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 |
|---|---|---|---|---|
47,890 | <p>hey so i'm programming something that finds an angle of a line between 0 and 180 degrees based on two points.... </p>
<p>the equation to find the answer is <code>Angle = sin-1((1/Hypotenuse)*B)</code>where B is the vertical side of the triangle formed and the hypotenuse is the distance between point 1 and 2. </p>
... | Nick Dixon | 12,462 | <p>You would calculate the answer in radians, and then convert to degrees. Inside of the inverse sin should just be (1/hypotenuse*B) since its a ratio of side lengths.</p>
<p>You're overthinking it, I'm guessing.</p>
|
3,691,692 | <p>Find all real values of a such that <span class="math-container">$x^2+(a+i)x-5i=0$</span> has at least one real solution. </p>
<p><span class="math-container">$$x^2+(a+i)x-5i=0$$</span></p>
<p>I have tried two ways of solving this and cannot seem to find a real solution.</p>
<p>First if I just solve for <span cla... | PythonSage | 596,065 | <p>Use AM-GM. <span class="math-container">$\frac{ad + bc}{2} \ge \sqrt{abcd}$</span>. Squaring both sides, you get the answer. A tiny tip: If everything is positive, and you have an inequality, think about AM-GM once at least. </p>
|
3,631,903 | <p>A Calculus A level trigonometry problem:</p>
<blockquote>
<p>Solve <span class="math-container">$\tan x = \dfrac{p}{q}$</span> where <span class="math-container">$p,q\in\mathbb{Z}$</span> such that <span class="math-container">$$3\cos x\ - 4\sin x = -5$$</span></p>
</blockquote>
<p>I tried moving terms to one s... | Zarrax | 3,035 | <p>For <span class="math-container">$x < -1$</span>, you have <span class="math-container">$|x + 1| = x + 1$</span> and for <span class="math-container">$x \geq -1$</span> you have <span class="math-container">$|x + 1| = -x - 1$</span>. So if you split into <span class="math-container">$x < -1$</span> and <span c... |
3,743,743 | <p>I have this condition:</p>
<p><strong>(A is true OR B is true OR C is true) OR (A is false AND B is false AND C is false)</strong></p>
<p><em>(edit: It's been pointed out that this formula is wrong for what I want)</em></p>
<p>So as the title says, I want the condition to be true if only 1 of A, B or C is true, or i... | Robert Israel | 8,508 | <p>If the change in <span class="math-container">$\mu$</span> is small, you might use the series</p>
<p><span class="math-container">$$ (B+\lambda I)^{-1} = (B+\mu I)^{-1} - (\lambda - \mu) (B + \mu I)^{-2} + (\lambda - \mu)^2 (B + \mu I)^{-3} - \ldots $$</span></p>
|
3,743,743 | <p>I have this condition:</p>
<p><strong>(A is true OR B is true OR C is true) OR (A is false AND B is false AND C is false)</strong></p>
<p><em>(edit: It's been pointed out that this formula is wrong for what I want)</em></p>
<p>So as the title says, I want the condition to be true if only 1 of A, B or C is true, or i... | jsievers | 805,998 | <p>Do you need to have the matrix in explicit form, or are you happy for it to be factored? If the latter, then note that adding a constant to the diagonal of a matrix leaves the eigenvectors unchanged and shifts all the eigenvalues by that constant. That's easy to see since if A<em>x</em>=<span class="math-container... |
2,079,822 | <p>I am asked to find the maximum velocity of a mass. </p>
<p>I know that the equation for maximum acceleration is </p>
<p>$$a = w^2A$$</p>
<p>However I do not know how to find the maximum velocity. Is velocity just the same as acceleration? </p>
| Anthony Storace | 403,229 | <p>acceleration depends on the applied force and the mass</p>
<p>acceleration = force/mass</p>
<p>for uniform acceleration velocity = acceleration X time</p>
<p>if acceleration varies you must integrate to find
velocity = integral of acceleration over time</p>
|
1,893,540 | <p>I've been asked to prove the following,
if $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$.
I tried proof by contrapositive, but I keep having trouble choosing the right $ε$. Can you guys help me out? </p>
| Paramanand Singh | 72,031 | <p>First of all, understand that most of the problems dealing with $\epsilon, \delta$ are an exercise in inequalities and use very very elementary ideas. The difficulty of such problems arises mainly because of use Greek letters and partially because students don't really understand inequalities as much as they underst... |
167,946 | <p>I seek to replace derivatives like <code>D[u[x, y], x, x]</code> which are evaluated as $u^{(2,0)}[x,y]$ by variables with names like <code>uxx</code>. Derivatives that I work with are denoted by their "order-vector", for example <code>{2,0}</code> is the vector for this particular derivative and <code>{1,1}</code> ... | J. M.'s persistent exhaustion | 50 | <p>Here is one method:</p>
<pre><code>derivativeToSymbol[expr_] /; Nest[Head, expr, 3] === Derivative := Block[{args, tmp},
args = List @@ expr; tmp = Head[expr];
Symbol[(ToString @@ tmp) <>
MapThread[ConstantArray, {ToString /@ args, List @@ Head[tmp]}]]]
</code></pre>
<p>... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Behrouz Maleki | 343,616 | <p>Set $y=\tan u$ we have
$$\int \frac{1-y^2}{(1+y^2)^2} dy=\int\frac{1-\tan^2u}{1+\tan^2 u}du=\int \cos 2u\,du=\frac{1}{2}\sin 2u+C= \frac{\tan u}{1+\tan^2 u}+C$$</p>
|
2,913,974 | <p>In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism
$$
\coprod_{i} \text{Hom} \left( A, X_{i} \right) \longrightarrow \text{Hom} \left( A, \coprod_{i} X_{i} \right)
$$
is a bijection. Suppose $A \oplus B$ is compact. ... | Hetong Xu | 749,743 | <p>In what follows, I shall add some detail to user14972's answer from the perspective of retract. Some more details and conventions are introduced in my question (post) <a href="https://math.stackexchange.com/questions/4155082/complete-compactness-and-retract">Complete compactness and retract</a> .</p>
<p>Let <span cl... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Kuba | 5,478 | <blockquote>
<h1>Status Partially Completed</h1>
</blockquote>
<h2>1. Invalid number of arguments for some of built-in functions e.g.</h2>
<ul>
<li><p><code>Which</code> called with an odd number of arguments</p>
</li>
<li><p><code>Switch</code> called with an even number of arguments</p>
</li>
<li><p><code>Module/With... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Jason B. | 9,490 | <p>These are really common mistakes that go unnoticed because they are valid syntax:</p>
<ul>
<li><code>Which</code> statement using <code>_</code> as the final predicate</li>
<li><code>Switch</code> statement using <code>True</code> as the final pattern</li>
<li>A list with a random dot product in the middle, usually... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Szabolcs | 12 | <blockquote>
<h1>Status Partially Completed</h1>
</blockquote>
<h2>Unused variables</h2>
<pre><code>fun[x_, y_] := x^2
^ y is unused
</code></pre>
<pre><code>Module[{x, y}, <-- y is unused
x+1
]
</code></pre>
<pre><code>y=1; <-- this instance of y is effectively unused
y=2;
</code></pre>
<... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | b3m2a1 | 38,205 | <p>Things like </p>
<pre><code>f[a_Association]:=...
</code></pre>
<p>Should ask if you want to convert to </p>
<pre><code>f[a_Association?AssociationQ]:=...
</code></pre>
<p>And similarly</p>
<pre><code>f[a_?AssociationQ]:=...
