with the goal of proving this statement, but then again, usually in group... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Mare | 61,949 | <p>A simple module <span class="math-container">$S$</span> over a finite dimensional algebra <span class="math-container">$A$</span> over an algebraically closed field <span class="math-container">$K$</span> such that there exists a non-split short exact sequence <span class="math-container">$0\rightarrow S \rightarrow... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Richard Stanley | 2,807 | <p>There exists a finitely-generated infinite group with only two conjugacy classes, a difficult result of <a href="https://arxiv.org/pdf/math/0411039.pdf" rel="noreferrer">Osin</a>.</p>
|
3,444,262 | <blockquote>
<p><span class="math-container">$a_n$</span> is a non-decreasing sequence of positive integers. If an positive integer <span class="math-container">$k$</span> appears in <span class="math-container">$a_n$</span> exactly <span class="math-container">$k$</span> times and <span class="math-container">$S_n$<... | Daniel McLaury | 3,296 | <p>That closed form gives you a factorization of the sum, unless the six on the bottom can eat at least one of the two terms. This can only happen for small <span class="math-container">$n$</span>, so you should be able to get an upper bound and then check the remaining cases directly. </p>
|
4,518,734 | <p>Is there a simple formula or distribution curve to answer a question like this?</p>
<p>Assume there are K buckets and we want to randomly assign N balls to them. Each ball has an equal chance of being assigned to any of the buckets. There are more balls than buckets. When all balls have been assigned, what is the li... | Ross Millikan | 1,827 | <p>If the number of balls and buckets is reasonably large, the number of balls in a specific bucket will approximate a Poisson distribution. The linearity of expectation then gives the expected number of empty buckets.</p>
<p>For low chances you can calculate the chance that a given set of buckets is empty and multipl... |
2,136,859 | <p>Is there a rational solution for the following equation?
<span class="math-container">$$\tan (\pi x)=y\\y\neq-1,0,1$$</span></p>
<p>I guess there is none, but I have no idea how to solve/prove it.</p>
<p>EDIT: I think also that if <span class="math-container">$y$</span> is rational, then <span class="math-containe... | Hagen von Eitzen | 39,174 | <p>Let $v_2$ denote the <a href="https://en.wikipedia.org/wiki/P-adic_order" rel="noreferrer">$2$-adic valuation on $\Bbb Q$</a>.</p>
<p>Let $$A=\{\,x\in\Bbb Q\mid \tan \pi x\in\Bbb Q\setminus\{-1,0,1\}\,\},$$
Suppose $x\in A$ with $\tan \pi x=\frac ab$. Then $a\ne \pm b$ and by the addition theorem, $$\tag 1\tan(2\pi... |
20,540 | <p>How can I combine or separate sums in <em>Mathematica</em> in the way that <code>Together</code> or <code>Expand</code> work for rational expressions？</p>
<p>For example, how does one transform from </p>
<p>$$\text{Sum}\left[\frac{a}{\sqrt{n!}},\{n,0,\infty \}\right]+\text{Sum}\left[\frac{b}{\sqrt{n!}},\{n,0,\inft... | j-- | 4,831 | <p>This solution works for combining any number of separate sums:</p>
<pre><code>Sum[a/Sqrt[n!], {n, 0, Infinity}] +
Sum[b/Sqrt[n!], {n, 0, Infinity}] +
Sum[c/Sqrt[n!], {n, 0, Infinity}] //. Sum[x_, z_] + Sum[y_, z_] :> Sum[Simplify[x + y], z]
(*outputs: Sum[(a + b + c)/Sqrt[n!], {n, 0, Infinity}]*)
</code></pre... |
1,251,457 | <p>I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up.</p>
<p>On Wikipedia it says:</p>
<blockquote>
<p>Parametrization is... the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a va... | Travis Willse | 155,629 | <p>The idea of parameterization is that you have some equation for a subset $X$ of a space (often $\mathbb{R}^n$), e.g., the usual equation
$$x^2 + y^2 = 1$$
for the unit circle $C$ in $\mathbb{R}^2$, and you want to describe a function $\gamma(t) = (x(t), y(t))$ that traces out that subset (or sometimes, just part of ... |
208,485 | <p>Two years ago, I made a conjecture <a href="https://math.stackexchange.com/questions/334448/1-a21-b21-c2-8abc-a-b-c-in-mathbbq-has-infinitely-many-sol/336803#336803">on stackexchange</a>:</p>
<p>Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$... | math110 | 38,620 | <p>@Allan methods it's nice! here is my answer:
since
$$\left(\dfrac{1-a^2}{2a}\right)\left(\dfrac{1-b^2}{2b}\right)\left(\dfrac{1-c^2}{2c}\right)=1$$
so let
$$\dfrac{x}{y}=\dfrac{1-a^2}{2a},\;\dfrac{y}{z}=\dfrac{1-b^2}{2b},\;\dfrac{z}{x}=\dfrac{1-c^2}{2c}$$
and solving for $a,b,c$,
$$a = \frac{-x+\sqrt{x^2+y^2}}{y},\;... |
993,132 | <p>I have been looking into this question :
we have two surfaces :</p>
<p>$$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$</p>
<p>we need to draw or describe the "shape" that we get .
