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 |
|---|---|---|---|---|
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | Community | -1 | <p>If there's a specific addition operation one wants to block, one can just apply a custom function instead, for example</p>
<pre><code>{x, y} ~plus~ v
(* plus[{x, y}, v] *)
</code></pre>
<p>and replace it with <code>Plus</code> when one is ready:</p>
<pre><code>{x, y} ~plus~ v /. v -> {vx, vy} /. plus -> Plu... |
2,072,059 | <blockquote>
<p>Let $x$ be an element of the group $G$. Prove that $x^2 = 1$ <em>iff</em> $|x|$ is either $1$ or $2$</p>
</blockquote>
<p>Now I know that $G$ is implicitly a group under multiplication, therefore $G = (A, \ \cdot \ )$, for some underlying set $A$.</p>
<p>The problem I'm facing is that $A$ could be a... | Eric Wofsey | 86,856 | <p>In the context of this problem, $|x|$ presumably means the order of $x$ as an element of $G$, not the absolute value of $x$. That is, $|x|$ is the least positive integer $n$ such that $x^n=1$ (where $x^n$ means $x$ multiplied with itself $n$ times using the group operation of $G$, and $1$ means the identity element... |
951 | <p>I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?</p>
| Kevin H. Lin | 263 | <p>For Moebius transformations, check out <a href="http://www.youtube.com/watch?v=JX3VmDgiFnY">this nice YouTube video</a>.</p>
|
951 | <p>I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?</p>
| Isaac | 72 | <p>The graphs in the middle of <a href="http://mathworld.wolfram.com/ConformalMapping.html" rel="nofollow">the MathWorld page on Conformal Mapping</a> show examples of the first method in Michael Lugo's answer as well as something somewhat similar to the second method in that answer.</p>
|
3,443,082 | <p>Find all the positive integral solutions of, <span class="math-container">$\tan^{-1}x+\cos^{-1}\dfrac{y}{\sqrt{y^2+1}}=\sin^{-1}\dfrac{3}{\sqrt{10}}$</span></p>
<p>Assuming <span class="math-container">$x\ge1,y\ge1$</span> as we have to find positive integral solutions of <span class="math-container">$(x,y)$</span>... | Certainly not a dog | 691,550 | <p>Your method seems decent. Here I propose another way to ensure the integer-ness of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<p>Note from a rough sketch and limits that the range of <span class="math-container">$f:\mathbb{R^+} \mapsto X,~f(y)= {3y-1\over y+3}$</span> i... |
209,869 | <p>I am very interested in the maximum number of triangles could a connected graph with $n$ vertices and $m$ edges have. For example, if $m\leq n−1$, this number is $0$, if $m=n$, this number is $1$, if $m=n+1$, this number is $2$, and if $m=n+2$, this number is $4$. </p>
| Igor Rivin | 11,142 | <p>This question (together with massive generalizations) is answered in <a href="http://arxiv.org/abs/math/0111106" rel="nofollow">I. Rivin's 2001 paper.</a></p>
|
3,700,938 | <p>I know that the equations are equivalent by doing the math with the same value for x, but I don't understand the rules for changing orders or operations.<br>
When it is not the first addition or subtraction happening in the equation, parentheses make the addition subtraction and vice versa? Are there any other rules... | J. W. Tanner | 615,567 | <p>Think of subtracting <span class="math-container">$(x+1)$</span> as adding <span class="math-container">$-1\cdot(x+1)$</span>. </p>
<p>Then apply the distributive property, so it's adding <span class="math-container">$-1\cdot x + -1\cdot 1=-x-1$</span>.</p>
<p>When you add <span class="math-container">$(x-x)$</sp... |
243,336 | <p>This question was asked to me in an interview, I still cannot think of its solution. Can anyone help? Following is the question:</p>
<blockquote>
<p>Given an infinite number of ropes of length $R$, you have to calculate the
minimum number of cuts required to make $N$ ropes of length $C$. You can
append ropes ... | Hagen von Eitzen | 39,174 | <p>Wlog. $R=1$.
If $C$ is an integer, no cuts are required.
Otherwise let $c=C-\lfloor C\rfloor$, a real number between $0$ and $1$.
Each rope must finally contain at least one cut end, thus the number of cuts is at least $\frac N2$ and it is easily solvable with $N$ cuts.</p>
<p>Indeed, $\lceil \frac N2 \rceil$ is en... |
1,750,592 | <p>Can anyone help with this? I got a wrong answer. </p>
<p>Problem: Joe's French poodle, FooFoo, is tied to the corner of the barn which measures 40 x 30. FooFoo's rope is 50 long. In terms of π, over how many square feet can FooFoo wander?</p>
<p><a href="https://i.stack.imgur.com/Nmrc4.jpg" rel="nofollow noreferr... | Ross Millikan | 1,827 | <p>Hint: If there weren't a building in the way, it would be a circle of radius $50$. If FooFoo is left of the right wall of the building and above the bottom wall, the rope will be flat against the right wall, reducing its effective length. You should get an area that is parts of three circles. If you show your re... |
729,054 | <p>Let $f$ be continuous and $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.</p>
<p>Suppose $|f(x)-x| \leq 2$ holds for all $x$. Is $f$ surjective?</p>
| Marc van Leeuwen | 18,880 | <p>You can do this by Brouwer's fixed-point theorem. Let $a\in\Bbb R^2$ be arbitrary, and define $g:\Bbb R^2\to\Bbb R^2$ by $g(x)=x-f(x)+a$. By hypothesis, $g$ is continuous and has image contained in the disk of radius$~2$ and center$~a$. Restricting to that disk, $g$ has a fixed point $p$ in it by mentioned theorem. ... |
207,266 | <p>I am looking for the integrability condition of the following system of pde:</p>
<p>$$\partial_{[\nu}\Gamma^\kappa_{\mu]\lambda}+\Gamma^\kappa_{[\nu|\rho|}\Gamma^\rho_{\mu]\lambda}=\frac{1}{2}R_{\mu\nu\lambda}{}^{\kappa},\,\,\,\,\,\,\,\,\,(1)$$</p>
<p>given sufficiently smooth functions $R_{\mu\nu\lambda}{}^{\kapp... | Zurab Silagadze | 32,389 | <p>It seems a partial answer (the solution is found with some additional restrictions)
is given in the paper <a href="http://link.springer.com/article/10.1007%2FBF00759087" rel="nofollow">http://link.springer.com/article/10.1007%2FBF00759087</a> (Determination of the metric from the curvature, by Hernando Quevedo)</p>
... |
207,266 | <p>I am looking for the integrability condition of the following system of pde:</p>
<p>$$\partial_{[\nu}\Gamma^\kappa_{\mu]\lambda}+\Gamma^\kappa_{[\nu|\rho|}\Gamma^\rho_{\mu]\lambda}=\frac{1}{2}R_{\mu\nu\lambda}{}^{\kappa},\,\,\,\,\,\,\,\,\,(1)$$</p>
<p>given sufficiently smooth functions $R_{\mu\nu\lambda}{}^{\kapp... | Robert Bryant | 13,972 | <p>Part of the difficulty in providing an answer to your question is the fact that the expression "the integrability condition" is a somewhat vague notion, and it's used in slightly different senses in different contexts. </p>
<p>The usual, somewhat imprecise, sense is that, for a given system of PDE, its 'integrabil... |
3,415,266 | <p>I cannot find how or why this,</p>
<p><span class="math-container">$$5\sin(3x)−1 = 3$$</span></p>
<p>Has one of two solutions being this,</p>
<p><span class="math-container">$$42.29 + n \times 120^\circ$$</span></p>
<p>I am lost on how to get this solution. I have found the other one, so I will not mention it.</... | Quanto | 686,284 | <p>Note that <span class="math-container">$\sin 3x =\sin (180^\circ - 3x)= \frac45=\sin53.1^\circ$</span>, which gives two sets of solutions, </p>
<p><span class="math-container">$$3x=53.1^\circ + n\>360^\circ$$</span></p>
<p><span class="math-container">$$3x=(180-53.1)^\circ + n\>360^\circ$$</span></p>
<p>wit... |
2,017,993 | <p>Is $0$ an eigenvalue for a compact normal operator?</p>
<p>Many texts mention that compact normal operators have a complete orthonormal basis of eigenvectors. If they do, then what about the kernel of the operator? The elements in the kernel, may not be eigenvectors.</p>
<p>Where is the mistake in my understanding... | Fred | 380,717 | <p>If $X$ is a normed space and $\dim X < \infty$, then each linear operator $K:X \to X$ is compact. Thus $0$ may or may not be an eigenvalue.</p>
|
2,017,993 | <p>Is $0$ an eigenvalue for a compact normal operator?</p>
<p>Many texts mention that compact normal operators have a complete orthonormal basis of eigenvectors. If they do, then what about the kernel of the operator? The elements in the kernel, may not be eigenvectors.</p>
<p>Where is the mistake in my understanding... | Gyu Eun Lee | 52,450 | <p>A compact normal operator need not have zero as an eigenvalue. But nothing rules that possibility out.</p>
<p>For easy examples, we have the operator <span class="math-container">$A$</span> represented by the following matrix in the standard basis on <span class="math-container">$\mathbb{R}^2$</span>:
<span class="... |
643,851 | <p>Let the ODE
$$\frac{dy}{dx}=\frac{y+x-2}{y+x-4}$$</p>
<p>I got the general (implicit) solution:
$$y=\ln|x+y-3|+x+A$$ A is arbitrary constant.</p>
<p>My question is:
is $3=y+x$ a solution of this ODE? I know it's not contained in the general solution.</p>
| LecSka | 122,425 | <p>Let $n\geq 100$ an even number. Consider the quantities $n-91$, $n-93$ and $n-95$, one of these is a multiple of 3, and not exactly 3 cause $100-95>3$, then is a composite odd number.
