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 |
|---|---|---|---|---|
29,823 | <p>Just a random thought here: Can cohomology theories (e.g. sheaf cohomology) on the Stone space $S_n(T)$ (the space of complete n-types) of a first-order theory $T$ tell us anything interesting (e.g. the classification of theories)? Is there any result in model theory that is obtained (probably most easily) by this k... | Antongiulio Fornasiero | 7,986 | <p>You "only" need to change the topology of the stone space to make it interesting.
In o-minimality, the "spectral" topology is often used: see e.g. many papers by Edmundo.
A similar approach can be used in other topological structures, as long as the structure is definably connected; for structures that are (totally... |
1,053,683 | <p>How to show that
$$\sum_{n=1}^\infty\frac{1}{n^2+3n+1}=\frac{\pi\sqrt{5}}{5}\tan\frac{\pi\sqrt{5}}{2}$$
?</p>
<p><strong>My try:</strong></p>
<p>We have
$$n+3n+1=\left(n+\frac{3+\sqrt{5}}{2}\right)\left(n+\frac{3-\sqrt{5}}{2}\right),$$
so
$$\frac{1}{n^2+3n+1}=\frac{2}{\sqrt{5}}\left(\frac{1}{2n+3-\sqrt{5}}-\frac{1... | xpaul | 66,420 | <p>Note
$$ n^2+3n+1=(n+\frac{3}{2})^2+\left(\frac{\sqrt 5i}{2}\right)^2 $$
and
and hence
\begin{eqnarray}
\sum_{n=0}^\infty\frac{1}{n^2+3n+1}&=&\sum_{n=0}^\infty\frac1{(n+\frac{3}{2})^2+\left(\frac{\sqrt 5i}{2}\right)^2}\\
&=&\frac12\sum_{n=-\infty}^\infty\frac1{(n+\frac{3}{2})^2+\left(\frac{\sqrt 5i}{2... |
3,905,331 | <p>I need to prove that <span class="math-container">$\lim_\limits{n\to \infty}$$\sqrt{\frac{n^2+3}{2n+1}} = \infty$</span> (series) by using the definition:</p>
<p>"A sequence <span class="math-container">$a_n$</span> converges to <span class="math-container">$\infty$</span> if, for every number <span class="math... | fleablood | 280,126 | <p>Your proof is mostly okay but you do three things wrong. One is linguistically wrong. You say, and quote,</p>
<p>"I'll find an <span class="math-container">$N$</span> such that whenever <span class="math-container">$n≥N$</span> it follows that <span class="math-container">$a_n>M$</span> for every <span cla... |
3,461,531 | <p>I have to determine differentiability at <span class="math-container">$(0,1)$</span> of the following function:
<span class="math-container">$$f(x,y)=\frac{|x| y \sin(\frac{\pi x}{2})}{x^2+y^2}$$</span>
The partial derivatives both have value <span class="math-container">$0$</span> at <span class="math-container">$(... | Randall | 464,495 | <p>Take elements of <span class="math-container">$\mathbb{Z}$</span>, then remove elements of <span class="math-container">$\mathbb{Q}$</span>. What's left? Nothing. And for sure, <span class="math-container">$\varnothing$</span> is a subset of <span class="math-container">$\mathbb{N}$</span>. Along the same lines, ... |
3,516,776 | <p>I've been trying to solve this for limit comparison test with <span class="math-container">$a_n=a^\frac{1}{n}+a^{-\frac{1}{n}}-2 , b_n= \frac{1}{n}$</span>,
but
<span class="math-container">$\frac{a_n}{b_n}\rightarrow\ln{a}(a^{\frac{1}{x}}-a^{-\frac{1}{x}})\rightarrow 0$</span>.
Any help appreciated.</p>
| Luca Goldoni Ph.D. | 264,269 | <p>Let be <span class="math-container">$a>0$</span>. Since
<span class="math-container">$$
a^x + a^{ - x} - 2 = 2\left[ {\cosh (x\ln a) - 1} \right]
$$</span>
and since
<span class="math-container">$$
\mathop {\lim }\limits_{t \to 0} \left[ {\frac{{\cosh (t) - 1}}
{{t^2 }}} \right] = \frac{(\ln a)^2}
{2}
$$</span>... |
3,135,386 | <p>Our teacher tells us to convert it this way <span class="math-container">$ 3^x = e^{\ln 3^x}= e^{x\cdot\ln 3}$</span> and then use the rule <span class="math-container">$e^u\cdot u'$</span> but I can't understand where <span class="math-container">$\ln$</span> comes from and how <span class="math-container">$\ln 3^x... | Michael Rybkin | 350,247 | <p>And are you familiar with this basic property of logarithms?
<span class="math-container">$$\log_{b}{x^y}=y\log_{b}{x}$$</span>
You can bring the power out front.</p>
<p>How about this fact?</p>
<p><span class="math-container">$$
a=e^{\ln{a}}, a >0
$$</span></p>
<p>Do you know what the derivative of an exponen... |
3,016,386 | <p>Hi I am struggling with this exercise, which may be perceived as simple. so I was trying to write tangents as follows:</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$$</span> and then <span class="math-container">$$z=a+bi$$</span>, which led me to <span class="math-container">... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that by the standard trick</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}\cdot\frac{e^{i\bar z}+e^{-i\bar z}}{e^{i\bar z}+e^{-i\bar z}}=\ldots$$</span></p>
<p>then use the well know identities for <span class="math-container">$\cos$</span... |
3,016,386 | <p>Hi I am struggling with this exercise, which may be perceived as simple. so I was trying to write tangents as follows:</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$$</span> and then <span class="math-container">$$z=a+bi$$</span>, which led me to <span class="math-container">... | MPW | 113,214 | <p>From the title, you're looking for values <span class="math-container">$w$</span> for which <span class="math-container">$\tan^{-1} w$</span> exists. So you want to solve the following equation for <span class="math-container">$z$</span> :</p>
<p><span class="math-container">$$w=\tan z\tag{requires $z\neq\tfrac{(2k... |
779,509 | <p>I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. When constructing the expansion you can multiply by conjugates (kind of), e.g. </p>
<p>$\sqrt 3 =1+\sqrt 3 -1 = 1+\... | Redu | 735,404 | <p>The accepted answer looks like based on Vincent's continued fractions method (1836). Downside is it's inefficiency. Say, the root is at <span class="math-container">$0.000001$</span> so <span class="math-container">$a_0=0$</span>. In order to calculate the next term <span class="math-container">$a_1$</span> you have... |
3,154,332 | <p>I have a calculus question which i will display here as an image:
<a href="https://i.stack.imgur.com/8xN3P.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8xN3P.png" alt="enter image description here"></a></p>
<p>I am interested to understand part (b) of this question.
I actually got the answer, ... | SNEHIL SANYAL | 636,469 | <p>The first derivative of any function <span class="math-container">$y=f(x)$</span> at point <span class="math-container">$x$</span> gives you the slope of the tangent at that point or the rate of change of the function <span class="math-container">$y=f(x)$</span> at the point <span class="math-container">$x$</span>. ... |
3,939,620 | <p>Given a polynomial of the form <span class="math-container">$R(z):=\frac{P(z)}{Q(z)}$</span> such that <span class="math-container">$R(z)$</span> has no real roots and <span class="math-container">$deg(Q) \geq deg(P) + 2$</span>, then the integral can be expressed as</p>
<p><span class="math-container">$$\int_{-\inf... | lab bhattacharjee | 33,337 | <p>What if <span class="math-container">$\sin x(\sin x+1)=0?$</span></p>
<p>Else we need <span class="math-container">$\sin x(\sin x+1)>0$</span></p>
<p>If <span class="math-container">$\sin x>0,\sin x+1>0\iff\sin x>-1\implies \sin x>$</span> max<span class="math-container">$(0,-1)$</span></p>
<p><span ... |
3,059,571 | <p><span class="math-container">$$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$</span></p>
<p>I only know of L'hopital method but that is very long. Is there a shorter method to solve this?</p>
| zipirovich | 127,842 | <p>Here's yet another approach. First of all, notice that the factor of <span class="math-container">$\left(1+\tan\dfrac{x}{2}\right)$</span> in the denominator is the only one that is not equal to zero. So it has nothing to do with the <span class="math-container">$\dfrac{0}{0}$</span> indeterminate form in this limit... |
2,699,942 | <p>I am confused about one thing during the lecture. </p>
<p>Let $x_n = n$ and $A_n = \{x_k | k \ge n\} = \{n, n+1, n+2, ...\}$.</p>
<p>Then, $\inf A_n = n $, and $\sup A_n = \infty$. </p>
<p>My lecturer also said that $\lim\inf x_n = \lim\inf A _n=\lim n$. </p>
<p>My thinking is that $\{x_n\}_{n=1}^{\infty}=\{1, ... | mechanodroid | 144,766 | <p>Indeed strange since we usually define$$\liminf_{n\to\infty} A_n = \bigcup_{n=1}^\infty \left(\bigcap_{k=n}^\infty A_n\right)$$</p>
<p>which is clearly a set.</p>
<p>It can be shown that </p>
<p>$$\liminf_{n\to\infty} A_n = \{x \in \mathbb{R} : x \in A_n \text{ for all except finitely many } n \in \mathbb{N}\}$$<... |
2,255,617 | <p>I am trying to learn how to do proofs by contradiction. The proof is,</p>
<p>"Prove by Contradiction that there are no positive real roots of $x^6 + 2x^3 +4x + 5$"</p>
<p>I understand that now I am attempting to prove that there is a positive real root of this equation, so I am able to contradict myself within the... | Angina Seng | 436,618 | <p>Certainly not. Take for instance $\Bbb Z$ and $\Bbb Q$.