</code></pre>
<p>Should ask if you want to do the same</p>
<p>The longer pattern is t... |
3,977,687 | <p>A coin of radius 1 cm is tossed onto a plane surface that has been tessellated by right triangles whose sides are 8 cm, 15 cm, and 17 cm long. Find the probability that the coin lands within a triangle.</p>
<p>I know that this has to do with similarity because the inner triangle that is formed by the area where the ... | achille hui | 59,379 | <p>You don't need to compute the sides of the new triangle, you only need to know the change in the inradius.</p>
<p>Given any triangle with area <span class="math-container">$\Delta$</span> and perimeter <span class="math-container">$\ell$</span>, its inradius <span class="math-container">$r$</span> equals to <span cl... |
86,755 | <p>I need to solve the following integral equations for a problem I'm working on - </p>
<p>$\displaystyle \frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\displaystyle \frac{-i}{2 \pi}$ $\int_{a}^{\infty} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$</p>
<p>where $\tau, k \in \mat... | Aegon | 30,377 | <p>So upon thinking about this more and playing with different assumptions when trying to integrate the above function(s), what I found was the following - </p>
<p>$\int_{a}^{\infty} dt \frac{e^{i k t}}{t + i \tau} = e^{k\tau}\,\Gamma\left(k(-ia + \tau) \right)$, where $\Gamma$ is the incomplete Gamma function. </p>
... |
1,336,506 | <p>We know that the usual $\leq$ is a partial order relation on the group of integers $\mathbb Z$ and $\mathbb Z$ is a totally ordered with this partial order relation. Is there any other partially order relation exist in $\mathbb Z$ which makes $\mathbb Z$ a partially ordered group (or totally ordered group)? </p>
| lhf | 589 | <p>If $0$ and $1$ are related, then the only possible partial orders that make $\mathbb Z$ into a partially ordered group are the usual order $\le$ and its reverse order $\ge$:</p>
<p>If $0 \prec 1$, then $1 = 0 + 1 \prec 1 + 1 = 2$, and so $n \prec n+1$ for all $n$ by induction.</p>
<p>If $0 \succ 1$, then $n \suc... |
2,300,613 | <p>I tried to calculate few derivatives, but I cant get $f^{(n)}(z)$ from them. Any other way? </p>
<p>$$f(z)=\frac{e^z}{1-z}\text{ at }z_0=0$$</p>
| Jack D'Aurizio | 44,121 | <p>Since, formally,
$$ \frac{1}{1-z}=1+z+z^2+z^3+\ldots $$
the multiplication by $\frac{1}{1-z}$ brings the power series $a_0 + a_1 z+ a_2 z^2 +\ldots $ into the power series $a_0+(a_0+a_1)z+(a_0+a_1+a_2) z^2+\ldots$. It follows that:</p>
<p>$$ \frac{e^{z}}{1-z}=\sum_{n\geq 0}z^n \left(\sum_{j=0}^{n}\frac{1}{j!}\right... |
3,913,244 | <p>My attempt :</p>
<p><span class="math-container">$A=2^3×5^2×7^3$</span></p>
<p>Let's determine number of numbers primes with A, and
Smaller than A</p>
<p><span class="math-container">$\rho (A) = 2^2 ×4×5×6×7^2 =23520$</span></p>
<p>23520 is a number of numbers primes with A and smaller than A</p>
<p><span class="mat... | Calvin Lin | 54,563 | <p><strong>Claim 1:</strong> For a fixed <span class="math-container">$ 1 \leq k \leq p-2$</span>, <span class="math-container">$p \mid \sum_{i=1}^{p-1} i^k$</span>.<br />
This almost corresponds to <span class="math-container">$ F(x) = x^k$</span> in the question.</p>
<p>Proof: Work mod <span class="math-container">... |
598,962 | <p>I have to determine the following:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)$$</p>
<p>I've got so far:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)= \lim_{x \rightarrow 0}\left(\frac{27}{x(x+3)^3}-\frac{1}{x}\right)=\lim_{x \righta... | Tī-Kong n̂g | 114,467 | <p>Just develop the nuumerator and you'll get
$$
\lim_{x \rightarrow 0} \frac{-x^3 - 9x^2 - 27x}{x(x+3)^3}
=\lim_{x \rightarrow 0} \frac{-x^2 - 9x - 27}{(x +3)^3} = -1
$$</p>
|
1,512,549 | <p>As the title says everything.</p>
<p>Thanks,</p>
| Community | -1 | <p>Let $s=\dfrac{S_n}{a_1}$ for conciseness.</p>
<p>You need to solve the polynomial equation</p>
<p>$$s(1-q)=1-q^n,$$
or
$$P_n(q)=q^n-sq+s-1=0.$$</p>
<p>There is no analytical solution for general $n$ and you need to resort to numerical methods. <a href="https://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollo... |
1,512,549 | <p>As the title says everything.</p>
<p>Thanks,</p>
| Lutz Lehmann | 115,115 | <p>Since $q=1+r$ with some small value $r$ for the interest rate (assuming no usury) one gets $q-1=r$ and $q^n-1=nr+\frac{n(n-1)}2r^2+\frac{n(n-1)(n-2)}{6}r^3+…$ so that approximately
$$
r_0=\frac{S}{na}-1\approx \frac{n-1}2r·\left(1+\frac{n-2}3r\right)
$$
Multiplying with $\left(1-\frac{n-2}3r\right)$ and ignoring hig... |
572,541 | <blockquote>
<p>Let $L$ be the set of all lines in the plane. Prove that $L$ is uncountable, but only countably many of the lines in $L$ contain more than one rational point.</p>
</blockquote>
<p><strong>Attempt</strong>: Well, I was trying to define $L$ using linear combinations of points since a line is a linear c... | user99680 | 99,680 | <p>You can List all lines in the plane this way:</p>
<p>Use the fact that a line is described uniquely once you know its slope, and one of its
intercepts with the axes. The slope is indexed by the Reals, and so is the intercept, say the x-intercept. So we count all the possible pairs ( slope, x-intercept), and show
it... |
457,977 | <p>I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but this ended up cancelling off the term I wanted. I wasn't sure if there was another contour I should use.... | Ron Gordon | 53,268 | <p>Consider the integral</p>
<p>$$\oint_C dz \frac{\log^2{z}}{(1+z)^3}$$</p>
<p>where $C$ is a keyhole contour in the complex plane, about the positive real axis. This contour integral may be seen to vanish along the outer and inner circular contours about the origin, so the contour integral is simply equal to</p>
... |
996,052 | <p>A disk of radius <span class="math-container">$5$</span> cm has density <span class="math-container">$10$</span> g/cm<span class="math-container">$^2$</span> at its center, density <span class="math-container">$0$</span> at its edge, and its density is a linear function of the distance from the center. Find the mass... | gniourf_gniourf | 51,488 | <p>You computed the integral:
$$\iint_{x\in[0,5],\,y\in[0,2\pi]}(10-2x)\,\mathrm{d}x\,\mathrm{d}y=50\pi\approx157.08.$$
Though your computation is correct, the result is not the total mass of the disc.</p>
<p>For a 2-dimensional object $\Sigma$ of density of mass $D$, the total mass $m$ of $\Sigma$ is
$$m=\iint_\Sigma... |
1,081,021 | <p>In what follows I'm only considering positive real valued functions.</p>
<p>Everywhere I look about the definition of the Lebesgue integral it is required to consider a measurable function. Why do we not define the integral for non-measurable functions? From what I see we require measurablility of the simple functi... | GEdgar | 442 | <p>Also, maybe it is useful to show the bad properties of this lower integral when applied to non-measurable funtions. Take your $\{\varnothing, \mathbb R\}$ example. Let $f = \chi_{[0,1]}$ and $g = 1-f$. Then $\int f = \int g = 0$ but $\int(f+g) = 1$. So even simple linearity fails.</p>
|
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | Quanto | 686,284 | <p><span class="math-container">\begin{align}
