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 |
|---|---|---|---|---|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Gray Taylor | 987 | <p>At the accessible end of the scale, <a href="http://www.youtube.com/user/Vihart">Vi Hart's</a> "doodling in math class" series and subsequent videos are a delight. </p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | expmat | 7,313 | <p>Videos recorded at IMPA:</p>
<p><a href="http://video.impa.br/index.php?page=download">http://video.impa.br/index.php?page=download</a></p>
<p>(some in English, some in Portuguese)</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Unknown | 5,627 | <p>I found the <a href="https://www.math.duke.edu/video/video.html" rel="nofollow">Graduate weekend</a> repository of lectures at the Mathematics Department of Duke's University very entertaining. There is more in the other folders(G.Tian, Langlands, just to name a few )$\ldots$</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Spice the Bird | 14,167 | <p>Stoney Brook math videos:</p>
<p><a href="http://www.math.sunysb.edu/Videos/dfest/" rel="noreferrer">http://www.math.sunysb.edu/Videos/dfest/</a></p>
<p><a href="http://www.math.sunysb.edu/html/videos.shtml" rel="noreferrer">http://www.math.sunysb.edu/html/videos.shtml</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Bill Kronholm | 5,000 | <p>This is an old thread, but this video was recently posted to the Don Davis topology list, and I have to share it. It was created by Niles Johnson at UGA and it illustrates the Hopf fibration.</p>
<p><a href="http://www.youtube.com/watch?v=AKotMPGFJYk">http://www.youtube.com/watch?v=AKotMPGFJYk</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Gil Kalai | 1,532 | <p>Marcus du-Sautoy's lecture - <a href="https://www.youtube.com/watch?v=uQFE6dYuQW4" rel="nofollow noreferrer">Music of The Prime Numbers</a>, is a very nice popular talk about prime numbers</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Edmund Harriss | 15,516 | <p>The amazing patterns that turn up in piece-wise isometries, like circles dancing in a rhomb:</p>
<p><a href="http://vimeo.com/23772888" rel="nofollow">http://vimeo.com/23772888</a></p>
|
51,341 | <p>I have a function that is a summation of several Gaussians. Working with a 1D Gaussian here, there are 3 variables for each Gaussian: <code>A</code>, <code>mx</code>, and <code>sigma</code>:</p>
<p>$A \exp \left ( - \frac{\left ( x - mx \right )^{2}}{2 \times sigma^{2}} \right )$</p>
<pre><code>A*Exp[-((x - mx)^2/... | Anthony Mannucci | 24,733 | <p>The solution by Apple was very helpful. However, I found it did not work for me (I had a similar situation). No function of x is created in that example. I found the following would work:</p>
<pre><code>f[data_] := Total[#1*Exp[-((x - #2)^2/(2 #3^2))] & @@@ data];
(* a specific example might be *)
g[x_] := Eval... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | BLAZE | 144,533 | <p><strong>Sets are self-defined</strong>; what you're asking here is equivalent to asking:
What is the <em>definition</em> of a definition? </p>
<p>In any case, here is the $\color{red}{\mathrm{old}}$ "definition" of a set:</p>
<pre><code>A set is a collection of 'things'.
</code></pre>
<p>There are <a href="https... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | goblin GONE | 42,339 | <blockquote>
<p><strong>Definition 0.</strong> A set is an $\infty$-groupoid in which every two parallel $1$-cells are equal.</p>
</blockquote>
<p>Okay, but what the heck is an $\infty$-groupoid? Well, we can define it like so:</p>
<blockquote>
<p><strong>Definition 1.</strong> An $\infty$-groupoid is an $\infty$... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Dan Christensen | 3,515 | <p>(Extensively edited)</p>
<blockquote>
<p>So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?</p>
</blockquote>
<p>In the ZFC axioms, there is no distinction made between objects that are sets and those that are not. Everything is a set. So it doesn't see... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Peter Smith | 35,151 | <blockquote>
<p>"So is it, for example, a right definition to say that a set is
anything that satisfies the ZFC list of axioms?"</p>
</blockquote>
<p>Well, that can't be right, or it would make a sheer nonsense of the idea that there are alternative theories of sets which deviate from ZFC -- like NF, for example.<... |
4,252,428 | <p>I encountered this system of nonlinear equations:
<span class="math-container">$$\begin{cases}
x+xy^4=y+x^4y\\
x+xy^2=y+x^2y
\end{cases} $$</span></p>
<p>My ultimate goal is to show that this has only solutions when <span class="math-container">$x=y$</span>. I didn't find any straight forward method to solving this.... | José Carlos Santos | 446,262 | <p>The approach is fine, but since you did not show us your computations, I cannot tell you whether or not the full solution is correct.</p>
<p>Here's how I would do it. Note that<span class="math-container">\begin{align}x+xy^2=y+yx^2&\iff x-y=yx^2-xy^2\\&\iff x-y=xy(x-y)\end{align}</span>and so if <span class=... |
4,252,428 | <p>I encountered this system of nonlinear equations:
<span class="math-container">$$\begin{cases}
x+xy^4=y+x^4y\\
x+xy^2=y+x^2y
\end{cases} $$</span></p>
<p>My ultimate goal is to show that this has only solutions when <span class="math-container">$x=y$</span>. I didn't find any straight forward method to solving this.... | Donald Splutterwit | 404,247 | <p>Square the equation <span class="math-container">$x(1+y^2)=y(1+x^2)$</span> and we have (Note that <span class="math-container">$2x^2y^2$</span> will cancel)
<span class="math-container">\begin{eqnarray*}
x^2(1+y^4)=y^2(1+x^4).