Or $\Bbb R$ and $\Bbb C$.</p>
|
21,372 | <blockquote>
<p>Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.</p>
</blockquote>
<p>How to solve this problem?</p>
| Ross Millikan | 1,827 | <p>Hint: if $x$ is quite large and positive, which of the two choices will be smaller? If $x$ is large and negative, which will be smaller? Can you find the crossover point?</p>
|
1,989,259 | <p><strong>Can modus tollens be statement of proof by contradiction or is it just a specific case of contradiction?</strong></p>
<p>i.e we know that in general, proof by contradiction stated as follows</p>
<p><span class="math-container">$[P' \implies (q \land q')] \implies P$</span></p>
<p>And by modus tollens, we hav... | Mauro ALLEGRANZA | 108,274 | <p>The two are equivalent.</p>
<p>For example, having <em>proof by contradiction</em> we get :</p>
<p>1) $(\lnot P \to Q) \land \lnot Q$ --- premise</p>
<p>2) $\lnot P \to Q$ --- from 1) by conjunction-elimination</p>
<p>3) $\lnot Q$ --- from 1) by conjunction-elimination</p>
<p>4) $\lnot P$ --- assumed [a]</p>
<... |
50,736 | <p>Hi guys,</p>
<p>I have recently started looking at polynomials $q_n$ generated by initial choices $q_0=1$, $q_1=x$ with, for $n\geq 0$, some recurrence formula</p>
<p>$$q_{n+2}=xq_{n+1}+c_n q_n$$</p>
<p>where $c_n$ is some function in $n$. The first few of these are</p>
<p>$$q_2=x^2+c_0$$
$$q_3=x^3+(c_0+c_1)x$$
... | Ira Gessel | 10,744 | <p>These polynomials are closely related to continuants, which arise in studying continuing fractions. The $n$th continuant of a sequence $a_0$, $a_1$, $\ldots$ is defined by $K(0)=1$, $K(1)=a_1$, $K(n)=a_n K(n-1) + K(n-2)$. They are sums of products of $a_1,\dots, a_n$ in which
consecutive pairs are deleted. (See, for... |
396,794 | <p>Let's say that a (right) module <span class="math-container">$M$</span> is <em>well complemented</em> if every non-zero submodule of <span class="math-container">$M$</span> has an indecomposable direct summand (by the way, is there a better or more standard name for this property?). For instance, every module of fin... | Benjamin Steinberg | 15,934 | <p>The answer is no. Take a compact totally disconnected space <span class="math-container">$X$</span> with no isolated points, like the Cantor set. Let <span class="math-container">$K$</span> be any field and let <span class="math-container">$R$</span> be the ring of locally constant functions <span class="math-cont... |
2,551,233 | <p>There are 4 fair coins and 1 unfair coin that has only heads. We choose a coin and flip it three times. The result is HHH. What is the probability that the fourth flip is H? </p>
| Mario | 508,026 | <p>$ p(H|HHH) = p(H|unfair) + 4 \cdot p(H|fair)= p(unfair) + p(fair) \cdot p(H) = \dfrac{1}{5} + \dfrac{4}{5} \dfrac{1}{2} = \dfrac{6}{10}$</p>
|
2,551,233 | <p>There are 4 fair coins and 1 unfair coin that has only heads. We choose a coin and flip it three times. The result is HHH. What is the probability that the fourth flip is H? </p>
| Remy | 325,426 | <p>We would expect the probability to be greater than $\frac{1}{2}$. We must solve for $P(\text{Heads})$. We have</p>
<p>$$P(\text{Heads})=P(\text{Heads}|\text{fair})\cdot P(\text{fair})+P(\text{Heads}|\text{unfair})\cdot P(\text{unfair})$$</p>
<p>However, we cannot just say $P(\text{fair})=\frac{4}{5}$ because we ar... |
1,533,362 | <p>I need to prove this identity:</p>
<p>$\sum_{k=0}^n \frac{1}{k+1}{2k \choose k}{2n-2k \choose n-k}={2n+1 \choose n}$</p>
<p>without using the identity:</p>
<p>$C_{n+1}=\sum_{k=0}^n C_kC_{n-k}$.</p>
<p>Can't figure out how to.</p>
| Marko Riedel | 44,883 | <p>Suppose we seek to prove that</p>
<p><span class="math-container">$$\sum_{k=0}^n \frac{1}{k+1} {2k\choose k}
{2n-2k\choose n-k} = {2n+1\choose n}.$$</span></p>
<p>We get for the LHS</p>
<p><span class="math-container">$$[z^n] (1+z)^{2n} \sum_{k=0}^n \frac{1}{k+1} {2k\choose k}
\frac{z^k}{(1+z)^{2k}}.$$</span></p>
<p... |
1,141,074 | <p>I need help with this integral: $$\int\frac{\sqrt{\tan x}}{\cos^2x}dx$$ I tried substitution and other methods, but all have lead me to this expression: $$2\int\sqrt{\tan x}(1+\tan^2 x)dx$$ where I can't calculate anything... Any suggestions? Thanks!</p>
| clathratus | 583,016 | <p>As you have noted, your integral simplifies to
<span class="math-container">$$2\int\sqrt{\tan x}\ \sec^2x\ dx$$</span>
If one makes the substitution <span class="math-container">$u=\tan x$</span>, one gets <span class="math-container">$du=\sec^2x dx$</span>, which reduces our integral to
<span class="math-container"... |
62,790 | <p>Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which captures the intuition, ... | Community | -1 | <p>How one models ordered pairs is not particularly important; what matters is the existence of a pairing function $(-,-)$ along with functions $\text{first}(-)$ and $\text{second}(-)$ satisfying the requisite properties.</p>
<p>Which definition you use only matters for a brief period between the definition and the po... |
1,144,141 | <p>I have a question for my exam and I find it hard to understand.</p>
<p>I have to prove that the following formula is logically valid:</p>
<p><img src="https://i.stack.imgur.com/kdELq.jpg" alt="Example"></p>
<p>The professor told me to "push" all the symbols inside the brackets, and use the deduction theorem.</p>
... | dtldarek | 26,306 | <p>Perhaps he meant something like this:</p>
<p>\begin{align}
\exists x\ \big(p(x) &\to \forall y\ p(y)\big) \\
\exists x\ \big(\neg p(x) &\lor \forall y\ p(y)\big) \\
\big(\exists x\ \neg p(x)\big) &\lor \big(\forall y\ p(y)\big) \\
\neg\big(\forall x\ p(x)\big) &\lor \big(\forall y\ p(y)\big) \\
\big... |
257,623 | <p>Consider the following ellipse, generated by the bounding region of the following points</p>
<pre><code>ps = {{-11, 5}, {-12, 4}, {-10, 4}, {-9, 5}, {-10, 6}};
rec = N@BoundingRegion[ps, "FastEllipse"];
Graphics[{rec, Red, Point@ps}]
</code></pre>
<p><a href="https://i.stack.imgur.com/gvtUB.png" rel="nofol... | Michael E2 | 4,999 | <p>Stolen from @J.M.'s answer, <a href="https://mathematica.stackexchange.com/a/239797/4999">https://mathematica.stackexchange.com/a/239797/4999</a>, with one correction (is that enough to make it not a duplicate?):</p>
<pre><code>Nodes = ps;
ellipsoidBR = BoundingRegion[Nodes, "FastEllipse"]; (* not "Fa... |
2,107,854 | <p>What is the limit when $n \to \infty$?</p>
<p>$$\lim_{n \to \infty} \frac{1}{n^4} \sum_{J=0}^{2n-1} J^3=?$$</p>
| Community | -1 | <p><strong>Hint:</strong> $$1^3+2^3+3^3+\cdots+k^3=\left(\frac{k(k+1)}{2}\right)^2.$$</p>
|
9,437 | <p>I love the way that Mathematica allows me to type in of formulas. It is really easy to type complicated expressions with shortcuts on the keyboard. It would be great if I could use Mathematica completely to publish my articles. The biggest reason I don't already do this is:</p>
<p>I can't find the proper tutorial f... | Silvia | 17 | <p>You can adjust page size, page number style, headers, footers, etc from items under <strong>File</strong> -> <strong>Printing Settings</strong> menu. Or you can programmatically modify them by manipulating <code>Notebook</code>'s options: <code>PrintingCopies</code>, <code>PrintingStartingPageNumber</code>, <code... |
2,040,041 | <p>I was able to think that the numerator will always be positive and will overpower the denominator as well. But couldn't proceed from there.</p>
| Black-horse | 170,518 | <p>Hint:</p>
<p>Let $f(x)=x^2e^x-2(e^x-(1+x)),x>0.$ Then
$$f^{'}(x)=x^2e^x+2xe^x-2(e^x-1)$$
and
$$f^{''}(x)=x^2e^x+2xe^x+2xe^x+2e^x-2e^x=x^2e^x+4xe^x>0.$$</p>
|
3,086,758 | <p>I know that if <span class="math-container">$\mathbb{E}[X]=\mathbb{E}[X|Y] , \mathbb{E}[Y]=\mathbb{E}[Y|X]$</span>, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> can be dependent, for example a ‘uniform’ distribution in a unit circle.