Observing thet 91,93 and 95 are composite, you conclude that every $n\geq 100$ works. Now check directly the remaining numbers, an... |
3,999,652 | <p>Let triangle <span class="math-container">$ABC$</span> is an equilateral triangle. Triangle <span class="math-container">$DEF$</span> is also an equilateral triangle and it is inscribed in triangle <span class="math-container">$ABC \left(D\in BC,E\in AC,F\in AB\right)$</span>. Find <span class="math-container">$\cos... | heropup | 118,193 | <p>Note that there are two distinct solutions given the conditions in the problem. This is because there are two admissible orientations of <span class="math-container">$\triangle DEF$</span>: one in which <span class="math-container">$AF > BF$</span>, and one in which <span class="math-container">$AF < BF$</sp... |
3,999,652 | <p>Let triangle <span class="math-container">$ABC$</span> is an equilateral triangle. Triangle <span class="math-container">$DEF$</span> is also an equilateral triangle and it is inscribed in triangle <span class="math-container">$ABC \left(D\in BC,E\in AC,F\in AB\right)$</span>. Find <span class="math-container">$\cos... | Saeed | 858,459 | <p>Here I will show how you can draw the figure. The calculations that follow are somewhat similar to other answers.
<a href="https://i.stack.imgur.com/5ciiq.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5ciiq.jpg" alt="enter image description here" /></a></p>
<p>We can build triangle <span class="... |
584,975 | <p>I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did:</p>
<p><a href="http://puu.sh/5w3XQ.png" rel="nofollow">http://puu.sh/5w3XQ.png</a></p>
<p>What I'm wondering is about the 2^4 - 2^3. I know this is is 2^3 because 2^3 is the... | hhsaffar | 104,929 | <p><em>Hint</em>:</p>
<p>Consider $b_n=a_n-\frac{1}{2}$</p>
<p>Can you write a recursive equation for $b_n$?</p>
|
584,975 | <p>I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did:</p>
<p><a href="http://puu.sh/5w3XQ.png" rel="nofollow">http://puu.sh/5w3XQ.png</a></p>
<p>What I'm wondering is about the 2^4 - 2^3. I know this is is 2^3 because 2^3 is the... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
1,507,076 | <p>I'm supposed to give a constructive proof of the following claim, but to get some hints on what to do I tried to prove the inverse and see where the proof breaks down. I have the following.</p>
<p>Claim: Define $X * Y = \{ xy \mid x \in X, y \in Y\}$. There is a ring $R$ with ideal $U$ such that $R / U$ is not clos... | Rob Arthan | 23,171 | <p>You are being asked to produce an example, not a proof: take $R = \Bbb{Z}$ and $U = 2\Bbb{Z}$. Then $U \in R/U$, but $U*U = 4\Bbb{Z} \not\in R/U$.</p>
|
420,294 | <p>While reading Bill Thurston's <a href="http://www.ams.org/publications/journals/notices/201601/rnoti-p31.pdf" rel="noreferrer">obituary</a> in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):</p>
<blockquote>
<p>Bill’s enthusiasm during the early stages of mathematical discovery was ... | Lee Mosher | 20,787 | <p>I wrote that quote, and I'll take the <a href="https://mathoverflow.net/questions/420294/group-theory-with-grep#comment1079616_420294">hint of @SamNead</a> and try to write an answer, although the best I can do is to write a somewhat speculative extension of the story behind the quote, laced with some mathematical m... |
1,382,479 | <p>I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. </p>
<p>Please keep in mind that I have little background in math, and I am trying to understand these theorems to understand the math... | Samir Khan | 183,661 | <p>The statement that $a^p\equiv a\mod p$ is the same as $a^{p-1}\equiv 1\mod p$ when $a$ and $p$ are relatively prime, because in this case we can divide both sides of the congruence by $a$, and obtain one from the other. </p>
<p>Euler's theorem says that </p>
<p>$$a^{\phi(m)}\equiv 1\mod m,$$</p>
<p>where $\phi(m)... |
358,184 | <p>We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
<span class="math-container">$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[X]) \geq c, 0<\theta < 1
$$</span></p>
<p>However, It would be great to know if t... | Iosif Pinelis | 36,721 | <p>Let <span class="math-container">$Y:=X-EX$</span>. We need to obtain a lower bound on <span class="math-container">$P(Y>0)$</span>. </p>
<p>Suppose that <span class="math-container">$-a\le Y\le b$</span> for some real <span class="math-container">$a>0$</span> and <span class="math-container">$b>0$</span>, ... |
634,929 | <p>How can I evaluate this integral?</p>
<p>$$
\int{x^{3}\,{\rm d}x \over \left(x - 1\right)^{2}\sqrt{x^{2} + 2x + 4}}$$</p>
<p>I would be grateful for any tips.</p>
| Mike | 17,976 | <p>Let me provide an alternate method of solving the last 2 integrals from lab bhattacharjee's answer. The first step is to rewrite</p>
<p>$$\sqrt3\sin\psi-2\cos\psi\text{ as }A\cos(\psi+B)=A\cos\psi\cos B-A\sin\psi\sin B$$</p>
<p>So we have</p>
<p>$$A\cos B=-2,A\sin B=-\sqrt3$$
$$\tan B=\frac{\sqrt3}2,B=\tan^{-1... |
1,842,365 | <p>I recently got acquainted with a theorem:</p>
<p>If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$.</p>
<p>I am having a difficulty in understanding this theorem. Does this theorem mean that $f(ax+b)=f(ax+b+ \dfrac{P}{a})$? </p>
<p>If the above mean... | Arthur Sinulis | 269,465 | <p>Just as another remark: a neat corollary which follows from this statement and the strong law of large numbers is that whenever $X_i$ is an iid sequence of r.v. such that the SLLN holds (e.g. $X_i \in L^1$ by Kolmogorov's SLLN), then $\mathbb{E}(X_1 \mid S_n) \to \mathbb{E}(X_1)$ $\mathbb{P}$-almost surely.</p>
|
2,178,428 | <p><strong>Goal</strong>: I need help to intuitively understand division by a fraction. </p>
<p><strong>Background</strong>: I've read <a href="https://math.stackexchange.com/questions/248385/what-is-the-physical-explanation-of-a-division-by-a-fraction">this</a> and <a href="http://www.mathsisfun.com/division.html" re... | Robert Israel | 8,508 | <p>The <strong>definition</strong> of division is: $ a \div b$ is $c$ where $a = b \times c$.