\end{eqnarray*}</span>
Now divide by the first equation and we have <span class="math-con... |
2,948,118 | <p>I understand that for a function or a set to be considered a vector space, there are the 10 axioms or rules that it must be able to pass. My problem is that I am unable to discern how exactly we prove these things given that my book lists some weird general examples.</p>
<p>For instance: the set of all third- degre... | Boshu | 257,404 | <p>First of all, note two things in your solution</p>
<ul>
<li>your choice of <span class="math-container">$\sigma$</span> does not give the difference <span class="math-container">$U-L$</span> to be less than <span class="math-container">$\epsilon$</span>, for that it needs to be <span class="math-container">$10-\eps... |
1,464,747 | <p>I am trying to solve this question:</p>
<blockquote>
<p>How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?</p>
</blockquote>
<p>I am very lost at trying to solve this one. My attempt to start this problem involved drawing 5 boxes, an... | Community | -1 | <p>This is same as number of ways of writing as a sum of 5 positive integers (order does not matter since the boxes are indistinguishable). In short we want the 5 partitions of 8, $P(8,5)$. Using the recurrence relation $$P(n,k) = P(n-k,k)+P(n-1,k-1)$$ we get $P(8,5) = P(3,5)+P(7,4)$. Clearly, $P(3,5) = 0$. Using the r... |
2,764,221 | <p>Let $A$ be a symmetric invertible $n \times n$ matrix, and $B$ an antisymmetric $n \times n$ matrix. Under what conditions is $A+B$ an invertible matrix? In particular, if $A$ is positive definite, is $A+B$ invertible? </p>
<p>This isn't homework, I am just curious. Assume all matrices have entries in $\mathbb{R}$... | Ivo Terek | 118,056 | <p>Nah. Take $$A = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \quad \mbox{and} \quad B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$Then $A$ is invertible and $$A+B = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$has determinant $\det(A+B) = 0$.</p>
<p>(Every matrix is the sum of a sym... |
3,169,142 | <p>The question goes like this:</p>
<p>If <span class="math-container">$f(x)$</span> is a non-constant, continuous function defined on a closed interval <span class="math-container">$[a,b]$</span> Then by the Extreme Value Theorem, there exist an absolute minimum <span class="math-container">$m$</span> and an absolute... | Paul Hurst | 149,898 | <p>First show that the range is contained in the interval <span class="math-container">$[m,M]$</span>. Then use the Intermediate Value Theorem to show that if <span class="math-container">$y \in [m,M]$</span> then there exists an <span class="math-container">$x \in [a,b]$</span> with <span class="math-container">$y=f(x... |
2,622,092 | <p>I want to study the convergence of the improper integral $$ \int_0^{\infty} \frac{e^{-x^2}-e^{-3x^2}}{x^a}$$To do so I used the comparison test with $\frac{1}{x^a}$ separating $\int_0^{\infty}$ into $\int_0^{1} + \int_1^{\infty}$.</p>
<p>For the first part, $\int_0^{1}$, I did $$\lim_{x\to0} \frac{\frac{e^{-x^2}-e^... | Community | -1 | <p>It’s easy to see that your integral can be written as:
$$I=\int_0^\infty e^{-x^2}x^{-a}dx-\int_0^\infty e^{-3x^2}x^{-a}dx$$
Then you compute the two parts of the integral separately:
$$\int_0^\infty e^{-x^2}x^{-a}dx\overbrace{=}^{x^2=z}\frac 12\int_0^\infty e^{-z}z^{-\frac{a-1}2}dz=\frac 12\Gamma\left(\frac{1-a}2\ri... |
912,217 | <p>Let $X$ be a R.V whose pdf is given by
$$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+
(1-p)\frac{1}{\sqrt{2\pi\sigma_2^2}}\exp\left(-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right)$$</p>
<p>clearly $X\sim pN(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)=N(p\mu_1+(1-p)\mu_2,p^2\sigma_... | AlexR | 86,940 | <p>You are using the fact that $f$ is increasing to see that if $\Delta x > 0$, $f(x+\Delta x) \ge f(x) \Rightarrow f(x+\Delta x) - f(x) \ge 0 \Rightarrow \frac{f(x+\Delta x) - f(x)}{\Delta x} \ge 0$.<br>
Analogously for $\Delta x < 0$ you still have $\ge 0$.<br>
Now using that if $g(x) \ge 0 \forall x \in U$ whe... |
912,217 | <p>Let $X$ be a R.V whose pdf is given by
$$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+
(1-p)\frac{1}{\sqrt{2\pi\sigma_2^2}}\exp\left(-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right)$$</p>
<p>clearly $X\sim pN(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)=N(p\mu_1+(1-p)\mu_2,p^2\sigma_... | Community | -1 | <p>If $f$ is differentiable and non-decreasing, then we know that:</p>
<p>$f=\int f' dx$. However, we can define $f'(x)=g(x)\mathbf{I}_{(-\infty,a)}(x) + h(x)\mathbf{I}_{(a,\infty)}(x) + \alpha{I}_{a}(x)$, where $g(x),h(x)>0$. Since the lebesgue measure of $\alpha{I}_{a}(x)=0$, we can make $\alpha$ be anything we w... |
2,832,374 | <p>I'm given the following definition asked to prove the following theorem:</p>
<p>Definition: Let $X$ be a set and suppose $C$ is a collection of subsets of $X$. Then $\cup \mathbf{C}=\{x \in X : \exists C\in \mathbf{C}(x\in C)\}$</p>
<p>Theorem: Let $\mathbf{C,D}$ be collections of subsets of a set $X$. Prove that ... | Fred | 380,717 | <p>Let $f \in V$ be a limit point of $A$. Then there is a sequence $(f_n)$ in $A$ such that $||f_n-f|| \to 0$ as $(n \to \infty)$.</p>
<p>Then we have that $||f_n|| \to ||f||$ as $(n \to \infty)$.</p>
<p>Can you proceed to show that $||f|| \le 1$ ?</p>
|
84,034 | <p>Vieta's theorem states that given a polynomial
$$ a_nx^n + \cdots + a_1x+a_0$$
the quantities
$$\begin{align*}s_1&=r_1+r_2+\cdots\\
s_2&=r_1 r_2 +r_1 r_3 + \cdots \end{align*}$$
etc., where $r_1,\dots, r_n$ are the roots of the given polynomial, are given by
$$s_i = (-1)^i \frac{a_{n-i}}{a_n} .$$</p>
<p>S... | lhf | 589 | <p>You cannot use the symmetric functions to find the roots despite a natural urge to do so. It is instructive to try it and find yourself back where you started from. Try it with a quadratic equation.</p>
|
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | JonathanZ supports MonicaC | 275,313 | <p>Well, I hope this doesn't come off as snarky, but why should we expect that <span class="math-container">$$x^2 +1 =0$$</span> should have solutions? And why should we abandon the meaning of "squaring" that we all first learned for real numbers and adopt <span class="math-container">$$(a,b)^2 = (a^2-b^2, 2ab)$$</span... |
2,398,215 | <p>If $f$ is continuous on $\mathbb{R}$ any of the following conditions are satisfied then $f$ must be a constant.</p>
<p>(1).$f(x)=f(mx),\forall x\in \mathbb{R},|m|≠1,m\in \mathbb{R}$</p>
<p>(2).$f(x)=f(2x+1),\forall x\in \mathbb{R}$</p>
<p>(3).$f(x)=f(x^2),\forall x\in \mathbb{R}$.</p>
<p>Suppose $f$ satisfy (1).... | Guillermo Angeris | 125,757 | <p>For each of these, you have something of the form $f(x) = f(g(x))$ where $g^n(x) \to 0$ (where $g^n$ is the repeated application of $g$) or $g^n(x) \to 1$ (e.g. $g$ is a contraction to some point, let's call it $y$ in general), you can use this and continuity to show that we have
$$
f(x) = f(g^n(x)) \to f(y), \foral... |
2,398,215 | <p>If $f$ is continuous on $\mathbb{R}$ any of the following conditions are satisfied then $f$ must be a constant.</p>
<p>(1).$f(x)=f(mx),\forall x\in \mathbb{R},|m|≠1,m\in \mathbb{R}$</p>
<p>(2).$f(x)=f(2x+1),\forall x\in \mathbb{R}$</p>
<p>(3).$f(x)=f(x^2),\forall x\in \mathbb{R}$.</p>
<p>Suppose $f$ satisfy (1).... | Jihoon Kang | 452,346 | <p>For 1: Suppose $|m|=0$, then$$f(x)=f(0) \ \forall \ x \in \mathbb{R}$$ So $f$ is constant.</p>
<p>Otherwise, suppose $|m| > 1$.</p>
<p>Take any $x, y \in \mathbb{R}$</p>
<p>Since $f$ is continuous on $\mathbb{R}$, $$\forall p \in \mathbb{R}, \forall \epsilon>0, \exists \delta>0:\forall q \in \mathbb{R} \... |
809,336 | <p>If G is abelian then factor group G/H is abelian.</p>
<p>How about the converse of this statement? </p>
<p>Is it true?</p>
| Patrick Da Silva | 10,704 | <p>Well, depends. If you say "abelian for all subgroups $H$", then $\{1\}$ is of course a subgroup of $G$ and $G/\{1\} \simeq G$, so there is nothing to say there. </p>
<p>If you say "abelian for some subgroup $H$", then of course there are counter-examples. An easy one is take any group $G$ and $H = G$ : the group $G... |
3,931,807 | <p>I need to find max and min of <span class="math-container">$f(x,y)=x^3 + y^3 -3x -3y$</span> with the following restriction: <span class="math-container">$x + 2y = 3$</span>.</p>
<p>I used the multiplier's Lagrange theorem and found <span class="math-container">$(1,1)$</span> is the minima of <span class="math-conta... | PierreCarre | 639,238 | <p>In this case, it is really not a great idea to use Lagrange multipliers. We can write <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span> (or vice-versa) using the restriction and reduce this question to a one variable optimisation problem. Substituting <span class="math-conta... |
287,129 | <p>The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$.</p>
<p>I'm looking for strengthenings of this notion; for example, in the above definition it's not decidable whether there is a $1$ in $s$; or, given $i$, whether there i... | Arno | 15,002 | <p>What a <em>computable sequence</em> is essentially follows from what computability is, and from what a sequence is.</p>
<p>Let us first agree that a sequence over $\mathbf{X}$ is a function $s : \mathbb{N} \to \mathbf{X}$. Then asking that whether or not a sequence over $\{0,1\}$ is constant be decidable amounts to... |
255,827 | <p>I've had trouble coming up with one.</p>
<p>I know that if I could find </p>
<p>an irreducible poly $p(x)$ over $\mathbb{Q}$
which has roots $\alpha, \beta, \gamma\in Q(\alpha)$,</p>
<p>then $|\mathbb{Q}(\alpha) : \mathbb{Q}| $ = 3 and would be a normal extension,
as $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha,\beta,\g... | i. m. soloveichik | 32,940 | <p>The splitting field of $x^3-7x+7$ is of degree 3 over the rationals, since the discriminant is a rational square and the polynomial is irreducible.</p>
|
2,536,163 | <p>How to integrate using contour integration?
