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 |
|---|---|---|---|---|
712 | <p>I read <a href="https://math.stackexchange.com/questions/625/why-is-the-derivative-of-a-circles-area-its-perimeter-and-similarly-for-spheres">this question</a> the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle. </p>
... | Larry Wang | 73 | <p>Actually, it is also true for squares (and for regular polygons in general!). The problem you ran into is what the <a href="http://en.wikipedia.org/wiki/Apothem" rel="noreferrer">equivalent of "r"</a> is. The side length of a square is actually more comparable to the circle's diameter.</p>
<p>Instead, the correct a... |
786,187 | <p>I can't seem to figure this one out:</p>
<p>$\int \frac{t^5}{\sqrt{t^2+2}}dt$</p>
<p>$t^5 / \sqrt{t^2+2}$</p>
<p>I know I need to substitute t for a trigonometric function, $\tan$ I think</p>
<p>Any hints are greatly appreciated!</p>
| Samrat Mukhopadhyay | 83,973 | <p>Let $$I_n=\int \frac{t^n}{\sqrt{t^2+a^2}} dt$$ Put $t=a\tan u$ to get $$I_n=a^{n}\int {\tan^nu}\ {\sec u du}$$ If $n$ is odd then, by replacing $\sec u=z$ you will get $$I_n=a^n\int (z^2-1)^{n/2}dz$$ which you can then expand and integrate. </p>
<p>If $n$ is even then by integration by parts $$I_n=\tan^{n-1}u\sec u... |
4,340,560 | <p>I have to teach limits to infinity of real functions of one variable.
I would like to start my course with a beautiful example, not simply a basic function like <span class="math-container">$1/x.$</span> For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the bas... | Ahmed zeribi | 716,266 | <p>"John Napier" and how he get to "e=2.7182818284..." might be a good real life limit.
(<a href="https://en.wikipedia.org/wiki/E_(mathematical_constant)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/E_(mathematical_constant)</a> )</p>
|
4,340,560 | <p>I have to teach limits to infinity of real functions of one variable.
I would like to start my course with a beautiful example, not simply a basic function like <span class="math-container">$1/x.$</span> For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the bas... | Piquito | 219,998 | <p>COMMENT.-You can consult about examples of mathematical modeling (there are many!) and then give them the appropriate tone for your purposes. For example, suppose that a severe drought has eliminated all mammals of a certain species in a region and that a board of ecologists wants to repopulate the area in question ... |
3,009,351 | <p>Consider the vectors <span class="math-container">$q_1=(1,1,1)$</span> and <span class="math-container">$q_3=(1,1,-2)$</span>. I need to find a third vector <span class="math-container">$q_2$</span> such that <span class="math-container">$\{q_1,q_2,q_3\}$</span> is a arthogonal basis for <span class="math-container... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Since <span class="math-container">$q_1$</span> and <span class="math-container">$q_3$</span> are orthogonal it suffices to find <span class="math-container">$q_3$</span> by</p>
<p><span class="math-container">$$q_2=q_1\times q_3$$</span></p>
<p>As an alternative by GS we have</p>
<p... |
366,103 | <p>As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in <a href="http://www-personal.umich.edu/%7Ewangzuoq/635W12/Notes/Lec%2024.pdf" rel="noreferrer">these notes</a> (I have cut ... | Yuri Vyatkin | 2,002 | <p><strong>Update</strong>. Apparently the question is now clear to the OP, but anyway I've decided to summarize the discussion that occurred here for those who are not interested in reading the whole lot.</p>
<p>As the OP commented below @Ted's answer,</p>
<blockquote>
<p>But isn't $\nabla$ the connection on M, wh... |
1,326,652 | <p>What would be the nature of the roots of the equation $$2x^2 - 2\sqrt{6} x + 3 = 0$$</p>
<p>My book says that as the discriminant is 0 so the roots are rational and equal.
But discriminant can be used for determining the nature of roots only when the roots are rational numbers. Is the answer in the book wrong becau... | Hagen von Eitzen | 39,174 | <p>The discriminant can always be used. Also, your suspicion that the roots are irrational does not conflict with the claim that they are real.
Note that
$$ 2x^2-2\sqrt 6x+3=(\sqrt 2 x-\sqrt 3)^2$$</p>
|
475,267 | <p>Suppose $f(0) =0 $ and $0<f''(x)<\infty (\forall$ $x>0)$, then $\frac{f(x)}{x}$ strictly increases as $x$ increases. </p>
<p>I have shown that $f'(x)-\frac{f(x)}{x} = \frac{1}{2}xf''(c)$, for some $c\in (0,x)$. How do I proceed from here? </p>
| Alexy Vincenzo | 33,821 | <p>Consider $g(t)=tf'(t)-f(t)-\frac{1}{2}t^2A $ defined on $[0,x]$ , where $A=2\frac{xf'(x)-f(x)}{x^2}$. Then, $g(0)=g(x)=0. $ By Rolle's theorem, $\exists c\in(0,x)$ such that $g'(c)=0$. It follows that $cf''(c)+f'(c)-f'(c)-cA=0$ and upon simplification, we get: $f'(x)-\frac{f(x)}{x}=\frac{1}{2}xf''(c)$</p>
|
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| George Frank | 30,674 | <p>I hope Von Neumann was one of the very few of us who realized that remembering and familiarity are not at all the same as understanding. Understanding means that we see the need, the origin, the purpose of definitions, the REASON a definition is made. The a priori purpose. And it means that when we encounter a "theo... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Carlos Medina | 4,698 | <p>I would just like to add a reformulation, an isomorphism if you will, of von Neumann’s quote, in the shape of a story/anecdote, by a rather perspicacious fella that goes by the name of John Cleese, who, from what I have read so far of his autobiography, is either a professional thief, a bible salesman, a professiona... |
1,578,940 | <p>I am asked to find $[T]_{\alpha}^{\alpha}$ for linear map $T: M_{2\times 2} \rightarrow M_{2\times 2}$ where $\alpha$ is the standard basis and </p>
<p>$T(x) =
\begin{bmatrix}
1 & 1 \\
0 & -1
\end{bmatrix}
x$</p>
<p>How can I approach this... I tried applying T to every vector in the standard ba... | seeker | 267,945 | <p>Hint :-</p>
<p>First of all since $M_{2\times 2}$ is $4$ dimensional hence the matrix for $T$ must be a $4\times 4$ matrix. So if we denote by $E_{ij}$ the basis of $M_{2\times 2}$, where $E_{ij}$ is $2\times 2 $ matrix having $1$ as the $ij$-th entry and the rest of the entry is $0$, then </p>
<p>$T(E_{11})$ = $\... |
303,944 | <p>Interesting problem I spotted while learning:</p>
<blockquote>
<p>Let <span class="math-container">$X=\left\{1,..,n\right\}$</span>. We randomly select subset of <span class="math-container">$X$</span> and name it <span class="math-container">$A$</span>. Each subset if equally likely.</p>
<p>a) Find the expected val... | Ewan Delanoy | 15,381 | <p>All your guesses are correct. What happens here is that you initially work
on the probability space $\Omega={\cal P}(X)$, the set of all subsets of $X$.</p>
<p>On that probability space, you can define the random variable $V_i$, equal to $1$
if $i\in A$ and $0$ otherwise.</p>
<p>Denote by $E_i=\lbrace A \in \Omega... |
303,944 | <p>Interesting problem I spotted while learning:</p>
<blockquote>
<p>Let <span class="math-container">$X=\left\{1,..,n\right\}$</span>. We randomly select subset of <span class="math-container">$X$</span> and name it <span class="math-container">$A$</span>. Each subset if equally likely.</p>
<p>a) Find the expected val... | passerby51 | 7,202 | <p>Regarding your first question, note that a subset has an equivalent representation as a binary sequence. For example, for $n = 4$, the subset $\{1,3\}$ can be identified with $(1,0,1,0)$. Now, picking a subset uniformly at random, is like picking a sequence uniformly at random out of the $2^n$ possibilities. You sho... |
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | Jean-Philippe Burelle | 521,944 | <p>This is really a linear algebra question.</p>
<p>You can show that the directional derivative depends linearly on the direction vector $v$, that is, it satisfies the relation:</p>
<p>$$\nabla_{a v + b u} f = a\,\nabla_v f + b\,\nabla_u f.$$</p>
<p>For scalars $a,b$ and vectors $v,u$. The gradient is the vector of... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Henry Towsner | 62 | <p>Questions like this, or variants (from students, the notorious "when will I use this in real life") seem to be pretty common, and I'm always a little surprised, because the unstated premise - that high school is supposed to teach students things narrowly tailored to their future career - is so obviously false.</p>
... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Community | -1 | <blockquote>
<p>Why is Mathematics a compulsory subject for high school students, especially those who are clearly studying in Humanities streams?</p>
</blockquote>
<ol>
<li><p>A kid at age 14 is not ready to make irrevocable decisions that will affect them for the rest of their life. That's why we don't let them ge... |
3,043,699 | <p>I'm having difficulty understanding how below integral is evaluated : </p>
<p><span class="math-container">$$\int_0^1\frac{(1-y)^2}{2}+(1-y)y dy = \frac{-1(1-y)^3}{6}+\frac{y^2}{2}-\frac{y^3}{3}\bigg|^1_0$$</span></p>
<p>What are steps involved in this evaluation ?</p>
<p>For <span class="math-container">$\frac{(... | Kavi Rama Murthy | 142,385 | <p>Antiderivative if <span class="math-container">$(1-y)^{2}$</span> is not <span class="math-container">$(1-y)^{3} /3$</span>. If you differentiate the latter you get <span class="math-container">$-(1-y)^{2}$</span>. Hence the antiderivative is <span class="math-container">$-(1-y)^{3} /3$</span>.</p>
|
3,043,699 | <p>I'm having difficulty understanding how below integral is evaluated : </p>
<p><span class="math-container">$$\int_0^1\frac{(1-y)^2}{2}+(1-y)y dy = \frac{-1(1-y)^3}{6}+\frac{y^2}{2}-\frac{y^3}{3}\bigg|^1_0$$</span></p>
<p>What are steps involved in this evaluation ?</p>
<p>For <span class="math-container">$\frac{(... | Rhys Hughes | 487,658 | <p>Remember that where <span class="math-container">$f(x)$</span> is linear we have that:</p>
<p><span class="math-container">$$\int[f(x)]^n dx=\frac{[f(x)]^{n+1}}{(n+1)f'(x)}+C$$</span></p>
|
4,349,748 | <p>I would appreciate some help with this.</p>
<p>Is:
<span class="math-container">$$
\nabla \cdot \left( \nabla \vec{v}\right)^T= \nabla \left( \nabla \cdot \vec{v}\right)
$$</span></p>
<p>How can I show this? Is the gradient of a vector mathematically defined?</p>
<p>Best regards</p>
| Henno Brandsma | 4,280 | <p>Your boundedness proof does not use nets at all and can also be simplified: let <span class="math-container">$A$</span> be compact and let <span class="math-container">$a_0 \in A$</span>. Consider the open cover <span class="math-container">$\{\mathbb{B}(a_0,n)\mid n \in \Bbb N\}$</span> of <span class="math-contain... |
64,022 | <p><strong>Edit</strong></p>
<p>(As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question)</p>
<p>For infinite independent Bernoulli trials with probability $p$ to success, define a random variable N which equals to the number of successful tri... | Robert Israel | 8,508 | <p>As long as $p > 0$, $N$ will be $\infty$ with probability 1. The first mistake is in
$$ \sum_{i=1}^\infty b(i;\infty,p) = \lim_{m \to \infty} \sum_{i=1}^m b(i; m,p)$$</p>
|
2,243,646 | <p>Suppose a, b, c are positive real numbers such that </p>
<p>$$(1+a+b+c)\left(1+\frac 1a+\frac 1b+\frac 1c\right)=16$$
Then is it true that we must have $a+b+c=3$ ?</p>
<p>Please help me to solve this. Thanks in advance. </p>
| Dr. Sonnhard Graubner | 175,066 | <p>Remember the inequality between the arithmetic and the harmonic mean
$$\frac{1+a+b+c}{4}\geq \frac{4}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}$$ for all $$a,b,c>0$$
the equality holds if $$a=b=c=1$$ thus $$a+b+c=3$$</p>
|
2,243,646 | <p>Suppose a, b, c are positive real numbers such that </p>
<p>$$(1+a+b+c)\left(1+\frac 1a+\frac 1b+\frac 1c\right)=16$$
Then is it true that we must have $a+b+c=3$ ?</p>
<p>Please help me to solve this. Thanks in advance. </p>
| lulu | 252,071 | <p>Expand your product to get $$4+\left(a+\frac 1a\right)+\left(b+\frac 1b\right)+\left(c+\frac 1c\right)+\left(\frac ab+\frac ba\right)+\left(\frac ac+\frac ca\right)+\left(\frac bc+\frac cb\right)=16$$</p>
<p>Now, the arithmetic-geometric inequality tells us that (for $x>0$): $$\left(x+\frac 1x\right)≥2$$ and tha... |
2,534,369 | <p>I am trying to work my through the exercises in Spivak's <em>Calculus on Manifolds.</em> I am currently working on the exercises in Chapter 3 which deals with Integration. I am having trouble with the following question:</p>
<blockquote>
<p>Let:</p>
<p>\begin{equation}
f(x,y)=\begin{cases}
0, & \te... | WannaBeRealAnalysist | 319,730 | <p>So here is an attempt at a solution:</p>
<p>So for any partition $P$, </p>
<p>$u(f,P) = 0$, so it should be enough to show that $U(f,P)$ is arbitrarily close to $0$. For a natural number $q$, consider the partition,</p>
<p>$P = \bigl((0,1/q,2/q,\cdots,(q-1)/q,1),(0,1)\bigr)$.</p>
<p>Let $x \in [\frac{p}{q}, \fra... |
4,090,775 | <p>Let <span class="math-container">$\mu$</span> be the Lebesgue measure and, <span class="math-container">$E$</span> and <span class="math-container">$F$</span> compact subsets of <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$E\subset F$</span> and <span class="math-container... | HallaSurvivor | 655,547 | <p>Hint:</p>
<p>Consider the sets <span class="math-container">$[-r,r] \subseteq \mathbb{R}$</span>.</p>
<p>Can you show <span class="math-container">$r \mapsto \mu(E \cup [-r,r] \cap F)$</span> is continuous? Can you show for any <span class="math-container">$r$</span> the set we're measuring is compact? What does the... |
664,220 | <p>If $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$? is this statement true?</p>
| dato datuashvili | 3,196 | <p>if $A\cdot B=B\cdot A$</p>
<p>then $e^{A+B}=e^{A}\cdot e^{B}$</p>
<p>now multiply both side by $t$</p>
<p>we have</p>
<p>$e^{At+Bt}$,</p>
<p>you can apply same rule for this too</p>
<p>there is question related to matrix expoenntial</p>
<p><a href="https://math.stackexchange.com/questions/370817/commuting... |
664,220 | <p>If $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$? is this statement true?</p>
| Robert Lewis | 67,071 | <p>It is in fact the case. For a thorough explanation, please see my answer to <a href="https://math.stackexchange.com/questions/568450/m-n-in-bbb-r-n-times-n-show-that-emn-emen-given-mn-nm">this question</a>. It's only a mouse click away!</p>
<p>Hope this helps. Cheerio,</p>
<p>and as always,</p>
<p><strong><em>... |
2,008,437 | <p>Given four points in the plane, there exists a one-dimensional family of conics through these, often called a pencil of conics. The locus of the centers of symmetry for all of these conics is again a conic. What's the most elegant way of computing it?</p>
<p>I know I could choose five arbitrary elements from the pe... | Narasimham | 95,860 | <p>Excuse me, I am unable to understand , since fifth point fixes the conic and its center, what is the variable fifth point $P_5$ to determine a locus of corresponding conic centers?</p>
<p>To find a locus we need to define our object property <em>independently</em>, but that property should <em>not again depend</em>... |
1,085,164 | <p><strong>I am currently taking a course on Numerical PDE. The course covers the following topics listed below.</strong> </p>
<p><strong>Chapter 1: Solutions to Partial Dierential Equations:</strong></p>
<p><strong>Chapter 2: Introduction to Finite Elements:</strong></p>
| Cookie | 111,793 | <p><a href="http://link.springer.com/book/10.1007%2F978-1-4899-7278-1" rel="nofollow">Numerical PDEs by J.W. Thomas</a> might be a good book. If you check its table of contents, they have the majority of the topics you listed. </p>
<p>Here is one example of professor's <a href="http://www.tat.physik.uni-tuebingen.de/~... |
2,810,755 | <p>I'm trying to set up a double integral on $e^{-(x+y)}$ and the range listed is: $0 < x < y < \infty.$</p>
<p>I'm interpreting this as $0 < x < y$ for $x$'s range and $x < y < \infty$ as $y$'s range. I've put those bounds on my two integrals and proceeded with the mission. However, $y$ is stil... | Doug M | 317,162 | <p>You look great right up through here.</p>
<p>Assume:
$1\cdot2+2\cdot3+3\cdot4+...+k(k+1)=\frac{(k(k+1)(k+2))}{3}$</p>
<p>We must show that:</p>
<p>$1\cdot2+2\cdot3+3\cdot4+...+k(k+1)+(k+1)(k+2)=\frac{(k+1)(k+2)(k+3)}{3}$</p>
<p>$1\cdot2+2\cdot3+3\cdot4+...+k(k+1)+(k+1)(k+2) = \frac{(k(k+1)(k+2))}{3}+(k+1)(k+2)\ ... |
1,289,834 | <p>Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$
$$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$
I cannot seem to find any such triplets. Any hints on how to prove it?</p>
| Domenico Vuono | 227,073 | <p>Let $y=k^2$, $z=r^2$ and $x=c^2$(with $k,r,c$ integers) then the expression becomes: $$(\sqrt {2\cdot (k+r)(c+r)}-k-r)^2$$
Now for $a$ to be an integer $$2(k+r)(c+r)=n^2$$
with $n$ integer.