In the case of positive integers $b$ you can interpret this as even distribution of an amount $a$ into $b$ equal parts. This is because if $b$ is a positive integer, $b \times c$ is the result of adding together $b$ copies o... |
2,178,428 | <p><strong>Goal</strong>: I need help to intuitively understand division by a fraction. </p>
<p><strong>Background</strong>: I've read <a href="https://math.stackexchange.com/questions/248385/what-is-the-physical-explanation-of-a-division-by-a-fraction">this</a> and <a href="http://www.mathsisfun.com/division.html" re... | Timothy | 137,739 | <p>I don't have a math research job. I don't know everything with certainty. This is just my guesswork that the following is a simplifying approximation of reality.</p>
<p>There's no universal law that a binary operation whose steps include only multiplication and not division of integers cannot be called division. A ... |
549,159 | <p>How to simplify this:</p>
<p>$$(5-\sqrt{3}) \sqrt{7+\frac{5\sqrt{3}}{2}}$$</p>
<p>Dont know how to minimize to 11.</p>
<p>Thanks in advance!</p>
| DonAntonio | 31,254 | <p>Justify each step in the following:</p>
<p>$$(5-\sqrt3)\sqrt{7+\frac{5\sqrt3}2}=(5-\sqrt3)\frac{\sqrt{14+5\sqrt3}}{\sqrt2}=$$</p>
<p>$$=\frac{22}{\sqrt2}\frac{\sqrt{14+5\sqrt3}}{5+\sqrt3}=\frac{11\sqrt2}{5+\sqrt3}\sqrt{14+5\sqrt3}=$$</p>
<p>$$=\frac{11}{5+\sqrt3}\sqrt{28+10\sqrt3}=11\frac{\sqrt{(5+\sqrt3)^2}}{5+\... |
5,253 | <p>I have an acyclic digraph that I would like to draw in a pleasing way, but I am having trouble finding a suitable algorithm that fits my special case. My problem is that I want to fix the x-coordinate of each vertex (with some vertices having the same x-coordinate), and only vary the y. My aesthetic criteria are (... | David Eppstein | 440 | <p>If the x-coordinates are compatible with the acyclic structure of your DAG (that is, for an edge u->v, the x coordinate of u should always be less than that of v) then this is a standard problem in graph drawing, known as Sugiyama-style layered drawing. (Usually it is the y coordinates that are fixed but that makes ... |
1,189,814 | <p>Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?</p>
<p>It's not difficult to show that there is no such module if the ring is a integral domain. For general, I guess the answer is yes. But I failed to find a desired... | Manos | 11,921 | <p>Here is a proof that no such examples exist over integral domains:</p>
<p>Let $M$ be finitely generated over an integral domain $R$.
Let $x_1,\dots,x_n$ be generators of $M$. Take $0 \neq a_i \in \operatorname{ann}(x_i)$ for
every $i=1,\dots,n$. Define $a=a_1 \cdots a_n$. Since $R$ is an integral
domain, $a\neq 0$... |
2,859,463 | <blockquote>
<p>Prove or disprove. All four vertices of every regular tetrahedron in $ \mathbb{R}^3$ have at least two irrational coordinates.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/hYrWv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hYrWv.png" alt="enter image description here"... | John WK | 551,440 | <p>I think you want to prove that there doesn't exist a regular tetrahedron such that all its vertices have only rational coordinates. Assume the contrary, let $ABCD$ be such a tetrahedron. Consider the plane $(ABC)$ as a complex plane with origin in $A$ and let the points $B$ and $C$ have associated complex numbers $a... |
500,973 | <p>I am presented with the question:</p>
<blockquote>
<p>The photoresist thickness in semiconductor manufacturing has a mean of
10 micrometers and a standard deviation of 1 micrometer. Assume that
the thickness is normally distributed and that the thicknesses of
different wafers are independent.</p>
<p>(a... | copper.hat | 27,978 | <p>Suppose $t_k$ are the wafer thicknesses, each has distribution ${\cal N} (\mu, \sigma^2)$, and the $t_k$ are independent.</p>
<p>Then $t_1+\cdots +t_n$ will then have distribution ${\cal N} (n\mu, n\sigma^2)$, and so the quantity $\frac{t_1+\cdots + t_n-n \mu}{\sqrt{n} \sigma}$ will have distribution ${\cal N}(0,1)... |
3,015,149 | <p>I’m using the formula that the number of conjugacy class is given to be <span class="math-container">$\frac{1}{|G|}\sum|C_{G}(g)|$</span>, where <span class="math-container">$C_{G}(g)=\{h \in G ; gh=hg\}$</span>, which is a special result by Burnside’s theorem.</p>
<p>I found that the number of conjugacy class in <... | Curious student | 415,928 | <p><span class="math-container">$D_n$</span>
In general its class equation given by following </p>
<p>Case 1: n is odd</p>
<p><span class="math-container">$Z(D_n)=$</span>{<span class="math-container">$e$</span>}</p>
<p><span class="math-container">$(n-1)/2$</span> classes of {<span class="math-container">$r^i,r^-i$... |
315,386 | <p>I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in this format -
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Ellipse_Properties_of_Directrix_a... | Narasimham | 95,860 | <p>EDIT1:</p>
<p>What you at first proposed as ellipse looks like:</p>
<p><a href="https://i.stack.imgur.com/eOj2D.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/eOj2D.png" alt="enter image description here" /></a></p>
<p>The Ellipse parametrization is done differently. To more clearly distinguish b... |
4,020,464 | <p>So <span class="math-container">$Ax = λx$</span>;</p>
<p><span class="math-container">$A(Ax) = λ(Ax) \to (A^2)x = (λ^2)x$</span></p>
<p>I kind of dont know how to get the <span class="math-container">$1$</span> and <span class="math-container">$I$</span> here...</p>
<p>Any help is appreciated</p>
| Michael Cohen | 651,310 | <p>Given independent random samples <span class="math-container">$X_{1}, X_{2}, \dots, X_{m}$</span> taken from a normal random variable with mean <span class="math-container">$\mu$</span> and variance <span class="math-container">$\sigma^2$</span>, it is a standard result that the sample mean <span class="math-conta... |
1,639,232 | <p>A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$?</p>
<p>Thank you very much.</p>
| DanielV | 97,045 | <pre><code>Number[] a = new Number[p + 1]; // range from 1 to p
... some stuff to initialize a ...