Now we add the variance, if
<span class="mat... | jmerry | 619,637 | <p>Here's a discrete example:
<span class="math-container">$$100P(x,y)=\begin{array}{c|ccccc}x\backslash y&-2&-1&0&1&2\\ \hline -2&1&0&6&0&1\\-1&0&9&0&9&0\\0&6&0&36&0&6\\1&0&9&0&9&0\\2&1&0&6&0&1\end{a... |
9,508 | <p>I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are watermarked with another code.</p>
<p>They should like kind of fancy and random. Currently they look like this:</p>
<p>1: ... | Yuval Filmus | 1,277 | <p>Take any odd $a$ and calculate $x \mapsto ax + b \pmod{2^{27}}$.</p>
<p>EDIT: Here's a more sophisticated suggestion. The following functions are all invertible and easy to implement:</p>
<ul>
<li> Multiplication by an odd number modulo $2^{27}$.
<li> Addition of an arbitrary number modulo $2^{27}$.
<li> XOR of an... |
4,127,149 | <p>I understand that the addition and subtraction of complex number is the same as vector addition and subtraction. But what is the vector equivalent of multiplication and division of complex numbers?</p>
| Somos | 438,089 | <p>You asked</p>
<blockquote>
<p>But what is the vector equivalent of multiplication and division of complex numbers?</p>
</blockquote>
<p>The Wikipedia article on <a href="https://en.wikipedia.org/wiki/William_Rowan_Hamilton" rel="nofollow noreferrer">William Rowan Hamilton</a> states</p>
<blockquote>
<p>Hamilton was ... |
247,553 | <p>Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex function on $[0,+\infty)$?</p>
<p>Updates:</p>
<p>1) It was pointed out by @user44191 that, observing $\binom{x}{i}=... | Liviu Nicolaescu | 20,302 | <p><em>This is not an answer to your question, is only an equivalent reformulation that seems promising. I write it as an answer only because of space constraints.</em></p>
<p>For any nonnegative integer $n$ and any $\newcommand{\bR}{\mathbb{R}}$ $x\in\bR$ we define</p>
<p>$$ a_n(x)=\sum_{k=0}^n \binom{x}{2k}, \;\;... |
247,553 | <p>Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex function on $[0,+\infty)$?</p>
<p>Updates:</p>
<p>1) It was pointed out by @user44191 that, observing $\binom{x}{i}=... | fedja | 1,131 | <p>Since it looks like the best we have at the moment is a casework with some brute force estimates, I'll post it just to set the upper bound for the proof clumsiness. </p>
<p><strong><em>Case 1: $0\le x\le 1$.</em></strong></p>
<p>We have to show that the even coefficients of $F(t)=\frac{(1+t)^x\log^2(1+t)}{1-t^2}$ ... |
853,659 | <p>Evaluate the integral:</p>
<p>$$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$</p>
<p>After a lot of help I have reached this point:</p>
<p>$x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$</p>
<p>But now I don't really know how to solve for $A, B, C$, and $D$. Please help!</p>
| amWhy | 9,003 | <p>First: Divide! Use polynomial division to get $$\frac{x^6}{x^4 - 1} = 1 + \frac{x^2}{x^4 - 1}$$ $\int 1\,dx = x + C$</p>
<p>For the second term:</p>
<p>Now factor the denominator, and decompose: $$x^4 - 1 = (x^2 + 1)(x^2-1) = (x^2 + 1)(x-1)(x+1)$$</p>
<p>So the set up we want for the second term is:</p>
<p>$$\fr... |
4,473,543 | <p>For example, given <span class="math-container">$\color{green}{l_1:5x-2y-8=0}$</span> and <span class="math-container">$\color{blue}{l_2:3x+8y-8=0}$</span>,</p>
<p><a href="https://i.stack.imgur.com/sxZJ1m.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sxZJ1m.jpg" alt="img1" /></a></p>
<p>We can ... | AlexSp3 | 963,724 | <p>Let's say that
<span class="math-container">$$L_1 : \{Ax+By+C=0\} \Longrightarrow \vec{L_1} =(A, B, C)$$</span>
<span class="math-container">$$L_2 : \{A'x+B'y+C'=0\} \Longrightarrow \vec{L_2} =(A', B', C')$$</span>
The intersection point can be found by solving the system by Cramer rule:
<span class="math-container"... |
168,819 | <p>I was looking for a free PDF from which I can review MV calculus.</p>
<p>Specifically:</p>
<ol>
<li>MV Limits, Continuity, Differentiation.</li>
<li>Differentiation of vector and scalar fields</li>
<li>Surface/Multiple Integrals</li>
</ol>
<p>A succinct book would be great, (coherent) course notes and presentatio... | Brian M. Scott | 12,042 | <p>You might try Paul Dawkins’ on-line <a href="http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx" rel="nofollow">Calculus III notes</a>, which can be downloaded in PDF format. I’ve not looked at them, but I’ve taught Calculus I and II from his notes for those courses and found them quite usable, though there... |
2,137,332 | <p>On the MathWorld page: </p>
<p><a href="http://mathworld.wolfram.com/FermatPseudoprime.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/FermatPseudoprime.html</a></p>
<p>in the first table, I expect to see $561$ on every line, but it is not on the line for base $3$.</p>
<p>When you click on the link ... | Matematleta | 138,929 | <p>For $z=x+iy\ $ write $z=|z|e^{it}, $ where $|z|=\sqrt{x^{2}+y^{2}}\ $ and $\tan t=y/x.\ $Then, $z\mapsto |z|^2e^{2it},\ $ so $|z|\mapsto |z|^2$ and $t\mapsto 2t,\ $ which means that each point in the complex plane doubles its angle, and squares its modulus. </p>
<p>Therefore, the transformed triangle will be the re... |
157,876 | <p>Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?</p>
<p>In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?</p>
| Arturo Magidin | 742 | <p>In any group, a subgroup is normal if and only if it is a union of conjugacy classes. </p>
<p>In $S_n$, the conjugacy classes are very easy: a conjugacy class consists exactly of all permutations of a given cycle structure. These corresponds to all possible partitions of $n$.</p>
<p>So, consider $S_4$. The conjuga... |
3,458,154 | <p>I came across this exercise in my Measure Theory workbook and I've been stuck on it. This is the question :</p>
<p>Let F be the set of all non-decreasing right-continuous functions <span class="math-container">$f : \mathbb{ R} \rightarrow \mathbb{R}$</span> with
<span class="math-container">$f(0) = 0$</span>.