I tried to solve it by drawing the two surfaces and imagining the intersection whic... | Ángel Mario Gallegos | 67,622 | <p>The surface $S_2$ can be described by $(x,\,y,\,z)=(\cos u,\sin u,v)$, $u\in[0,2\pi]$ and $v\in\mathbb{R}$. Then, the curve we are looking is $(x,y,z)=(\cos u, \sin u, 1-\cos u)$, $u\in[0,2\pi]$.</p>
<p>In order to prove that the curve is an ellipse let $A$ be the point $(\frac{1}{\sqrt{2}},0,1-\frac{1}{\sqrt{2}})$... |
102,186 | <p>Suppose that $f$ is differentiable on $\mathbb{R}$. If $f(0)=1$ and $|f^{'}(x)|\leq1$ for all $x\in\mathbb{R}$, prove that $|f(x)|\leq|x|+1$ for all $x\in\mathbb{R}$. </p>
<p>I tried: </p>
<p>Let $g(x)=|f(x)|-|x|-1$. Then I tried to find $g^{'}(x)$ but I'm not sure where to start.</p>
| Bill Cook | 16,423 | <p>Integrate?</p>
<p>$|f'(x)| \leq 1$ implies that $-1 \leq f'(x) \leq 1$ integrate from 0 to $x$ and get
$-x \leq f(x)-f(0) \leq x$ thus $-x+1 \leq f(x) \leq x+1$. If $f(x) \geq 0$, then $|f(x)|=f(x) \leq x+1 \leq |x|+1$. If $f(x)<0$, then $-x+1 \leq f(x)<0$ so $0<|f(x)|=-f(x)\leq x-1\leq|x-1|\leq |x... |
107,882 | <p>Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes. I'm not interested (at this time) in a book that tie... | Mathemagician1234 | 7,012 | <p>My recommendations from the earlier thread : <a href="https://math.stackexchange.com/questions/34442/book-recommendation-on-plane-euclidean-geometry">Book recommendation on plane Euclidean geometry</a> - are still to me as good a set of recommendations as there are. </p>
<p>I'll also throw my support behind Robin ... |
2,339,398 | <p>I'm trying to get my calculus back up to scratch after not using it for 20 odd years. During my research, I've just seen this on <a href="https://physics.info/kinematics-calculus/" rel="nofollow noreferrer">https://physics.info/kinematics-calculus/</a>:</p>
<p>$$a = \frac{dv}{dt}$$</p>
<p>$$dv = a\ dt$$</p>
<p>$$... | Trevor Gunn | 437,127 | <p>You only need $a_i \ge 1$.</p>
<p>Define
$$\begin{bmatrix} p_{-1} & p_{0} \\ q_{-1} & q_{0} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. $$
Recursively, set
$$p_k = a_kp_{k-1} + p_{k - 2}, q_k = a_kq_{k-1} + q_{k - 2} \tag{1} $$
for $k \ge 1$. We will show that</p>
<p>$$ [a_1,\dots... |
3,209,497 | <p>The problem is:</p>
<p>Suppose <span class="math-container">$H$</span> is Hilbert and <span class="math-container">$\{e_n\}_{n = 1}^\infty$</span> is its orthonormal basis. Prove <span class="math-container">$x_n \rightharpoonup x_0$</span> if and only if:</p>
<ol>
<li><span class="math-container">$||x_n||$</span>... | David | 651,991 | <p>There are compactifications of <span class="math-container">$\mathbb{R}$</span> that only use one infinity point (<span class="math-container">$\infty$</span>) (the numberline would "look like" an infinite radius circle) with the positives and negatives connected at both <span class="math-container">$0$</span> and <... |
101,513 | <p>First of all, I don't really know how to formulate the question, so if you understand my question and know a better way to phrase it, please revise.</p>
<p>I have the following Matrixes:</p>
<pre><code>a[n_, m_] := Table[n + m - i - j, {i, 1, n}, {j, 1, m}]
b[n_, m_] := Round[Table[(n*m/2)*(1 + 2 j/m), {j, 1, m}]]... | eldo | 14,254 | <pre><code>a[n_, m_] := Table[n + m - i - j, {i, 1, n}, {j, 1, m}];
b[n_, m_] := Round[Table[(n*m/2)*(1 + 2 j/m), {j, 1, m}]];
fun[li_, rp_] :=
Module[{p = First@FirstPosition[li - rp, _?Positive]},
If[p === "NotFound", li, Join[li[[;; p - 1]], rp[[p ;;]]]]]
fun[#, b[3, 8]] & /@ Accumulate /@ a[3, 8] // Matri... |
764,389 | <p>Is there a geometric interpretation of the line integrals
:</p>
<p>$\int_{\gamma} f(x,y)\, dx$ </p>
<p>$\int_{\gamma} f(x,y)\, dy$</p>
<p>(which should not be confused with $\int_{\gamma} f(x,y)\, ds$)</p>
<p>where the function $f(x,y)$ to be integrated is evaluated along a curve $\gamma$ ?</p>
| Santiago Canez | 1,914 | <p>A line integral of the form $\int_\gamma f(x,y)\,dx$ is simply the line integral of the vector field $\vec F(x,y) = f(x,y)\vec i$ over $\gamma$, so if you understand the geometric interpretation of line integrals of vector fields you have your answer. Similarly for a line integral with respect to $dy$.</p>
|
1,245,499 | <p>In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover.</p>
<p>But, $(0,1)$ set also belongs to cover mentioned above.
We can say $\{(0,1)\}$ is a subcover of mentioned above cover.</p>
<p>I am not able to understand what I am doing wrong.</p>
| Harish | 181,936 | <p>I suggest you go the following way:
First try to show all compact sets are closed, then see why (0,1) is not compact.</p>
|
1,676,427 | <p>I was asked to use Romberg integration to evaluate the integral$$\int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n}$$ and compare the result with the result I get from the sum. And I also need to estimate how many function evaluation Romberg integration will require to achieve 12 digit accuracy. I looked it up on Wikipedia b... | Ian | 83,396 | <p>This is not really an answer, but it is <em>way</em> too long for a comment. Here is an example of Romberg-type quadrature:</p>
<p>The base rule is a left hand Riemann sum on a uniform partition with subintervals of length $h=1$. The problem of interest is to integrate $e^x$ on $[0,1]$ with a tolerance of $10^{-1}$... |
28,562 | <p>In the course of my research I have come across the following integral:</p>
<p>$\int_{0}^{\infty} e^{- \Lambda \sqrt{(z^2+a)^2+b^2}}\mathrm{d}z$.</p>
<p>This initially looks like it should be solvable by some suitable change of variable which will allow you to get it into a gaussian form. Unfortunately after tryi... | Community | -1 | <p>Would a double expansion, once for the square root and once for the exponential, work? It seems like you're interested in the $b\rightarrow 0$ behavior so I tried it and the answer is of the form: $I=\sqrt{\frac{\pi}{4\Lambda}}e^{-\Lambda a}-\frac{b^{2}\pi}{4\sqrt{a}} erf(\Lambda a)+ O(b^{6})$ (The 4th order cancels... |
3,870,392 | <p>Taking the real numbers to be a complete ordered field, why do we believe that they model distances along a line? How do we know (or why do we believe) that any length that can be drawn is a real number multiple of some unit length?</p>