Number[] ceps = new Number[Q + 1]; // range from 0 to Q
ceps[0] = ln(G);
for (int q = 1; q <= p; q++) {
Number sum = a[q];
for (k = 1; k <= q - 1; k++) {
sum += (k - q) / q * a[k] * ceps[q - k];
}
cep... |
1,746,776 | <p>I am wondering how I could solve the integral </p>
<p>$$\iiint \frac{1-e^{-(x^2+y^2+z^2)}}{[x^2+y^2+z^2]^{2}}$$</p>
<p>over $\mathbb{R}^{3}$</p>
<p>I thought maybe I could break it up into three single integrals and multiply or something. I think it is not supposed to be difficult to solve. How should it be appro... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. By the use of a symetry and by use of spherical coordinates
$$
\begin{align}
x&=r\sin\theta\cos\phi\\
y&=r\sin\theta\sin\phi\\
z&=r\cos\theta
\end{align}
$$ one gets a jacobian equal to $r^2\sin \theta \:d\phi \:d\theta\: dr$ giving
$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\inft... |
3,203,678 | <blockquote>
<p>Find all prime numbers <span class="math-container">$p$</span>, for which there are positive integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> such that <span class="math-container">$p=m^2+n^2$</span> and <span class="math-container">$p \mid m^3+n^3-4$</span>... | W-t-P | 181,098 | <p>There are no primes with this property other than <span class="math-container">$p=2$</span> and <span class="math-container">$p=5$</span>; here is a proof.</p>
<p>Suppose that <span class="math-container">$m,n>1$</span> and <span class="math-container">$p=m^2+n^2$</span> is a prime dividing <span class="math-con... |
1,557,353 | <p>Context: I'm taking calc based physics, and we are supposed to be able to integrate moment of inertia for a cylinder. I referenced a mit vid, and though I have no education on multiple integrals, I got all but one thing. <a href="https://www.youtube.com/watch?v=iYFogDTPlRo" rel="nofollow">https://www.youtube.com/wat... | David | 119,775 | <p>Here is an intuitive (though not rigorous) geometrical argument.</p>
<p>Consider the volume element you get if you start at a point $(r,\theta,z)$ and increase $r$ by $dr$ and increase $\theta$ by $d\theta$ and increase $z$ by $dz$. You get a "curved box" where one side is a line with length $dr$, one is a line wi... |
2,812,314 | <p>For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ with the definition $\operatorname{rad}(1)=1$ (the Wikipedia's article dedicated to this multiplicative function is <a ... | José Carlos Santos | 446,262 | <p>You could note that<span class="math-container">$$\sum_{n=1}^\infty\frac{n-2n\sqrt n+n^2}{n^3}=\sum_{n=1}^\infty\frac{1-2\sqrt n+n}{n^2}$$</span>and that<span class="math-container">$$\lim_{n\to\infty}\frac{\frac{1-2\sqrt n+n}{n^2}}{\frac1n}=\lim_{n\to\infty}\frac{1-2\sqrt n+n}n=1.$$</span>Therefore, your series div... |
720,282 | <p>I was trying to come up with reasons, why we naturally consider the topology of uniform convergence on compact sets as the appropriate framework for spaces of holomorphic functions such as e.g. $H(\mathbb{C}^n)$ (which is the space of entire functions on $\mathbb{C}^n$).</p>
<p>I understand that e.g. by Weierstrass... | fgp | 42,986 | <p>I'd say being closed alone is sufficient reason. Usually the <em>main</em> reason for imbueing a topology onto a space is to be able to reason about convergence. Non-closed spaces are burdensome then, because there will be sequences which <em>look</em> like they converge (for metrizable spaces, think cauchy sequence... |
1,182,432 | <p>Is it possible, that everyone is a pseudo-winner in a tournament with 25 people?
(pseudo-winner means that either he won against everyone, or if he lost against someone, then he beated someone else, who beated the one who he lost to).</p>
<p>In the "language" of graph theory: Is it possible in the directed K(25) (a... | Leader47 | 214,065 | <p>We can prove that this is possible for any odd integer $2n+1$ by induction. </p>
<p>For $n=1$ it is possible. </p>
<p>If it is possible for $n\ge 1$ Let $G$ have $2(n+1)+1$ vertices. Let $u,v$ be some $2$ vertices and the others are $v_1,v_2,...,v_m$ where $m=2n+1$ Connect $v_1,...,v_m$ by induction. so that from ... |
2,010,434 | <p>Say we are given n piles of stones.
Sizes are $s_{1}, s_{2}, .. , s_{n}$, they can be any positive integer numbers.
The game is played by two players, they alternate their moves.
The allowed moves are:
1. Take exactly 1 stone from 1 pile.
2. Take all stones from 1 pile.</p>
<p>Wins the player who mades the last mo... | DLIN | 355,583 | <p>We can consider the positive case, i.e. $a,b,c>0$.</p>
<p>We can categorize into the 3 cases(from simple to copmlex):</p>
<p>1): $a=b=c$. Then $\frac1a+\frac1b+\frac1c=\frac3a\in\mathbb Z$, so $a|3$, hence $a=1,3$. </p>
<p>2): $0<a<b<c$. Then $1\leq\frac1a+\frac1b+\frac1c<\frac3a$, $1\leq a<3$, ... |
28,104 | <p>It occured to me that the Sieve of Eratosthenes eventually generates the same prime numbers, independently of the ones chosen at the beginning. (We generally start with 2, but we could chose any finite set of integers >= 2, and it would still end up generating the same primes, after a "recalibrating" phase).</p>
<p... | Theo Johnson-Freyd | 78 | <p>So, let's see if I can precisify your claim. We start with a finite "seed set" that we assert to be Ghi-Om-prime (the seed set must not contain 1). Numbers smaller than the largest seed we completely ignore. Now for numbers larger than any seed prime, we run the Seive. You claim that there is some cut-off, depen... |
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Ethan Bolker | 72,858 | <p>Not an answer - essentially a comment and too long for a comment that I don't want lost in the flurry of existing comments.</p>
<p>Many students try L'Hopital unthinkingly when faced with the limit of an indeterminate form like $0/0$. Often the application is incorrect. Even when it works it's often not the easiest... |
3,700,814 | <p>Consider the vector space <span class="math-container">$V=C(\mathbb{R},\mathbb{R})$</span> and <span class="math-container">$V\ni U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$</span>. I want to find a complement of <span class="math-container">$U$</span>, such that <span class="math-container">$V=U\oplus W$</span>. T... | Rick | 601,449 | <p>The issue is that the condition <span class="math-container">$V=U\oplus W$</span> requires <span class="math-container">$V$</span> and <span class="math-container">$W$</span> to be <em>vector subspaces</em>, as oposed to just <em>sets</em> as you say; in particular they have to be closed under vector addition. Havi... |
2,784,784 | <p>Let $f(x)=$$x-1 |x \in \mathbb{Q} \brace 5-x| x \in \mathbb{Q}^c$</p>
<p>Show that $\lim_{x \to a}f(x)$ does not exists for any $a \not= 3$</p>
<p>I first showed that $lim_{x \to 3}f(x)=2$. </p>
<p>I don't know how to approach this part. Can anyone please guide? I was thinking of using density theorem at first t... | Fred | 380,717 | <p>Let $a \ne 3$.</p>
<p>If $(x_n)$ is a sequence in $ \mathbb Q$ with $x_n \to a$, then $f(x_n)=x_n-1 \to a-1$.</p>
<p>If $(y_n)$ is a sequence in $ \mathbb Q^c$ with $y_n \to a$, then $f(x_n)=5-y_n \to 5-a$.</p>
<p>Since $a \ne 3$ we have $a-1 \ne 5-a$, hence $\lim_{x \to a}f(x)$ does not exist.</p>
|
2,495,555 | <p>I want to use Konig's theorem to show that the pictured graph $G$ has no perfect matching. By this theorem it suffices to find a vertex cover of size $|G|/2-1= 20$, but so far I have only been able to find vertex covers of size 21. I'm just doing this by inspection as opposed to using any algorithms, and it's not im... | Misha Lavrov | 383,078 | <p>If you don't want to just stumble around blindly until you get there, here's how you find the solution.</p>
<p>The solution you found with $21$ vertices probably looks something like this:</p>
<p><a href="https://i.stack.imgur.com/DNNc5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DNNc5.png" alt="Naiv... |
1,016,884 | <p>Four friends, Andrew, Bob, Chris and David, all have different heights. The sum of their heights is 670 cm.
Andrew is 8cm taller than Chris and Bob is 4cm shorter than David.
The sum of the heights of the tallest and shortest of the friends is 2cm more than the sum of the heights of the other two.