$$\int_1^{\infty}\frac{\sqrt{x-1}}{(1+x)^2}dx$$
I was putting $y = x-1$ then $\frac{dy}{dx}$= $1-0$, ${dy}={dx} $ then i get $$\int_1^{\infty}\frac{\sqrt{y}}{(2+y)^2}dy$$</p>
<p>I don't know how to take it from here. I would appreciate if someone could help me and give... | Dylan | 135,643 | <p>Ok, you want to do complex? Let's do complex.</p>
<p>The complex function</p>
<p>$$ f(z) = \frac{z^{1/2}}{(2+z)^2} $$</p>
<p>Has a pole of order $2$ at $z = -2$ and a branch point at $z=0$. Let's pick a branch cut on the positive real line so it doesn't interfere with the pole.</p>
<p>We pick a keyhole-shaped co... |
1,026,506 | <p>If $I_n=\int _0^{\pi }\:sin^{2n}\theta \:d\theta $, show that
$I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $</p>
<p>Hence calculate $\int _0^{\pi }\:\:sin^4tcos^6t\:dt$</p>
<p>I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$ ,, but I am... | Community | -1 | <p><strong>Hint</strong></p>
<p>Integrate by parts</p>
<p>$$I_n=\int_0^\pi \sin\theta\sin^{2n-1}\theta d\theta$$
and use the identity</p>
<p>$$\cos^2\theta+\sin^2\theta=1$$</p>
|
1,026,506 | <p>If $I_n=\int _0^{\pi }\:sin^{2n}\theta \:d\theta $, show that
$I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $</p>
<p>Hence calculate $\int _0^{\pi }\:\:sin^4tcos^6t\:dt$</p>
<p>I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$ ,, but I am... | Idris Addou | 192,045 | <p>My answer it attached in two pages</p>
<p><img src="https://i.stack.imgur.com/eUpbA.jpg" alt="integrationByParts"><img src="https://i.stack.imgur.com/D1Qmt.jpg" alt="Formula"></p>
|
4,488,991 | <p>Let <span class="math-container">$M,M'$</span> be oriented connected compact smooth manifolds of the same dimension, let <span class="math-container">$S$</span> be a smooth manifold,
and let <span class="math-container">$\nu : S\times M\rightarrow M'$</span> be some smooth map.
Let <span class="math-container">$\nu_... | heropup | 118,193 | <p>In a Pythagorean right triangle <span class="math-container">$\triangle ABC$</span>, we know that <span class="math-container">$a^2 + b^2 = c^2$</span> where <span class="math-container">$a, b, c$</span> are positive integers. We also know that <span class="math-container">$|\triangle ABC| = rs$</span>, where <span... |
3,888,766 | <p>I need to prove this identity:</p>
<p><span class="math-container">$$2\cos\left(2\pi ft + \phi\right)\cos(2\pi ft) = \cos(4\pi ft + \phi) + \cos(\phi)$$</span></p>
<p>I know I have to use some identity or property but I can't find any to do it.</p>
| uniquesolution | 265,735 | <p>You would set up two equations in the unknowns <span class="math-container">$a,b$</span>, by using the fact that vectors are equal if and only if their components are equal. Thus, you get
<span class="math-container">$$a+3b=1,\quad\hbox{and}\quad 4a-b=0$$</span>
which I am pretty sure you can solve.</p>
|
1,608,645 | <p>Is there supposed to be a fast way to compute recurrences like these?</p>
<p>$T(1) = 1$</p>
<p>$T(n) = 2T(n - 1) + n$</p>
<p>The solution is $T(n) = 2^{n+1} - n - 2$. </p>
<p>I can solve it with:</p>
<ol>
<li><p>Generating functions.</p></li>
<li><p>Subtracting successive terms until it becomes a pure linear re... | vonbrand | 43,946 | <p>Generating functions. Define $t(z) = \sum_{n \ge 0} T(n) z^n$, shift indices to $T(n + 1) = 2 T(n) + n + 1$; by the recurrence backwards $T(0) = 0$, and you get directly:</p>
<p>$\begin{align}
\frac{t(z) - T(0)}{z}
&= 2 t(z) + \sum_{n \ge 0} (n + 1) z^n \\
&= 2 t(z) + \frac{1}{(1 - z)^2} \\
t(z)... |
804,283 | <p>I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing? </p>
<p>Thanks!</p>
| Lee Mosher | 26,501 | <p>The best you can hope for in this situation is to be able to calculate the solution out to as many digits as you are asked for. There are numerical methods to do just that, for instance Newton's Method.</p>
|
14,612 | <p>For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ? </p>
| Gerald Edgar | 127 | <ol>
<li><p>in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points. </p></li>
<li><p>In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero</p></li>
<li><p>In general topology: sets homeomorphic to the Canto... |
14,612 | <p>For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ? </p>
| Michael Greinecker | 8,799 | <p>The Cantor set is quite useful in its own right. My preferred way to think of the Cantor set is as "the most general compact metrizable space." That it is the most general such space means that it is often good for counterexamples because it lacks the particulars. At the same time, one can construct things by specia... |
2,706,165 | <p>So if $y=\log(3-x) = \log(-x+3)$ then you reflect $\log(x)$ in the $y$ axis to get $\log(-x)$.</p>
<p>Then because it is $+3$ inside brackets you then shift to the left by $3$ giving an asymptote of $x=-3$ and the graph crossing the $x$ axis at $(-4,0)$. </p>
<p>However this does not work. The answer shows the $+3... | 2'5 9'2 | 11,123 | <p>Start with $y=\log(x)$. To shift this left three units, replace "$x$" with "$x+3$". Now you have $y=\log(x+3)$.</p>
<p>Now reflect over the $y$-axis. To do this, replace "$x$" with "$-x$". Now you have $y=\log(-x+3)$.</p>
<p>The order that the horizontal graph transformations happen is opposite from what you might... |
347,171 | <p>Let <span class="math-container">$\text{ppTop}$</span> denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor <span class="math-container">$FG: \text{ppTop}\to \text{Gp}$</span> where GP is the category of groups.</p>
... | SashaP | 39,304 | <p>For any such lift <span class="math-container">$\widetilde{FG}:\mathrm{pTop}\to \mathrm{Gp}$</span> the induced map <span class="math-container">$pTop(X,Y)\to Gp(\widetilde{FG}(X),\widetilde{FG}(Y))$</span> must factor through the set of homotopy classes of the maps between <span class="math-container">$X$</span> an... |
4,089,114 | <p>I'm a newbie for mathematics and now I'm learning PDE and stuck on that. Could anyone help me out to understand this elimination from PDE. The equation is similar to solve <span class="math-container">$$(D^2 -6DD'+9D'^2)u = y\cos x$$</span></p>
| mathcounterexamples.net | 187,663 | <p>I would proceed as follows.</p>
<p>As <span class="math-container">$\operatorname{Im}(A^{k+1}) \subseteq \operatorname{Im}(A^{k}) $</span>, one can find a linear subspace <span class="math-container">$G$</span> of <span class="math-container">$\operatorname{Im}(A^{k}) $</span> such that <span class="math-container">... |