We put $$k+r=2^{2p+1}\cdot t^s$$
And $$c+r=t^v$$
(or vice versa,with $v$ and $s$ both odd or even).
If $r=1$ $$k=2^{2p+1}\cdot ... |
1,289,834 | <p>Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$
$$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$
I cannot seem to find any such triplets. Any hints on how to prove it?</p>
| Barry Cipra | 86,747 | <p>Checking a solution given by Winther in comments beneath the OP led me to the following:</p>
<blockquote>
<p>In general, if $z=x+y$, then $a=x$.</p>
</blockquote>
<p>This is seen by writing</p>
<p>$$\sqrt{2(\sqrt x+\sqrt z)(\sqrt y+\sqrt z)}=\sqrt x+\sqrt y+\sqrt z$$</p>
<p>squaring both sides and expanding to... |
2,976,057 | <p>I’m doing partial fractions and need to factorize the denominator. They are quadratic. However there are some that aren’t so easy to factorize and my first choice was to use the quadratic equation to find the roots however comparing my answer with the correct one the signs are different. Is the quadratic formula onl... | Jean-Luc Bouchot | 24,153 | <p>I'm guessing, what you have is the following fraction:
<span class="math-container">$$
f(z) = \frac{N(z)}{D(z)},
$$</span>
where <span class="math-container">$D(z) = -3z^2-4z-1$</span>. </p>
<p>Basically, you may write <span class="math-container">$D(z) = -(3z^2+4z+1) = -D_2(z)$</span>. </p>
<p>Finally, you can d... |
1,620,795 | <p>Find function f(x), where:
$$f(3)=3$$
$$f'(3)=3$$
$$f'(4)=4$$
$$f''(3) = \nexists$$ </p>
<p>How to find function like this in <strong>general?</strong> What steps should I do?</p>
| Claudius | 218,931 | <p>You know that $\lvert x_i\rvert \le \max_{j=1,\dotsc,n}\lvert x_j\rvert$ for each $i=1,\dotsc, n$. Hence $\lvert x_i\rvert\le \lVert x\rVert_\infty$ for each $i=1,\dotsc,n$. Now,
\begin{align*}
\lvert x\cdot y\rvert &= \left\lvert\sum_{i=1}^nx_iy_i\right\rvert \le \sum_{i=1}^n \lvert x_i\rvert\cdot \lvert y_i\rv... |
2,523,581 | <p>I'm a bit rusty on my complex numbers, how would you solve the following problem on paper?</p>
<blockquote>
<p>Determine and sketch (graph) the set of all complex numbers of form:
$$z_n=\frac{2n+1}{n-i},n\in\mathbb R$$</p>
</blockquote>
<p>Rationalizing yields $$S=\left\{z_n\in\mathbb C : z_n=\frac{n + 2 n^2}{... | nonuser | 463,553 | <p>Say $$w= {2n+1\over n-i}\Longrightarrow n ={wi+1\over w-2}$$</p>
<p>Since $\overline{n}=n$ we have:</p>
<p>$$ (\overline{w}-2)(wi+1)= (-i\overline{w}+1)(w-2)$$</p>
<p>so</p>
<p>$$ |w|^2i-2wi+\overline{w}-2 = -i|w|^2+2i\overline{w}+w-2$$</p>
<p>so
$$ |w|^2 = (\overline{w}+w)+{w-\overline{w}\over 2i}$$</p>
<... |
2,523,581 | <p>I'm a bit rusty on my complex numbers, how would you solve the following problem on paper?</p>
<blockquote>
<p>Determine and sketch (graph) the set of all complex numbers of form:
$$z_n=\frac{2n+1}{n-i},n\in\mathbb R$$</p>
</blockquote>
<p>Rationalizing yields $$S=\left\{z_n\in\mathbb C : z_n=\frac{n + 2 n^2}{... | Hari Shankar | 351,559 | <p>$\dfrac{1}{\overline{z_n}} = \dfrac{n+i}{2n+1} = \dfrac{n}{2n+1}+\dfrac{i}{2n+1}$</p>
<p>Evidently $\dfrac{1}{\overline{z_n}}$ lies on the line $L: 2x+y=1$</p>
<p>$z_n$ hence lies on the curve obtained by inverting $L$ in the circle $|z|=1$. The result is a circle passing through origin $O$. The radius and centre ... |
3,681,254 | <p><span class="math-container">$$\int_{0}^{\frac{\pi}{2}}{e^{-x}} \cos3xdx$$</span>
using the intergral by parts formula
<span class="math-container">$$\int{f(x).g(x)'}\mathrm{d}x = (f(x).g(x)) - \int{f'(x)g(x)dx}$$</span>
so it should be
<span class="math-container">$${{{e^{-x}}.\frac{1}{3}\cos(3x)}} - \int_{0}^{\fra... | P. Lawrence | 545,558 | <p>There are two ways of doing this question. (i) First you can write <span class="math-container">$e^{-x}$</span> as <span class="math-container">$(-e^{-x})'$</span> and then apply the integration by parts forrmula, which gives other terms and, up to numerical factors, the integral of <span class="math-container">$e^{... |
3,681,254 | <p><span class="math-container">$$\int_{0}^{\frac{\pi}{2}}{e^{-x}} \cos3xdx$$</span>
using the intergral by parts formula
<span class="math-container">$$\int{f(x).g(x)'}\mathrm{d}x = (f(x).g(x)) - \int{f'(x)g(x)dx}$$</span>
so it should be
<span class="math-container">$${{{e^{-x}}.\frac{1}{3}\cos(3x)}} - \int_{0}^{\fra... | Anton Vrdoljak | 744,799 | <p><span class="math-container">$I=\int_0^{\frac{\pi}{2}}e^{-x}\cos {3x}dx = \text{int. by parts} =\left(-e^{-x}\cos{3x}\right)_0^{\frac{\pi}{2}}-3\int_0^{\frac{\pi}{2}}e^{-x}\sin {3x}dx\\
= 1-3\int_0^{\frac{\pi}{2}}e^{-x}\sin {3x}dx\\
= \text{int. by parts} = 1-3\left( \left( -e^{-x}\sin{3x} \right)_0^{\frac{\pi}{2}} ... |
1,207,134 | <p>Given the image:
<img src="https://i.stack.imgur.com/EJ3ax.jpg" alt="enter image description here"></p>
<p>and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i
, i = 1, 2, 3, · · ·$ can be arbitrarily picked.</p>
<p>How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$?</p>
<p>I actually know what... | Blue | 409 | <p>If $P_i = (x_i,y_i)$, then $P_{n+1}$ is obtained by rotating $P_n$ about the origin by angle $\theta_{n+1}$, and scaling by a factor of $\frac{|OP_{n+1}|}{|OP_n|} = \sec\theta_{n+1}$. Using a rotation matrix, we can write</p>
<p>$$P_{n+1} \;=\; \sec\theta_{n+1}\;\left[\begin{array}
\;\cos\theta_{n+1} & -\sin\t... |
3,467,790 | <p>I saw this in the following question: <a href="https://math.stackexchange.com/questions/24413/is-there-a-function-with-infinite-integral-on-every-interval">Is there a function with infinite integral on every interval?</a></p>
<p>I already understood all other steps on the first answer, however, I don't know how to ... | Kavi Rama Murthy | 142,385 | <p>I think you have to take <span class="math-container">$(q_n)$</span> to be an ennumeration of rationals in a finite interval instead of the whole line. </p>
<p><span class="math-container">$\int_a^{b} \sum \frac {2^{-n}} {|x-q_n|} dx = \sum \int_a^{b}\frac {2^{-n}} {|x-q_n|} dx$</span> and <span class="math-contai... |
3,467,790 | <p>I saw this in the following question: <a href="https://math.stackexchange.com/questions/24413/is-there-a-function-with-infinite-integral-on-every-interval">Is there a function with infinite integral on every interval?</a></p>
<p>I already understood all other steps on the first answer, however, I don't know how to ... | acreativename | 347,666 | <p>You could define the sets</p>
<p><span class="math-container">$A_{q_j , \epsilon} := \{ y |$</span> <span class="math-container">$ $</span> <span class="math-container">$|y-q_j| \leq \epsilon \cdot (1.5)^{-j} \}$</span></p>