Find the height of... | Jimmy R. | 128,037 | <p>You are given that $$\begin{cases}A=C+8\\B=D-4\\A+B+C+D=670\\A\neq B\neq C\neq D\end{cases}$$ Substituting the first two in the third you obtain $$C+D=333$$ therefore
$$\begin{cases}A=C+8\\B=329-C\\D=333-C\\A\neq B\neq C\neq D\end{cases}$$</p>
<p>Now compare the sums by pairs to check which pairs satisfy the condit... |
2,570,008 | <p>For what value of a will this equation have only one real root: </p>
<p>$$(2a−5)x^2−2(a−1)x+3=0$$</p>
<p>Note: $x$ is a variable</p>
<p>If found that $a=4$ works, but there seems to be another solution. Any help?</p>
| Andrei | 331,661 | <p>You can have only one solution if the discriminant is $0$, or if the coefficient of the $x^2$ term is $0$. $a=5/2$</p>
|
3,748,392 | <blockquote>
<p>Let <span class="math-container">$G=\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$</span> with <span class="math-container">$|\ker\rho| = 2$</span>. How many <span class="math-container">$3$</span>-Sylow subgroups are there in <span class="math-container">$G$</span>?</p>
</blockquote>
<p>I know that the nu... | markvs | 454,915 | <p>If the number of Sylow <span class="math-container">$3$</span>-subgroups is <span class="math-container">$1$</span>, then the Sylow <span class="math-container">$3$</span>-subgroup is normal and trivially intersects the normal Sylow <span class="math-container">$7$</span>-subgroup, whence it commutes with the Sylow ... |
3,254,290 | <p>Consider two random variables <span class="math-container">$X,Y$</span>. <span class="math-container">$Y$</span> has support <span class="math-container">$\mathcal{Y}\equiv \mathcal{Y}_1\cup \mathcal{Y}_2$</span>. <span class="math-container">$X$</span> has support <span class="math-container">$\mathcal{X}$</span>. ... | Gautam Shenoy | 35,983 | <p>Hint: When faced with a random variable <span class="math-container">$X$</span> and an event <span class="math-container">$A$</span>, the following may help:
<span class="math-container">$$E[X|A] = \frac{E[X1_{A}]}{P(A)}$$</span>
where <span class="math-container">$1_{()}$</span> is the indicator function.
In your c... |
405,648 | <p>Is there a sensible characterization of groups <span class="math-container">$G$</span> with the following property?</p>
<blockquote>
<p>Every extension of groups <span class="math-container">$1\to G\to H\to K\to 1$</span> is split.</p>
</blockquote>
<p>A complete group <span class="math-container">$G$</span> has tha... | Brauer Suzuki | 332,108 | <p>Since my late comment to YCor's answer is easily overlooked, I allow myself to repeat it here: The question was answered in [J. S. Rose, Splitting properties of group extensions, Proc. London Math. Soc. (3) 22 (1971), 1–23] with exactly the same outcome as in YCor's answer.</p>
|
2,936,608 | <blockquote>
<p>Let <span class="math-container">$a_n$</span> where <span class="math-container">$n \in \mathbb {N}$</span> be a sequence of rational numbers converging to <span class="math-container">$a$</span>. Suppose <span class="math-container">$a \neq 0$</span>, for <span class="math-container">$k = 1, 2, ...$<... | TheSilverDoe | 594,484 | <p>It is not necessary to use Cauchy sequences.</p>
<p>Let <span class="math-container">$\varepsilon = \frac{|a|}{2} > 0$</span> (because <span class="math-container">$a \neq 0)$</span>. The sequence <span class="math-container">$(a_k)$</span> converges to <span class="math-container">$a$</span>, so there exists <s... |
3,855,736 | <p>Mathematics is not my primary discipline, but I know enough about both it and academics in general to know that many to most mathematical researchers do what they do because they enjoy doing it. This would seem to make "recreational mathematics" a rhetorical tautology, yet the term is used as if it were a ... | J.G. | 56,861 | <p>The history of mathematics is replete with examples of something novel being done to address a problem at the time. It might have been "applied" in the sense of helping with a technological innovation, or it might have been "pure", e.g. mathematicians trying to make sense of something unrigorous ... |
3,003,672 | <p>Say I have an infinte 2D grid (ex. a procedurally generated world) and I want to get a unique number for each integer coordinate pair. How would I accomplish this?</p>
<p>My idea is to use a square spiral, but I cant find a way to make a formula for the unique number other than an algorythm that just goes in a squa... | Ross Millikan | 1,827 | <p>You need the Cantor <a href="https://en.wikipedia.org/wiki/Pairing_function" rel="nofollow noreferrer">pairing function</a>, tuned up to accept integers instead of naturals. The basic function takes a pair of naturals (including zero) <span class="math-container">$x,y$</span> and returns a natural <span class="math... |
4,128,510 | <p>Let <span class="math-container">$(V, ||.||)$</span> be a normed space and define <span class="math-container">$d(x,y) = ||x|| + ||y|| $</span> if <span class="math-container">$ x \neq y$</span> and as <span class="math-container">$0$</span> if <span class="math-container">$x=y$</span>. Describe all convergent seque... | alphaomega | 775,794 | <p><span class="math-container">$d(x_n, x)\to 0$</span> iff <span class="math-container">$\|x_n\| +\|x\|\to 0$</span> iff <span class="math-container">$\|x_n\|\to 0 \ \& \ \|x\|\to 0$</span> iff <span class="math-container">$x_n\overset{\tiny\| \cdot \|}{\to} 0 \ \& \ x=0$</span></p>
|
4,128,510 | <p>Let <span class="math-container">$(V, ||.||)$</span> be a normed space and define <span class="math-container">$d(x,y) = ||x|| + ||y|| $</span> if <span class="math-container">$ x \neq y$</span> and as <span class="math-container">$0$</span> if <span class="math-container">$x=y$</span>. Describe all convergent seque... | Alessandro | 671,329 | <p>Let <span class="math-container">$x_n \to x$</span>, by definition: <span class="math-container">$\forall \varepsilon > 0 \ \exists \ N \in \mathbb{N}: \ n \geqslant N \Rightarrow ||x||+||x_n||=d(x_n,x)<\varepsilon.$</span></p>
<p>Note that <span class="math-container">$|| x || \leqslant ||x|| + ||x_n||< \v... |
4,128,510 | <p>Let <span class="math-container">$(V, ||.||)$</span> be a normed space and define <span class="math-container">$d(x,y) = ||x|| + ||y|| $</span> if <span class="math-container">$ x \neq y$</span> and as <span class="math-container">$0$</span> if <span class="math-container">$x=y$</span>. Describe all convergent seque... | Henno Brandsma | 4,280 | <p>If <span class="math-container">$x_n \to x$</span> in <span class="math-container">$d$</span>, there are two cases:</p>
<p><span class="math-container">$x=0$</span>. Then <span class="math-container">$d(x_n,x)= \|x_n\| + \|x\| = \|x_n\|\to 0$</span> and so <span class="math-container">$x_n \to x$</span> under <span ... |
1,585,772 | <p>I am finding this problem confusing :</p>
<blockquote>
<p>If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$.</p>
</blockquote>
<p>When $x=1$ I have that $f(1)=f(2a)$ using the first identity.</p>
<p>Then when $x=2a$ I have by the second identity that $f(2a+2)=27f(2a)$,after that I simple stare at ... | Steven Stadnicki | 785 | <p>In fact, you barely even need the second relation. Hint: first set $f(2a)=c$. Now you know that $f(x)$ is an exponential function, $f(x)=c^x$ (and the second relation implies that $c\neq 1$ — this is all it does; 27 could as easily be 0.27 or $10^{27}$ and it wouldn't change the answer); this function is one... |
327,291 | <p>I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: <i>Malcev, A. I. On a class of homogeneous spaces. Amer. Math. Soc. Translation 1951, (1951). no. 39, 33 pp.</i> (<a href="https://mathscine... | Ben McKay | 13,268 | <p><a href="http://www.mathnet.ru/links/f4bbcad8dff13663fdc1e07b7d350dd1/im3161.pdf" rel="nofollow noreferrer">The Russian original</a>:</p>
<p>You can use <a href="https://www.deepl.com/en/translator" rel="nofollow noreferrer">DeepL</a> to translate it, for free.