2,792,751 | <p>Prove that $p_2(n) = \left \lfloor{\frac{n}{2}}\right \rfloor+1$ using the identity $$\frac{1}{(1-x)(1-x^2)}=\frac{1}{2}\left(\frac{1}{(1-x)^2}\right)+\frac{1}{2}\left(\frac{1}{1-x^2}\right)$$</p>
<p>where $p_k(n)$ is the number of partitions of an integer $n$ into a most $k$ parts. The generating function $P_k(x)$... | Servaes | 30,382 | <p><strong>HINT:</strong> If the $7$ vertices are divided into two sets of size $a$ and $b$, then there are at most $a\cdot b$ edges between them. So then $a\cdot b\geq11$ and $a+b=7$. Now see if you can draw such a graph.</p>
|
1,846,168 | <p>Let <span class="math-container">$R$</span> be a commutative ring, <span class="math-container">$a \in R$</span>, and <span class="math-container">$\forall i = 1, ...,r \ \ f_i(x) \in R[x]$</span>.</p>
<p>Prove the equality of ideals</p>
<p><span class="math-container">$(f_1(x), ..., f_r(x), x-a ) = (f_1(a), ...f_r(... | Tsemo Aristide | 280,301 | <p>$(f_i(x)-f_i(a))(a)=0$. Write $f_i(x)-f_i(a)=q(x)(x-a)+c$ where $c\in R$, you have $f_i(a)-f_i(a)=q(a)(a-a)+c=0$, implies that $c=0$, thus $f_i(x)\in (f_1(a),...,f_r(a),x-a)$.</p>
<p>On the other hand, $f_i(x)=q(x)(x-a)+c$ this implies that $c=f_i(a)=f_i(x)-q(x)(x-a)$ thus $f_i(a)\in (f_1(x),...,f_r(x),x-a)$.</p>
|
2,539,693 | <p>A number theory textbook asked us to compare $\tan^{-1}(\frac{1}{2})$ and $\sqrt{5}$. In fact, these are rather close:</p>
<p>\begin{eqnarray*}
\tan^{-1} \frac{1}{2} &=& 0.46364 \\ \\
\frac{1}{\sqrt{5}} &=& 0.44721
\end{eqnarray*}</p>
<p>So at least numerically I think we have the answer that the ... | Acccumulation | 476,070 | <p>This seems like more of a real analysis question than number theory. The first method I thought of was taking the Taylor series for tangent($\sqrt{5}$) and comparing that to $\frac{1}{2}$ (and noting that tangent is an increasing function), but arctan has a nicer Taylor series than tangent does. Either way, what you... |
2,102,124 | <p>If $p$, $q$ and $r$ are positive integers and $p + \displaystyle\frac{1}{q + \displaystyle\frac{1}{r}} = \frac{129}{31}$ then what is the value of $p + q + r$?</p>
<p>I tried getting a common denominator, but nothing seems to work as the correct answer is an actual number</p>
| ajotatxe | 132,456 | <p>The fraction
$$\frac1{q+\dfrac1r}$$
is lesser than $1$, so
$$p=\left\lfloor\frac{129}{31}\right\rfloor=4$$
(The brackets $\lfloor\quad\rfloor$ mean "integer part").</p>
<p>Now,
$$\frac1{q+\dfrac1r}=\frac{129}{31}-4=\frac5{31}$$
Therefore
$$q+\frac1r=\frac{31}5$$</p>
<p>Proceed similarly to find $q$ and $r$.</p>
|
2,102,124 | <p>If $p$, $q$ and $r$ are positive integers and $p + \displaystyle\frac{1}{q + \displaystyle\frac{1}{r}} = \frac{129}{31}$ then what is the value of $p + q + r$?</p>
<p>I tried getting a common denominator, but nothing seems to work as the correct answer is an actual number</p>
| Brevan Ellefsen | 269,764 | <p><strong>One Liner:</strong>
$$\frac{129}{31} = 4+\frac{5}{31} = 4+\frac{1}{\frac{31}{5}} = 4+\frac{1}{6+\frac{1}{5}}$$
<em>Fundamentally, this just comes down to writing fractions in simplest form, where denominator exceeds the numerator</em></p>
|
623,428 | <blockquote>
<p>Suppose $$
Y = X^TAX,
$$ where $Y$ and $A$ are both known $n\times n$, real, symmetric matrices. The unknown matrix $X$ is restricted to $n\times n$.</p>
</blockquote>
<p>I think there should be at least one real valued solution for $X$. How do I solve for $X$? </p>
| Algebraic Pavel | 90,996 | <p>The solution of the problem exists if and only if the (symmetric) matrices $Y$ and $A$ have the same <em>inertia</em>, that is, they have the same number of positive, zero, and negative eigenvalues.</p>
<p>Every symmetric matrix $S$ can be transformed by a (nonsingular) congruent transformation to a diagonal matrix... |
1,591,311 | <p>I've been thinking about this problem which I think is interesting, but can't solve it.</p>
<p>There are $n$ distinguishable items, and $b$ distinguishable bins. Each bin has to include at least one item. But, once some set of items are placed in a bin, they become indistinguishable. How many ways are there to plac... | robjohn | 13,854 | <p>The number of functions from $n$ items to $b$ bins is $b^n$</p>
<p>The number of functions from $n$ items to $b$ bins that miss a particular $k$ bins is $(b-k)^n$. There are $\binom{b}{k}$ ways to choose the $k$ bins to miss.</p>
<p>Thus, inclusion-exclusion says that the number of surjective functions from $n$ it... |
26,152 | <p>In my textbook, they said:</p>
<p>$$2x^{3} + 7x - 4 \equiv 0 \pmod{5}$$</p>
<p>The solution of this equation are the integers with $x \equiv 1 \pmod{5}$, as can be seen by testing $x = 0, 1, 2, 3,$ and $4.$</p>
<p>And I have no clue how do they had $x \equiv 1 \pmod{5}$. I tested as they suggested:</p>
<p>Let $y... | yunone | 1,583 | <p>Any integer is going to be congruent to one of $0,1,2,3,4$ modulo $5$. As the testing shows, only those integers which are congruent to $1$ modulo $5$ are solutions, and any integer congruent to $1$ mod $5$ is a solution.</p>
<p>Those tests are using an arbitrary $x$ each time, so you shouldn't think of them as a s... |
26,152 | <p>In my textbook, they said:</p>
<p>$$2x^{3} + 7x - 4 \equiv 0 \pmod{5}$$</p>
<p>The solution of this equation are the integers with $x \equiv 1 \pmod{5}$, as can be seen by testing $x = 0, 1, 2, 3,$ and $4.$</p>
<p>And I have no clue how do they had $x \equiv 1 \pmod{5}$. I tested as they suggested:</p>
<p>Let $y... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\rm\ \ x\ p(x)\ =\ 2\ x^4 + 7\ x^2 - 4\ x\ \equiv\ 2\ (x-1)^2\ \ (mod\ 5)\ $ for $\rm\ x\not\equiv 0\ $ since then $\rm\ x^4\equiv 1\ $</p>
|
679,904 | <p>The question is let $a \in \mathbb{R} $ does not contain 0. Prove that $|a+\frac{1}{a}| \ge 2$. I have no idea how to start this problem and any help on it would be greatly appreciated.</p>
| BaronVT | 39,526 | <p>Assume for a moment that $a >0$, just to make the intuition clearer. Then you want to prove</p>
<p>$$
a + \frac{1}{a} \geq 2
$$</p>
<p>which is true if and only if</p>
<p>$$
a^2 + 1 \geq 2a
$$</p>
<p>i.e. if</p>
<p>$$
a^2 - 2a + 1 = (a - 1)^2 \geq 0
$$</p>
|
119,636 | <p>I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.</p>
<p>For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
according to <a href="http://www.wolframalpha.com/input/?i=partial+sum+of+n+2%5En" rel="noreferrer">http://www.wolfr... | Tom Cooney | 8,580 | <p>I see that one of the tags is pre-calculus, so here is a way to answer the question that does not use differentiation:</p>
<p>$S = r + 2r^2 +3r^3 +\dots + (m-1)r^{m-1}+mr^m $<br>
$rS = \ \ \ \ r^2 +2r^3 +\dots + (m-2)r^{m-1}+(m-1)r^m + mr^{m+1} $ </p>