\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx
=&\int_0^1 \frac {(1-x^2)\ln x}{x^4+1}\ dx\\
\overset{ibp}=&-\frac1{2\sqrt2}\int_0^1\frac1x
\ln\frac{x^2+\sqrt2 x+1}{x^2-\sqrt2 x+1}dx\\
=&-\frac1{2\sqrt2}\int_0^1\int_{-\pi/4}^{\pi/4}
\frac{2\cos y}{x^2+2x\sin y+1}dy... |
4,609,833 | <p>As far as I can tell using Mathematica, the following identity seems to hold:
<span class="math-container">$$(n+2)^n=(n+1)\sum_{k=0}^n\binom{n}{k}\frac{(k+1)^{k-1}(n-k)^{n-k}}{n+1-k},$$</span>
where we define <span class="math-container">$0^0=1$</span>. However, I am having trouble proving it. I thought that this lo... | Marko Riedel | 44,883 | <p>We seek to prove that</p>
<p><span class="math-container">$$(n+2)^n = (n+1) \sum_{k=0}^n {n\choose k}
\frac{(k+1)^{k-1} (n-k)^{n-k}}{n+1-k}.$$</span></p>
<p>Observe that the RHS is</p>
<p><span class="math-container">$$\sum_{k=0}^n {n+1\choose k}
(k+1)^{k-1} (n-k)^{n-k}
\\ = (n+2)^n +
\sum_{k=0}^{n+1} {n+1\choose... |
4,350,695 | <p>My book was introducing the concept of integrals and wrote this:</p>
<p><span class="math-container">$$\text{Area under the curve of $f(x)$}=\lim_{\Delta x\to0}\sum_{n=1}^{N}f(x)\Delta x\tag{1}$$</span></p>
<p>My problem with <span class="math-container">$(1)$</span> is that there is no <span class="math-container">... | Saksham Paliwal | 1,012,372 | <p>Yeah actually your book wants to write <span class="math-container">$$\sum_{n=1}^{N}f(x_n)\Delta x_n$$</span></p>
<p><span class="math-container">$\Delta x_n$</span> is the width of the elemental rectangle and <span class="math-container">$f(x_)$</span> is the value of the function at the near neighbor of <span clas... |
1,417,286 | <p>So I'm trying to learn about RSA and have come across various subtopics, including the discrete logarithm problem. </p>
<p>This mentions primitive roots, which I do not understand.</p>
<p>Essentially all I want is an answer in simple terms of what a primitive root actually is.</p>
<p>Thanks</p>
| Peter | 82,961 | <p>Let $n>1$ be a natural number.</p>
<p>Let $S$ be the set of numbers coprime to $n$. A number $m$ is called a primitive
root in $\mathbb Z_n$, if the Set $\{m,m^2,m^3,...,m^{\phi(n)}\}$ modulo $n$ contains
every element of $S$.</p>
<p>$\phi(n)$ is the Euler-Phi-Function : The number of $m's$ with $gcd(m,n)=1$</p... |
1,989,253 | <p>I am trying to evaluate: </p>
<p>$$\lim_{x \to 4}\frac{\sqrt{5-x} - 1}{2-\sqrt{x}}.$$</p>
<p>Even though I tried rationalizing both denominator and numerator, I still end up with the functioning being undefined.</p>
<p>How can I solve this without rationalizing?</p>
| Nicolas FRANCOIS | 288,125 | <p>Rationalizing gives :
$$\frac{\sqrt{5-x}-1}{2-\sqrt x} = \frac{(4-x)(2+\sqrt x)}{(4-x)(\sqrt{5-x}+1)}$$
I think you can find the limit from here :-)</p>
|
1,358,002 | <p>My son did something quite impressive the other day. It was shear luck but I don't think I'll ever see it duplicated again in my lifetime. </p>
<p>I brought my kids to the boardwalk and my son wanted to play an amusement game. It was the arrow spin wheel game. It had 90 different names or possibilities to win. You ... | Tryss | 216,059 | <p>With 3 names, the probability to win a game on the first try is $\frac{1}{30}$</p>
<p>So the probability to win 3 games in a row is<br>
$$\frac{1}{30} \times \frac{1}{30} \times \frac{1}{30} = \frac{1}{27000} \simeq 0.003\%$$</p>
|
646,109 | <p>For function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$
and is not the zero-function I can prove that $f\left(1\right)>0$
and $f\left(x\right)=f\left(1\right)^{x}$ for each $x\in\mathbb{Q}$.
Is there a way to prove that for $x\in\mathbb{R}$?</p>
<p>This ... | Andreas Caranti | 58,401 | <p>No, because if $f$ is any of your functions, you may take any additive function $g : \mathbb{R} \to \mathbb{R}$ (that is, a function such that $g(x+y) = g(x) + g(y)$), and $f \circ g$ will still satisfy your assumption, as $f \circ g (x + y) = f(g(x+y)) = f(g(x) + g(y)) = f(g(x)) f(g(y)) = f \circ g(x) f \circ g(y)$... |
1,178,080 | <p>How to calculate the number of solutions of the equation $x_1 + x_2 + x_3 = 9$ when $x_1$, $x_2$ and $x_3$ are integers which can only range from <code>1</code> to <code>6</code>.</p>
| Mayank Jha | 220,999 | <p>A very simple way first count no.of
Positive solutions by $(n-1)C(r-1)$.
I hope you understand the meaning of $C$.Then if $x_1$ is $7$ then no. of solutions is $1$ similarly for $x_2$ & $x_3$
is also $1$, so number of solution is $28-3=25$.</p>
|
48,077 | <p>First, I'm a beginner.</p>
<p>I can compute the sum of roots with the follwing:</p>
<pre><code>Roots[x^7 + 5 x^6 + x^5 + x + 1 == 0, x]
Plus @@ (x /. {ToRules[%]}) // Simplify
</code></pre>
<p>Of course I get, except the sign, the coefficient of x^6.</p>
<p>Now, is there a way to compute more elaborate symmetric... | Wouter | 7,680 | <p>It is not clear if this is what you were asking, but it might help:
from the link to <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial" rel="nofollow">https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial</a> we extract </p>
<pre><code>Product[\[Lambda]+Subscript[x, k],{k,7}]==Sum[\[La... |
944,948 | <p>$\textbf{QUESTION-}$
Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.</p>
<p>If $P=Z(P)$ it is true. Now let $n > 1$, then</p>
<p>If I see $P$ as a nilpotent group and construct its upper central series, it will end , so let it be,</p>
<p>$e=Z_0<Z_1<Z_2<......<Z_r=P... | egreg | 62,967 | <p>You can write your equation as
$$
f(x+y)=f(x)+x^2y+xy^2+f(y)
$$
so
$$
f'(x)=\lim_{y\to0}\frac{f(x+y)-f(x)}{y}=
\lim_{y\to0}\frac{f(x)+x^2y+xy^2+f(y)-f(x)}{y}=
\lim_{y\to0}\frac{x^2y+xy^2+f(y)}{y}
$$
I bet you can take on from here.</p>
|
590,817 | <p>(I'm a software developer so excuse me)</p>
<p>I'm building an application for a client and one of the formulas that has been provided in the spec is <code>value1 = value2 * (1 + 5%)</code>. When I asked about it I was told that it's some kind of notation for <code>value1 = value2 * 0.15</code>.</p>
<p>Also, they ... | alexjo | 103,399 | <p>Note that $ a\%$ means $a/100$, so $5\%=5/100=0.05$ and $1+5\%=1+0.05=1.05$ and $1-5\%=1-0.05=0.95$. Thus value1=value2*(1+5%) is the same as value1=value2*1.05 and value1=value2*(1+ -5%) is the same as value1=value2*0.95.</p>
<p>For more explanations you can read <a href="http://en.wikipedia.org/wiki/Percentage"... |
1,211,287 | <p>Given that the angles between the consecutive lateral edges AB, AC & AD meeting at the vertex A of a tetrahedron ABCD are $ α, β, γ$ (as shown in the diagram below). Is there any set-formula to find out the solid angle subtended by the tetrahedron at the same vertex? </p>
<p>Note: A tetrahedron is a solid havin... | AB Balbuena | 147,730 | <p>Denote the solid angle by $\omega$ and let $v_1,v_2,v_3$ be the vectors from vertex $A$ along the edges $AB, AD, AC$. Then we have (using the usual cross product, dot product, and Euclidean norm):</p>
<p>$$(4 \pi)\omega + \pi = \cos ^{-1} \left( \frac{ (v_1 \times v_2) \cdot (v_1 \times v_3)}{||v_1 \times v_2|| ||v... |
1,878,573 | <p><a href="https://i.stack.imgur.com/3iZQ8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3iZQ8.png" alt="enter image description here"></a></p>