| Noah Schweber | 28,111 | <p>Completeness often feels a bit technical at first: we show that there is exactly one complete ordered field up to isomorphism, but why should that confluence of properties correspond to line-ness?</p>
<p>I think it's more intuitive to focus instead on <strong>connectedness</strong>. This is really the same thing in ... |
3,870,392 | <p>Taking the real numbers to be a complete ordered field, why do we believe that they model distances along a line? How do we know (or why do we believe) that any length that can be drawn is a real number multiple of some unit length?</p>
| j4nd3r53n | 446,918 | <p>I think steven gregory's answer is probably the best so far: we simply choose to believe that real numbers describe distances - it has turned out to be a useful notion, but it is simply a choice we made, it isn't absolute truth.</p>
<p>This really about the fundamental nature of mathematics: there are certain things... |
1,470,879 | <p>I want to solve this system of advective-diffusive-reactive equations analytically:</p>
<p>$$\left(\alpha - k_0c_B\right)c_A+v\frac{dc_A}{dx}-D\frac{d^2c_A}{dx^2} = f_A $$
$$\left(\alpha - k_0c_A\right)c_B+v\frac{dc_B}{dx}-D\frac{d^2c_B}{dx^2} = f_B $$
$$k_0c_Ac_B+\alpha c_C+v\frac{dc_C}{dx}-D\frac{d^2c_C}{dx^2} = ... | E.H.E | 187,799 | <p>Hint
$$\frac{1}{\sqrt{a}+\sqrt{a+1}}=\sqrt{a+1}-\sqrt{a}$$</p>
|
264,061 | <p>I need help with the following problem. Suppose $Z=N(0,s)$ i.e. normally distributed random variable with standard deviation $\sqrt{s}$. I need to calculate $E[Z^2]$. My attempt is to do something like
\begin{align}
E[Z^2]=&\int_0^{+\infty} y \cdot Pr(Z^2=y)dy\\
=& \int_0^{+\infty}y\frac{1}{\sqrt{2\pi s}}e^{... | Jon | 98,140 | <p>This is old, but I feel like an easy derivation is in order.</p>
<p>The variance of any random variable $X$ can be written as
$$
V[X] = E[X^2] - (E[X])^2
$$</p>
<p>Solving for the needed quantity gives
$$
E[X^2] = V[X] + (E[X])^2
$$</p>
<p>But for our case, $E[X] = 0$, so the answer of $\sigma^2$ is immediate.</... |
1,149,602 | <p>The parametric curve $r=(-2t^2+8t-2,cos({\pi}t),t^3-28t)$ crosses itself at one and only one point. The point is $(x,y,z)$. I found $t=-2$ to be the answer and $(x,y,z)=(-26,-1,48)$ to be the correct answer. </p>
<p>However it asks for the acute angle between the two tangent lines to the curve at the crossing point... | Vinícius Ferraz | 213,537 | <p>Confering your text,</p>
<p>$-2t^2 +8t - 2 = -26$</p>
<p>$t^2 -4t - 12 = 0$</p>
<p>$t = -2$ or $t = 6 \Rightarrow r(t) = (-26, +1, +48)$</p>
<p>$r'(t) = ...$</p>
<p>$u = r'(-2) = ...$</p>
<p>$v = r'(6) = ...$</p>
<p>$u \cdot v = |u| \cdot |v| \cdot cos \theta \Rightarrow \theta = ...$</p>
|
2,132,994 | <p>I need to prove that absolute value of any real number is greater than or equal to that real number, where $|a| = a ; a\ge0 , |a| = -a ; a<0 $</p>
<p>I came across this on real analysis.
I need this proven Filed and Order Axioms and basic definitions.</p>
| Community | -1 | <ul>
<li>$a\ge0\implies -a\le0\le a\implies -a,a\le a=|a|$.</li>
<li>$a\le0\implies a\le0\le-a\implies -a,a\le-a=|a|$.</li>
</ul>
<hr>
<p>Alternatively, you can establish that the defintion of the absolute value is equivalent to</p>
<p>$$|a|=\max(a,-a),$$ from which the claim follows.</p>
<hr>
<p>Alternatively,</p... |
1,740,981 | <blockquote>
<p>Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Metric space $C[a,b]$ defined by $d(x(t),y(t))=\max\limits_{a\leqslant t\leqslant b} \mid... | Hagen von Eitzen | 39,174 | <p>If $f\notin B$ then all $g$ with $d(f,g)<|f(0)|$ are also $\notin B$.</p>
|
3,000,052 | <p>Given <span class="math-container">$Z_n=\arg{\frac{i^n}{n}}$</span>, how do I show that it has no limit?</p>
| user | 505,767 | <p>Recall that</p>
<ul>
<li><span class="math-container">$i^2=-1$</span></li>
<li><span class="math-container">$i^3=-i$</span></li>
<li><span class="math-container">$i^4=1$</span></li>
<li><span class="math-container">$i^5=i$</span></li>
</ul>
<p>then we have that</p>
<ul>
<li><p>for <span class="math-container">$n=... |
4,501,398 | <p>First of all forgive my very poor and not to scale drawing. Also for the not so good looking maths formatting</p>
<p>Essentially I am looking for the area of the shaded part.
This is what I've gotten so far</p>
<p>Area of larger circle with radius being <span class="math-container">$20+w$</span> minus area of inner ... | Jean-Armand Moroni | 1,064,750 | <p>(Assuming "<span class="math-container">$Z+$</span>" means <span class="math-container">$\mathbb{N}$</span>).</p>
<p>"Countable union of countable sets is countable" (in the comments) is a good proof.</p>
<p>For a more direct proof, you can build an injection <span class="math-container">$f$</spa... |
399,135 | <p>I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.</p>
<p>Once again, I'm stuck at the setup (this happens a lot with me). I know that I need to parameterize F, but how would I go about ... | rurouniwallace | 35,878 | <p>Another method is to use Green's theorem:</p>
<p>$$\oint_C \vec{F}\cdot d\vec{r}=\iint_D|\nabla\times \vec{F}|dA$$</p>
<p>Where D is the area within the loop.</p>
<p>$$|\nabla\times \vec{F}|=y - 2x$$</p>
<p>Convert to cylindrical coordinates:</p>
<p>$$y-2x = r\sin{\theta}-2r\cos{\theta}$$</p>
<p>And evaluate t... |
793,930 | <p>I was asked to evaluate the integral </p>
<p>$$\int_{-1}^{1} \frac{\sin{x}}{1+x^2}dx$$</p>
<p>if it exists.</p>
<p>This is a problem from Calculus and the student has been taught how to use trigonometric substitution. My intuition was to do trig sub with $$x=\tan{\theta}$$ and eliminating $$\frac{dx}{1+x^2}$$ b... | Leucippus | 148,155 | <p>The integral is
\begin{align}
I &= \int_{-1}^{1} \frac{\sin(x)}{1+x^{2}} dx \\
&= \int_{-1}^{0} \frac{\sin(x)}{1+x^{2}} dx + \int_{0}^{1} \frac{\sin(x)}{1+x^{2}} dx \\
&= - \int_{0}^{1} \frac{\sin(x)}{1+x^{2}} dx + \int_{0}^{1} \frac{\sin(x)}{1+x^{2}} dx \\
&= 0
\end{align}
where the change of varia... |
763,499 | <p>This seems trivial, and yet after a bit of thinking, I couldn't supply a simple proof.</p>
<p>Is the following true?</p>