<p>Subtracting the bottom line from the top, we get<br>
$$(1-... |
792,356 | <p>There is a dark night and there is a very old bridge above a canyon. The bridge is very weak and only 2 men can stand on it at the same time. Also they need an oil lamp to see holes in the bridge to avoid falling into the canyon.</p>
<p>Six man try to go through that bridge. They need 1,3,4,6,8,9(first man, second ... | Asimov | 137,446 | <p>Obviously one person will need to go back and forth, taking people across, and then bringing the lamp back. This person will take the most trips, and should be the fastest person (the 1). What he does is guide person 2 across, runs back, guides person 3 across etc. until all are across.</p>
<p>Take 3 across (3 MIN... |
792,356 | <p>There is a dark night and there is a very old bridge above a canyon. The bridge is very weak and only 2 men can stand on it at the same time. Also they need an oil lamp to see holes in the bridge to avoid falling into the canyon.</p>
<p>Six man try to go through that bridge. They need 1,3,4,6,8,9(first man, second ... | Phil | 337,472 | <p>A minimal solution is:</p>
<p>1 and 3 cross the bridge; 1 comes back (4)
12 and 8 cross the bridge, 3 comes back (15)
1 and 6 cross the bridge; 1 comes back (7)
1 and 3 cross the bridge (3)
Total is 4+15+7+3=29. </p>
|
239,136 | <p>I was given this question and I'm not really sure how to approach this...</p>
<p>Assume $(r,s) = 1$. Prove that If $G = \langle x\rangle$ has order $rs$, then $x = yz$, where $y$ has order $r$, $z$ has order $s$, and $y$ and $z$ commute; also prove that the factors $y$ and $z$ are unique.</p>
| Marc van Leeuwen | 18,880 | <p>The cyclic group $G$ is isomorphic to the additive group of $\Bbb Z/rs\Bbb Z$, so this is just the <a href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem#Theorem_statement" rel="nofollow">Chinese remainder theorem</a> for the coprime moduli $r$ and $s$ (in the statement for rings $\Bbb Z/n\Bbb Z$, but only ... |
2,497,216 | <p>I have this exercise, but I feel like something is wrong. As far as I know, if $m$ is a maximal ideal, then $m \subsetneq A$. But with this hypothesis, I think to take $I = \lbrace 1_A \rbrace$, so the only ideal that contains $I$ is $A$, so $I$ is not maximal. </p>
<p>EDIT:</p>
<p>I wrote something wrong. I didn'... | Jesko Hüttenhain | 11,653 | <p>The set $I=\{1\}$ is usually not an ideal, unless $1$ is the only element of $A$. That is because the ideal axioms require that for any $a$, if $1\in I$, we also have $a=a\cdot 1\in I$. Hence, $A\subseteq I$ and consequently, $A=I$. So this is not a contradiction because only <em>proper</em> ideals are contained in ... |
3,428,668 | <p>How to prove this</p>
<p><span class="math-container">$$S = \{(x, y) | Ax + By ≥ c, x ≥ 0, y ≥ 0\}$$</span>
where <span class="math-container">$A$</span> is an <span class="math-container">$m \times n$</span> matrix, <span class="math-container">$B$</span> is a positive semi-definite <span class="math-container">$m... | Bill Dubuque | 242 | <p><span class="math-container">$\!\!\bmod 7\!:\,\ 2y\equiv 3\equiv 10\iff y\equiv 5\iff y = 5\!+\!7n,\,$</span> hence</p>
<p><span class="math-container">$x = 2\! +\! 27y = 2\!+\!27(5\!+\!7n) =\, \bbox[5px,border:1px solid #c00]{137+ 189n}\ $</span> is our solution.</p>
<p><strong>Remark</strong> <span class="math-c... |
3,428,668 | <p>How to prove this</p>
<p><span class="math-container">$$S = \{(x, y) | Ax + By ≥ c, x ≥ 0, y ≥ 0\}$$</span>
where <span class="math-container">$A$</span> is an <span class="math-container">$m \times n$</span> matrix, <span class="math-container">$B$</span> is a positive semi-definite <span class="math-container">$m... | zwim | 399,263 | <p>Do you know about modular inverse ?</p>
<p>Here you have <span class="math-container">$\quad 2y\equiv 3 \pmod 7$</span></p>
<p>Notice that when we multiply by <span class="math-container">$4$</span> we get <span class="math-container">$\quad 8y\equiv 12\pmod 7$</span> </p>
<p>And since <span class="math-container... |
4,337,820 | <p>We know that during projection 3D space points <span class="math-container">$(x, y, z)$</span> projects to projection plane which has 2D points <span class="math-container">$(x, y).$</span> But during matrix calculation we use homogenous coordinates is of the form <span class="math-container">$(x, y, 1).$</span>
And... | Lee Mosher | 26,501 | <p>Each point of the projective plane <span class="math-container">$P^2$</span> can be represented in the form <span class="math-container">$[x : y : z] \in P^2$</span> for some point <span class="math-container">$(x,y,z) \ne (0,0,0)$</span> in <span class="math-container">$\mathbb R^3$</span>. Using this representatio... |
3,180,914 | <p>Let <span class="math-container">$G$</span> be a cyclic group of order <span class="math-container">$n$</span>. Let <span class="math-container">$G_k$</span> the subgroup
<span class="math-container">$$G_k=\left\{x^k: x\in G\right\}.$$</span>
Is it true that <span class="math-container">$[G:G_k]\in\{1,k\}$</span>?</... | Chinnapparaj R | 378,881 | <p>Here is a similar one using lhf hint:</p>
<p>Take <span class="math-container">$G=\Bbb Z_{6}$</span> and <span class="math-container">$$G_4=\{x^4=4x=x+x+x+x:x \in \Bbb Z_{6}\}=\{0,4,2\}$$</span> and <span class="math-container">$[G:G_4]=2\neq1,4$</span></p>
<hr>
<p>Your result is clearly true if <span class="math... |
3,547,529 | <p>I did the following: I set <span class="math-container">$3^m+3^n+1=x^2$</span> where <span class="math-container">$x\in\Bbb{N}$</span> and assumed it was true for positive integer exponents and for all whole numbers x so that I can later on prove it's invalidity with contradiction. Since <span class="math-container"... | Ali Taghavi | 143,009 | <p><strong>Proof</strong>: It is sufficient to prove the result when <span class="math-container">$b=0$</span>. Otherwise we set the change of variable <span class="math-container">$x:=x-b/2na$</span> to obtain a polynomial with <span class="math-container">$b=0$</span>.</p>
<p>Assume that <span class="math-contain... |
2,571,909 | <p>$$\left|\frac{-10}{x-3}\right|>\:5$$</p>
<ul>
<li>Find the values that $x$ can take. </li>
</ul>
<p>I know that</p>
<p>$$\left|\frac{-10}{x-3}\right|>\:5$$
and
$$\left|\frac{-10}{x-3}\right|<\:-5$$</p>
| Dr. Sonnhard Graubner | 175,066 | <p>simplifying and multiplying by $$|x-3|$$ for $$x\ne 3$$ we get $$2>|x-3|$$ this is equivalent to
$$2>x-3$$ if $$x\geq 3$$ and $$2>-x+3$$ if $$x<3$$</p>
|
54,496 | <p>If there a group G acting on a variety V.