<p>and the set <span class="math-container">$A_{\epsilon} := \bigcup_{j = 1}^{\infty} A_{q_... |
1,990,865 | <blockquote>
<p>Given set $\{3 - x \mid x < 0\}$, find if it is bounded.</p>
</blockquote>
<p>So, from intuition it should be bounded from below, but how to show it?</p>
| Beans on Toast | 257,517 | <p><a href="https://i.stack.imgur.com/KwOUh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KwOUh.png" alt="Equation of straight line for real x"></a></p>
<p>Proof without words in this plot.</p>
|
4,200,164 | <blockquote>
<p>How do you show that for <span class="math-container">$x \in (0,2 \pi)$</span> the following series converges?
<span class="math-container">$$\sum_{n=1}^\infty \frac{\cos(xn)}{n}$$</span></p>
</blockquote>
<p>Of course, this series doesn't converge absolutely. For <span class="math-container">$x= \pi$</... | QC_QAOA | 364,346 | <p>We have</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{\cos(nx)}{n}=\Re \left[\sum_{n=1}^\infty \frac{e^{i nx}}{n}\right]$$</span></p>
<p>This then <a href="https://www.wolframalpha.com/input/?i=sum%28exp%28i%20x%20n%29%2Fn%2C%28n%2C1%2Cinfty%29%29" rel="nofollow noreferrer">simplifies</a> to</p>
<p><... |
4,619,641 | <p>I need to find the image of the function <span class="math-container">$f(z) = z^2$</span> whose domain is <span class="math-container">${z: Re(z) > 0}$</span>. I first let <span class="math-container">$z = x + iy$</span>. Then <span class="math-container">$$w = f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + 2xyi$$</span> He... | Ben | 650,264 | <p>You can indeed infer a Wick's theorem for spheres from the usual one for Gaussians.</p>
<p>As we know that only an even number of contractions gives a non-zero result, let us consider a <span class="math-container">$2k$</span>-point function for the standard Gaussian in <span class="math-container">$\mathbb R^n$</sp... |
3,906,403 | <p>Let be two vectors <span class="math-container">$u$</span> and <span class="math-container">$v$</span> of <span class="math-container">$\Bbb R^n$</span>. We assume that <span class="math-container">$||u|| = ||v||$</span>.</p>
<ul>
<li><p>The vectors <span class="math-container">$u$</span> and <span class="math-conta... | Mike | 544,150 | <p>Two vectors <span class="math-container">$y$</span> and <span class="math-container">$z$</span> are orthogonal if <span class="math-container">$y \cdot z$</span> is 0. Now setting <span class="math-container">$y=2(u+v)$</span> and <span class="math-container">$z=2(u-v)$</span>, it follows that <span class="math-cont... |
3,906,403 | <p>Let be two vectors <span class="math-container">$u$</span> and <span class="math-container">$v$</span> of <span class="math-container">$\Bbb R^n$</span>. We assume that <span class="math-container">$||u|| = ||v||$</span>.</p>
<ul>
<li><p>The vectors <span class="math-container">$u$</span> and <span class="math-conta... | Souza | 468,313 | <p>A nice way to see that is using the <a href="https://en.wikipedia.org/wiki/Polarization_identity" rel="nofollow noreferrer">Polarization Identity</a>, that is, <span class="math-container">$\langle x,y\rangle=\dfrac14\left(|x+y|^2-|x-y|^2\right)$</span>. Setting <span class="math-container">$x=2(u+v)$</span> and <sp... |
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| Allure | 511,061 | <p>There's an error in your computation - the $y^2$ term doesn't match on both sides of the equation.</p>
<p>Having said that, you can still argue that the RHS is positive because $(x+z)^2 \geq z^2$ if $x\geq0$.</p>
|
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| user | 505,767 | <p>As an alternative at completing the square you can look at the <strong>matrix associated to the quadratic form</strong>:</p>
<p>$$x^TAx$$</p>
<p>$$A=\begin{bmatrix}
6 & 0 & 2 \\
0 & 4 & -2 \\
2 & -2 & 2 \\
\end{bmatrix}$$</p>
<p>and note that its <strong>signature</strong> ... |
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| Will Jagy | 10,400 | <p>Your quadratic form is positive definite. I do not know what the eigenvalues of the matrix are. This method is the same as "completing the square." The diagonal entries of the diagonal matrix $D$ are all positive.</p>
<p>$$ P^T H P = D $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
-... |
2,880,377 | <p>As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on.</p>
<p>I believe the answer is 0. I think that to start, I can show that $\frac{1}{\sin(\frac{1}{1+x})} \rightarrow x + 1$ for large x. By loo... | user | 505,767 | <p>To avoid Taylor's expansion by $y=\frac{1}{1+x}\to 0$ we have that</p>
<p>$$x\left(x + 1 - \frac1{\sin\left(\frac{1}{1+x}\right)}\right)=\frac{1-y}{y}\left(\frac1y - \frac1{\sin y}\right)=$$$$=\frac{1-y}{y}\left(\frac{\sin y -y}{y\sin y}\right)=(1-y)\frac{y}{\sin y}\left(\frac{\sin y -y}{y^3}\right)\to 1 \cdot 1 \c... |
25,412 | <p>This is hopefully a simpler version of this previous unanswered <a href="https://mathematica.stackexchange.com/questions/23496/solving-recursion-relations-using-mathematica">question</a> of mine. </p>
<p>Let me just focus on the two expressions $F_2^{(s)}$ and $F_3^{(s)}$ given in A.3 and A.4 of page 19 of <a href... | Simon Woods | 862 | <p>If you are using <em>Mathematica</em> version 9, the best approach is probably to use the new symbolic tensor functionality as suggested by zentient.</p>
<p>However for this problem it may be sufficient to explicitly specify a rule to convert expressions like <code>Norm[-q]</code> into <code>Norm[q]</code>:</p>
<p... |
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Sam Hopkins | 25,028 | <p>Let <span class="math-container">$P$</span> be a (not necessarily finite) poset such that each element covers and is covered by a finite number of elements. Then we can define the operators <span class="math-container">$U,D\colon \mathbb{Q}P\to\mathbb{Q}P$</span> on the vector space of formal linear combinations of ... |
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Tri | 51,389 | <p>Find a nice expression for the number of down-sets (order ideals) of a product of four finite chains (totally ordered sets), <span class="math-container">$|\bf2^{(\bf k\times \bf m\times \bf n\times \bf r)}|$</span>, where <span class="math-container">$k,m,n,r$</span> are positive integers and "<span class="math-con... |
1,363,407 | <p>If $x,y,z$ are elements of a group such that $xyz=1,$ then which of the following are true?</p>
<ol>
<li>$yzx=1$</li>
<li>$yxz=1$</li>
<li>$zxy=1$</li>
<li>$zyx=1$</li>
</ol>