<p>The series $$\underset{n=1}{\overset{\infty}\sum}\cos(nx)$$ is divergent for almost every $x\in[-\pi,\pi]$ (or at least for a positive measure subset, though I believe a.e., or likely everywh... | Lutz Lehmann | 115,115 | <p>The sequence $\cos(nx):n\ge N$ is dense in $[-1,1]$ for any $N$ whenever $\frac x\pi$ is irrational. I think t was the name of Kronecker or Minkowski that is connected to the fact that the set
$$
\{m+nw:m,n\in\Bbb Z\}
$$
is either discrete or dense, depending on $w$ being rational or irrational </p>
<p>And even if ... |
2,478,695 | <p>So I need to solve the integral </p>
<blockquote>
<p>$$\int \frac { \tan { x } }{ \left( \sin { x } \right) ^{ 2 }+2\left( \cos { x } \right) ^{ 2 } } dx$$</p>
</blockquote>
<p>I saw some exercises that suggest I need to use the secant function to solve it but can I do it without it?</p>
| Michael Rozenberg | 190,319 | <p>$$\int \frac{\tan x}{(\sin x)^2+2(\cos x)^2}dx=-\int\frac{1}{\cos{x}(1+\cos^2x)}d(\cos{x})=$$
$$=\int\left(\frac{\cos{x}}{1+\cos^2x}-\frac{1}{\cos{x}}\right)d(\cos{x})=\frac{1}{2}\ln(1+\cos^2x)-\ln|\cos{x}|+C.$$</p>
|
1,710,469 | <p>If $p_1 = 2$ and $p_{n+1} = \frac{p_n}{2}+ \frac{1}{p_n}$, determine $p_n$ is decreasing or increasing.</p>
<p>Here are the first few terms:
$$p_2 = \frac{3}{2}, p_3 = \frac{3}{4} + \frac{2}{3} = \frac{17}{12}, p_4 = \frac{17}{24} + \frac{12}{17} = \frac{577}{408}$$</p>
<p>The sequence seems decreasing to me so I ... | Laurent Duval | 257,503 | <p>A calculus approach. Let $$f(x) = \frac{x}{2}+\frac{1}{x}.$$ So $$f(x)-x = -\frac{x}{2}+\frac{1}{x}$$ is the sum of two decreasing functions on $]0,+\infty]$. Hence $f(p_{n})-p_n<0$, the sequence is decreasing.</p>
<p>$f(x)$ is strictly increasing on $]\sqrt{2},2]$ (easy with the derivative), thus if $x \in ]\s... |
3,713,900 | <p>I have the function:</p>
<p><span class="math-container">$$f : \mathbb{R} \rightarrow \mathbb{R} \hspace{2cm} f(x) = e^x + x^3 -x^2 + x$$</span></p>
<p>and I have to find the limit:</p>
<p><span class="math-container">$$\lim\limits_{x \to \infty} \frac{f^{-1}(x)}{\ln x}$$</span></p>
<p>(In the first part of the ... | Erik Satie | 698,573 | <p>Hint : <span class="math-container">$$\forall x\geq 1\quad 2e^x \geq e^x+x^3-x^2+x\geq e^x+1$$</span></p>
<p>So <span class="math-container">$$\ln(\frac{x}{2})\leq f^{-1}(x)\leq \ln(x-1)$$</span></p>
<p>So we deduce that your limit is a constant by the sandwich theorem . </p>
|
1,547,097 | <p>So I have</p>
<p><strong>1.</strong> $$\frac{r}{3\tan \theta} = \sin \theta$$</p>
<p><strong>2.</strong> $$r=3\cos \theta$$</p>
<p>What would be the Cartesian equation???</p>
| Hosein Rahnama | 267,844 | <p>First note that </p>
<p>$$\left\{ \matrix{
x = r\cos \theta \hfill \cr
y = r\sin \theta \hfill \cr} \right.$$</p>
<p>the general approach will be to solve for $r$ and $\theta$ and replace in your polar equation. However, in most times there are some shortcuts. See the following for the second one</p>
<p>$... |
1,841,644 | <p>I have a square matrix called A. How can I find $A ^ {-1/2}$. Should I compute $a_{ij} ^ {-1/2}$ for all of its elements?</p>
<p>Thanks</p>
| Qwerty | 290,058 | <p>It exists only if <span class="math-container">$A$</span> is invertible, i.e. <span class="math-container">$A^{-1}$</span> exists. Now this matrix have to <a href="https://en.wikipedia.org/wiki/Square_root_of_a_matrix" rel="nofollow noreferrer">follow certain properties , then a square root can be computed.</a></p>
... |
64,716 | <p>We know that by using Stirling approximation:
$\log n! \approx n \log n$</p>
<p>So how to approximate $\log {m \choose n}$?</p>
| Chris Taylor | 4,873 | <p>A better approximation for the logarithm of a factorial can be found by using $\log n! \approx n \log n - n$. Interestingly, the additional terms in the approximation of the binomial coefficient cancel out, and the result is the same as if you used the simpler approximation $\log n! \approx n\log n$:</p>
<p>$$\begi... |
3,491,595 | <blockquote>
<p>Evaluate the sum <span class="math-container">$$\frac{1}{3} + \frac{1}{3^{1+\frac{1}{2}}}+\frac{1}{3^{1+\frac{1}{2}+\frac{1}{3}}}+\cdots$$</span></p>
</blockquote>
<p>It seems that <span class="math-container">$1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}$</span> approaches <span class="ma... | Community | -1 | <p>A very direct and simple way of getting quite close to the sum value is replacing the sum with an integral and using the simplest approximation for n-th Harmonic number</p>
<p><span class="math-container">$$H(n) \approx \ln(n) + \gamma $$</span></p>
<p><span class="math-container">$$ \int\limits_{x=1}^{+\infty} \f... |
3,037,921 | <p>Find all critical points of the system</p>
<p><span class="math-container">$y_1'= y_1(10-y_1-y_2)$</span></p>
<p><span class="math-container">$y_2'= y_2(30-2y_1-y_2)$</span></p>
<p>then classify them as stable, asymptotically stable, or unstable. </p>
<p>I need help with this particular question, as you may see,... | caverac | 384,830 | <p>The line going from <span class="math-container">$(0, 0)$</span> to <span class="math-container">$(2, 2)$</span> has equation <span class="math-container">$y = x$</span>, so a parametrization could be </p>
<p><span class="math-container">\begin{eqnarray}
x(t) &=& t \\
y(t) &=& x(t) = t
\end{eqnarray... |
125,630 | <p>Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to </p>
<p>$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$ </p>
<p>which can be rewritten as
$$
h^0(X, L)+h^0(X, L^\ast)=\frac{1}{2}(L\cdot L)+2+h^1(X, L)
$$</p>
<p>(I use Serre's duality to identify $H^2(X, L)... | Sándor Kovács | 10,076 | <p>Noam and Francesco have already pointed out that in order to get such a line bundle you can always take either a multiple of a $(-2)$-curve or the union of disjoint $(-2)$-curves, both of which is possible on many $K3$'s. </p>
<p>On the other hand, if you were looking for an $L$ that has a non-zero section whose ze... |
3,355,722 | <p>In the book I am reading I have been given multiple definitions of the identity relation, they are,</p>
<blockquote>
<ol>
<li><p>a=b if a and b are identical</p>
</li>
<li><p>a=b if a and b refer to the same object</p>
</li>
</ol>
</blockquote>
<p>and for the negation of identity I was given,</p>