<p>I cannot get the $f'(0)$ by using L'Hôpital's rule, because it appears recurrence item. Can you help me?</p>
| DanielWainfleet | 254,665 | <p>For $x\ne 0$ we have $|\sin \frac {1}{x^3}|\leq 1.$ We have $e^x=1+x +O(x^2)$ as $x\to 0$. So $e^{x^2}-e^{-x^2}=2x^2 +O(x^4)$ as $x\to 0.$ So for $x\ne 0$ we have $$|\frac {f(x)-f(0)}{x-0}|=|\frac {f(x)}{x}|\leq |\frac {e^{x^2}-e^{-x^2}}{x}|=|\frac {2x^2+O(x^4)}{x}|=$$ $$=|2x+O(x^3)|.$$ So $f'(0)=0 .$</p>
|
3,227,215 | <p><a href="https://i.stack.imgur.com/7pJ4t.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7pJ4t.png" alt="enter image description here" /></a></p>
<blockquote>
<p><span class="math-container">$(O, R)$</span> is the circumscribed circle of <span class="math-container">$\triangle ABC$</span>. <span c... | Lê Thành Đạt | 654,749 | <p>Firstly,</p>
<p><span class="math-container">$$\frac{IM}{AM} + \frac{IN}{BN} + \frac{IP}{CP} = \frac{A_{CIB}}{A_{CAB}} + \frac{A_{AIC}}{A_{ABC}} + \frac{A_{BIA}}{A_{BCA}} = 1$$</span></p>
<p>where <span class="math-container">$A_m$</span> denotes the area of shape <span class="math-container">$m$</span>.</p>
<p><... |
821,875 | <p>A school director must randomly select 6 teachers to participate in a training session. There are 30 teachers at the school. In how many different ways can these teachers be selected, if the order of selection does not matter?</p>
| Asinomás | 33,907 | <p>If the order mattered it would be $30\cdot29\cdot28\cdot27\cdot26\cdot25$ since there would be $30$ option for the first to be picked, $29$ for the second to be piecked and so on...</p>
<p>However if order does not matter each unordered selection appears $6\cdot5\cdot4\cdot3\cdot2$ in the ordered selections.</p>
<... |
120,067 | <p>The <em>theta function</em> is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.</p>
<blockquote>
<p><strong>Theta reciprocity</strong>: $\theta... | Jonah Sinick | 683 | <p>Hecke generalized the argument that you mention to prove quadratic reciprocity relative to any given number field $K$ (see, e.g. his <a href="http://books.google.com/books/about/Lectures_on_the_Theory_of_Algebraic_Numb.html?id=QEocn1kbfxAC" rel="nofollow noreferrer">Lectures on the Theory of Algebraic Numbers</a>).<... |
3,950,463 | <blockquote>
<p>What is <span class="math-container">$100$</span>th derivative of <span class="math-container">$y=\ln(2x-x^2)$</span> at <span class="math-container">$x=1$</span>?</p>
<p><span class="math-container">$a)2\times99!$</span></p>
<p><span class="math-container">$b)-2\times99!$</span></p>
<p><span class="mat... | dezdichado | 152,744 | <p>You split them first:</p>
<p><span class="math-container">$$y = \ln(2-x) + \ln(x)\implies y' = \frac{1}{x-2}+\frac 1x\dots{}{} $$</span></p>
|
10,505 | <p>I've only done a few questions here but already it's grinding on me. Why can't we have the writing-answer panel and the preview panel side by side, rather than below, this means for big answers I can't make use of the preview! It'd be great if side by side, two scroll-bars, or even a pop out (I have a window manager... | Antonio Vargas | 5,531 | <p>I've been contributing to a userscript which adds side-by-side editing functionality to Stack Exchange. It can be toggled on and off by clicking a new button in the editor and works when writing new as well as editing existing questions and answers.</p>
<h2>Screenshot</h2>
<p><img src="https://i.stack.imgur.com/RSP8... |
35,463 | <p>I was asked the following vector calculus problem:</p>
<blockquote>
<p>Let <span class="math-container">$D$</span> be the unit ball and let <span class="math-container">$S$</span> be the unit sphere in <span class="math-container">$\mathbb{R}^3$</span>. Suppose that <span class="math-container">$F:\mathbb{R}^3\righ... | Hans Engler | 9,787 | <p>Your second proof is a good one and entirely appropriate within vector calculus. The first attempt has a gap (I think). </p>
<p>A third proof relies on Hopf's Lemma (commonly taught in graduate level classes on partial differential equations) which implies here that if a function $u$ satisfying $\Delta u \le 0$ in ... |
2,317,867 | <p>It is said that asymmetry implies antisymmetry, but how come so?</p>
<p>If $aRb \Rightarrow \neg(bRa)$, isn't this also the case for $a = b$, which indicates $aRa \Rightarrow \neg(aRa)$, which is a contradiction.</p>
| user454135 | 454,135 | <p>Asymmetry says that given $aRb$, we can't have $bRa$.
Antisymmetry has both $aRb$ and $bRa$ as hypothesis, so in this case it is true by vacuity, since both hypothesis can never be satisfied simultaneously.</p>
|
2,317,867 | <p>It is said that asymmetry implies antisymmetry, but how come so?</p>
<p>If $aRb \Rightarrow \neg(bRa)$, isn't this also the case for $a = b$, which indicates $aRa \Rightarrow \neg(aRa)$, which is a contradiction.</p>
| luifrancgom | 768,068 | <p>First some definitions:</p>
<ul>
<li><p><span class="math-container">$R \text{ is antisymmetric in } A \iff (\forall x)(\forall y)((x \in A \land y \in A \land xRy \land yRx) \implies x = y)$</span></p></li>
<li><p><span class="math-container">$R \text{ is asymmetric in } A \iff (\forall x)(\forall y)((x \in A \lan... |
3,985,177 | <p>When it comes to proving that two sets are equal, say <span class="math-container">$A = B$</span>, we're usually told that we have to prove that <span class="math-container">$A \subset B$</span> and <span class="math-container">$B \subset A$</span>. However, I'm under the impression that this strategy isn't unique. ... | Strichcoder | 655,660 | <p>Yes, Mobius transformations send generalized circles to generalized circles. A generalized circle is a cirlce or a line. If there was a Mobius transformation sending <span class="math-container">$\mathbb{C}$</span> to the disk, then its inverse would also be a Mobius transformation and would send the unit circle to ... |
3,064,501 | <p>So I was trying to find the <strong>time complexity</strong> of an algorithm to find the <span class="math-container">$N$</span>th prime number (where <span class="math-container">$N$</span> could be any positive integer).</p>
<p>So is there any way to exactly determine how far <span class="math-container">$(N+1)$<... | S Spring | 622,021 | <p>A good method for the operation of a prime number program is to output only the primes between an upper-bound and the upper-bound - k . Then the program just appears to be jumping from one section of a list to the other on each computation. Of course required consecutive primes are obviously found within the range o... |
2,298,665 | <p>*prior to the body, note that title might be insufficient or inappropriate. Please edit it if it's needed. </p>
<p>I am proving the claim below: </p>
<p>Let $f: [a,b] \to \Bbb R$ be of bounded variation. </p>
<p>$f(x) \ge c \gt 0$ for all $x \in [a, b]$ where $c$ is a constant</p>
<p>$\Rightarrow$ $h(x)$ = $1 \o... | Arthur | 15,500 | <p>For instance, for any $p, q\in \Bbb Q$ with $p\neq 0$, $\pi\mapsto p\pi+q$ is an automorphism.</p>
|