Let ... | kam | 514,050 | <p>I am also trying to show the same problem. It is straightforward to show injectivity:</p>
<p>firstly take two functions <span class="math-container">$f_1, f_2$</span> that are not equal, that is to say they differ at some point <span class="math-container">$x\in{\mathbb{R}}$</span> s.t. <span class="math-container"... |
3,458,154 | <p>I came across this exercise in my Measure Theory workbook and I've been stuck on it. This is the question :</p>
<p>Let F be the set of all non-decreasing right-continuous functions <span class="math-container">$f : \mathbb{ R} \rightarrow \mathbb{R}$</span> with
<span class="math-container">$f(0) = 0$</span>.
Let ... | Henry Garrett | 506,609 | <p>for surjectivity I think try:</p>
<p>Let <span class="math-container">$\mu \in M$</span> then set <span class="math-container">$f(x)= \mu ((0,x])$</span> for positive <span class="math-container">$x$</span> similar for negative. then show f must be increasing and right cts. </p>
|
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | StasK | 97,144 | <p>This is most likely a non-answer, but my (personal, strong, heavily biased, another math culture infused) opinion is that the confusion stems from things being presented in calculus classes in a bizarre illogical order, starting with complicated things (limits of change in functions, i.e., derivatives) and then, as ... |
2,355,852 | <p>Given are $m$ bins with equal probability of choosing one of them. Unknown number of balls $n$ is placed into the bins, and, at the end of placement, we observe number of empty bins $m_e$ and non-empty bins $m_{n}$.</p>
<p>Given $m$, $m_e$, $m_n$, what is the most likely number of balls $n$, which have been placed ... | G Cab | 317,234 | <p>1) <em>Premise</em> </p>
<p><em>Randomly throw (put) $n$ balls into $m$ bins (of unlimited capacity)</em><br>
means that you consider as equiprobable and indipendent events the<br>
<em>launch the $k$-th ball into the $j$-th bin</em><br>
i.e., a sequence of $n$ independent events, each having $m$ equiprobable resu... |
2,330,196 | <p>The question asks me to draw a Hasse diagram for the given set of rules.
$$ (\{n\in \mathbb N: n\mid 100\ \lor\ n = 75 \}, {}\mid{} ) $$</p>
<p>My approach is to write down the set satisfying for $n\mid 100$, but I dont get what's with "or" $n =75.$</p>
<p>Could someone help me figure out what that means? is it s... | Francesco Polizzi | 456,212 | <p>Riemann-Roch implies that the degree of a canonical divisor on a compact Riemann surface of genus $g$ is $2g-2$. </p>
<p>On the other hand, a direct computation using differential forms shows that any canonical divisor on the Riemann sphere has degree $-2$, hence $g=0$. </p>
|
2,897,785 | <blockquote>
<p>Fix a $2\times 2$ real matrix $A$. Let $V$ be the set of all $2\times 2$ real matrices $X$ such that $AX=XA$. Show that $V$ is a vector space of dimension of at least 2.</p>
</blockquote>
<p>I'm struggling to see a good way to approach this problem. There's the brute force style method of algebraical... | user | 505,767 | <p>We can consider the following cases</p>
<ul>
<li>$A=0$ then X can be any matrix</li>
<li>$A\neq 0$ singular then $X$ can be $kI$ and $kA$</li>
<li>$A=kI$ then X can be any matrix</li>
<li>$A\neq kI$ not singular then $X$ can be $kI$ and $kA$</li>
</ul>
|
2,778,575 | <p>Given the equation: $\sin^2{x}+\cos{x}=0$</p>
<p>How is it solved?</p>
<p>I think: $\sin^2{x}=1-\cos^2{x}$, but even if I get a quadratic equation with one function (cos), how can I solve it?</p>
| Mohammad Riazi-Kermani | 514,496 | <p>$$\sin^2{x}+\cos{x}=0$$</p>
<p>$$1-\cos ^2 x +\cos x =0$$</p>
<p>$$\cos ^2 x -\cos x -1 =0$$</p>
<p>$$ \cos x = \frac {1-\sqrt 5 }{2}$$</p>
<p>$$x= \cos ^{-1} (\frac {1-\sqrt 5 }{2}) \approx 128.17 \text { degrees.}$$</p>
|
818,169 | <p>The variational distance is defined by,
$$
V(P,Q)=\sum _{i}|p_{i} -q_{i} |
$$
where $P=(p_{1} ,...,p_{n})$ and $Q=(q_{1} ,...,q_{n} )$ are discrete distributions.</p>
<p>It is fairly easy to see that $V$ is a metric, and in particular that it satisfies the triangle inequality. I now introduce class prior, $0<\... | Omri | 21,373 | <p>Actually, I've recently realized that the inequality is false. It is true only for equal class priors. Sorry about that. </p>
|
1,853,464 | <p>I am using the Lorentz Force Equation and the electric-cross-magnetic field velocity equation] to solve for the $E$ and $B$ fields given the known path of a particle moving in 3D. </p>
<p>So with that I have the following equations where a and v are known:
<a href="https://i.stack.imgur.com/oL7fg.gif" rel="nofollow... | Eugenio | 305,569 | <p>Well, I'm not sure if they are sufficient, but you could add <a href="https://en.wikipedia.org/wiki/Magnetostatics#Magnetostatics_as_a_special_case_of_Maxwell.27s_equations" rel="nofollow">magnetostatic equations</a> if your magnetic field is constant over time, or instead <a href="https://en.wikipedia.org/wiki/Fara... |
1,369,641 | <p>I do not know how to set this problem up. Any insight as to how to get the equation would be great. </p>
<p>It is John's birthday and his parents want to make him a cake in the shape of a rectangular box. The height of the cake will be $15$ centimeters, and $2$ times the width plus $2$ times the length will be $180... | Community | -1 | <p>Only the base are matters, as the height is fixed.</p>
<p>Then, $w+l=90$. You need to maximize $wl=w(90-w)$.</p>
<p>By canceling the derivative, $90-2w=0$, or $w=45$, and $l=45$. The largest area (volume) is given by a square (square prism).</p>
|
129 | <p>Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not sufficient. Are there any other special homotopical properties of manifolds?</p>
| Mike | 1,579 | <p>I'm relying on memory here. A good example, which is discussed in Madsen and Milgram's book on surgery and classifying spaces for topological, $PL$ and smooth manifolds is the set of $1$-connected Poincare duality spaces of dimension 5 with 4-skeleton h. e. to the $4$-skeleton of $S^2\times S^3$, which is $S^2\vee S... |
202,699 | <p><a href="https://i.stack.imgur.com/UqPw4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UqPw4.png" alt="enter image description here"></a></p>
<p>I try to solve for "t" at the various "x" from the function of </p>
<pre><code>f[t_, x_] =
0.5 Erfc[(x - 0.0236454911650369 t)/Sqrt[4*0.010827497... | Roman | 26,598 | <p>Here's a code snippet that I wrote a long time ago, specifically for the Hammer projection.</p>
<pre><code>HammerPlot::usage=
"HammerPlot[f] shows a Hammer projection (see http://en.wikipedia.org/wiki/Hammer_projection) of a function f[θ,φ]. "~~
"The option ViewPoint->{Θ,Φ,χ} places the viewer over the poin... |
456,892 | <p>Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$.</p>
<p>I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the problem.</p>
| Sachin | 68,597 | <p>$$
\begin{align}
4\cos^2x -4\sin x -5 & = 0 \\
\Rightarrow 4(1-\sin^2x)-4\sin x -5 &=0 \\