The action is algebraic.
What is the definition of algebro-geometric quotient of this action?</p>
<p>I hope you can give a very basic explanation.</p>
<p>Thanks.</p>
| Steven Landsburg | 10,503 | <p>You are looking for the theory of stacks. The <a href="http://en.wikipedia.org/wiki/Algebraic_stack" rel="nofollow">Wikipedia article</a> will get you started. Or see the <a href="http://cel.archives-ouvertes.fr/docs/00/39/21/43/PDF/tomas_notes.pdf" rel="nofollow">article</a> by Tomas, which is quite readable (if ... |
54,496 | <p>If there a group G acting on a variety V.
The action is algebraic.
What is the definition of algebro-geometric quotient of this action?</p>
<p>I hope you can give a very basic explanation.</p>
<p>Thanks.</p>
| Dan Petersen | 1,310 | <p>It is certainly possible to give the <em>definition</em> of a quotient of a variety by an algebraic group without mentioning algebraic stacks or spaces. </p>
<p>There are two distinctly different situations here, depending on whether the group is finite (and discrete) or a general algebraic group. </p>
<p>In the f... |
590,205 | <p>I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times [0,1]/((0,1)\times\{1\}).$$ But how do I show that it can not be embedded in an Euclidean space? Cause for me it looks like it is possible... | Niels J. Diepeveen | 3,457 | <p>Euclidean spaces are metric spaces, so in order for the cone to be embeddable
in one of them, it must be a metrizable space. As far as I can make out,
the cone of a space is metrizable if and only if the space itself is
metrizable and compact. Obviously $(0, 1)$ is metrizable but not compact.</p>
<p>I have been loo... |
590,205 | <p>I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times [0,1]/((0,1)\times\{1\}).$$ But how do I show that it can not be embedded in an Euclidean space? Cause for me it looks like it is possible... | Damien L | 59,825 | <p>Let me add a little bit of context and change the terminology such that
‘automatic thinking’ gets it right.</p>
<p>Let <span class="math-container">$\mathrm{C}(X)$</span> denote the (real, geometric, true...) <strong>cone</strong> on <span class="math-container">$X$</span>.
Its points are the pairs <span class="math... |
1,406,796 | <p>Can someone please show me how they would work it out as I have never come across this before.</p>
<p>$$(x^2-5x+5)^{x^2-36} =1$$</p>
| hmakholm left over Monica | 14,366 | <p>It's kind of a trick question; there's no <em>general</em> way to solve that kind of equations (save for numerically), if the right-hand side had been anything else than $1$.</p>
<p>However, you should know that $a^b=1$ only if $a=1$ or $b=0$ (or $a=-1$ and $b$ even), so you can break it into three ordinary quadrat... |
1,808,258 | <p>I was reading about orthogonal matricies and noticed that the $2 \times 2$ matrix
$$\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} $$
is orthogonal for every value of $\theta$ and that every $2\times 2$ orthogonal matrix can be expressed in this form. I then wonder... | thecat | 338,383 | <p>(3) is only sometimes true, and it is at only the times that (3) is true that A is orthogonal.</p>
<p>More particularly, the reason A is orthogonal when A is some multiple of the rotation matrix is that (x(t), y(t)) describes a circle, so position is orthogonal to velocity.</p>
<p>For A where position is not ortho... |
2,672,097 | <p>What are the must-know concepts and best resources for preparing the <strong>mathematical background for advanced machine learning studies</strong>?</p>
<p>Currently, looking into the book <strong>What is Mathematics? by Richard Courant</strong> to strengthen my fundamentals. Are there any better references that ca... | bthmas | 222,365 | <p>Machine Learning as a whole is incredibly diverse. Likewise, the type of math seen largely depends on the certain kind of questions you're interested in. Regardless of your interests, a strong background in linear algebra and probability/statistics is a must.</p>
<p>Assuming that you're interested in <strong>Deep L... |
3,182,802 | <p>Show that if <span class="math-container">$ \sigma $</span> is a solution to the equation <span class="math-container">$ x^2 + x + 1 = 0 $</span> then the following equality occurs:</p>
<p><span class="math-container">$$ (a +b\sigma + c\sigma^2)(a + b\sigma^2 + c\sigma) \geq 0 $$</span></p>
<p>I looked at the solu... | Mark Bennet | 2,906 | <p>Hint: look at the <span class="math-container">$ab$</span> terms, where you have <span class="math-container">$ab\sigma^2+ab\sigma=ab(\sigma^2+\sigma)$</span>: now what can you do with your existing hint to simplify this part of your expression?</p>
<p>That should get you a start.</p>
|
1,304,344 | <p>How do I find the following:</p>
<p>$$(0.5)!(-0.5)!$$</p>
<p>Can someone help me step by step here?</p>
| Community | -1 | <p>Factorial of any real number $n$ is defined by Gamma function as follows:
$$\Gamma (n) = (n-1)!$$
$$\quad \Rightarrow ( \dfrac{1}{2} )! ( -\dfrac{1}{2} ) ! = ( \dfrac{3}{2}-1 ) ! ( \dfrac{1}{2}-1 ) ! = \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} )$$
It is also known that:
$$\Gamma {(1+z)} = z\Gamma {(z)}$$
$$\q... |
1,640,217 | <p><em>Use the dot/scalar product to solve the problem</em></p>
<p>Line 1 has vector equation $(2\mathrm{i}-\mathrm{j}) + \lambda(3\mathrm{i} + 2\mathrm{j})$
Find the vector equation of the line perpendicular to Line 1 and passing through the point with position vector $(4\mathrm{i} + 3\mathrm{j})$.</p>
<p>I can solv... | user296856 | 296,856 | <p>If two cartesian lines are perpendicular, the product of their slopes are -1.</p>
<p>So if you have $y = kx + b$</p>
<p>Then $y_p = -\frac{1}{k}x+b_p$, assuming $k$ is not zero</p>
<p>Plug in the position vector to find $b_p$.</p>
|
158,720 | <p>By induction I can prove :
$$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} = \frac{(D+M)!}{D!M!} $$</p>
<p>However, I couldn't derive the right hand side directly.</p>
<p>It would be of great help if anyone can solve it!!</p>
| Community | -1 | <p>Start with the binomial coefficient relation
$${n\choose D}={n-1\choose D}+{n-1\choose D-1}$$
rewritten as
$${n\choose D}-{n-1\choose D}={n-1\choose D-1}.$$ </p>
<p>Adding these increments gives
$$ {D+M\choose D}-{D-1\choose D}=\sum_{n=D}^{D+M}{n-1\choose D-1}.$$
Since ${D-1\choose D}=0$ this gives you the sum t... |
158,720 | <p>By induction I can prove :
$$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} = \frac{(D+M)!}{D!M!} $$</p>
<p>However, I couldn't derive the right hand side directly.</p>
<p>It would be of great help if anyone can solve it!!</p>
| tulasi | 33,411 | <p>$(1+x)^{M} (1+x)^{D} = (1+x)^{M+D} $</p>
<p>compute coefficient of $x^{D}$ on both sides. $$$$
on LHS, it is $$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} $$
and on it is RHS $$ \frac{(D+M)!}{D!M!} $$</p>
|
205,080 | <p>I have a problem that I cannot figure out how to do. The problem is:<br>
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?<br><br>
I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have tried to find the inverse of function s but I got stuck ... | marty cohen | 13,079 | <p>Going back to your original problem statement, you have to understand that the range of a function is the set of values that the function takes on for arguments in the function's domain. This avoids worrying about functions that are not 1-1. (I don't see the need for the "domain of $s^{-1}$" statement.) I assume tha... |
1,445,913 | <p>Given 2 lines r and s. </p>
<ul>
<li>r and s don't have an intersection point</li>
<li>none of them touch the origin (0,0,0)
What approach should I use to find the equation of the line that cross the origin and also cross r and s?</li>
</ul>
<p>if necessary, we can consider r and s as:</p>
<pre><code> x = at +... | Ross Millikan | 1,827 | <p>Hint: can you factor $n^2-1$?</p>
|
2,315,739 | <p>I have an irregular quadrilateral.