<blockquote>
<ol st... | fleablood | 280,126 | <p>"Can two different objects be identical?"</p>
<p>No.</p>
<p>Just no.</p>
<p>At least according to your book.</p>
<blockquote>
<p>With definition 1 you could have two different objects, one named 'a' and one named 'b', but with the exact same properties so that they are identical</p>
</blockquote>
<p>I don't b... |
847,093 | <p>Given that $-2\pi≤\theta≤0$ and $\theta$ has a reference angle of $\cfrac{\pi}{6}$ , find $\theta$ if it is in the</p>
<p>a) 1st quadrant</p>
<p>b) 2nd quadrant</p>
<p>c) 3rd quadrant</p>
<p>d) 4th quadrant</p>
<p>I need help on this problem which i'm unfamiliar with negative in radian..</p>
| Wonder | 27,958 | <p>Number of edges in graph = $k|x| = k|y|$. Now divide by k.</p>
|
773,591 | <p>I'm having trouble proving that $|A^{B×C}| = |(A^B)^C|$ , where $M^N$ is the set of all the functions $f:N \to M$. </p>
<p>My thoughts: to prove this, I need to find a bijection between $|A^{B×C}|$ and $|(A^B)^C|$, so I need a bijection between the set of all functions $g:B×C \to A$ and the set of all functions ... | Ryan Sullivant | 20,727 | <p>First, an element of $A^{B\times C}$ is a function $f:B\times C \to A$ and we want to map this to a function $g: C \to A^B$ (which is a function which outputs functions). </p>
<p>So we need a map $h:A^{B \times C} \to (A^B)^C$ that inputs functions $f: B\times C \to A$ and whose output is a function $h(f): C \to A... |
3,012,042 | <p>We learned expression of deduce, i.e. =>.</p>
<p>But now I dont have capable reason for I agree I represent True if assumption is False.</p>
<p>Anybady there having to explain reason for its deduce?</p>
<p>Best regards,</p>
| Bill Dubuque | 242 | <p>Below are <span class="math-container">$\,4\,$</span> simple ways to compute it using about <span class="math-container">$10$</span> seconds of mental arithmetic, by using the Binomial Theorem, which reduces to the first <span class="math-container">$\,2\,$</span> terms by <span class="math-container">$\,11^{\large ... |
3,480,050 | <p>I've been struggling to solve the following exercise:</p>
<p>For <span class="math-container">$x\in\mathbb{R}$</span>, find the radius of convergence of the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{x^n}{n+\sqrt{n}}$</span>.</p>
<p>My approach so far: Compute <span class="math-container">$\lims... | cardinalRed | 735,716 | <p>You could compute the limit of <span class="math-container">$\frac{(n+1)+\sqrt{n+1}}{n+\sqrt{n}}=\frac{1+1/n+\sqrt{1/n+1/n^2}}{1+\sqrt{1/n}}$</span>. This tends to <span class="math-container">$1$</span>. <a href="https://math.stackexchange.com/q/69386/735716">Therefore</a>, <span class="math-container">$\sqrt[n]{n+... |
1,372,558 | <p>$y=\sqrt{x^x}$</p>
<p>How do I convert this into a form that is workable and what indicates that I should do so? </p>
<p>Anyway, I tried this method of logging both sides of the equation but I don't know if I am right.</p>
<p>$\ln\ y=\sqrt{x} \ln\ x$</p>
<p>$\frac{dy}{dx}\cdot \frac{1}{y}=\sqrt{x}\ \frac{1}{x} +... | quid | 85,306 | <p>There are a couple of things that are not quite right, moreover taking the logarithm might not be the best choice (but it seems you are supposed to do this). </p>
<ul>
<li><p>Note $\ln \sqrt{u}= \frac{1}{2} \ln u$, so your first line is incorrect.</p></li>
<li><p>In the second you dropped an $x$ after the $\ln$. </... |
1,266,507 | <p>Let $K_{a,b}$ be the complete bipartite graph. Show that
$K_{a,b}$ is a tree if and only if $a = 1$ or $b = 1$.</p>
<p>The way my professor showed us for a complete graph is as below. I just don't know how to start for a complete bipartite graph. </p>
<blockquote>
<p>$K_a$ is a tree if and only if $a=2$ or $a=1$... | Adhvaitha | 228,265 | <p>Number of edges in a complete bipartite graph $K_{m,n}$ is $mn$.</p>
<p>Number of edges in a tree with $v$ vertices is $v-1$.</p>
<p>Hence, if we want $K_{a,b}$ to be a tree, we need</p>
<p>$$ab = a+b-1 \implies ab-a-b+1 = 0\implies (a-1)(b-1) = 0 \implies a = 1 \text{ or }b=1$$</p>
|
2,732,220 | <p>A friend of mine asked me to help him evaluate the series</p>
<p>$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{\sin (n \pi y) \sin \left ( n \pi x \right )}{n^2 \pi^2} \quad , \quad x , y \in (0, 1)$$</p>
<p>It does not ring any bells as to what it could be behind. The only thing I see is Fourier series and probably a... | user | 293,846 | <p>Let us start with the well-known Fourier series for a repeated parabola:
$$
x^2=\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos{nx};\quad -\pi\le x\le\pi,
$$
which upon substitution $x=\pi(1-t)$ transforms to:
$$
\pi^2(1-t)^2=\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{\cos\pi n t}{n^2}
\Rightarrow (1-t)^2=\f... |
3,720,083 | <p><strong>My attempt</strong>
<span class="math-container">$$1-\frac{1}{3\cdot 3}+\frac{1}{5\cdot 3^2}-\frac{1}{7\cdot 3^3}+\cdots=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$$</span></p>
<p>By Leibniz alternative test for convergence. It is a convergent alternative series. How do I evaluate this limit?</p>
| Integrand | 207,050 | <p><span class="math-container">$$
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}=\sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)3^{n}}=\sqrt{3}\sum_{n=0}^\infty \frac{(-1)^{n}(1/\sqrt{3})^{2n+1}}{(2n+1)}
$$</span>
<span class="math-container">$$
=\sqrt{3}\arctan(1/\sqrt{3}) = \frac{\sqrt{3}\pi}{6}
$$</span></p>
|
76,299 | <p>This is a bit embarrassing, but I can't seem to solve for $x$ in
$$2x=\frac{x}{y}-\frac{1}{1-y}.$$
Could someone please give me a hand!</p>
| J. M. ain't a mathematician | 498 | <p>I present here a rather elaborate coordinate geometry proof. (I'd sure like to see a simpler way of going about this!)</p>
<p>Take the ellipse with eccentricity $\varepsilon\in(0,1)$ and semilatus rectum $p$ to have the polar equation</p>
<p>$$r=\frac{p}{1-\varepsilon\cos\,\theta}$$</p>
<p>such that one of the el... |
3,493,203 | <p>I would like to understand why if <span class="math-container">$P$</span> is positive semidefinite, then <span class="math-container">$x^{\text{T}}Px=0$</span> if and only if <span class="math-container">$Px=0$</span>. How can I prove this?