\Rightarrow 4\sin^2x+4\sin x +1 &=0
\end{align}
$$</p>
<p>Let $\sin x =t$ </p>
<p>$$\Rightarrow 4t^2+4t+1=0$$ </p>
<p>Solving this equation you get </p>
<p>$$\begin{align}
t & =\frac{-1}{2} \\
\Rightarrow ... |
204,365 | <p>Consider a positive matrix <code>M</code> and a positive vector <code>b</code>, e.g.</p>
<pre><code>nn = 1000;
M = Table[RandomReal[{0, 100}], {i, 1, nn}, {j, 1, nn}];
b = Table[RandomReal[{0, 100}], {i, 1, nn}];
</code></pre>
<p>I would like to find a positive vector <code>X</code></p>
<pre><code>X = Array[x,... | Carl Woll | 45,431 | <p>You can use the new in M12 function <a href="http://reference.wolfram.com/language/ref/QuadraticOptimization" rel="noreferrer"><code>QuadraticOptimization</code></a>, which minimizes functions of the form:</p>
<p><span class="math-container">$$\frac{1}{2} x . q. x + c . x$$</span></p>
<p>subject to linear constrai... |
2,304,379 | <p>My textbook give the following definition.</p>
<blockquote>
<p>Let $G$ be any topological group. A representation of $G$ on a nonzero complex Hilbert space $V$ is a group homomorphism $\phi$ of $G$ into the group of bounded linear operators on $V$ with bounded inverses, such that the resulting map $ G\times V\to ... | Alex | 293,781 | <p>Here is an example of such a set with infinitely many elements: </p>
<p>The <a href="https://math.stackexchange.com/questions/1270822/">set of integers in $\mathbb R$</a> is closed, and any nonempty subset of integers is closed as well.</p>
|
1,102,638 | <p>Let $n\in \mathbb{N}$. Can someone help me prove this by induction:</p>
<p>$$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$</p>
| Alex Silva | 172,564 | <p><strong>Hint:</strong></p>
<p>Multiply the numerator and the denominator by $$\sqrt{1+x+x^2}+1.$$</p>
|
3,065,818 | <blockquote>
<p>If <span class="math-container">$$z=\dfrac{\sqrt{3}-i}{2}$$</span> then <span class="math-container">$$(z^{95}+i^{67})^{94}=z^n$$</span> then, <span class="math-container">$\text{find the smallest positive integral value of}$</span> <span class="math-container">$n$</span> <span class="math-container">... | Bill Dubuque | 242 | <p>Mimic <span class="math-container">$\rm\color{#c00}{subtractive}$</span> Euclidean algorithm on <span class="math-container">$\color{#c00}{(a,b)}.\,$</span> Clear if <span class="math-container">$\,a\!=\!b\,$</span> by <span class="math-container">$\,a,b\,$</span> coprime <span class="math-container">$\Rightarrow\, ... |
638,875 | <p>Let $P$ be a $p$-group and let $A$ be maximal among abelian normal subgroups of $P$. Show that $A=C_P(A)$.</p>
<p>This is the second part of a problem in which I successfully proved the following:
Let $P$ be a finite $p$-group and let $U<V$ be normal subgroups of $P$. Show that there exists $W \triangleleft P$... | DonAntonio | 31,254 | <p>It is an easy-to-prove fact that if $\;|P|=p^n\;,\;\;p\;$ a prime, then for any $\;0\le k\le n\;$ there exists a normal subgroup of $\;P\;$ of order $\;p^k\;$ (induction + the center of finite $\;p-$groups is non-trivial).</p>
<p>The above proves your part one without problem.</p>
<p>Let's now try the following (B... |
4,326,073 | <p><a href="https://i.stack.imgur.com/RTyOy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTyOy.jpg" alt="enter image description here" /></a>
I came across questions in the free module section of my abstract algebra text. In the text, the notation <span class="math-container">$End_{R}(V)$</span> d... | Zieac | 991,422 | <p>For any <span class="math-container">$\varepsilon > 0$</span> we have <span class="math-container">$N$</span> such that any <span class="math-container">$n \ge N$</span>, <span class="math-container">$a - \varepsilon \le a_n \le a + \varepsilon$</span> holds.
Let <span class="math-container">$M_1 = \sum_{i = 1}^{... |
4,326,073 | <p><a href="https://i.stack.imgur.com/RTyOy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTyOy.jpg" alt="enter image description here" /></a>
I came across questions in the free module section of my abstract algebra text. In the text, the notation <span class="math-container">$End_{R}(V)$</span> d... | Theo Bendit | 248,286 | <p>An application of <a href="https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="noreferrer">Stolz-Cesaro</a> works:
<span class="math-container">\begin{align}
\lim_{n \to \infty} \frac{\sum_{i=1}^n ia_i}{n^2} = \frac{a}{2} &\color{red}\impliedby \lim_{n \to \infty} \frac{\sum_{i=1}^{n+1} ia_i - ... |
352,849 | <p>I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ </p>
<hr>
<p>I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}$$ and $$\lim \limits_{... | dtldarek | 26,306 | <p><strong>Hint:</strong></p>
<p>$$ 0 \leq \lim_{n\to \infty}\frac{n!}{(2n)!} \leq \lim_{n\to \infty} \frac{n!}{(n!)^2} = \lim_{k \to \infty, k = n!}\frac{k}{k^2} = \lim_{k \to \infty}\frac{1}{k} = 0.$$</p>
|
1,380,402 | <p>I'm developing a C++ program and I need to find a formula that given a number to reduce and a limit number, get a value between 0 and this limit number.</p>
<p>I don't know if it is allow to put C++ code here, but I want to show you my function:</p>
<pre><code>double Utils::reduceNumber(double numberToReduce, doub... | mathlove | 78,967 | <p>Let $a$ be a given number. Also, suppose that the limit is between $0$ and $N$.</p>
<p>If you want an integer $b$ such that
$$0\le a+Nb\le N\iff -\frac{a}{N}\le b\le \frac{N-a}{N},$$
then $$b=\left\lfloor\frac{N-a}{N}\right\rfloor$$
works where $\lfloor x\rfloor$ is the largest integer not greater than $x$.</p>
... |
1,380,402 | <p>I'm developing a C++ program and I need to find a formula that given a number to reduce and a limit number, get a value between 0 and this limit number.</p>
<p>I don't know if it is allow to put C++ code here, but I want to show you my function:</p>
<pre><code>double Utils::reduceNumber(double numberToReduce, doub... | anak | 133,414 | <p>Just setting up some notation:</p>
<p>For your limit number $l > 0$ and the number you want to "reduce", $n$, you want to find another number $x$ such that it satisfies: $$(n + l\cdot x) \in (0,l).$$</p>
<p>So for example, we know that $\frac{l}{2} \in (0,l)$ obviously, so we will say you want to satisfy the eq... |
51,752 | <p>Can someone give an argument, if possible using only the axioms of set theory, because I'm <strong>very</strong> weak there and have virtually no background, except the usual knowledge of the operation with sets one has to have when doing non-set theoretic non-research mathematics, why $\emptyset \in \emptyset$ or $... | Zev Chonoles | 264 | <p>An axiomatic argument (as ccc points out, we must assume that the ZF axioms are in fact consistent) would proceed as follows: By the <a href="http://en.wikipedia.org/wiki/Axiom_of_empty_set">axiom of the empty set</a>, $\forall x(\neg x\in\emptyset)$. So in particular, it is false that $\emptyset\in\emptyset$. </p>
... |
1,019,078 | <p>Let $\alpha_1=[ 2,1,3,0] $