I know the length of three sides (a, b and c) and the length of the two diagonals (e and f).
All angles are unknown
How do I calculate the length of the 4th side (d)?</p>
<p>Thank you for your help.
Regards,</p>
<p>Mo</p>
<p><a href="https://i.stack.imgur.com/Jtdzv.jpg" rel="nof... | Toby Mak | 285,313 | <p>You could use <a href="https://en.wikipedia.org/wiki/Brahmagupta%27s_formula" rel="nofollow noreferrer">Brahmagupta's formula</a> which is $\sqrt{s(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter $(\frac{1}{2})(a+b+c+d)$ only if the quadrilateral is cyclic, in which $e$ and $f$ are equal. </p>
<p>The proof use... |
2,087,235 | <p>I have a question about this question. Find all complex numbers $z$ such that the equation
$$t^2 + [(z+\overline z)-i(z-\overline z)]t + 2z\overline z\ =\ 0$$
has a real solution $t$.</p>
<p><strong>Attempt at a solution</strong></p>
<p>The discriminant is</p>
<p>$[(z+\overline z) - i(z-\overline z)]^2 - 4(2z\ove... | Dr. Sonnhard Graubner | 175,066 | <p>with $$z=x+iy$$ we get $$\bar z=x-iy$$ thus our equation is given by
$$t^2+2(x+y)t+2(x^2+y^2)=0$$
can you proceed?</p>
|
2,163,067 | <p>Prove that $\mathbb{Z}_5[x]$ is a unique factorization domain.</p>
<p>My approach is to prove that $\mathbb{Z}_5[x]$ is a PID, which implies that it is a UFD.</p>
<p>Proof:</p>
<p>Suppose there exists an ideal $I$ in $\mathbb{Z}_5[x]$ such that it is generated by two or more elements of $\mathbb{Z}_5[x]$. That is... | Joshua Ruiter | 399,014 | <p>By $\mathbb{Z}_5$ I assume you mean $\mathbb{Z}/5\mathbb{Z}$. This is a field, since 5 is prime. Any field is a PID. See <a href="https://math.stackexchange.com/questions/415081/why-any-field-is-a-principal-ideal-domain">Why any field is a principal ideal domain?</a></p>
<p>EDIT: After comments, I realized that whi... |
3,154,212 | <p>I'm working a lot with series these days, and I would like to know if there are any texts, papers, articles that might suggest a general outline for finding <span class="math-container">$n$</span>th partial sums of convergent series. Most of my searching turns up methods for finding the sums of geometric/telescopin... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>Let <span class="math-container">$u=cw+v_1$</span> with <span class="math-container">$v_1\in U$</span> (i.e. <span class="math-container">$v_1\cdot w=0$</span>) therefore <span class="math-container">$$u\cdot w=cw\cdot w+v_1\cdot w=cw\cdot w=c|w|^2\implies c={u\cdot w\over |w|^2}$$</spa... |
775,265 | <p>Please help me get the answer to this question.</p>
<p>Prove $f(x)=\sqrt{2x-6}$ is continuous at $x=4$ by using precise definition. ($\epsilon-\delta$ definition of limits.)</p>
| Alberto Takase | 146,817 | <p>Let $f:[3,\infty)\to \mathbb{R}$ be defined by
$$f(x)=\sqrt{2x-6}$$
for every $x\in[3,\infty)$. (It is important to define the function explicitly; that is, the domain is relevant for this proof.)</p>
<p>$f$ is continuous at $4$.</p>
<p>Proof. To prove $f$ is continuous at $4$, we will show that for each $\epsilon... |
10,880 | <p>I am posting to formally register my disapproval of <a href="https://math.stackexchange.com/users/93658/anti-gay">this user's</a> name.</p>
<p>I believe it constitutes hate speech. If you look at the comments on this user's answers, you will see that many others do too. The name is already causing a lot of trouble, ... | anon | 11,763 | <p>It seems overboard to classify the mere labeling of oneself anti-gay as hate speech. Would this apply for example to the username <code>anti-x</code> for various labels <code>x</code> that can be applied to people, for example christian, atheist, homophobe, etc? At any rate, the more unarguably germane issue is that... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Will Dana | 14,967 | <p>In a sense, the quotient group is indeed a measurement of how many copies of your normal subgroup are within the larger group. In the simple example of $\mathbb{Z}/3\mathbb{Z}$, the group has three elements: one for the subgroup $3\mathbb{Z}$ itself and one for each of its two cosets, which, if you were to plot them... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Michael Hardy | 11,667 | <p>$$
\begin{array}{c|c|c}
A & B & C \\ D & E & F \\ G & H & I \\ J & K & L
\end{array}
$$</p>
<p>$12\div 4 = 3$ because $3$ is how many sets of $4$ it takes to make a set of $12.$</p>
<p>A normal subgroup of order $4$ in a group of order $12$ has $3$ cosets; thus the quotient group ha... |
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Ciro Santilli OurBigBook.com | 53,203 | <p><strong>The quotient group is the result of a simplification done by an homomorphism</strong></p>
<p><a href="https://math.stackexchange.com/a/69063/53203">https://math.stackexchange.com/a/69063/53203</a> mentions that the quotient subgroup is a type of subgroup but "with less information".</p>
<p>This is ... |
4,539,043 | <p>I have tried to prove this statement by utilizing the proof by cases method. My cases are (1)<span class="math-container">$x=6$</span>, (2)<span class="math-container">$x>6$</span> and (3)<span class="math-container">$x<6$</span>.</p>
<p>For (3) for some reason it's not true</p>
<p>Case (1):
For <span class="m... | lone student | 460,967 | <p>Your attempt seems incorrect to me when you analyze the case <span class="math-container">$3$</span>.</p>
<p>Note that, if <span class="math-container">$x≤6$</span>, then you have:</p>
<p><span class="math-container">$$
\begin{align}
&x^2-x+6-5>0\wedge x≤6\\
\implies &\left(x-\frac 12\right)^2+\frac 34 &g... |
105,750 | <p>Given a <code>ContourPlot</code> with a set of contours, say, this:</p>
<p><a href="https://i.stack.imgur.com/cKoyo.jpg"><img src="https://i.stack.imgur.com/cKoyo.jpg" alt="enter image description here"></a></p>
<p>is it possible to get the contours separating domains with the different colors in the form of lists... | MarcoB | 27,951 | <p>You ask whether it is "possible to define the areas with the same color as separate geometric regions in the sense of the computation geometry, and then work with these domains separately". This seems to me to be a great task for <code>ImplicitRegion</code>:</p>
<pre><code>regions = Table[
ImplicitRegion[i < x*... |
881,282 | <p>Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.</p>
| Kaster | 49,333 | <p>If you do a substitution $t = x \sqrt 3$, you get
$$
\frac 3 {t^2} + \frac 1{(4 - t)^2} = 1 \implies t^4 - 8t^3 + 12 t^2 + 24t - 48=0
$$
You can check that $t = 2$ is a solution, so $P_4(t) = (t-2)P_3(t)$, therefore $x = \frac 2{\sqrt 3}$ is a solution of the initial equations.</p>
|
2,872,807 | <p>I was browsing through facebook and came across this image: <a href="https://i.stack.imgur.com/jozSg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jozSg.png" alt="enter image description here"></a></p>
<p>I was wondering if we can find more examples where this happens?</p>
<p>I guess this redu... | Batominovski | 72,152 | <p>Let $S$ be the set of allowed values of $a$, $b$, and $c$ ($S=\mathbb{Z}$ in this OP's setting, but $S$ can be something else like $\mathbb{Q}$, $\mathbb{Q}_{>0}$, $\mathbb{R}$, or even $\mathbb{F}_p$, where $p$ is a prime natural number). If $b=-a$, then $c$ can be any number not equal to $-a$. That is, $(a,b,... |
3,294,082 | <p>The exercise is to prove that the minimum value between <span class="math-container">$a^{1/b}$</span> and <span class="math-container">$b^{1/a}$</span> is no greater than <span class="math-container">$3^{1/3}$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are posit... | rtybase | 22,583 | <p><strong>Hints</strong>. From (assuming)
<span class="math-container">$$a<b \Rightarrow \frac{1}{a}>\frac{1}{b}$$</span> then (f(x)=<span class="math-container">$c^x$</span> is ascending for <span class="math-container">$c>1$</span>, <span class="math-container">$g(x)=x^t$</span> is ascending for <span class... |
619,477 | <blockquote>
<p>Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob.