I can say that <span class="math-container">$x'Px=x'M'Mx$</span>, where <sp... | rschwieb | 29,335 | <p>This is only possible if you ring lacks an identity.</p>
<p>Perhaps you didn't notice, but this is trivially derived from <a href="https://math.stackexchange.com/a/452338/29335">this solution</a>, where
<span class="math-container">$R=2\mathbb Z$</span> and <span class="math-container">$M=4\mathbb Z$</span> is give... |
1,391,573 | <blockquote>
<p>If <span class="math-container">$f(x)$</span> is finite at <span class="math-container">$x$</span> and <span class="math-container">$\lim\limits_{h\to 0}\frac{1}{h}\int_x^{x+h} |f(t)-f(x)|dt = 0$</span> then <span class="math-container">$x$</span> is called a Lebesgue point of function <span class="math... | coldnumber | 251,386 | <p>Maybe there's a nicer way, but you could consider the intermediate extension: </p>
<p>$$\Bbb Q \subseteq \Bbb Q(1+\sqrt 5)\subseteq \Bbb Q(\sqrt{1+\sqrt 5})$$</p>
<p>$1+\sqrt 5$ has degree $2$ over $\Bbb Q$ because it's not rational and it's a root of $x^2-2x-6 \in \Bbb Q[x]$.</p>
<p>Because of the multiplicative... |
1,391,573 | <blockquote>
<p>If <span class="math-container">$f(x)$</span> is finite at <span class="math-container">$x$</span> and <span class="math-container">$\lim\limits_{h\to 0}\frac{1}{h}\int_x^{x+h} |f(t)-f(x)|dt = 0$</span> then <span class="math-container">$x$</span> is called a Lebesgue point of function <span class="math... | André Nicolas | 6,312 | <p>It is enough to show that $(x^2-1)^2-5$, that is, $x^4-2x^2-4$ is irreducible over the rationals. There are no rational roots, so we only need to rule out factorization as a product of quadratics with integer coefficients. </p>
<p>Since there is no $x^3$ term. we can confine attention to factorizations $(x^2+ax+b)(... |
2,768,996 | <p>I am trying to proof some identies from concrete mathmatics p 265. But i cant get nowhere. What i have found out its a recurrence, in the Stirling Numbers Triangle, its a vertical one. I get what it means, e.g. n=4 m=2 the Number of Partitioning 5 Elements into 4 Partitions ${5 \brace 3}$ can be created from from $... | Phicar | 78,870 | <p>Let see, $n+1\brace m+1$ means that you take $n+1$ elements and you partition them in $n+1$ disjoint and non empty blocks. Denote the partitions in the following way
$${[n+1]\brace m+1}=\{\pi \vdash [n+1]:|\pi|=m+1\},$$ the set of partitions. Given a partition $\pi ,$ let $j$ the maximum integer such that there is n... |
2,768,996 | <p>I am trying to proof some identies from concrete mathmatics p 265. But i cant get nowhere. What i have found out its a recurrence, in the Stirling Numbers Triangle, its a vertical one. I get what it means, e.g. n=4 m=2 the Number of Partitioning 5 Elements into 4 Partitions ${5 \brace 3}$ can be created from from $... | Marko Riedel | 44,883 | <p>Starting from the LHS we write with $n\ge m$ and an Iverson bracket</p>
<p>$$\sum_{j=m}^n {j\brace m} (m+1)^{n-j}
= \sum_{j\ge m} {j\brace m} (m+1)^{n-j}
[[j\le n|j\ge 0]]
\\ = \sum_{j\ge m} {j\brace m} (m+1)^{n-j} [z^n] \frac{z^j}{1-z}
\\ = [z^n] \frac{1}{1-z} \sum_{j\ge m} {j\brace m} (m+1)^{n-j} z^j
\\ = [z^n] \... |
168,163 | <p>I wonder what kind of functions satisfy </p>
<p>$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$
I suppose all functions must be continuous.</p>
| Robert Israel | 8,508 | <p>Your equation can be rewritten as
$$\lim_{n \to \infty} (n+1) \int_0^1 x^n (f(x) - f(1))\ dx = 0 $$
(I'd rather use $n+1$ than $n$, because $(n+1) \int_0^1 x^n \ dx = 1$)</p>
<p>Note that as $n \to \infty$, $(n+1) x^n \to 0$ uniformly on $[0,1-\delta]$ for any $\delta > 0$, implying that $(n+1) \int_0^{1-\delta}... |
168,163 | <p>I wonder what kind of functions satisfy </p>
<p>$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$
I suppose all functions must be continuous.</p>
| Kasun Fernando | 35,030 | <p>Certainly all the continuous functions <strong>do</strong> satisfy this property. But I think we will have to see whether there are classes of functions which are not continuous but satisfy this property. </p>
<p>To show that continuous functions satisfy this, we shall use the well-known Stone Weierstrass Theorem (... |
1,406,219 | <p>Hi I am reviewing partial derivatives.
For the question below, I am not sure why $(x-1)$ appears. Could anyone give me a explanation on this?</p>
<p>$y = x\sin(z)e^{-x}$ </p>
<p>$\partial y/\partial x = -e^{-x}(x-1)\sin(z)$</p>
| Community | -1 | <p>It goes like this:
$$\frac{\partial}{\partial z} (f^{p/2}(\bar f)^{p/2}) = \frac{p}{2} f' f^{p/2-1}(\bar f)^{p/2} $$
since the antiholomorphic term is treated as a constant. Then differentiate in $\bar z$ similarly, getting
$$ \frac{p}{2} f' f^{p/2-1}\frac{p}{2}\bar f' (\bar f)^{p/2-1}$$
Simplify to $ \dfrac{p^2}{4... |
2,121,863 | <blockquote>
<p>Show that a subring of a division ring must be a domain.</p>
</blockquote>
<p>Let $S$ be a subring of $R$ and let $R$ be a division ring.