$\alpha_2=[ 1,1,1,-1] $, $\alpha_3=[ 2,-1,5,4] $, $\alpha_4=[ 1,2,0,-3] $, $\alpha_5=[ 3,1,6,1] $
be vectors from $\mathbb{R}^4$ . From vectors system ($\alpha_1,\alpha_2, \alpha_3, \alpha_4, \alpha_5 $) choose basis of vector space $V=lin(\alpha_1,\alpha_2, \alpha_3, \alpha_4, \alpha_5)\s... | egreg | 62,967 | <p>Consider the matrix
\begin{bmatrix}
2 & 1 & 2 & 1 & 3\\
1 & 1 & −1 & 2 & 1\\
3 & 1 & 5 & 0 & 6\\
0 & −1 & 4 & −3 & 1
\end{bmatrix}
and perform Gaussian elimination on it:
\begin{align}
\begin{bmatrix}
2 & 1 & 2 & 1 & 3\\
1 & 1 & ... |
184,824 | <p>I have two piecewise function</p>
<pre><code>equ1 = Piecewise[{{0.524324 + 0.0376478x, 0.639464 <= x <= 0.839322}}]
equ2 = Piecewise[{{-0.506432 + 1.48068x, 0.658914 <= x <= 0.77085}}]
</code></pre>
<p>Now, I am trying to solve <code>equ1 = equ2</code>.</p>
<p>Firstly I tried <code>FindRoot</code>: </... | Michael E2 | 4,999 | <p>Either:</p>
<pre><code>FindRoot[equ1 == equ2, {x, 0.7, 0.639464, 0.839322}]
(* {x -> 0.714299} *)
</code></pre>
<p>Or:</p>
<pre><code>NSolve[equ1 == equ2 && 0.639464 <= x <= 0.839322, x]
</code></pre>
<blockquote>
<p><code>NSolve::ratnz</code>:.... <em>[Unimportant warning.]</em></p>
</block... |
1,113,760 | <blockquote>
<p>$\frac{4}{3} e^{3x} + 2 e^{2x} - 8 e^x$</p>
</blockquote>
<p>I have some confusion especially because of the e </p>
<p>how can I approach the solution?</p>
<p>The solution of the x-intercept is 0.838</p>
<p>Many thanks</p>
| Eff | 112,061 | <p><strong>Hint:</strong> The $x$-intercept is when $\frac43 e^{3x}+2e^{2x}-8e^{x} = 0$. Now set $y = e^{x}$, so your equation is </p>
<p>$$\frac{4}{3}y^3+2y^2-8y = 0 $$</p>
<p>which means</p>
<p>$$y\left(\frac{4}{3}y^2+2y-8\right) = 0. $$</p>
|
4,530,792 | <p>I have the following sequence <span class="math-container">$\left \{k \sin \left(\frac{1}{k}\right) \right\}^{\infty}_{1}$</span>. I don't know how to show that this is monotonically increasing.</p>
<p>I tried taking the derivative of the corresponding function <span class="math-container">$f(x) = x \sin \left(\frac... | DonAntonio | 31,254 | <p>Observe that</p>
<p><span class="math-container">$$\sin\frac1x-\frac1x\cos\frac1x\ge0\iff\tan\frac1x\ge\frac1x$$</span></p>
<p>Put now <span class="math-container">$\;t:=\frac1x\;$</span>, so that you want to find about</p>
<p><span class="math-container">$$f(t):=\tan t-t\;,\;\;f'(t)=\frac1{\cos^2t}-1\ge0\;\;\forall... |
4,394,247 | <p>I know how to represent the sentence “there is exactly one person that is happy”,</p>
<p>∀y∀x((Happy(x)∧Happy(y))→(x=y))</p>
<p>Edit: ∃x∀y(y=x↔Happy(y))
(NOW, I actually know how to represent it)</p>
<p>Where x and y represent a person.</p>
<p>However, my problem is that I can’t figure out how to say “there are exac... | Tom Sharpe | 342,007 | <p>You haven't actually asserted that anyone <em>is</em> happy. So, You need to introduce a happy person, and then assert that anyone else who is happy is actually that person:
<span class="math-container">$$\exists x\left(\mathrm{Happy}(x)\wedge\left(\forall y\left(\mathrm{Happy}(y)\to (y=x)\right)\right)\right).$$</s... |
71,184 | <p>I'm solving some problems for practice, and I've come across a something I don't quite understand... So here's the deal:</p>
<blockquote>
<p>$A = \{x \in \mathbb{N}: -1 \leq x < 2\}$</p>
<p>$B = \{x \in \mathbb{Z}: -10 < x \leq 0\}$</p>
<p>$C = \{n \in \mathbb{Z}: n = 2k + 1, k \in \mathbb{Z}\}$</... | Zev Chonoles | 264 | <p>The two examples you gave are both <a href="http://en.wikipedia.org/wiki/Algebraic_functions" rel="nofollow">algebraic functions</a>, i.e. functions that satisfy a polynomial equation whose coefficients are rational functions. For example,
$$f=9-\sqrt{x}$$
satisfies the polynomial equation
$$(y-9)^2-x=y^2-18y+(81-x)... |
71,184 | <p>I'm solving some problems for practice, and I've come across a something I don't quite understand... So here's the deal:</p>
<blockquote>
<p>$A = \{x \in \mathbb{N}: -1 \leq x < 2\}$</p>
<p>$B = \{x \in \mathbb{Z}: -10 < x \leq 0\}$</p>
<p>$C = \{n \in \mathbb{Z}: n = 2k + 1, k \in \mathbb{Z}\}$</... | Gerry Myerson | 8,269 | <p>I suppose you could call them Puiseux polynomials by analogy with <a href="http://en.wikipedia.org/wiki/Puiseux_series" rel="nofollow">Puiseux series,</a> though I'm not sure anyone has ever done so. </p>
|
3,794,101 | <blockquote>
<p>Show that <span class="math-container">$f: \mathbb{R^3} \to \mathbb{R}$</span> <span class="math-container">$$f(x, y, z) = xy + z^2$$</span> is continuous.</p>
</blockquote>
<p>One could just deduce that since it's a polynomial it's continuous, but how would I show this using <span class="math-container... | José Carlos Santos | 446,262 | <p>Note that if <span class="math-container">$(x,y,z),(x_0,y_0,z_0)\in\Bbb R^3$</span>, then<span class="math-container">\begin{align}\bigl|f(x,y,z)-f(x_0,y_0,z_0)\bigr|&=|xy-x_0y_0+z^2-z_0^{\,2}|\\&\leqslant|xy-x_0y_0|+|z^2-z_0^{\,2}|\\&=\bigl|(x-x_0)y_0+(y-y_0)x_0+(x-x_0)(y-y_0)\bigr|+\\&\phantom{=}+\... |
979,432 | <p>i was recently watching a single variable calculus video of mit 18.01, lecture 23. in that it is said that average height of a point on semicircle with respect to arc length is 2/pi.I have a hard time to understand that point. i understand why average height of point on semi circle with respect to x is pi/4. but i d... | aschepler | 2,236 | <p>The "average value" of any formula $\varphi$ with respect to any increasing variable $\xi$ is defined as</p>
<p>$$ \frac{\int \varphi \, d\xi}{\int d\xi} .$$</p>
<p>For a unit semicircle, arc length is equal to the angle $\theta$, so we can write the average of height $y$ with respect to arc length as</p>
<p>$$ \... |
3,379,837 | <p>I know that the two semigroups <span class="math-container">$(\{0,1,2,\dots \},\times)$</span> and <span class="math-container">$(\{0,1,2,\dots \},+)$</span> are not isomorphic because if we want to map identity elements together then it can be see that we can't have injective function between them,but what can we ... | J.-E. Pin | 89,374 | <p>Suppose there is an isomorphism <span class="math-container">$f:(\Bbb{N},+) \to ((\Bbb{N}-\{0\}, \times)$</span>. Then since <span class="math-container">$f$</span> preserves idempotents, one has <span class="math-container">$f(0) = 1$</span>. Let <span class="math-container">$a = f(1)$</span>. Then for every <span ... |
617,927 | <p>Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.</p>
<p><strong>Taylor Series</strong></p>
<p>$$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$</p>
<p>I've got my first term...</p>
<p>$f(a) = \sin(-1+1)\sin(-1+2)=\sin(0)\sin(1)=0$</p>
<p>Now, I... | Thomas Belulovich | 831 | <p>Hint: </p>
<p>$f′(x)=\sin(x+1)\cos(x+2)+\sin(x+2)\cos(x+1) = \sin(2x+3).$</p>
|
3,646,911 | <p>Exercise 14.7.4 from Dummit and Foote</p>
<blockquote>
<p>Let <span class="math-container">$K=\mathbb{Q}(\sqrt[n]{a})$</span>, where <span class="math-container">$a\in \mathbb{Q}$</span>, <span class="math-container">$a>0$</span> and suppose <span class="math-container">$[K:\mathbb{Q}]=n$</span>(i.e., <span cl... | Gareth Ma | 623,901 | <p>It can be written as