268,676 | <p>It is not hard to check that the three roots of $x^3-2=0$ is $\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}$, hence the splitting field for $x^3-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}]$. However, since $\sqrt[3]{2}\zeta_3^{2}$ can be compute through $\sq... | pritam | 33,736 | <p>Note that the roots of $x^n-a$ are $\alpha ,\alpha\zeta_n,\alpha\zeta_n^2,\ldots,\alpha\zeta_n^{n-1}$ where $\alpha$ is as you said. Then the splitting field of $x^n-a$ is $F:=\mathbb{Q}(\alpha ,\alpha\zeta_n,\ldots,\alpha\zeta_n^{n-1})$. Now $\alpha\in F$ and $\zeta_n=\alpha^{-1}(\alpha\zeta_n)\in F$, hence $\mathb... |
1,930,901 | <p>I need to prove that:</p>
<p>$$f(z) = \frac{Re(z)}{|z|}$$</p>
<p>and </p>
<p>$$g(z) = \frac{Im(z^2)}{|z^2|}$$</p>
<p>both have limit at $z=0$</p>
<p>If I see $z$ as $z = x+iy$ I have:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{x}{\sqrt{x^2+y^2}}$$</p>
<p>but if I take this limit at $y = x$ we have:</p>
<p>$$\lim_{(... | Olivier Moschetta | 369,174 | <p>Let $\varepsilon>0$. Write first:
$$\left|\dfrac{1}{j}\sum_{k=1}^j a_k-l\right|\leq\dfrac{1}{j}\sum_{k=1}^j |a_k-l|=\dfrac{1}{j}\sum_{k=1}^{N}|a_k-l|+\dfrac{1}{j}\sum_{k=N+1}^{j}|a_k-l|\,(\ast)$$
where $1\leq N\leq j$. Since $a_j\rightarrow l$ we can choose $N$ such that
$$|a_k-l|<\dfrac{\varepsilon}{2}\text{ ... |
706,546 | <p>If we have a non-zero real $n$ by $m$ matrix $M$, then there may exist a non-zero unit vector $v$ of $m$ elements so that $Mv = 0$. I understand we can't call this an eigenvector with eigenvalue $0$. </p>
<blockquote>
<p>Why is this not a sensible definition of an eigenvector of a rectangular matrix?</p>
</bloc... | Gabriel Romon | 66,096 | <p>Let <span class="math-container">$F=\mathrm{span}({I_n,A,A^2,...})$</span>.
It suffices to prove that <span class="math-container">$(I_n, A,..., A^{n-1})$</span> is a family with cardinality <span class="math-container">$n$</span> that spans <span class="math-container">$F$</span>.</p>
<p>By Cayley-Hamilton, one can... |
5,711 | <p>I am teaching abroad to non-native English speakers with a large variance of language skills. </p>
<p>I teach both pre-calculus and AP calculus (AB & BC). For both of those classes I define the new terms. I use word problems, have them read from the textbook, take notes in English during class, have the stude... | Pablo B. | 3,260 | <p>Well, apart from what you are already doing, there are not too many things you can do. You teach mathematics in English during Math class. And your students learn the grammar, general vocabulary, etc. during English class. And this is something your principal should understand. Depending on the level of your student... |
207,515 | <p>Suppose I have the following list, </p>
<pre><code>l = {{"b", "c", "d"}, {"e", "b"}, {"a", "b", "d", "e"}}
</code></pre>
<p>and further suppose I have the following association, </p>
<pre><code>l1=<|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>
</code></pre>
<p>I wonder how can I replac... | kglr | 125 | <p>For the specific numbering in OP, you can also use <a href="https://reference.wolfram.com/language/ref/LetterNumber.html" rel="noreferrer"><code>LetterNumber</code></a>:</p>
<pre><code>LetterNumber[l]
</code></pre>
<blockquote>
<p>{{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} </p>
</blockquote>
|
2,354,467 | <p>I am trying to evaluate the following
\begin{equation}
I(a,b) = \int_{a}^{\frac{a+b}{2}} (x-a)^{\alpha-1} \, x^n \, dx + \int_{\frac{a+b}{2}}^{b} (b-x)^{\alpha-1} \, x^n \, dx,
\end{equation}
where $0<\alpha<1$. Wolfram alpha gives no solution. I tried integration by parts without success. My problem is that ... | JJacquelin | 108,514 | <p>\begin{equation}
I(a,b) = \int_{a}^{\frac{a+b}{2}} (x-a)^{p-1} \, x^n \, dx + \int_{\frac{a+b}{2}}^{b} (x-a)^{p-1} \, x^n \, dx \qquad 0<p<1.
\end{equation}</p>
<p>I wonder if where is a typo in the equation : Why you don't simply wrote
\begin{equation}
I(a,b) = \int_{a}^{b} (x-a)^{p-1} \, x^n \, dx \qquad... |
62,539 | <p>I am using two books for my calculus refresher.</p>
<ol>
<li>Thomas' Calculus </li>
<li>Higher Math for Beginners by Ya. B. Zeldovich</li>
</ol>
<p><strong>My question is :</strong> When applying Integral Calculus for calculation of volumes of solids, generated by curves revolved around an axis, we use slices of '... | joriki | 6,622 | <p>I like this question; what you're trying to understand is important to understand.</p>
<p>In this answer I'll be talking loosely about infinitesimal quantities linear or quadratic in $\mathrm dx$; I think this is the best way to get a feel for this sort of thing, but similar arguments could also be presented more r... |
1,300,853 | <p>Could somebody tell me the exact value of this series?
$$
\sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k}
$$
where
$$
H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n}
$$</p>
<p>Thanks!</p>
| Przemo | 99,778 | <p>Denote ${\bf H}^{(q)}_n(t) := \sum\limits_{m=1}^\infty H_m^{(q)}/m^n \cdot t^m$. By using the formula on the bottom of the answer to <a href="https://math.stackexchange.com/questions/2169507/calculating-alternating-euler-sums-of-odd-powers/2251637#2251637">Calculating alternating Euler sums of odd powers</a> we have... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | M. Mueger | 75,007 | <p>I second Brandsma's answer. Still such proofs tend to be 10 pages long if everything is spelled out. (Engelking spends 7 on proving Brouwer's fixed point theorem using Sperner's lemma, then several pages on dimension theory.)</p>
<p>I wrote down a fairly simple self-contained proof of invariance of dimension (for c... |
1,858,297 | <p>Suppose the diameter of a nonempty set $A$ is defined as </p>
<p>$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$</p>
<p>where $d(x,y)$ is a metric.</p>
<p>Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable additivity for this particular case?</p>
| chi | 207,328 | <p>Let $A$ be a circle of diameter 1, and let $B$ a circle of diameter 2, having the same center of $A$. Note that $A \subseteq B$.</p>
<p>Now, $\sigma(B) = 2$, yet $\sigma(B \setminus A) + \sigma(A) = 2 + 1 = 3$.</p>
|
3,717,932 | <p>How can this identity convolution be shown?</p>
<p><span class="math-container">$$\int^\infty_{-\infty} f(\tau)\delta(t-\tau)d\tau=f(t)$$</span></p>
<p>I keep getting stuck in traps when trying to show this and need a bit of assistance</p>
| Daniel Robert-Nicoud | 60,713 | <p>First of all, one needs to understand that the Dirac delta is <em>not</em> a function, but a distribution, ie an element of the dual space of some space of test functions. Thus, one <em>defines</em> the Dirac delta by its action on smooth, compactly supported function
<span class="math-container">$$\delta(f):=f(0)\ ... |
280,393 | <p>I want to show that $(3, \sqrt 15)$ is not a principal ideal in the ring $ R = \mathbb{Z}[\sqrt{15}]$ with norm $N(a + b \sqrt 15) = a^2 - 15b^2$.</p>
<p>My attempt:</p>
<p>Suppose $(3, \sqrt 15) = (x) $</p>
<p>Then $3 = x * r1$ and $\sqrt 15 = x * r2$ , $r1,r2 \in R$.</p>
<p>$N(3) = 9 = N(x) N(r1)$ and $N(\sqrt... | Community | -1 | <p>Here is a proof using some results from algebraic number theory. Suppose your ideal $I = (3,\sqrt{15})$ is a principal generated by some $(\alpha)$ with $\alpha = a + b\sqrt{15}$. Then firstly your ideal $I$ is prime since
$$\begin{eqnarray*} R/I &\cong& \Bbb{Z}[x]/(x^2 - 15)/(3,x)/(x^2 - 15)\\
&\cong&a... |
765,404 | <p>Can anyone explain the partial derivative below:</p>
<p>$\frac{\partial a^tX^{-1}b}{\partial X} = -X^{-t}ab^tX^{-t}$</p>