<p>I have found options 1 and 3 to be correct, but how to prove that options 2 and 4 are wrong (that is, what the given an... | Asinomás | 33,907 | <p>$1$ is always true. We are told $x$ is the inverse of $yz$ and so $yzx=1$. </p>
<p>analogously $3$ is always true. We are told $xy$ is the inverse of $z$ and so $zxy=1$</p>
<hr>
<p>The others are not true, for $2$ notice if $xy\neq yx$ then they can't have the same inverse. (examples of $xy\neq yx$ exist in any ... |
313,508 | <p>If anyone is familiar with Horowitz and Hill... its exercise 1.19</p>
<p>Show that all the average power delivered to the preceding circuit winds up in the resistor. Do this by computing the value of $V^2/R$. What is the power, in watts for a series circuit of a 1$\mu F$ capacitor and a 1k resistor placed across th... | copper.hat | 27,978 | <p>(I am not sure what part is throwing you, please ask for elaboration in the comments and I will do my best to explain.)</p>
<p>The usual convention is to use $j = \sqrt{-1}$ to avoid confusion with the symbol for current, $i$.</p>
<p>If the supply voltage is $v(t) = \sqrt{2} \operatorname{Re} V_{\text{sup}} e^{j ... |
367,638 | <p>Reading over a book on computability, it asserts that in P.C., if A is a theorem, then A has arbitrarily many proofs. I can't see how that would work, would you do an infinite loop in the sequence of well-formed-formulae?</p>
| Doug Spoonwood | 11,300 | <p>There exist several ways to indicate this. I'll consider classical logical systems which only have detachment and universal substitution as primitive inference rules. I use Polish notation. Classical logic has CpCqp as a theorem. So, if "a" is a theorem, and "b" is a theorem we can prove "a" as follows:</p>
<p>... |
1,598,255 | <p>Consider the system
$$
\dot{x}=y,\qquad\dot{y}=x+x^2-y.
$$
It has two equilibria, namely $(0,0)$ and $(-1,0)$.</p>
<p>I would like to linearize the system in both equilibria.</p>
<p>My start is to set $\Delta x=x-x_0,\qquad \Delta y=y-y_0$. Then
$$
\dot{\Delta x}=\dot{x}-\dot{x_0}=y-y_0=\Delta y
$$
and
$$
\dot{\De... | Daniel Robert-Nicoud | 60,713 | <p>I will do the general case.</p>
<p>Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth map. Consider the differential equation on $\mathbb{R}^n$ given by
$$\dot{z} = F(z).$$
An <em>equilibrium point</em> is a point $z_0\in\mathbb{R}^n$ where $F(z_0)=0$. This implies that $z(t)=z_0$ is a solution of the differential equ... |
11,299 | <p>How to write this small piece in a functional way (ie. without state variables)?:</p>
<pre><code>test[oldJ_List, newJ_List] := Total[Abs[oldJ - newJ]] > 1;
relax[j_List, x_?NumericQ] := Mean[Nearest[j, x, 4]];
j = Range[100]; (* any numeric list *)
j1 = j/2; (*some initial value for the While[] test to return T... | kglr | 125 | <p>This also works:</p>
<pre><code>fold = Function[{lst},Fold[(ReplacePart[#1, #2 ->relax[#1, #1[[#2]]]]) &,
lst, Range@Length@lst]];
fxpnt = FixedPoint[fold, #, SameTest -> (Not[test[#1, #2]] &)] &;
fxpnt@j
</code></pre>
|
221,137 | <p>What's the difference between Fourier transformations and Fourier Series? </p>
<p>Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?</p>
| Community | -1 | <p>If you have a locally compact Abelian group <span class="math-container">$G$</span> you can define a group called the <a href="https://en.wikipedia.org/wiki/Pontryagin_duality" rel="nofollow noreferrer">Pontryagin dual group</a> - <span class="math-container">$\widehat{G}$</span>. You can define a <a href="https://e... |
1,511,753 | <p>Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence,</p>
<p>$$(1) \quad C_0=0.1, \ C_1=0.101, \ C_2=0.101000101, 0.101000101000000000101000101,...,C_n$$</p>
<p>The astute reader will notice this is analog... | fleablood | 280,126 | <p>It depends on the real number, but most "normal" irrational numbers will not have such constructions. This is intuitive as any such method would finitely expressed and there would only be countably many of them. $\pi$ and $\sqrt 2$ have several series expansions that allow as to calculate the decimal expansions. ... |
3,691,747 | <p>Consider a random sample of 11 letters chosen from the alphabet with replacement. What is the probability that the letters can be arranged to spell ‘mississippi’.</p>
<p>For this question, the number of ways to choose 11 letters would be (26)^(11).
To spell 'mississippi',we need:
1M, 4I. 4S and 2P.</p>
<p>Hence, t... | twosigma | 780,083 | <p>Your denominator of <span class="math-container">$26^{11}$</span> is correct, but your numerator should use the multinomial coefficient <span class="math-container">$\displaystyle \binom{11}{1,4,4,2} = \frac{11!}{1!4!4!2!}$</span>.</p>
<p>Why? Well, the multinomial coefficient <span class="math-container">$\display... |
1,378,769 | <p>The equation is given by</p>
<p>$$ \sum_{n=1}^N \min(\gamma, \beta a_n)=N$$
where $\beta$ is the variable with $\beta\in[0,\sqrt\gamma/\min(a_n\mid a_n>0)]$, $ \gamma $ is a constant with $1\le\gamma\le N$, and $a_n$ is an nonnegative constant.</p>
<p>I know that the function $f(\beta)=\sum_{n=1}^N \min(\gamm... | Steven Alexis Gregory | 75,410 | <p>A sequence is a function $f:\mathbb Z^+ \to \mathbf S$.</p>
<p>$f:\mathbb Z^+ \to \mathbf S$ is a special case of a relation
$f \subset \mathbb Z^+ \times \mathbf S.$</p>
<p>The cartesian product $\mathbb Z^+ \times \mathbf S$ is the set
$\{(x,y): x \in \mathbb Z^+$ and $y \in \mathbf S \}$</p>
<p>I'm not sure wh... |
2,581,593 | <p>I am trying to compute the hessian from a linear mse (mean square error) function using the index notation. I would be glad, if you could check my result and tell me if the way that I use the index notation is correct ?</p>
<p>The linear MSE:
$$L(w) = \frac{1}{2N} e^T e$$where $e=(y-Xw)$,</p>
<p>$y \in R^{Nx1} (ve... | frank | 506,630 | <p>For ease of typing, I'll represent the differential operator $\frac{\partial}{\partial w_k}$ by $d_k$</p>
<p>The known relationships are
$$\eqalign{
e_i &= X_{ij}w_j - y_i \cr
d_ke_i &= X_{ij}\,d_kw_j =X_{ij}\,\delta_{jk} = X_{ik} \cr
}$$
Use this to find the derivatives of the objective function
$$\eqali... |
2,581,593 | <p>I am trying to compute the hessian from a linear mse (mean square error) function using the index notation. I would be glad, if you could check my result and tell me if the way that I use the index notation is correct ?</p>
<p>The linear MSE:
$$L(w) = \frac{1}{2N} e^T e$$where $e=(y-Xw)$,</p>
<p>$y \in R^{Nx1} (ve... | V. Vancak | 230,329 | <p>Matrix notations:
$$
\frac{\partial}{\partial w} (Y - Xw)'(Y-Xw) = 2X'(Y-Xw).