Here is a DeepL translation of the title and the abstra... |
654,198 | <p>$6x^3 -11x^2 + 6x + 5 \equiv (Ax-1)(Bx - 1)(x - 1) + c$</p>
<p>Find the value of A, B and C.</p>
<p>I started it like this: </p>
<p>$6x^3 -11x^2 + 6x + 5 \equiv (Ax-1)(Bx - 1)(x - 1) + c$</p>
<p>Solving the right hand side:</p>
<p>$ (ABx^2 - Ax - Bx + 1)(x - 1) + C$</p>
<p>$ ABx^3 - ABx^2 - Ax^2 + Ax - Bx^2 + ... | Devgeet Patel | 116,131 | <p>Equating x coefficients you will get A+B=5.You already have AB=6.Solving you will get A=2 or 3.This will be the easier approach.</p>
|
1,503,958 | <p>In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?</p>
<p>I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!</p>
| Ángel Mario Gallegos | 67,622 | <p><strong>Hint:</strong></p>
<p>Notice that $$x+3-4\sqrt{x-1}=x-1-4\sqrt{x-1}+4=(\sqrt{x-1}-2)^2$$</p>
<p>and</p>
<p>$$x+8-6\sqrt{x-1}=x-1-6\sqrt{x-1}+9=(\sqrt{x-1}-3)^2=(3-\sqrt{x-1})^2$$</p>
<p>After, you can try by cases.</p>
|
1,503,958 | <p>In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?</p>
<p>I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!</p>
| Zach466920 | 219,489 | <p><a href="https://i.stack.imgur.com/qVT0Y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qVT0Y.png" alt="enter image description here"></a></p>
<p>I found that to be odd. So the solutions are $x \in [5,10]$. I guess it wouldn't be too hard to formally prove that. </p>
<blockquote>
<p><em>Hint:... |
1,503,958 | <p>In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?</p>
<p>I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!</p>
| Ennar | 122,131 | <p>\begin{align}
&\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}} = 1\\ \implies &\sqrt{(x-1)-4\sqrt{x-1} + 4}+\sqrt{(x-1)-6\sqrt{x-1}+9}=1\\ \implies &\sqrt{(\sqrt{x-1}-2)^2}+\sqrt{(\sqrt{x-1}-3)^2}=1\\ \implies &|\sqrt{x-1}-2| + |\sqrt{x-1}-3| = 1\tag{1}
\end{align}</p>
<p>This calls for casework:</p>
... |
746,750 | <p>This is a consequence of the exponential rule, but how do I actually prove it to be true?</p>
| Ant | 66,711 | <p>How do you define $e^x$?</p>
<p>The most common way to define it is</p>
<p>$$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$</p>
<p>If you take this definition, then if $x > 0$, $e^x$ is the sum of an infinite number of positive terms. </p>
<p>One should show that the series converges, but if you take that for g... |
1,179,195 | <p>Good day everyone. </p>
<p>I need to know automata theory. Can you advice me the best way to study math?
What themes will I need to know to understand automata theory. What a sequence of study? What level will I need to study intermediate themes? Maybe can you say something yet, what can help me quickly learn autom... | John Gowers | 26,267 | <p>You seem to be a bit confused about how to think of a contraction as a line on a graph. Let's assume that we are looking for a contraction on $[0,1]$, though we could use any complete subset of the reals (or even the real line itself.</p>
<p>Intuitively, we have:</p>
<ul>
<li>The line $y=x$ plotted on $[0,1]$ (so... |
123,587 | <p>I always wonder how many inbuilt functions does Mathematica have (of course you can google for it) and how they are connected with each other! So I tried this (v10.1).</p>
<pre><code>SetDirectory[$InstallationDirectory<>"/Documentation/English/System/ReferencePages/Symbols"]
comms = FileNames[];
ncomms = Leng... | Szabolcs | 12 | <p>Something like this:</p>
<pre><code>g = SimpleGraph@Graph[
Catenate[
Thread /@
Normal@DeleteMissing@
WolframLanguageData[WolframLanguageData[], "RelatedSymbols", "EntityAssociation"]
],
DirectedEdges -> False
];
</code></pre>
<p>It's slow, like most new <code>*Data</code> fu... |
123,587 | <p>I always wonder how many inbuilt functions does Mathematica have (of course you can google for it) and how they are connected with each other! So I tried this (v10.1).</p>
<pre><code>SetDirectory[$InstallationDirectory<>"/Documentation/English/System/ReferencePages/Symbols"]
comms = FileNames[];
ncomms = Leng... | BoLe | 6,555 | <pre><code>(* docs on my system (10.3, Windows) *)
base = FileNameJoin[{$InstallationDirectory, "Documentation",
"English", "System", "ReferencePages", "Symbols"}];
FileNames[FileNameJoin[{base, "*"}]] // Length
</code></pre>
<blockquote>
<p>4811</p>
</blockquote>
<pre><code>(* symbols in See Also section *)
a... |
1,349,654 | <p>Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$</p>
<p>Note :By mathematica,the result is :
$\frac{Gamma\left(\frac1 4\right)Gamma\left(\frac5 4\right)}{\sqrt{\pi}}-\sqrt{2} Hypergeometric2F1\left(\frac1 4,\frac3 4,\frac5 4,\frac1 4\right).$
and ... | Claude Leibovici | 82,404 | <p><em>This is not an answer but it is too long for a comment.</em></p>
<p>As I wrote in comment, there is something wrong somewhere since $$\int\frac{\sqrt{\tan x}}{\sin x}dx=-2 \sqrt{\cos (x)} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\cos
^2(x)\right)$$ and $$I=\int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{... |
387,202 | <p>What is the smallest 3-regular graph to have a unique perfect matching?</p>
<p>With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in this question <a href="https://mathoverflow.net/questions/98385/cubic-graphs-without-a-perfect-matching-and-a-ve... | Matthieu Latapy | 158,328 | <p>Regarding your second <em>even better</em> question, I warmly suggest the <a href="https://users.cecs.anu.edu.au/%7Ebdm/data/" rel="nofollow noreferrer">Brendan McKay page on combinatorial objects</a>, that gives many kinds of graph examples.</p>
|
55,965 | <p>I'm a games programmer with an interest in the following areas:</p>
<ul>
<li>Calculus</li>
<li>Matrices</li>
<li>Graph theory</li>
<li>Probability theory</li>
<li>Combinatorics</li>
<li>Statistics</li>
<li>More linguistic related fields of logic such as natural language processing, generative grammars</li>
</ul>
<... | Community | -1 | <ul>
<li><span class="math-container">$\textbf{Theorem.}$</span> Let <span class="math-container">$|G| = 2^{n} \cdot m$</span> where <span class="math-container">$2 \nmid m$</span>. If <span class="math-container">$G$</span> has a cyclic <span class="math-container">$2$</span>- Sylow subgroup, then <span class="math-co... |
3,019,506 | <p>I am stuck on this problem during my review for my stats test. </p>
<p>I know I have to use the convolution formula, and I understand that:</p>
<p><span class="math-container">$f_{U_1}(U_1) = 1$</span> for <span class="math-container">$0≤U_1≤1$</span> </p>
<p><span class="math-container">$f_{U_2}(U_2) = 1$</span... | Zeno Rogue | 265,219 | <p>You would need a surface of negative curvature.</p>
<p><a href="https://i.stack.imgur.com/m7Erq.png" rel="noreferrer"><img src="https://i.stack.imgur.com/m7Erq.png" alt="illustration of a hexagon in the hyperbolic plane"></a></p>
<p>It is best to use a hyperbolic plane for this, where you can easily fit any regula... |
93,099 | <p>Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?</p>
| kaleidoscop | 16,934 | <p>For any convex set $K$ in dimension $d$ with volume $V(K)$, it is aymptotically $$V(K)-\frac{T(K)}{(d+1)^{d-1}(d-1)!}n^{-1}ln(n)^{d-1}+O(n^{-1}ln(n)^{d-2}ln(ln(n)))$$ (see {New perspectives in stochastic geometry} by W. Kendall and I. Molchanov, p. 49) where $T(K)$ is the number of "flags", i.e. of sequences $f_0\su... |
1,998,391 | <p>To define a <strong>real</strong> exponential function
$$f(x)=a^x=e^{x \,\mathrm{lg} a}$$</p>
<p>It is strictly necessary that $a>0$.</p>
<p>But is the same true if the exponent is a <strong>natural</strong> number?</p>
<hr>