<p>I was trying to derive this equation using the below formula, but failed.</p>
<p><img src="https://i.stack.imgur.com/apR2q.png" alt="enter image description here"></p>
| Hyperplane | 99,220 | <p>A totally mechanical approach. By the chain rule:</p>
<p><span class="math-container">$$\frac{∂a^⊤ X^{-1} b}{∂ X} = \frac{∂a^⊤ X^{-1} b}{∂ X^{-1}}∘\frac{∂X^{-1} }{∂ X}$$</span></p>
<p>Consider the first term <span class="math-container">$\frac{∂a^⊤ X^{-1} b}{∂ X^{-1}}$</span>. Note that the nominator is linear in <s... |
650,710 | <p>How would I go about simplifying $4(a-2(b-c)-(a-(b-2)))$. Show working out and steps please.</p>
<p>I'd show my working out but I'm not really sure where to start. Firstly, I would want to get rid of the 4 so I'd times everything else by 4 right? No idea. </p>
| Karolis Juodelė | 30,701 | <p>Basic algebra is all about term rewriting. You need to identify complete parts of the expression, extract them, solve them and place the results back where they were taken from.</p>
<p>You can start by separating $4(a-2(b-c)-(a-(b-2)))$ into $4(a+?-(a-(b-2)))$ and $? = -2(b-c)$. Then solve $? = -2(a-c) = -2a+2c$ an... |
1,913,873 | <p>If $a,b,c,d,e>1$, then prove that:
$$\frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 20. $$</p>
<p>I don't know how to begin. What should be the approach?</p>
| Joey Zou | 260,918 | <p>Note that $x^2-4(x-1) = (x-2)^2\ge 0$ for all $x\in\mathbb{R}$, and hence $x^2\ge 4(x-1)$ for all $x\in\mathbb{R}$. As such, we have
$$ \frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 4\left(\frac{a-1}{b-1} + \frac{b-1}{c-1} + \frac{c-1}{a-1} + \frac{d-1}{e-1} + \frac{e-1}... |
1,913,873 | <p>If $a,b,c,d,e>1$, then prove that:
$$\frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 20. $$</p>
<p>I don't know how to begin. What should be the approach?</p>
| zyx | 14,120 | <p>The functions $x^2$ and $(x-1)$ are both increasing in $x$. Therefore, by the rearrangement inequality, the sum is $ \geq \sum \frac{a^2}{a-1}$ which is easier to control, in fact each term of this sum is $\geq 4$.</p>
|
1,913,873 | <p>If $a,b,c,d,e>1$, then prove that:
$$\frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 20. $$</p>
<p>I don't know how to begin. What should be the approach?</p>
| Landon Carter | 136,523 | <p>By Cauchy Schwarz, $($given expression$)(a+b+c+d+e-5)\geq(a+b+c+d+e)^2$</p>
<p>Now this shows $($given expression$)\geq \dfrac{(a+b+c+d+e)^2}{a+b+c+d+e-5}$</p>
<p>Now $(a+b+c+d+e)^2=(a+b+c+d+e-5)^2+2\times(a+b+c+d+e-5)\times 5+25$ hence on division by $a+b+c+d+e-5$ we get $a+b+c+d+e-5+10+\dfrac{25}{a+b+c+d+e-5}=(a... |
3,027,528 | <p>I am trying to resolve an exercise and there are 2 point that are missing in order to finalize:</p>
<p>Suppose <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, and <span class="math-container">$P$</span> are <span class="math-container">$R$</... | Servaes | 30,382 | <p>For the first point consider for <span class="math-container">$\phi$</span> the quotient morphism <span class="math-container">$\pi:\ C\ \longrightarrow\ \operatorname{coker}g$</span>.</p>
<p>For the second point consider for <span class="math-container">$\phi$</span> the identity morphism <span class="math-contain... |
1,230,112 | <p>Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by $\mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x}=r^2$.</p>
<p>I tried to do this with tensor algebra but I'm pretty new ... | mrp | 134,447 | <p>As 5xum mentioned in the comments, union starts with a u and has a symbol $\cup$ that looks very much like a $u$. Then intersection is simply the same symbol flipped $\cap$.</p>
<p>As for what they mean, you can think of union $A \cup B$ as a cup (indeed, the LaTeX command for it is <code>\cup</code>) in which you ... |
19,865 | <p>What does this set look like? $A = \{ (x, \frac{1}{x}) |  x \in \mathbb{R} \backslash 0 \}$, where $A$ is a subset of $\mathbb{R}^2$ with the Euclidean topology.</p>
<p>I thought $A = (-\infty, 0] \cup [0, \infty)$ but I know that its projection $(x,y) \rightarrow x$ is not closed, so this must be wrong.</p>
... | Willie Wong | 1,543 | <p>Interpreting your notation to mean that $A$ is a certain subset of $\mathbb{R}^2$...</p>
<p><strong>Hint</strong> $A$ can be treated as the graph of some function.</p>
|
19,865 | <p>What does this set look like? $A = \{ (x, \frac{1}{x}) |  x \in \mathbb{R} \backslash 0 \}$, where $A$ is a subset of $\mathbb{R}^2$ with the Euclidean topology.</p>
<p>I thought $A = (-\infty, 0] \cup [0, \infty)$ but I know that its projection $(x,y) \rightarrow x$ is not closed, so this must be wrong.</p>
... | user6495 | 6,495 | <p>@Matt: The set $A$ is closed so it is equal to its own closure. To show it is closed you can use the characterization of a closed set (in metric spaces) in terms of sequences.</p>
|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| paw88789 | 147,810 | <p>Hint: Could you uniquely associate to a sequence of $1$s and $2$s a strictly increasing sequence of natural numbers?</p>
|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Siong Thye Goh | 306,553 | <p>Given a sequence $(a_0, a_1, \ldots, )$, map it to the sequence $(a_1-a_0, a_2-a_1, \ldots)$.</p>
<p>The image of such map would be the sequence of all natural numbers . Indeed the map is surjective since given a sequence of natural number $(b_0, b_1, \ldots)$, we can find a preimage $(a_0, a_1, \ldots)$ which sati... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| fleablood | 280,126 | <p>Let $\{{s_i}_j\}$ be a countable list of strictly increasing sequences; define $\{c_i\}$ via $c_i = \max (c_{i-1}, {s_i}_i)+1$ and .... presto, Cantor!</p>
<p>But it could be simpler (depending on one's idea of simple) to reduce to things we already know are uncountable.</p>
<p>For simplicity, we can consider the... |
3,425,373 | <p>Consider:</p>
<p><span class="math-container">$$ 1+1/2^2+2/3^2+1/4^2+2/5^2+1/6^2+...$$</span></p>
<p>Does this sum have a closed form?</p>
<p>If all the numerators are <span class="math-container">$1$</span> then it does have a closed form. </p>
| Calvin Lin | 54,563 | <p>Yes it does.</p>
<p><strong>Hint:</strong> <span class="math-container">$\sum \frac{1}{n^2} = \frac{\pi^2}{6}$</span></p>
<p><strong>Hint:</strong> <span class="math-container">$\sum \frac{1}{(2n)^2} = \frac{\pi^2}{24}$</span></p>
|
872,493 | <p>$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... =
\exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$</p>
<p>Now, from continuity we can evaluate each term separately.</p>
<p>$\lim_{x\to 0^+} \ln(\frac{\sin x}{x}) = 0$</p>
<p>Therefore, we have:<br>
$$\exp \left(\lim_{x\to 0^+} \fr... | Mark | 24,958 | <p>The L'Hôpital's rule isn't more rigorous than others, nay, mathematicians generally prefer not to use this rule because it is seen as a mechanical way to solve a limit. Furthermore the rule, if it is used mechanically, not always allows to solve a limit. For example, with the limit:
$$\lim_{x\rightarrow\infty}\frac{... |