$$
Using indices you are taking derivative of the sum of squares w.r.t. each of the $w_j$, i.e.,
$$
\frac{\partial}{\partial w_j} ( \sum_{i=1}^N(y_i - \sum_{j=1}^Dx_{ij} w_j))^2= -2 \sum_{i=1}^N(y_i - \sum_{j=1}^Dx_{ij} w_j)x_{ij}.
$$
Bac... |
313,597 | <p>Let $V$ be a finite-dimensional vector space with an ordered base $\beta$ over a field $F$.</p>
<p>Let $x\in V$ an $[x]_{\beta}$ be a coordinate vector of $x$ relative to $\beta$.</p>
<p>Is this an element of $M_{n\times 1}(F)$ or $F^n$?</p>
<p>Since we usually treat them as a matrix, i think it should be defined... | Community | -1 | <p>The purpose of a coordinate vector is so that you can turn problems into matrix algebra; $M_{n \times 1}(F)$ is the best choice for what a coordinate vector <em>should</em> be if you're into such fine detail.</p>
<p>But it really is a <em>fine</em> detail. Part of the power of linear and multi-linear algebra is how... |
3,336,592 | <p>I could prove the following result from my Real Analysis course:</p>
<blockquote>
<p>Let <span class="math-container">$f:[0,1] \rightarrow [0,1]$</span> be an increasing mapping. Then it has a fixed point.</p>
</blockquote>
<p>I understand that this is a very baby version of Tarski’s Fixed Point Theorem. Now, I ... | Chris Eagle | 693,182 | <p>Your summary seems accurate, with one exception: The theory of algebraically closed fields of characteristic 0 is complete. Perhaps you meant the theory of algebraically closed fields, without specifying the characteristic?</p>
|
3,336,592 | <p>I could prove the following result from my Real Analysis course:</p>
<blockquote>
<p>Let <span class="math-container">$f:[0,1] \rightarrow [0,1]$</span> be an increasing mapping. Then it has a fixed point.</p>
</blockquote>
<p>I understand that this is a very baby version of Tarski’s Fixed Point Theorem. Now, I ... | DanielV | 97,045 | <blockquote>
<p>We can have undecidable and incomplete theories. e.g Peano Arithmetic</p>
</blockquote>
<p>This is based on a very different definition of complete than what you wrote. Godel's Incompleteness Theorem uses the "if it is true then it is provable" pseudo definition of completeness. And he gets around ... |
1,997,407 | <p>What is the relationship between the characteristic polynomial of two square matrices and the characteristic polynomial of the product of these two square matrices? If I know the characteristic polynomials of each one of these matrices, what can I say about the characteristic polynomial of their product?
I can't see... | Dan Fox | 60,380 | <p>In general there is no nice relation. The product of nilpotent matrices need not be nilpotent (for example, consider the product of a nontrivial nilpotent Jordan block with its transpose), so the characteristic polynomial of the product of two matrices whose characteristic polynomials have only $0$ as a root can hav... |
1,997,407 | <p>What is the relationship between the characteristic polynomial of two square matrices and the characteristic polynomial of the product of these two square matrices? If I know the characteristic polynomials of each one of these matrices, what can I say about the characteristic polynomial of their product?
I can't see... | ManifoldFR | 254,063 | <p>Suppose the matrices are such that $AB=BA$: one can put both matrices in triangular form within the same basis $(e_i)$ should this condition be satisfied. Call $\lambda_i$ (resp. $\mu_i$) the eigenvalues of $A$ and (resp. $B$) associated with eigenvector $e_i$. Then the roots of the characteristic polynomial of $AB$... |
4,138,346 | <p>I would like someone to verify this exercise for me. Please.</p>
<p>Find the following limit:</p>
<p><span class="math-container">$\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\right)$</span></p>
<p><span class="math-container">$=\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac... | Z Ahmed | 671,540 | <p>Lastly, use <span class="math-container">$\;k/n\rightarrow x,\;\;1/n \to dx\;,\;$</span> then
<span class="math-container">$$L=\int_{0}^{2} \frac{dx}{1+x}=\ln(1+x)\big|_{0}^{2}=\ln 3\;.$$</span></p>
<p>Edit: the lower limit is <span class="math-container">$x_l=1/n, x_u=2n/n$</span>, when <span class="math-container"... |
70,158 | <p>This question is related to <a href="https://mathoverflow.net/questions/52825/coloring-mathbb-zk"> this one </a> but feels more Ramsey-type, so perhaps it is easier. Let $S$ be a finite set, $|S|=k$. Suppose we color all subsets of $S$ in $1000$ colors. What is the maximal (in terms of $k$) guaranteed length $l=l(k)... | Fedor Petrov | 4,312 | <p>It is much bigger for sure, even if we restrict two subsets of cardinality 2 (call them edges). You need monochromatic path of length $\ell$. Take the color with at least $k(k-1)/2000$ edges of this color, consider only them. Consider the maximal path in our graph. It has length at most $l-1$. Hence it endpoint (bot... |
2,790,910 | <blockquote>
<p>Let <span class="math-container">$X \sim N (0, 1)$</span> and <span class="math-container">$Y ∼ N (0, 1)$</span> be two independent random variables, and define <span class="math-container">$Z = \min(X, Y )$</span>. Prove that <span class="math-container">$Z^2\sim\chi^2(1),$</span> i.e. Chi-Squared with... | Wanshan | 530,208 | <p>$1-F_Z(t) = P(Z>t) = P(X>t)P(Y>t) =\frac{1}{2\pi}\left[ \int_t^{\infty}\exp(-x^2/2) \, dx \right]^2$. Take derivative w.r.t. $t$ and we can get
$$
f_Z(t) = -\frac{d}{dt}\frac{1}{2\pi} \left[ \int_t^\infty \exp(-x^2/2)\,dx \right]^2 = \frac{1}{\pi}\exp(-t^2/2)\left[\int_t^{\infty}\exp(-x^2/2)\,dx\right].