<span class="math-container">$$x^{-5}\sum_{n=0}^5 \binom{5}{n}(-1)^n x^{10-2n}$$</span>
<span class="math-container">$$x^{-5}\sum_{n=0}^5 \binom{5}{n}(-1)^n (x^2)^{5-n}$$</span>
<span class="math-container">$$=x^{-5}(x^2-1)^5$$</span>
<span class="math-container">$$=(\frac{x^2-1}{x})^5=32$$</spa... |
3,646,911 | <p>Exercise 14.7.4 from Dummit and Foote</p>
<blockquote>
<p>Let <span class="math-container">$K=\mathbb{Q}(\sqrt[n]{a})$</span>, where <span class="math-container">$a\in \mathbb{Q}$</span>, <span class="math-container">$a>0$</span> and suppose <span class="math-container">$[K:\mathbb{Q}]=n$</span>(i.e., <span cl... | Condo | 409,795 | <p>Factoring out the <span class="math-container">$x^5$</span> (because it doesn't depend on <span class="math-container">$n$</span>) we obtain <span class="math-container">$$x^5\sum_{n=0}^5(-1)^n{5 \choose n}x^{-2n}=32.$$</span> Now remark that <span class="math-container">$x^{-2n}=(\tfrac{1}{x^2})^n$</span>. So by th... |
833,827 | <p>I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas.<br>
E.g.<br>
I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex e.g. something like $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}$$<br>
Is there a good refreshe... | vonbrand | 43,946 | <p>The other answers are right, but they assume the innermost loop does work that is proportional to $k$, while I believe you intend it to be constant. You are right, the total work done is $O(N^3)$. You can use the sums-of-powers formulas mentioned to get the precise value if needed.</p>
|
988,628 | <p>Problem : </p>
<p>For the series $$S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2+\frac{1}{(1+3+5+7)}(1+2+3+4)^2+\cdots $$ Find the nth term of the series. </p>
<p>We know that nth can term of the series can be find by using $T_n = S_n -S_{n-1}$ </p>
<p>$$S_n =1+ \sum \frac{(\frac{n(n+1)}{2})^2}{(2n-1)... | Deepak | 151,732 | <p>You don't need to bother with calculating $S_n - S_{n-1}$ here as the terms are explicitly given in the summation.</p>
<p>Here $T_n = \frac{(1+2+...+n)^2}{(1+3+5+...+(2n-1))}$</p>
<p>The numerator is the square of the first $n$ integers, so is equal to $(\frac{1}{2}n(n+1))^2$.</p>
<p>The denominator is the sum of... |
988,628 | <p>Problem : </p>
<p>For the series $$S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2+\frac{1}{(1+3+5+7)}(1+2+3+4)^2+\cdots $$ Find the nth term of the series. </p>
<p>We know that nth can term of the series can be find by using $T_n = S_n -S_{n-1}$ </p>
<p>$$S_n =1+ \sum \frac{(\frac{n(n+1)}{2})^2}{(2n-1)... | Leucippus | 148,155 | <p>Consider the series
\begin{align}
S_{n} = 1 + \frac{(1+2)^{2}}{1+3} + \frac{(1+2+3)^{2}}{1+3+5} + \cdots + \frac{(1+2+\cdots+n)^{2}}{1+3+\cdots+(2n-1)}.
\end{align}
This series is seen as
\begin{align}
S_{n} &= 1 + \frac{1}{2^2}\binom{3}{2}^{2}+ \frac{1}{3^{2}} \binom{4}{2}^{2}+ \cdots + \frac{1}{n^{2}} \binom{n... |
3,735,798 | <blockquote>
<p><strong>QUESTION:</strong> Given a square <span class="math-container">$ABCD$</span> with two consecutive vertices, say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the positive <span class="math-container">$x$</span>-axis and positive <span class="math-containe... | toronto hrb | 802,748 | <p><a href="https://i.stack.imgur.com/qZVTM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qZVTM.jpg" alt="enter image description here" /></a></p>
<p>The area you want is the difference between the large square and 4 triangles. If <span class="math-container">$u\ge v$</span>, you have another case.... |
3,735,798 | <blockquote>
<p><strong>QUESTION:</strong> Given a square <span class="math-container">$ABCD$</span> with two consecutive vertices, say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the positive <span class="math-container">$x$</span>-axis and positive <span class="math-containe... | Narasimham | 95,860 | <p>Let <span class="math-container">$a,b $</span> be x and y intercepts. Draw lines parallel to x-axis and y-axis.</p>
<p><span class="math-container">$$ a= v- u,\; b=u $$</span></p>
<p>The diagram will be helpful.</p>
<p><a href="https://i.stack.imgur.com/8lq5c.png" rel="nofollow noreferrer"><img src="https://i.stack... |
1,675 | <p>This is a follow-up to <a href="https://mathoverflow.net/questions/1039/explicit-direct-summands-in-the-decomposition-theorem">this post</a> on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.</p>
<p>My question is how does one use the De... | Ben Webster | 66 | <p>The short answer is that in general its very hard. For special classes of maps like semi-small ones, it's not so bad (see the book of Chriss and Ginzburg), but for an arbitrary projective map, I don't know any reliable way of dealing with it. If you know the IC sheaves downstairs well, you can use point counting (... |
2,793,384 | <p>I am taking a course in Algebraic Topology next semester, so I thought of starting to read about it on my own.</p>
<p><strong>My concern:</strong> I have read <em>Topology</em> by Munkres, and read the first chapter on algebraic topology from that book, so I have an idea about the fundamental group and covering sp... | Community | -1 | <p>I have a few suggestions:</p>
<p>1) Algebraic Topology by Hatcher is a very readable book that explains things moderately well in more of an informal manner - lots of diagrams for low dimensional things. If you already know about covering spaces and fundamental groups this book will be easily accessible. </p>
<p>2... |
4,236,878 | <p>Given a symmetric matrix <span class="math-container">$S$</span> and positive definite matrix <span class="math-container">$B$</span>, with <span class="math-container">$S,B \in \mathbb{R}^{n \times n}$</span> can one prove that</p>
<p><span class="math-container">\begin{align*}
\text{tr}((S-B)B) \le -\mu(S) \text{t... | march | 852,914 | <p>Using the additive and cyclic properties of the trace, we can write
<span class="math-container">$$
\operatorname{Tr}((S-B)B) = \operatorname{Tr}(SB) -\operatorname{Tr}(B^2) = \operatorname{Tr}(BS) - \operatorname{Tr}(B^2).
$$</span>
Provided the matrix <span class="math-container">$B$</span> has all real eigenvalue... |
2,107,685 | <p><a href="https://i.stack.imgur.com/GtU6e.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GtU6e.png" alt="laaa"></a></p>
<p>I have to represent the function on the left as a power series, and this is the solution to it but I don't know how to calculate this for example when n=1?</p>
| Disintegrating By Parts | 112,478 | <p>Whenever you apply a spectral projection $E(S)\ne I$ to $A$, you end up with $0$ in the point spectrum of $E(S)A=AE(S)$ because $\{AE(S)\}E(\sigma\setminus S)=0$ and $E(\sigma\setminus S) \ne 0$. So that special case always requires special attention.</p>
<p>If $(a,b)\subseteq\sigma(A)$, then $E(a,b) \ne 0$; otherw... |
432,964 | <p>Let $X\in \mathbb{R}^{n \times n}$.