<p>In function series and moreover in power series I see things like
$$\sum_{n\geq0} ... | user90369 | 332,823 | <p>Of course if you use $e^{x\ln a}$ instead of $a^x$ then $a>0$ because otherwise the exponent $\ln a$ is not real any more. </p>
<p>(1) It's $e^{x\ln a}=a^x$ for $a>0$ ;</p>
<p>(2) it's only $a^x$ (and not $e^{x\ln a}$) valid for $a\in\mathbb{R}\setminus\{0\}$ with $x\in\mathbb{Z}$. </p>
<p>$e^{x\ln a}$ and ... |
2,262,661 | <p>The question is in how many ways can we select 20 different items from the empty set?</p>
<h3>ans:</h3>
<p>Obviously in 0 ways since the empty set has no items. I mean, this seem obvious, but maybe there is a trick to this question.</p>
| JMJ | 295,405 | <p>If you wanted to sound more mathematical you could make an induction argument:</p>
<p><em>There are exactly zero ways to select $n>0$ items from the empty set.</em></p>
<p><em>Proof:</em> Take $n =1$. Since the empty set has no members it is impossible to select a member of the empty set, thus zero ways to sele... |
1,476,982 | <p>I'm trying to understand why the volume of a parallelepiped whos sides are $s,u,w$ is $ V = s \cdot(u \times w)$.</p>
<p>Even the units of measurement don't add up. The length of the vectors $s,u,w$ is measured in centimeters, the volume is measured in cubic cm.</p>
<p>$u\times w$ is a vector. It is a vector that ... | vadim123 | 73,324 | <p>I think the apparent paradox in this question derives from the mistaken belief that a vector has associated units of length. This is incorrect. In three-dimensional space, the vector $(1,2,3)$ could be thought of as having units of ordered triples of centimeters (not the same as centimeters cubed). We could also ... |
4,173 | <p>I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead -- </p>
<p>Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?</p>
<p>$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$</p>
<p>Where $\mathbf{v}$ is a $d\times ... | Community | -1 | <p>One way is to find the householder matrix Q that maps v to a multiple of e_1 (first coordinate basis vector). Then (since Q is symmtric and orthogonal) w_2=Q*e_2 ... will be a basis of the orthogonal complement of v.
Explicitly, I get
w_k = e_k - u
where u = (v+sqrt(d)*e_1)/(d+sqrt(d))</p>
|
2,109,197 | <p><strong>Update:</strong><br>
(because of the length of the question, I put an update at the top)<br>
I appreciate recommendations regarding the alternative proofs. However, the main emphasis of my question is about the correctness of the reasoning in the 8th case of the provided proof (with a diagram).</p>
<p><stro... | Community | -1 | <p>We know that $z_1,z_2,z_3$ are collinear iff there exists some $t\in\mathbb{R}$ such that $$t(z_1-z_3)=z_1-z_2.$$
The problem becomes trivial if $z_1=z_3$ so we can suppose this is not the case, and then it is legal to write
$$
t=\frac{z_1-z_2}{z_1-z_3}.
$$
Multiplying both sides of this equation by $|z_1-z_3|^2=(\... |
2,679,153 | <p>Let $A\in\mathbb{R}^{n\times n}$ be a generic <em>lower triangular</em> matrix and let $P\in\mathbb{R}^{n\times n}$ be a symmetric <em>positive definite</em> matrix.</p>
<blockquote>
<p><strong>True or false.</strong> Does $AP + PA^\top=0$ imply $AP=0$? </p>
</blockquote>
| user | 505,767 | <p>Let $a^2+b^2=c^2$ then</p>
<ul>
<li>$\frac12 ab = 51.2$</li>
<li>$\frac{a}b=\frac{1}{1.6}$</li>
</ul>
<p>$$\implies b=1.6 a \implies0.8a^2=51.2\implies a=8 \quad b=12.8 \quad c=\sqrt{8^2+12.8^2}$$</p>
|
1,025,671 | <p>When we have something in this form</p>
<p>$$\sqrt{x + a} = \sqrt{y + b},$$</p>
<p>a common technique to solve is to square both side so that:</p>
<p>$$(\sqrt{x + a})^2 = (\sqrt{y + b})^2 \implies x + a = y + b.$$</p>
<p>I'm an engineer and not a mathematician. As I understand it engineers do lots of things that... | Ojas | 154,392 | <p>This is true because square root is not a Many$\to$1 function. i.e., 2 numbers can't possibly have the same square root. Thus, if square root of 2 numbers is the same, it means that the 2 numbers must be equal.</p>
|
866,808 | <p>In my lecture notes:</p>
<p>Let $m,n\in \mathbb{N}$ be relatively prime. The fundamental theorem of arithmetic implies that each divisor of $mn$ is the product of two unique positive relatively prime integers $d_1|m$ and $d_2|n$.</p>
<p>Please could someone help me understand how this is implied? I have no idea</p... | Adam Hughes | 58,831 | <p>Simply write</p>
<blockquote>
<p>$$m=\prod_{i=1}^rp_i^{e_i},\quad n=\prod_{j=1}^sq_j^{f_j}$$</p>
</blockquote>
<p>Since they are coprime, no $p_i=q_j$.</p>
<p>A divisor of $mn$ is determined by taking each $p_i$ to some power $0\le n_{p_i}\le e_i$ and $q_j$ to power $0\le n_{q_j}\le f_j$. Grouping the primes $\... |
16,733 | <p>This is a variant on the question <a href="https://matheducators.stackexchange.com/questions/14492/small-real-numbers">small real numbers</a>.</p>
<p>I have a disagreement with someone about the meaning of "bigger" real numbers.</p>
<p>Say we have the real number <span class="math-container">$-1.$</span> Is <span ... | Tommi | 2,083 | <p>Since "bigger" and "smaller" are ambiguous, it is best to avoid them, as you mention. The methods you mention seem reasonable, though I am not native English speaker.</p>
<p>Someone may be able to refer to a credible source or official standard, but regardless, you can only know how someone else understands the ter... |
1,030,274 | <p>$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?</p>
| user26857 | 121,097 | <blockquote>
<p>The ideal $I=(2,X)$ of $\mathbb Z[X]$ is not a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules.</p>
</blockquote>
<p>Let's suppose that $I=(2,X)$ is a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules. Then there exists a family $(N_{\alpha})_{\alpha\in A}$ of (non-zero) cyclic submodules... |
267,051 | <p>Games appear in pure mathematics, for example, <a href="https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game" rel="noreferrer">Ehrenfeucht–Fraïssé game</a> (in mathematical logic) and <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game" rel="noreferrer">Banach–Mazur game</a> (in topo... | Michael Greinecker | 35,357 | <p>The purity of math is in the eye of the beholder, but maybe some of the following examples qualify:</p>
<p>There is a proof of Kolmogorov's strong law of large numbers in the book <a href="http://www.probabilityandfinance.com/" rel="noreferrer">Probability and Finance</a> by Shafer and Vovk that uses the determinac... |
52,802 | <p>This is partly a programming and partly a combinatorics question.</p>
<p>I'm working in a language that unfortunately doesn't support array structures. I've run into a problem where I need to sort my variables in increasing order.</p>
<p>Since the language has functions for the minimum and maximum of two inputs (b... | Gerry Myerson | 8,269 | <p>Patrick's got it. Here's the same, in a somewhat different presentation. </p>
<p>The symmetric difference of two sets is the stuff that's in exactly one of the two sets. If the symmetric difference of $A$ and $B$ is $A$, then $B$ can't have any elements that are in $A$ (since those elements of $A$ wouldn't be in th... |
1,315,199 | <p>From the wikipedia article on sine waves:</p>
<blockquote>
<p>The sine wave is important in physics because it retains its wave
shape when added to another sine wave of the same frequency and
arbitrary phase and magnitude. It is the only periodic waveform that
has this property. This property leads to its i... | Kishan | 247,422 | <p>Well you already gave the answer that when you add 2 sinusoids you get a sinusoid again.</p>