$$
... |
2,666,390 | <p>How do I show that for $x>0$:
$$\frac{x-(x^2+1)\arctan(x)}{x^2(x^2+1)} < 0$$</p>
<p>I tried to do it somehow using the fact that $$\frac{\arctan(x)}{x} < 1$$ but still didn't figure it out...</p>
| TheSimpliFire | 471,884 | <p>We have $$\begin{align}\frac{x-(x^2+1)\arctan x}{x^2(x^2+1)} < 0&\iff x<(x^2+1)\arctan x\\&\iff (1+x^2)\arctan x-x>0\end{align}$$ This is true iff $$\frac d{dx}\left[(1+x^2)\arctan x-x\right]=2x\arctan x+\frac{1+x^2}{1+x^2}-1=2x\arctan x>0$$ which is true since both $x$ and $\arctan x$ are greate... |
3,669,961 | <p>Let <span class="math-container">$f:[a,b] \longrightarrow \mathbb{R}$</span> be a integrable function. I know if <span class="math-container">$ f> 0 $</span> then
<span class="math-container">$$\int_{a}^{b} f(x)\; dx >0.$$</span> The converse is true? That is, if <span class="math-container">$$\int_{a}^{b} f(x... | Saptak Bhattacharya | 734,601 | <p>If you do not restrict <span class="math-container">$f$</span> to be non negative,take <span class="math-container">$f=2\chi_{(1,2]} - \chi_{[0,1)}$</span> on <span class="math-container">$[0,2]$</span>.Otherwise,even if you restrict <span class="math-container">$f$</span> to be non negative,this is not always the c... |
3,669,961 | <p>Let <span class="math-container">$f:[a,b] \longrightarrow \mathbb{R}$</span> be a integrable function. I know if <span class="math-container">$ f> 0 $</span> then
<span class="math-container">$$\int_{a}^{b} f(x)\; dx >0.$$</span> The converse is true? That is, if <span class="math-container">$$\int_{a}^{b} f(x... | Saket Gurjar | 769,080 | <p>What does <span class="math-container">$\int_{a}^{b} f(x)\; dx >0$</span> show?....It says that the area covered by above x-axis is more that area covered by the part below x-axis.</p>
<p>For eg. a function which is always positive will always lie above the x-axis, so the area above x-axis will be always more t... |
2,426,450 | <p>Today when I was solving problems from GRE Manhattan I ran into a strange word problem.</p>
<p><a href="https://i.stack.imgur.com/nqzes.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nqzes.png" alt="enter image description here"></a></p>
<p>The first line of the problem already seems weird sinc... | Especially Lime | 341,019 | <p>You're right. It works out that $105$ students took both, $52.5$ took French but not Geography, $63$ took Geography but not French, and $199.5$ took neither. So the answer intended is presumably that only C is true, but in fact the situation described is impossible.</p>
|
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Mare | 61,949 | <p>Question 1 is a special case of the following statement for finite dimensional algebras:
Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be a finite dimensional algebras over a field <span class="math-container">$K$</span> such that <span class="math-container">$A/rad(A)$</spa... |
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Maxime Ramzi | 102,343 | <p>This is not quite what you're looking for, but here's a theorem (followed by a reference) which justifies the heuristic that the representation theory of a finite group over an algebraically closed field of characteristic <span class="math-container">$0$</span> "doesn't depend on the field" :</p>
<blockquo... |
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Benjamin Steinberg | 15,934 | <p>To summarize the situation given in the other answers (no real new content here) it is classical theory going back to Young that the complex irreducible representations of <span class="math-container">$S_n$</span> can be defined over <span class="math-container">$\mathbb Q$</span> (i.e., written with <span class="ma... |
370,151 | <p>Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ? </p>
| P.. | 39,722 | <p>If $f(f(c))=c$ and $f(c)=r$ then $f(r)=c$.<br>
Since $c\neq r$ it follows that there is an $s$ between $r$ and $c$ such that $f(s)=s$ (in the case $c<r\Rightarrow f(r)-r=c-r>0$ and $f(c)-c=r-c<0$...) $\Rightarrow\Leftarrow$.</p>
|
207,433 | <p>(When one says several things in a question, then several things may get answered and others neglected. Hence this posting overlaps with one of my earlier ones, but (I hope) this one will be short, simple, and narrowly focused.)</p>
<p><b>In what contexts in mathematics, at any level of sophistication, do products... | val11 | 5,856 | <p>In computer science, the notion of <a href="http://en.wikipedia.org/wiki/Polylogarithmic" rel="nofollow">polylogarithmic growth</a> appears naturally when analyzing time/space complexity of algorithms. </p>
<p>There also exists the notion of <a href="http://en.wikipedia.org/wiki/Quasi-polynomial_time#Quasi-polynom... |
4,005,381 | <p>Why the probability of a single element event is "zero" on the continuous model?, the explanation given is based on probability additivity axioms. but how? some more explanation with the source would be helpful.</p>
<p>cool thanks!</p>
| Joe | 623,665 | <p>Suppose that there is a biased random number generator that can produce numbers between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. The probability density function corresponding to this generator is <span class="math-container">$f(x)=2x$</span>:</p>
<p><a href="https://i.sta... |
1,363,967 | <p>A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in X$.</p>
<p>Since the action is transitive $\mathcal{O}_{x}=\{g\cdot x : g \in G,~ x \in X\}=X$ for any $x \in G$. </p... | Matt Samuel | 187,867 | <p>Suppose the normal subgroup $H$ fixes $x$, let $y$ be in the set and let $g$ be such that $y=gx$. Then $gHg^{-1}=H$ fixes $y$, so $H$ fixes every point.</p>
|
3,827,650 | <p>Does the generalised integral</p>
<p><span class="math-container">$\int_{0}^{\pi}\frac{\sqrt x}{\sin x}dx$</span></p>
<p>converge or diverge?</p>
<p>The first thing I would do here is split it into two integrals</p>
<p><span class="math-container">$$\int_0^\pi \frac{\sqrt x}{\sin{x}}dx=\int_0^{\frac{\pi}{2}} \frac{\... | Claude Leibovici | 82,404 | <p>For <span class="math-container">$x$</span> close to <span class="math-container">$0$</span>,
<span class="math-container">$$\frac{\sqrt x}{\sin (x)}=\frac{1}{\sqrt{x}}+\frac{x^{3/2}}{6}+O\left(x^{7/2}\right)$$</span> so no problem.</p>
<p>But close to <span class="math-container">$x=\pi$</span>
<span class="math-co... |
3,117,139 | <p>I am learning to calculate the arc length by reading a textbook, and there is a question</p>
<p><a href="https://i.stack.imgur.com/Zigqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Zigqv.png" alt="enter image description here"></a></p>
<p>However, I get stuck at calculating</p>
<p><span cla... | Claude Leibovici | 82,404 | <p><em>This is just a personal opinion.</em></p>
<p>I must confess that, when I started working on Mathematics Stack Exchange, I have been surprised to see how the "u" substitution was used (and then taught).</p>
<p>When I was young (that is to say long time ago !), the way we were taught was quite different. It was... |
4,552,988 | <p>I want to prove by diagonalization that the set of surjective total computable functions from N to N is not recursively enumerable. I know that the result is trivial using Rice's theorem, but I am trying to prove it only by a direct diagonalization argument. However, supposing that we can enumerate the functions of ... | Z Ahmed | 671,540 | <p>We have <span class="math-container">$f(x)=\int_0^{x} \frac{dt}{\sqrt{1+t^4}}$</span>, So by Lebnitz we have <span class="math-container">$f'(x)=\frac{1}{\sqrt{1+x^4}}$</span></p>
<p>Let <span class="math-container">$y=f(x)\implies x=f^{-1} (y)=g(y).$</span></p>
<p>Start with <span class="math-container">$f(g(x))=x$... |
4,552,988 | <p>I want to prove by diagonalization that the set of surjective total computable functions from N to N is not recursively enumerable. I know that the result is trivial using Rice's theorem, but I am trying to prove it only by a direct diagonalization argument. However, supposing that we can enumerate the functions of ... | Joseph Fox | 1,106,755 | <p>From what you have, <span class="math-container">$g'(0) = \sqrt{1+g^4(0)}$</span>.</p>
<p>Now, <span class="math-container">$g(0) = f^{-1}(0)$</span>, and by definition, <span class="math-container">$f(g(0)) = 0$</span>. Note that <span class="math-container">$f(x) = 0$</span> only when <span class="math-container">... |
199,889 | <p>What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$</p>