Then, is the function</p>
<p>$$ \text{Tr}\left( (X^T X )^{-1} \right)$$ </p>
<p>convex in $X$? ($\text{Tr}$ denotes the trace operator)</p>
| user1551 | 1,551 | <p>As pointed out in the above by user "1015" (who keeps changing his username ^_^ ), the set of all invertible matrices is not convex. Therefore your question does not make sense. However, $\operatorname{tr}\left((X^TX)\right)^{-1}$ is locally convex at every invertible matrix $X$ and this can be proved using the tric... |
426,499 | <p>Let <span class="math-container">$X$</span> be a separable metric space which is <em>homogeneous</em>, i.e. for every two points <span class="math-container">$x,y\in X$</span> there is a homeomorphism <span class="math-container">$h$</span> of <span class="math-container">$X$</span> onto itself such that <span class... | YCor | 14,094 | <p>Since you want a connected example:</p>
<p>A surface of infinite genus has no homogeneous compactification.</p>
<p>Indeed first observe a dense locally compact subset has to be open.</p>
<p>So the surface has to be open, and by homogeneity the compactification is a closed surface. But an open subset of a closed surf... |
4,545,300 | <p>Find the number of all <span class="math-container">$n$</span>, <span class="math-container">$1 \leq n \leq 25$</span> such that <span class="math-container">$n^2+15n+122$</span> is divisible by 6.</p>
<p><strong>My attempt</strong>. We know that:
<span class="math-container">\begin{align*}
n^2+15n+122 & \equiv ... | Mike | 544,150 | <p><strong>If <span class="math-container">$n$</span> must be an integer</strong>: HINT: First, <span class="math-container">$n^2+15n+122$</span> is even for all integers <span class="math-container">$n$</span>. Then <span class="math-container">$n^2+15n+122$</span> will be divisble by <span class="math-container">$6$<... |
4,545,300 | <p>Find the number of all <span class="math-container">$n$</span>, <span class="math-container">$1 \leq n \leq 25$</span> such that <span class="math-container">$n^2+15n+122$</span> is divisible by 6.</p>
<p><strong>My attempt</strong>. We know that:
<span class="math-container">\begin{align*}
n^2+15n+122 & \equiv ... | B. Goddard | 362,009 | <p>If <span class="math-container">$n$</span> is even then <span class="math-container">$n^2+15n$</span> is even. If <span class="math-container">$n$</span> is odd, then <span class="math-container">$n^2+15n$</span> is still even. So <span class="math-container">$n^2+15n+122$</span> is even for every <span class="mat... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | mau | 89 | <p>The first example which came to my mind is the <a href="http://en.wikipedia.org/wiki/Skewes%27_number">Skewes' number</a>, that is the smallest natural number n for which π(n) > li(n). Wikipedia states that now the limit is near e<sup>727.952</sup>, but the first estimation was much higher.</p>
|
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | J. W. Tanner | 615,567 | <p>Fermat conjectured that <span class="math-container">$F_n=2^{2^n}+1$</span> is prime for all <span class="math-container">$n$</span>, </p>
<p>but Euler showed that <span class="math-container">$ F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.$</span></p>
|
1,246,522 | <p>What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ corresponds to a <strong>prime</strong> ideal $P$ of $\mathbb F_p[x]$ containing $(x^2)$. And then $P$ contains $(x)$ since i... | rschwieb | 29,335 | <blockquote>
<p>and I still can't seem to prove that if they do [correspond], $P$ can't be some non-principal ideal properly larger than $(x)$.</p>
</blockquote>
<p>Assuming you convince yourself of the correspondence of prime ideals (which is just fine) here's an elementary way to see that there can only be one pri... |
26,451 | <p>I am trying to solve the following:</p>
<p>$\begin{align*}
&X \sim N(1,1)\\
&\mathrm{cov}(X, X^3) = \text{?}
\end{align*}$</p>
<p>where $\mathrm{cov}$ is the covariance.</p>
<p>How would you do this in <em>Mathematica</em>?</p>
<p>I have tried</p>
<pre><code>X = NormalDistribution[1, 1]
cov[x_, y_] := ... | wolfies | 898 | <p>Given $X$ ~ $N(\mu, \sigma^2)$ with pdf $f(x)$:</p>
<p>$$f=\frac{1}{\sqrt{2 \pi } \sigma } {\text{Exp} \left[-\frac{(x-\mu )^2}{2 \sigma ^2}\right]}; \text{ domain}[f]=\{x,-\infty ,\infty \}\land \{\mu \in \text{Reals},\sigma >0\};$$</p>
<p>Then, using the <code>mathStatica</code> package for <em>Mathematica</... |
3,595,622 | <p><strong>Problem: Give an example of a linear continuum which is not the real line <span class="math-container">$\mathbb{R}$</span>, nor
topologically equivalent to a subspace of <span class="math-container">$\mathbb{R}$</span>.</strong></p>
<p><strong>Definition of Linear Continuum:</strong> Let X be a linearly ord... | Henno Brandsma | 4,280 | <p>In fact, the lexicographically ordered square <span class="math-container">$I \times I$</span> is a classic example of a linear continuum, even compact, that is not a subspace of the reals, e.g. because it is not separable (and all subspaces of <span class="math-container">$\Bbb R$</span> are second countable hence ... |
3,805,989 | <p>I'm doing Exercise 4 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.</p>
<p><a href="https://i.stack.imgur.com/JQww8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JQww8.png" alt="enter image description here" /></a></p>
<blockquote>
<p>Show that, if <span class="math-container">$F$... | Wuestenfux | 417,848 | <p>Hint: If <span class="math-container">$\sigma$</span> is such an automorphism, then <span class="math-container">$f = \sum_i a_ix^i$</span> maps to <span class="math-container">$f^\sigma = \sum_i a_i^\sigma (x^i)^\sigma = \sum_i a_i (x^\sigma)^i =\sum_i a_i g^i$</span>, where <span class="math-container">$x^\sigma =... |
3,506,659 | <p>Let <span class="math-container">$\sigma_i$</span> denote the <a href="https://en.wikipedia.org/wiki/Pauli_matrices" rel="nofollow noreferrer">Pauli matrices</a>:
<span class="math-container">$$
\sigma_1\equiv \begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad
\sigma_2\equiv \begin{pmatrix}0&-i\\i&0\end{pma... | glS | 173,147 | <p>While <a href="https://math.stackexchange.com/a/3506882/173147">the other answer</a> is definitely what I was looking for, I will also add how to find the explicit form of <span class="math-container">$B$</span>, for future reference.</p>
<p>The idea is to find what <span class="math-container">$U\sigma_i U^\dagger... |
4,235,607 | <p>Let <span class="math-container">$x_1,\dots, x_n \geq 0$</span> be a sequence of numbers such that <span class="math-container">$\sum_{i=1}^n x_i = 1$</span>. For every <span class="math-container">$k \geq 1$</span>, I conjecture (and need to prove) that
<span class="math-container">$$
\frac{\sum_{1\leq i\neq j\leq ... | Clement C. | 75,808 | <ul>
<li><strong>A failed attempt, which gives the <span class="math-container">$2/k$</span> dependence but "loses" the denominator.</strong></li>
</ul>
<p>We add back the diagonal terms of the double sum, and bound the numerator <span class="math-container">$N_k(x)$</span> as
<span class="math-container">\b... |
4,235,607 | <p>Let <span class="math-container">$x_1,\dots, x_n \geq 0$</span> be a sequence of numbers such that <span class="math-container">$\sum_{i=1}^n x_i = 1$</span>. For every <span class="math-container">$k \geq 1$</span>, I conjecture (and need to prove) that
<span class="math-container">$$
\frac{\sum_{1\leq i\neq j\leq ... | fedja | 12,992 | <p>If you don't care about the constant, the inequality is rather simple. For all practical purposes (i.e., up to an absolute constant factor) <span class="math-container">$1-(1-x_i)^k\asymp\min(kx_i,1)=y_i$</span>. Also, we trivially have <span class="math-container">$\sum_i y_i\le k\sum_i x_i=k$</span>. Now we note t... |
374,105 | <p>Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.</p>
| Incnis Mrsi | 168,952 | <p>Hint: a regular quadrilateral can have <em>any</em> value of interior angles between 0 and π/2, and a regular pentagon can have <em>any</em> value of interior angles between 0 and 3π/5.</p>
|
16,802 | <p>In an attempt to squeeze more plots and controls into the limited space for a demo UI, I am trying to remove any extra white spaces I see.</p>
<p>I am not sure what options to use to reduce the amount of space between the ticks labels and the actual text that represent the labels on the axes.</p>
<p>Here is a smal... | kglr | 125 | <p>The most convenient way I found is to wrap <code>Plot</code> (without <code>FrameLabels</code> and <code>PlotLabel</code> and with appropriate <code>ImagePadding</code> and <code>ImageMargins</code>) inside <code>Labeled</code> and use the <code>Spacings</code> option to position the labels:</p>
<pre><code> Labeled... |
16,802 | <p>In an attempt to squeeze more plots and controls into the limited space for a demo UI, I am trying to remove any extra white spaces I see.</p>
<p>I am not sure what options to use to reduce the amount of space between the ticks labels and the actual text that represent the labels on the axes.</p>
<p>Here is a smal... | Szabolcs | 12 | <p>When using <a href="http://scidraw.nd.edu/" rel="noreferrer">SciDraw</a>, we have full control over all label positions. The key option names are variation on <code>TextNudge</code>. The value specified in this option will be added to the label position.</p>
<p>Example with frame labels:</p>
<pre><code>Figure[
{... |
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