Can we just say that since every element has an inverse in $R$, then every element also has an inverse in $S$, then we can deduce that since for all elements in $S... | Asinomás | 33,907 | <p>Subrings of domains are domains. Suppose $R$ is a domain and $S$ is a subring. We have to prove every $r\in S\neq 0$ is not a zero divisor, notice that $r$ is not a zero divisor in $R$, so it is not zero divisor in $S$ either, we are done.</p>
<p>In particular notice that division rings are domains.</p>
|
287,203 | <blockquote>
<p>Prove the statement:</p>
<p><span class="math-container">$\forall n \in \mathbb{N}$</span>,<span class="math-container">$\forall m \in \{2, 3,...,floor(\sqrt{n})\}$</span>, <span class="math-container">$m$</span> does not divide <span class="math-container">$n \implies n$</span> is prime</p>
</blockquot... | Mark Bennet | 2,906 | <p>Suppose $n$ is not prime, then $n=rs$ with $r,s>1; r\ge s$</p>
<p>Then $s^2\leq rs=n$.</p>
<p>So if $n$ is not prime it has a factor $s$ with $s\leq \sqrt n$.</p>
<p>And therefore if $n$ has no such factor, it must be prime.</p>
|
2,172,949 | <p>I have the integral:
$$\int _{\partial D(a, r)} \frac{e^z}{z^3 + 2z^2 + z} dz$$</p>
<p>which I have to find for different cases:</p>
<p>1 - $a = 0$ and $r =1/2$</p>
<p>2 - $a = -i - 1$ and $r = 1/2$</p>
<p>3 - $a = -1$ and $r = 1/2$</p>
<p>4 - $a = 0$ and $r = 2$</p>
<p>My attempt is this:</p>
<p>$$ \frac{e^z... | Ángel Mario Gallegos | 67,622 | <p>For $2$) observe that $f(z)=\frac{e^z}{z(z+1)^2}$ is analytic in the disc $D(a,r)$, then from the Cauchy's Theorem we get
$$\int_{\partial D(a,r)}\frac{e^z}{z(z+1)^2}\,dz=0.$$</p>
<p>In the case $3$) we can take the function $g(z)=\frac{e^z}{z}$, which is analytic in the disc $D(-1,\frac12)$. Then from the Cauchy's... |
3,940,818 | <p>Can anyone help how to find the eigenvalues of the following matrix in a simple way? I expand the characteristic polynomial being,
<span class="math-container">$$
\lambda(\lambda-3)(\lambda - 2k) = 0
$$</span>
and get the answer but intuition is that there must be a simple way to find it.
<span class="math-conta... | Doug M | 317,162 | <p>All three rows sum to 3. That means <span class="math-container">$(1,1,1)$</span> is an eigenvector with eigenvalue 3.</p>
<p><span class="math-container">$(1,0,-1)$</span> looks to also be an eigenvector. The eigevalue for this one is <span class="math-container">$2k.$</span></p>
<p>And the sum of the eigenvalues... |
144,567 | <p>Fix a homomorphism $f:A\rightarrow B$.
Choose $\{b_1,\dots,b_n\}$, $\{b'_1,\dots,b'_m\}$ subsets of elements in $B$. Suppose that $B$ is algebraic over $f(A)[b_1,\dots,b_n]$ and $\{b_1,\dots,b_n\}$ are algebraically independent over $f(A)$. Suppose that $\{b'_1,\dots,b'_m\}$ satisfies the same condition. Does it imp... | darij grinberg | 2,530 | <p>This is a sidenote to Sasha's answer. The "yes" part can be proven in a completely elementary way without prime ideals and Krull dimension. Here is a sketch, as I have to prepare a talk for Monday and finish a paper for very soon:</p>
<blockquote>
<p><strong>Theorem 1.</strong> Let <span class="math-container">$A... |
1,026,975 | <p>Given a topology $(X,T)$, $A\subset X$, $x \in X$ is a limit point of A if $\forall$ open $U$ that contains $x$, $(U\cap A)$\ {$x$} $\neq \emptyset$. $x \in X$ is in $cl(A)$ if $\forall$ open $U$ that contains $x$, $U\cap A$ $\neq \emptyset$. Is there any example that a point in the the closure of $A$ is not a limit... | lhf | 589 | <p>$X=\mathbb R$, $A=\{0,1\}$. Then $A$ is closed but none of its points is a limit point of $A$.</p>
|
266,738 | <p>Say I have an expression that has multiple subvalues how do I return the definition that would be applied to it.</p>
<p>Say I define for example:</p>
<pre><code>fun[y_][x_] := {x, y};
fun[3][x_] := 2;
</code></pre>
<p>Then I would like <code>findSubValue[fun[3][5]]</code> to return <code>HoldPattern[fun[3][x_]] :>... | Kvothe | 45,020 | <p>I think the following works because SubValues are automatically sorted (according to specificity, see <a href="https://mathematica.stackexchange.com/a/106093/45020">https://mathematica.stackexchange.com/a/106093/45020</a>) in the list returned by SubValues (or DownValues etc...):</p>
<pre><code>SetAttributes[outerm... |
20,634 | <p>From the <a href="https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem" rel="nofollow noreferrer">Peter–Weyl theorem in Wikipedia</a>, this theorem applies for compact group. I wonder whether there is a non-compact version for this theorem.</p>
<p>I suspect it because the proof of the Peter–Weyl theorem heavily ... | Zoran Skoda | 35,833 | <blockquote>
<p>What I want to ask is there any other
way to define quantized flag variety?
In the classical case, it is well
known that flag variety can be defined
as <span class="math-container">$G/B$</span>, say <span class="math-container">$G$</span> is general linear group
and B is Borel subgroup. Is there any
ana... |
3,561,242 | <blockquote>
<p>Determine if the following integral will converge.