<p>You thought you gave a counter example by saying when 2 square wave get added it produces a square wave. But what you missed out is that a square wave can be represented by a summation of many sinusoids, which is the Fou... |
1,225,359 | <p>Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is:</p>
<p>What I tried to do was make pairs of two terms and the expand it as a binomial expression and then again expand the binomial in the resulting series which gave me an expression with lot of un... | RE60K | 67,609 | <blockquote>
<p>Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is:</p>
</blockquote>
<p>Sum of terms with no y : $3^6$ (y=0 rest all 1)<br>
Sum of terms with no y and no x: $2^6$ (x,y=0 rest all 1)<br>
Sum of terms with no y but x: $3^6-2^6=665$ (sub... |
1,683,414 | <p>$$\begin{cases}
x^2 = yz + 1 \\
y^2 = xz + 2 \\
z^2 = xy + 4
\end{cases}
$$</p>
<p>How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got $(x - y)^2 + (y - z)^2 + (x - z)^2 = 14$, but it leads to nowhere.</p>
| André Nicolas | 6,312 | <p>Multiplying both sides of the first equation by $z$, of the second by $x$, and of the third by $y$, we get
$$x^2z=z^2y+z, \quad y^2x=x^2z+2x,\quad z^2y=y^2x+4y.$$
Adding up and cancelling, we get<br>
$$2x+4y+z=0.$$
Similarly,
$$x^2y=y^2z+y, \quad y^2z=z^2x+2z, \quad z^2x=x^2y+4x,$$
giving<br>
$$4x+y+2z=0.$$ </p>
... |
2,482,868 | <p>I am trying to find</p>
<p>$$ \lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}],$$</p>
<p>where $p>0$. I have tried to factor out as</p>
<p>$$(1+x^{p+1})^{\frac1{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} =x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... | Jack D'Aurizio | 44,121 | <p>For any $x>0$ and $\alpha>0$ we clearly have $(1+x^\alpha)^{\frac{1}{\alpha}}\geq x$. If $\alpha=p\in\mathbb{N}^+$,</p>
<p>$$ (1+x^p)^{\frac{1}{p}}=\text{GM}\left[\underbrace{x,\ldots,x}_{p-1\text{ times}},x+x^{1-p}\right]\leq \text{AM}\left[\underbrace{x,\ldots,x}_{p-1\text{ times}},x+x^{1-p}\right]=x+\frac{... |
685,642 | <p>I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.</p>
<p>I don't understand though how to generalize for the entries on th... | Ben Grossmann | 81,360 | <p>In order to finish constructing your basis, you could add the set of matrices for which the $(1,1)$ entry is $1$, the $(i,i)$ entry is $-1$ for some $i \neq 1$, and all other entries are zero.</p>
<p>Note that the space of $n\times n$ matrices with trace $0$ is $n^2 - 1$ dimensional, so you should have this many el... |
685,642 | <p>I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.</p>
<p>I don't understand though how to generalize for the entries on th... | Yulia Alexandr | 337,050 | <p>Since you have to find the dimension of the subspace of all matrices whose trace is $0$, having a linear transformation T: $M(n×n)→ ℝ$, all it really comes down to is finding the size of <em>ker(T)</em>. </p>
<p>In order to do so, notice that the standard matrix for the given transformation will have the dimension ... |
184,940 | <p>Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is embedded into space.</p>
<blockquote>
<p><strong>Question:</strong></p>
<p>is it also possible, to calculate the euclidean ... | Thomas Richard | 8,887 | <p>The answer is no.
The fact that you can bend without stretching a piece of paper should convince you of that. Consider for instance a piece of a plane and piece of a cylinder.</p>
<p>The question you ask can be rephrased in the following way: take a 2 dimensional Riemannian manifold $(M^2,g)$. Under which condition... |
1,437,073 | <p>Ok guys, I have to solve this ODE</p>
<p>$$
\frac{d^2y}{dx^2}=f(x), \quad
x>0,\quad y\left(0\right) = 0, \quad
\left.\frac{dy}{dx}\right\lvert_{x=0}=0
$$
The solution I should get is in the form of
$$y\left(x\right)=\int_0^x k\left(t\right)\, dt $$
Moreover, I should tell what the function $\,k\left(t\right... | Blazej | 155,834 | <p>You are given a form that will do the trick. The thing you need to do now is to differentiate it twice. By comparing result with your equation you will get condition on function $k$ that will make it work. This might involve double integration but you can try to simplify it by changing the order of integrals. This w... |
509,928 | <p>There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake.</p>
<p>I want to divide the cake into 2 non-empty rectangular pieces, by either a horizontal or a vertical cut, such that the nu... | san | 229,191 | <p>The lower bound $r=\lfloor \frac N4\rfloor$ is also an upper bound: </p>
<p>Assume that in a square with toppings one must cut at least $r>1$ toppings with every cut.
Number the edges of the square with $E_1,E_2,E_3,E_4$ clockwise, and set $\varepsilon_i=min\{d(E_i,R_j)\ :\ d(E_i,R_j)>0\}$,
where $R_j$ is any... |
463,619 | <p>let us consider following problem:</p>
<p>Roger sold a watch at a profit of $10$%. If he had bought it at $10\%$ less and sold it for $13$ dollar less,then he would have made a profit of $15$%. What is the cost price of the watch?</p>
<p>suppose that price of watch is $x$ dollar, $profit=sell -cost $</p>
<p>so ... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let Roger has bought the watch at $100x$ dollar</p>
<p>So, the current selling price $=110x$ dollar</p>
<p>If he had bought it at $10$% less, the buying price would have been $=90x$ dollar</p>
<p>If he had sold it for $13$ dollar less, selling price $=110x-13$ dollar</p>
<p>So, $$90x \left(1+\frac{... |
4,154,205 | <p>I study maths as a hobby. I am stuck on this question:</p>
<p>Find the values of c for which the line <span class="math-container">$2x-3y = c$</span> is a tangent to the curve <span class="math-container">$x^2+2y^2=2$</span> and find the equation of the line joining the points of contact.</p>
<p>I have established t... | Eric Wofsey | 86,856 | <blockquote>
<p>I can really only understand this through examples, though.</p>
</blockquote>
<p>There's nothing wrong with this! "Without loss of generality" is an imprecise term and can be used in a wide variety of contexts, so there's no fixed rules on exactly how it can be used or what exactly it means. ... |
3,892,583 | <p>I'm trying to find an example of a function <span class="math-container">$f: A \to B$</span> and <span class="math-container">$X \subset A$</span> so that <span class="math-container">$f^{-1}(f(X)) \ne X$</span>, and similarly where <span class="math-container">$Y \subset B$</span> so that <span class="math-containe... | peter.petrov | 116,591 | <p>You should apply the substitution: <span class="math-container">$x = t^{14}$</span></p>
<p>This will get you a rational function of <span class="math-container">$t$</span>.</p>
<p>And then... as we know all rational functions can be integrated,<br />
there is a well-known procedure for that.</p>
|
3,892,583 | <p>I'm trying to find an example of a function <span class="math-container">$f: A \to B$</span> and <span class="math-container">$X \subset A$</span> so that <span class="math-container">$f^{-1}(f(X)) \ne X$</span>, and similarly where <span class="math-container">$Y \subset B$</span> so that <span class="math-containe... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>As <span class="math-container">$[2,7,14]=14,$</span></p>
<p>Let <span class="math-container">$x^{1/14}=y, x=y^{14}, dx=14y^{13}dy$</span></p>
<p><span class="math-container">$$\int\dfrac{y^2+y^7}{y^{16}+y}\cdot14y^{13}dy$$</span></p>
<p><span class="math-container">$$=14\int\dfrac{y^{15}(1+y^5)}{y(1+y^... |
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