4,164,650 | <p>The outline of the exercise is that: Fix <span class="math-container">$b > 1, y > 0$</span> and show that there is a unique real <span class="math-container">$x$</span> such that <span class="math-container">$b^x = y$</span>. (For further specification see e.g. <a href="https://math.stackexchange.com/questions... | Jan Pulmann | 760,426 | <p>Let's set <span class="math-container">$\theta_0=0$</span>, since rotations preserve the quadratic form for conic sections. Then, you can see this quickly by rewriting the equation to
<span class="math-container">$$ r = \frac{1}{1+ex/r}$$</span>
which gives
<span class="math-container">$$ r+ex=1.$$</span>
Separating... |
2,995,643 | <p>Here is a thought experiment I have. </p>
<p>Say we flip a unique coin where we have a 99.99999999999% chance of it landing on heads, and a .000000000001% chance of it landing on tails (the two possibilities equal to 100%).</p>
<p>And say we have an <em>infinite</em> number of coins flipped all at once (and only o... | Bram28 | 256,001 | <p>Sure! In fact, this would be possible with a fair coin already. In fact, it would be possible with a coin that is largely biased to come up with tails: one that has a 99.99999999999% chance of it landing on tails, and a .000000000001% chance of it landing on heads: it is <em>always</em> possible that you get no tai... |
366,687 | <p>I am interested in the status of the conjecture about the minimum number of edge crossings <span class="math-container">$cr(K_{m,n})$</span> in a drawing of the complete bipartite graph <span class="math-container">$K_{m,n}$</span>.</p>
<p>The Wikipedia article <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_b... | ZsuzsannaMath | 161,836 | <p>It is a fascinating conjecture. The following might be a good reference for you: In 1997, Richter & Thomassen showed that
<span class="math-container">$$\lim_{n\to\infty}cr(K_{n,n})\left(\begin{array}{c} n \\ 2 \end{array}\right)^{-2}$$</span>
exists and is at most <span class="math-container">$1/4$</span>. If t... |
1,502,676 | <p>I'm aware of this similiar question:</p>
<p><a href="https://math.stackexchange.com/questions/1249308/what-is-the-difference-between-an-elliptical-and-circular-paraboloid-3d">what is the difference between an elliptical and circular paraboloid? (3D)</a></p>
<p>But I need help in a different way. In my calculus exa... | Community | -1 | <p>A circle is a special case of an ellipse; a circular paraboloid is a special case of an elliptic paraboloid. This is how I use these terms, and so do many others. </p>
<p>But it's possible that your teacher used them differently, when introducing these terms <em>in the context of the class</em>. That's her decision... |
1,850,258 | <p>From where can I learn mathematics from the basic blocks up? I feel like I have a lot of holes in the mathematics that I know and I would like to see where all those concepts come from. I would like to see what are the ideas that are took from granted, as foundation, and which ideas are made from this foundation.</p... | Peter | 82,961 | <p>$1$ It can be viewed as a lattice in the $xy$-plane.</p>
<p>$2$ Addition and multiplication is like in $\mathbb C$</p>
<p>$3$ Yes, $1+i$ , for example. The definition is analogue to the usual
primes. A Gaussian number is prime if it cannot be written as
a product of two Gaussian numbers, both of which are ... |
3,057,278 | <blockquote>
<p><strong>Question:</strong> Can we show that <span class="math-container">$$\sum_{n=0}^\infty(-1)^{n+1}\frac{(2n-3)!!}{(2n+3)!!}=\frac{\pi}{8} $$</span> ?</p>
</blockquote>
<hr>
<p>According to <a href="https://www.wolframalpha.com/input/?i=sum+(-1)%5E(n%2B1)+%5B(2n-3)!!%2F(2n%2B3)!!%5D+from+n%3D+0+t... | Claude Leibovici | 82,404 | <p><em>Considering the simplicity of Micah's answer, I am ashamed to provide this complex one.</em></p>
<p>Let
<span class="math-container">$$S_p(x)=\sum_{n=0}^p(-1)^{n+1}\frac{(2n-3)!!}{(2n+3)!!}x^p$$</span> It write
<span class="math-container">$$S_p(x)=\frac{\left(8 p^3+36 p^2+46 p+15\right) \,
_2F_1\left(-\frac... |
1,700,246 | <p>Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an element $g\in SO_{3}(q)$ defined by: $g: e\mapsto -e$, $f\mapsto \frac{1}{2}e -f +d$, $d\mapsto e+d$. I would like to de... | Andres Mejia | 297,998 | <p>This in general is not true. This is because $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)=x^2$ is not a surjective function. More specifically, for all $r<0$, this statement won't be true for some neighborhood around $r$.</p>
<p>For a more concrete example, pick any $x<0$ and try to find a squared rational ... |
1,182,684 | <p>Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is.</p>
<p><strong>Attempt at solution:</strong></p>
<p>We have that $\frac{\partial F_1}{\partial y} = 1 = \frac{\partial F_2}{\partial x} $, $\frac{\partial F_1}{\partial z} ... | kobe | 190,421 | <p>The function $\phi(x,y,z) = xy + \frac{z^3}{3}$ is a potential for $\mathbf{F}$ since $$\operatorname{grad} \phi = \phi_x \mathbf{i} + \phi_y \mathbf{j} + \phi_z \mathbf{k} = y\mathbf{i} + x \mathbf{j} + z^2\mathbf{k} = \mathbf{F}.$$</p>
<p>To actually derive $\phi$, we solve $\phi_x = F_1, \phi_y = F_2, \phi_z = F... |
621,461 | <p>I'm having trouble understanding division when the divisor is greater than the dividend, for ex 1/4.</p>
<p>I think of division as "how many times can the divisor fit into the dividend evenly". </p>
<p>Intuitively, when I see 1/4 in the context of slices of pizza, I think of it as 1 "out of" 4, but I can't seem to... | user110219 | 110,219 | <p>Suppose you have 1 bread and you want to distribute it among 4 people equally. The only solution is to cut the single bread into 4 equal pieces.</p>
<p>For the population suppose that you have collected 8 from A and B such that their donations are 5 and 3 respectively. So if someone asks you that what is the contri... |
621,461 | <p>I'm having trouble understanding division when the divisor is greater than the dividend, for ex 1/4.</p>
<p>I think of division as "how many times can the divisor fit into the dividend evenly". </p>
<p>Intuitively, when I see 1/4 in the context of slices of pizza, I think of it as 1 "out of" 4, but I can't seem to... | Zhoe | 99,231 | <p>Division is often represented algebraically as the <strong>fraction</strong> $\frac{a}{b}$ where "$a$ is divided by $b$". $a$ is the numerator (dividend) and $b$ the denominator (divisor). The numerator represents the number of equal parts and the denominator indicates how many of those parts make up the whole. Take... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Ben Millwood | 29,966 | <p>One natural use for smallish infinite ordinals, which makes clear that different infinite ordinals have distinct properties, is the length of games.</p>
<p>I'm playing a chess-like (turn-taking, deterministic, complete information) board game of some kind against an opponent. My teammate is also playing a separate ... |
3,792,683 | <p>Where <span class="math-container">$\alpha$</span> is a real constant, consider the sequence {<span class="math-container">$z_n$</span>} defined by <span class="math-container">$z_n=\frac{1}{n^\alpha}$</span>. For which value of <span class="math-container">$\alpha$</span> is {<span class="math-container">$z_n$</spa... | Bloodpolyhydron | 606,626 | <p><span class="math-container">$$\begin{array}\ \int^{+\infty}_0 \frac{e^{-x^2}}{(x^2+\frac{1}{2})^2} dx &= 2\bigg[\int^{+\infty}_0 \frac{e^{-x^2}}{x^2+\frac{1}{2}} dx - \int^{+\infty}_0 \frac{x^2e^{-x^2}}{(x^2+\frac{1}{2})^2} dx\bigg]\\ &= 2\bigg[\int^{+\infty}_0 \frac{e^{-x^2}}{x^2+\frac{1}{2}} dx + \frac{1}... |