<span class="math-container">$$\int_0^1\frac{e^{\sqrt x}-1}{x}dx$$</span></p>
</blockquote>
<p>My approach was something like this. I made the assumption that <span class="math-container">$e^{\sqrt{x}} − 1 ≈ x$</span> and then followed like this:</... | PierreCarre | 639,238 | <p>You can use the comparison criteria for improper integrals. Noting that
<span class="math-container">$$
\lim_{x\to 0}\frac{\frac{e^{\sqrt{x}}-1}{x}}{\frac{1}{\sqrt{x}}} = 1,
$$</span></p>
<p>you can establish that the given integral has the same nature as <span class="math-container">$\int_0^1 \frac{1}{\sqrt{x}} ... |
2,982,942 | <p>Our professor gave us definitions for closed and open intervals. </p>
<p>A set <span class="math-container">$U$</span> is open if <span class="math-container">$\forall x \in U$</span>, <span class="math-container">$\exists \epsilon \gt 0$</span> such that <span class="math-container">$(x- \epsilon,x+ \epsilon)\subs... | Olivier Oloa | 118,798 | <p>You are right.</p>
<p>To see the given limit exists and is zero, one may just employ <a href="https://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates" rel="nofollow noreferrer">polar coordinates</a>, obtaining
<span class="math-container">$$
\left|\frac{xy\sin{y}}{3x... |
216,268 | <p>Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$</p>
<p>The asymptotic expression for $(n!)^{-1}$ (<a href="https://stackoverflow.com/questions/3084937/how-to-calculate-the-inverse-factorial-of-a-real-... | Igor Rivin | 11,142 | <p>If you take logs of both sides, and expand the left hand side in a power series around infinity (to first order), you get:
$$\frac{\frac{j}{2}-\frac{j^2}{2}}{n}+j \left(-2 \log (j)-\log
\left(\frac{1}{n}\right)+2\right)-\frac{1}{6 j}+\log
\left(\frac{1}{j}\right)+\log \left(\frac{1}{2 \pi }\right) \approx \lo... |
2,571 | <p>Square of an irrational number can be a rational number e.g. $\sqrt{2}$ is irrational but its square is 2 which is rational.</p>
<p>But is there a irrational number square root of which is a rational number?</p>
<p>Is it safe to assume, in general, that $n^{th}$-root of irrational will always give irrational numbe... | kennytm | 171 | <p>Obviously, if p is rational, then p<sup>2</sup> must also be rational (trivial to prove).</p>
<p>$$ p \in \mathbb Q \Rightarrow p^2 \in \mathbb Q. $$</p>
<p>Take the <a href="http://en.wikipedia.org/wiki/Contraposition">contraposition</a>, we see that if x is irrational, then √x must also be irrational.</p>
... |
465,487 | <p>In Fraleigh, it said, </p>
<blockquote>
<p>"Consider the set N of all polynomials in x and y in F[x,y] having constant term 0. Then N is an ideal, but not a principal ideal." (p.399)</p>
</blockquote>
<p>Could you tell me why this is not a principal ideal?</p>
| Community | -1 | <p>Note that $N$ contains both the polynomials $x$ and $y$; is there any element of $F[x, y]$ dividing both $x$ and $y$ that is <em>not</em> a constant?</p>
|
19,069 | <p>In this weird pandemic school year, I'm doubly interested in technology integration to help my virtual (high school) students as much as my in-person students. I've been particularly eager to get that working with compass-and-straightedge constructions. Obviously, students need a little familiarity with holding a ... | Joseph O'Rourke | 511 | <p>This looks promising but it is not free:
<a href="https://www.mathspad.co.uk/i2/construct.php" rel="nofollow noreferrer">mathspad.co.uk</a>.</p>
<hr />
<img src="https://i.stack.imgur.com/sjImk.png" width="400" />
<hr />
You can experiment with the tools without creating (or paying for... |
216,025 | <p>Let $U \subset \mathbb{R}^n$ open. If $f: U \to \mathbb{R}$ attains a relative maximum ( or minimum) in the point $x \in U$, and $f$ is differentiable in point $x$, then $f'(x)=0$.</p>
| coffeemath | 30,316 | <p>This may be an approach: Take your function and create $n$ one variable functions by restricting $f$ to the coordinat axes near $x$. Apply the one variable result to each. Now you have all the partial derivatives zero; I seem to recall that "differentiable" in $n$ space implies the existence of a good tangent plane ... |
216,025 | <p>Let $U \subset \mathbb{R}^n$ open. If $f: U \to \mathbb{R}$ attains a relative maximum ( or minimum) in the point $x \in U$, and $f$ is differentiable in point $x$, then $f'(x)=0$.</p>
| Tomás | 42,394 | <p>I assume that you know this result is valid for one dimensional functions. So you can proceed like this:</p>
<p>Consider the set $U_{d}=\{x+td:t\in (-\epsilon,\epsilon)\}$ where $\epsilon$ is a small number and $d\in\mathbb{R}^{n}$. Consider the restriction of $f$ to $U_{d}$; lets denote it by $f_{d}$. </p>
<p>No... |
3,860,982 | <p>How do I prove that if:
<span class="math-container">$$\cos^3(x) + \sin^3(x) = 1$$</span>
then:
<span class="math-container">$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$</span></p>
<p>Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by <span class=... | TonyK | 1,508 | <p>Hint: <span class="math-container">$\cos^3x\le\cos^2x$</span>, with equality if and only if <span class="math-container">$\cos x=0$</span> or <span class="math-container">$1$</span>. And the same for <span class="math-container">$\sin$</span>.</p>
|
4,609,845 | <p>I am looking for a closed form for the integral <span class="math-container">$$\int_0^\infty \frac{t^s}{(e^t-1)^z}dt$$</span> valid for <span class="math-container">$s,z$</span> being both complex numbers, hopefully using complex analysis. I have already evaluated this integral when <span class="math-container">$s$<... | mils | 1,105,316 | <p>Here's my take on this integral.
<span class="math-container">$$
I(s, z) = \int_0^\infty t^s (e^t - 1)^{-z}dt
$$</span>
To start off, we'll use the following substitution.
<span class="math-container">$$u = e^{-t}\\ t = -\ln u \\ dt = -u^{-1} du$$</span>
Which then we'll have
<span class="math-container">$$
-\int_1^... |
2,737,823 | <p><strong>Background</strong></p>
<p>Hey everyone. I'm absolutely stumped on an exercise I am working on out of Axler's <em>Linear Algebra Done Right</em>, 3rd edition. Funnily enough, <a href="http://linear.axler.net/InnerProduct.pdf" rel="nofollow noreferrer">the sample chapter</a> available on his website is the... | Y. Hu | 395,803 | <p>Suppose $f\in U^{\perp}$. $\forall x_0\in[-1,1]$, if $f(x_0) \neq0$, say $f(x_0)>0$. By continuity, there is a neighborhood $V$ of $x_0$ such that $f(x) > f(x_0)/2$ in $V$. And we can always choose a non-negative function $g$ which lies in $U$ and is supported in $V$. Now $\langle f,g\rangle > 0$, which is... |
181,532 | <blockquote>
<p>Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field.</p>
</blockquote>
<p>How can one justify the answer in the shortest number of lines?</p>
| i. m. soloveichik | 32,940 | <p>Let $I=(3)$ then $Z[i]/I=Z_3[i]$ is a field (since $x^2+1$ is irreducible over $Z_3$) having 9 elements.</p>
|
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