| ACL | 10,696 | <p>I would first recommend the paper of Antoine Ducros (<a href="http://webusers.imj-prg.fr/%7Eantoine.ducros/asterisque.pdf" rel="noreferrer">Espaces analytiques <span class="math-container">$p$</span>-adiques au sens de Berkovich</a>, Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with appl... |
1,873,825 | <p>I can see why linear regression is linear, i.e., because it is represented by a line, but what does regression have to do with the term as a whole? </p>
<p>What is the meaning this word contributes to the term?</p>
| V. Vancak | 230,329 | <p>Semi-intuitive explanation: </p>
<p>Assume that we are interested in one's IQ. Not in a score that he might get in some IQ test, rather in his true IQ value. So, we have to assume that there exists such a value. Lets denote it by $\mu$. However, it is impossible to measure it directly. As such, we can use IQ tests ... |
1,833,495 | <p>I tried to solve this like this.</p>
<p>$x=1,y=1$ is solution.</p>
<p>And Let $x=a y=b\, (a\geq 1,b\geq2)$</p>
<p>Then, $11$ can divide $11^b = 10^a+1$</p>
<p>so $10^a = 10 \pmod{11}$ but order of $11(10) = 2$.</p>
<p>Then there is contradiction.</p>
<p>Can I solve this problem while I make $x=a y=b\, (a\geq2 ... | Ghartal | 83,884 | <p>Let's calculate evaluation of $5$ for RHS, we have$$x=v_5(11^y-1)=v_5(y)+v_5(11-1)=v_5(y)+1$$which gives $v_5(y)=x-1$ and we can write $y=5^{x-1}z$, where $\gcd(z, 5)=1$ is a positive integer, but for $x>1$ $$11^{5^{x-1}z}-1>11^{x}-1>10^x$$which is a contradiction. Hence $x=1$, which leads to $y=1$.</p>
|
123,102 | <p>As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both transitive and interpolative? (A relation $\rho$ is interpolative when $x\rho y\implies((\exists z)\ x\rho z \wedge z\rho y).$... | Benjamin Steinberg | 15,934 | <p>The answer seems to be in Butler, K. K.-H.,The number of idempotents in (0,1)-matrix semigroups, Linear Algebra and Its Applications 5 (1972), 233–246. I will see if I have access to the journal and will tell you more.</p>
<p><strong>Edit.</strong> The paper is <a href="http://ac.els-cdn.com/0024379572900055/1-s2... |
4,213,335 | <p>Consider this equation where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers.</p>
<p><span class="math-container">$$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$</span></p>
<p>This equation has the trivial solution <span class="math-container">$k=1, a=1, b=1$</span>.</... | Gottfried Helms | 1,714 | <p>Looking at old entries in my literature-database, I found an interesting limiting formula for a lower bound of <span class="math-container">$2^{a+b}-3^b$</span>. Some short tinkering with it seem to show, that you can prove your conjecture for all <span class="math-container">$a+b > 27$</span> with it.<br />
The ... |
2,588,271 | <blockquote>
<p>Show that $\mathcal{L}\{t^{1/2}\}=\sqrt{\pi}/(2s^{3/2}), \: s>0$</p>
</blockquote>
<p>By the definition of Laplace transform we get:</p>
<p>$$\mathcal{L}\{t^{1/2}\} = \int_0^\infty t^{1/2}e^{-st} \, dt = \{x = \sqrt{st} \} = \dfrac{2}{s^{3/2}} \int_0^\infty e^{-x^2} x^2 \, dx. $$</p>
<p>A known ... | Crescendo | 390,385 | <p>Start off with the substitution $u=x^2$ such that$$I=\frac 12\int\limits_0^{\infty}du\,\sqrt ue^{-u}$$The latter integral is just simply the factorial function.$$n!=\int\limits_0^{\infty}dt\,t^ne^{-t}$$So$$I=\frac 12\left(\frac 12\right)!=\frac {\sqrt{\pi}}4$$</p>
|
474,260 | <p>We know that irrational number has not periodic digits of finite number as rational number.<br>
All this means that we can find out which digit exist in any position of rational number.<br>
But what about non-rational or irrational numbers?<br>
For example:<br>
How to find out which digit exists in Fortieth position... | Hagen von Eitzen | 39,174 | <p>Let $\alpha$ be an irrational number. As long as there exists an algorithm the can decide whether $\alpha>q$ or $\alpha<q$ for any given rational $q$, you can obtain arbitrarily good rational approximations for $\alpha$. Especially, you can find upper and lower bounds good enough to uniquely determine any desi... |
474,260 | <p>We know that irrational number has not periodic digits of finite number as rational number.<br>
All this means that we can find out which digit exist in any position of rational number.<br>
But what about non-rational or irrational numbers?<br>
For example:<br>
How to find out which digit exists in Fortieth position... | name | 407,177 | <p>In general, no.</p>
<p>Suppose that for every irrational number $r$ there were an algorithm that takes a natural $n$ as input and returns the $n$-th digit of $r$.
The possible algorithms are countable, all the irrationals are not, hence it is not possible to have such algorithms for every irrational.</p>
<p>Howeve... |
2,764,586 | <p>Let $S$ be the set of all complex numbers $z$ satisfying the rule $$|z-i|=\sqrt{2}|\bar{z}+1|$$</p>
<p>Show that $S$ contains points on a circle.</p>
<p>My attempt,</p>
<p>By substituting $z = x + yi$, and squaring both sides. But I can't get the circle form. </p>
| Hypergeometricx | 168,053 | <p>Let $z=x+iy$. Note that $\overline{z}=x-iy$.</p>
<p>$$\begin{align}
|z-i|&=\sqrt{2}|\overline{z}+1|\\
|z-i|^2&=2|\overline{z}+1|^2\\
|x+i(y-1)|^2&=2|(x+1)-iy|^2\\
x^2+(y-1)^2&=2(x+1)^2+y^2)\\
x^2+4x+2+y^2+2y-1&=0\\
(x+2)^2+(y+1)^2&=2^2
\end{align}$$</p>
|
1,903,495 | <p>Suppose,$y_1=y_1(x_1,x_2)$ and $y_2=y_2(x_1,x_2)$,
such that,</p>
<p>$$dy_1=\frac{\partial y_1}{\partial x_1}dx_1+\frac{\partial y_1}{\partial x_2} \, dx_2$$</p>
<p>$$dy_2=\frac{\partial y_2}{\partial x_1}dx_1+\frac{\partial y_2}{\partial x_1} \, dx_2$$</p>
<p>Then,I've taken the product of the above two,but unab... | applyb | 170,197 | <p>(a) $C(25,5)$</p>
<p>(b) $C(15,5)$</p>
<p>(c) Three balls must be red which is equal to $C(15,3)$ and two balls must be white which is equal to $C(10,2)$. So, in total we have $C(15,3)C(10,2)$</p>
<p>(d) Here we must select samples with four red balls (and one white) and five red balls (and zero white). Apply t... |
1,972,079 | <p>I would like to split splines of DXF files to lines and arcs in 2D for a graphic editor. From DXF file, I have extracted the following data:</p>
<ul>
<li>degree of spline curve</li>
<li>number of knots and knot vectors</li>
<li>number of control points and their coordinates</li>
<li>number of fit points and their c... | fang | 180,792 | <p>Biarc fitting is to find two tangential connected arcs or one line and one arc that meet the given two end points and two end tangents. You can use it as a core algorithm to approximate a spline with lines and arcs (that are connected with G1 continuity). The algorithm would be something like this:</p>
<ol>
<li>Com... |
190,905 | <p>I want to make two general graphics something like the following.</p>
<p><a href="https://i.stack.imgur.com/1SNYt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1SNYt.png" alt="enter image description here"></a></p>
<p>I want to illustrate the approach to the function by means of a polynomial. ... | zhk | 8,538 | <p>Something like this?</p>
<pre><code> Plot[{Sin[x] + 2, Sin[2 x] + 2}, {x, 0, 8}, AxesLabel -> {x, y},
Ticks -> {{{1, "x[0]"}, {2, "x[1]"}, {3, ""}, {4, "x[3]"}, {5,
""}, {6, ""}, {7, "x[n]"}}, None},
PlotLabels -> Placed[{"y=f(x)", "y=p(x)"}, {Scaled[5], Above}],
AxesStyle -> Arrowheads[{... |
829,433 | <p><img src="https://i.stack.imgur.com/FgnTN.jpg" alt="enter image description here"></p>
<p>why is it when I get the average of the column "per hour", it's different from the total?</p>
| RandomUser | 142,278 | <p>Let's normalize some of the numbers. For cases, I'll divide by $1140$. For hours, I'll divide by $103.01$.</p>
<pre><code>ASC CASES HOURS
Feeder/Invoices 3.26 1.82
Manual 2.59 2.94
Audit/Processes 2.26 7.63
Other 1.00 1.00
</code></pre>
<p>As you can see, ... |
1,892,367 | <p>In Euclidean geometry two polygons are said to be similar if, by rotation and scaling, one can be transformed into the other, and vice versa. If we consider a general metric space, does this notion exist? I can think of how this works by scaling: you consider $d(a_i , a_{I+1} )$, where $a_i$ are the vertices of the ... | Eric Wofsey | 86,856 | <p>There are two different ways you can set this up in the language of metric spaces. If $X$ and $Y$ are metric spaces, say that a bijection $f:X\to Y$ is a <em>similarity</em> if there exists a constant $r>0$ such that $d(f(x),f(y))=rd(x,y)$ for all $x,y\in X$. There are then two definitions we could make:</p>
<... |
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