91,645 | <p>I asked a similar question previously, though this is more specific and directed. In the writing of mathematics research papers, when is information cited, such as definitions? I have read that if it is fairly recent, then cite it. But what is "fairly recent?" Also, should books from whence a definition came be ... | JRN | 18,398 | <p>Definitions that "every mathematician knows" need no citation. For example, if you define $\mathbb{N}$ as the set of positive integers (or, if you like, the set of non-negative integers), you do not need a citation.</p>
<p>"Fairly recent" means that there is currently no consensus, that is, other people have diffe... |
91,645 | <p>I asked a similar question previously, though this is more specific and directed. In the writing of mathematics research papers, when is information cited, such as definitions? I have read that if it is fairly recent, then cite it. But what is "fairly recent?" Also, should books from whence a definition came be ... | Carl Mummert | 630 | <p>The standard for citation in mathematics papers is very different than in (say) humanities. The <a href="http://www.ams.org/about-us/governance/policy-statements/sec-ethics">AMS Ethical Guidelines</a> say</p>
<blockquote>
<p>The correct attribution of mathematical results is essential, both because it encourages ... |
91,645 | <p>I asked a similar question previously, though this is more specific and directed. In the writing of mathematics research papers, when is information cited, such as definitions? I have read that if it is fairly recent, then cite it. But what is "fairly recent?" Also, should books from whence a definition came be ... | Stella | 117,795 | <p>I've always been told to use the APA citation style.</p>
<p>APA citation style refers to the rules and conventions established by the American Psychological Association for documenting sources used in a research paper
<a href="http://www.apastyle.org/" rel="nofollow">http://www.apastyle.org/</a></p>
|
3,701,582 | <p>I still struggle mighty with basic conceptions of truth and proof. </p>
<p>For example: The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds. Now, Goedel and Cohen proved that CH/~CH are independent from ZFC, so ZFC + CH and ZFC + ~CH are consistent (in case ZFC is consistent but mathe... | Noah Schweber | 28,111 | <p>As in <a href="https://math.stackexchange.com/q/3925373/28111">this other question of yours</a>, the issue comes from conflating <strong>truth/falsity</strong> (which are relevant to <strong>structures</strong>) with <strong>provability/disprovability</strong> (which are relevant to <strong>theories</strong>).</p>
<... |
2,544,755 | <p>How would I start to prove this set is bounded above and below? A hint would be appreciated!</p>
<p>$A:=[0,1)\cup\{2-\frac{1}{n^5}:n\in \mathbb{N}\}$</p>
| fleablood | 280,126 | <p>If $x \in [0,1)$ then $x \ge 0$. If $x \in \{2-\frac 1{n^5}\}$ then $x \ge 1 > 0$. (Because $\frac 1{n^5} \le 1$ for all $n$ so if $x= 2 - \frac 1{n^5}$ then $x \ge 2 -1 =1$.). So if $x$ is in $A$ then $0 \le x$.</p>
<p>So $0$ is a lower bound of $A$.</p>
<p>If $x \in [0,1)$ then $x < 1$. If $x \in \{2-... |
238,392 | <p>Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:</p>
<blockquote>
<p>There does not exist any Riemannian metric on the torus whose sectional curvature is $<0$.</p>... | Steve Huntsman | 1,847 | <p>Determining whether or not two simplicial complexes are homeomorphic is undecidable. Markov <em>fils</em> showed this in 1958 by reduction to the word problem for groups.</p>
|
3,961,131 | <p>Suppose we are given 2 predicates
<span class="math-container">$A(x)$</span> and <span class="math-container">$B(x)$</span> with domain <span class="math-container">$M$</span>.</p>
<p>Suppose next we are given the following predicate
<span class="math-container">$$\neg (A(x) \land B(x)) \land (\forall x(A(x) \righta... | yugikaiba | 865,997 | <p>1
<span class="math-container">$$\neg (A(x) \land B(x)) \land (\forall x(A(x) \rightarrow B(x)))$$</span></p>
<p>2
<span class="math-container">$$\neg (A(x) \land B(x)) \equiv (\neg A(x)) \lor (\neg B(x))$$</span></p>
<p><span class="math-container">$$(\forall x(A(x) \rightarrow B(x))) \equiv (\neg A(x)) \lor B(x)$$... |
401,389 | <p>I worked through some examples of Bayes' Theorem and now was reading the proof.</p>
<p>Bayes' Theorem states the following:</p>
<blockquote>
<p>Suppose that the sample space S is partitioned into disjoint subsets <span class="math-container">$B_1, B_2,...,B_n$</span>. That is, <span class="math-container">$S = B_1 ... | André Nicolas | 6,312 | <p>It is the sets $A\cap B_i$ that are pairwise disjoint, and that is precisely what you need to calculate the probability in the denominator.</p>
|
1,647,673 | <p>Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$</p>
<p>Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$.</p>
<p>EDIT: I was hoping there is a way without using a known inequality, ie to prove that $RHS-LHS\geq 0$</p>
| vrugtehagel | 304,329 | <p>Notice that $|a_i|^2=a_i^2$. Using the <a href="https://en.wikipedia.org/wiki/Generalized_mean" rel="nofollow">Generalized Mean Inequality</a>, we see $$\frac{|a_1|+\cdots+|a_n|}{n}\leq \sqrt{\frac{|a_1|^2+\cdots+|a_n|^2}{n}}$$
which we can rewrite to
$$|a_1|+\cdots+|a_n|\leq \sqrt{n}\sqrt{a_1^2+\cdots+a_n^2}$$
and ... |
869,506 | <p>In a paper I am reading, there is a step that seems to come from the following inequality:
$$(1+x)^\alpha \le 1+2^\alpha x,$$
where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the paper, but the above probably holds for more general $\alpha$, say, $\alpha\ge 1$.) It is stated with no explanation, a... | Semiclassical | 137,524 | <p>Here's a simple convexity argument. Let $g(x)=(1+x)^\alpha$, and note that $\alpha>1$ implies $g''(x)=\alpha(\alpha-1)(1+x)^{\alpha-2}>0$ for $0<x<1$; so $g(x)$ is concave up. In that case $gf(x)$ will be bounded above on the interval $[0,1]$ by $g(0)(1-x)+x g(1)$, i.e. the line segment connecting the en... |
614,962 | <blockquote>
<p>We have a continuous function <span class="math-container">$f:(a,b)\to \mathbb R$</span></p>
<p>Prove that: <span class="math-container">$\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$</span> such that:</p>
<p><span class="math-container">$$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$</span></p>
</blockquote>... | mathlove | 78,967 | <p>HINT : You can use $\lim_{x\to 0}\frac{\sin(2x)}{2x}=1.$</p>
|
614,962 | <blockquote>
<p>We have a continuous function <span class="math-container">$f:(a,b)\to \mathbb R$</span></p>
<p>Prove that: <span class="math-container">$\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$</span> such that:</p>
<p><span class="math-container">$$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$</span></p>
</blockquote>... | Claude Leibovici | 82,404 | <p>The Taylor expansion of Csc[y] built around y=0 is 1/y + y/6. You may forget the denominator and arrive to x (1/(2x) + (x/3)) = 1/2</p>
|
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