"That's funny, i was in it too, and i don't remember ever seeing you there."
"Well," admitted Alice sheepishly, "I guess i did skip class a lot." "Yeah, ... | Alex | 175,412 | <p>Let $C$ be the finite set of lectures.<br>
Let $A\subseteq C$ be the set of lectures attended by Alice.<br>
Let $B \subseteq C$ be the set of lectures attended by Bob.<br>
The intersection of $A$ and $B$ is empty since Alice and Bob never meet in a lecture.
We want to show that $|A|+|B| \le |C|$</p>
<p><strong>Proo... |
1,957,304 | <p>I'm proving the compact-to-Hausdorff lemma (probably not a universal name for it) which is stated as:</p>
<blockquote>
<p>If $X$ is compact, $Y$ Hausdorff, $f:X \rightarrow Y$ a continuous bijection, then $f$ is a homeomorphism.</p>
</blockquote>
<p>However, the following line has popped up in a proof of it:</p>... | Stefan | 375,579 | <p>$Y\setminus f(X\setminus U)=Y\setminus (f(X)\setminus f(U))=^*(Y\setminus f(X)) \cup (f(U)\cap f(X)\cap Y)=\emptyset \cup f(U)$</p>
<p>where the last equality holds because of bijectivity</p>
<p>$*$ holds because the elements in $f(U)\cap f(X) \cap Y$ are precisely the elements which are in $f(U) \cap f(X)$ so the... |
3,014,766 | <p>I am supposed to find the derivative of <span class="math-container">$ 2^{\frac{x}{\ln x}} $</span>. My answer is <span class="math-container">$$ 2^{\frac{x}{\ln x}} \cdot \ln 2 \cdot \frac{\ln x-x\cdot \frac{1}{x}}{\ln^{2}x}\cdot \frac{1}{x} .$$</span> Is it correct? Thanks. </p>
| Eleven-Eleven | 61,030 | <p>Let <span class="math-container">$y=2^\frac{x}{\ln{x}}$</span>. Then</p>
<p><span class="math-container">$$\ln{y}=\ln{2}^{\frac{x}{\ln{x}}}=\frac{x}{\ln{x}}\cdot\ln2$$</span></p>
<p>Now <span class="math-container">$$\frac{1}{y}\cdot\frac{dy}{dx}=\ln{2}\left[\frac{\ln{x}-1}{(\ln x)^2}\right]$$</span></p>
<p>Now ... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | user8675309 | 735,806 | <p>here's an approach that uses the hint. You have <span class="math-container">$f'(x)=[f(x)]^2\geq 0$</span>. If <span class="math-container">$f'(x)\gt 0$</span> everywhere, then the function is injective. </p>
<p>Thus not injective implies there is at lease one <span class="math-container">$a \in \mathbb R$</span... |
352,983 | <p>How to find this expression $(1000!\mod 3^{300})$?</p>
| Math Gems | 75,092 | <p><strong>Hint</strong> $\rm\ B^\color{#C00}A \mid (AB)!\, =\, 1\cdot\cdot\: B\,\cdot\cdot\: 2B\,\cdot\cdot\ 3B\,\cdots \color{#C00}AB$<br>
thus $\:3^{300}\mid 900!\mid 1000!$</p>
|
1,195,625 | <p>We have $X = R^n$ and the discrete metric:</p>
<p>$d(x,y) = 0$, if $x=y$ and $d(x,y) = 1$ in all other cases.</p>
<p>Is this space separable or not? I tried to prove, that the answer for that is no.</p>
<p>Let us have a random $x=(x_1, x_2, ..., x_n)$ vector from $R^n$. If $X$ is separable, then such $q$ exists, ... | Jolien | 133,535 | <p>Hint: You could prove this with induction to the number of points of your graph.</p>
<p>In the inductive step you leave one line out and all the lines connected to that one, apply the induction hypothesis to that new graph and then put all the lines back in.</p>
<p>Then you use the fact there are no triangles: the... |
244,241 | <p>How can I find minimum distance between cone and a point ?</p>
<p><strong>Cone properties :</strong><br/>
position - $(0,0,z)$<br/>
radius - $R$<br/>
height - $h$</p>
<p><strong>Point properties:</strong><br/>
position - $(0,0,z_1)$</p>
| Christian Blatter | 1,303 | <p>I find the use of complex numbers extremely helpful in problems of plane elementary geometry, in particular when there are symmetries present which have to be exploited. </p>
<p>In the "complex coordinate" $z$ of a point both real coordinates are encoded, you have the full vector algebra of the plane at your dispos... |
244,241 | <p>How can I find minimum distance between cone and a point ?</p>
<p><strong>Cone properties :</strong><br/>
position - $(0,0,z)$<br/>
radius - $R$<br/>
height - $h$</p>
<p><strong>Point properties:</strong><br/>
position - $(0,0,z_1)$</p>
| Lucian | 93,448 | <p>The trouble with maths is that, just like in the case of a living organism, all its various <em>apparently</em> unrelated parts are in reality <em>interconnected</em>. For instance, <a href="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan" rel="nofollow">Ramanujan</a>'s prime-counting function, belonging to the fie... |
244,241 | <p>How can I find minimum distance between cone and a point ?</p>
<p><strong>Cone properties :</strong><br/>
position - $(0,0,z)$<br/>
radius - $R$<br/>
height - $h$</p>
<p><strong>Point properties:</strong><br/>
position - $(0,0,z_1)$</p>
| CAGT | 119,244 | <p>I have used complex numbers to solve real life problems:
- Digital Signal Processing, Control Engineering: Z-Transform.
- AC Circuits: Phasors. This is a handful of applications broadly labeled under load-flow studies and resonant frequency devices (with electric devices modeled into resistors, inductors, capacitors... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.