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 |
|---|---|---|---|---|
269,474 | <p>If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that</p>
<p>$$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$</p>
<p>Also show when equality occurs.</p>
| Bobby | 10,809 | <p>We claim <span class="math-container">$\dim \ker\,A+\dim\ker B \geq \dim\ker AB$</span>.</p>
<p>Let <span class="math-container">$\beta=\{\alpha_1,\dots,\alpha_r \}$</span> be a basis for <span class="math-container">$\ker B$</span>.
It is not hard to see that <span class="math-container">$\ker B\subseteq \ker AB$</... |
250,397 | <p>If $(X,\mathcal{M})$ is a measurable space such that $\{x\}\in\mathcal{M}$ for all $x\in$$X$, a finite measure $\mu$ is called continuous if $\mu(\{x\})=0$ for all $x\in$$X$.</p>
<p>Now let $X=[0,\infty]$, $\mathcal{M}$ be the collection of the Lebesgue measurable subsets of $X$. Show that $\mu$ is continuous if an... | Patrick Da Silva | 10,704 | <p>To show that $f$ is continuous at $x \in [0,\infty]$, it suffices to see that the left/right limits exist and are equal to the value of the function. Take any sequence $x_n \searrow x$. Then
$$
\mu([0,x_n]) = \mu([0,x]) + \mu([0,x_n]) - \mu([0,x]) = \mu([0,x]) + \mu(]x,x_n]) \longrightarrow \mu([0,x]) + \mu(\varnot... |
3,864,442 | <p>I have a propositional formula:</p>
<p><span class="math-container">$$(\neg p \lor \neg q \lor r)$$</span></p>
<p>Can i rewrite it in this way?:</p>
<p><span class="math-container">$$(\neg p \lor \neg q \lor r) = (\neg p \lor (q \land \neg r))= (\neg p \lor q) \land (\neg p \lor \neg r))$$</span></p>
| Qiaochu Yuan | 232 | <p>Here's an abstract way to organize things. You can actually define metrics to take values in any <em>ordered monoid</em>, as follows:</p>
<p>An ordered monoid is a set <span class="math-container">$R$</span> equipped with both a partial order <span class="math-container">$\ge$</span> and a monoid operation <span cla... |
282,541 | <p>Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?</p>
| Dan Brumleve | 1,284 | <p>A random walk on $\mathbb{Z}$ or $\mathbb{Z}^2$ will return to the origin almost surely, but <a href="http://mathworld.wolfram.com/PolyasRandomWalkConstants.html" rel="nofollow">this fails for $\mathbb{Z}^3$</a>. It is not related to the reals specifically but it is a curious difference between two dimensions and t... |
282,541 | <p>Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?</p>
| Community | -1 | <ol>
<li><p>The unit sphere $\{x:\|x\|=1\}$ in $\mathbb R^3$ has the property that the area of each spherical slice
$\{x:a\le x_1\le b\}$, $-1\le a\le b\le 1$, depends only on $b-a$. In more technical terms, the pushforward of the
surface measure on a sphere under orthogonal projection to a line is a uniform measure ... |
448,581 | <p>Let $\{X_n\}_{n \geq 1}$ be an independent sequence of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $n \geq 1$. I want to prove that $X_1, \ldots, X_n$ is independent of $\limsup X_n$.</p>
<p>This result is incredibly obvious, at least intuitively: the limsup does not depend on the first $n$ element... | Stephen Herschkorn | 27,997 | <p>The value of $\text{inf}_{k\ge n}\, \text{sup}_{m \ge k}\, x_m$ is independent of $n$.</p>
|
3,364,447 | <p>Let <span class="math-container">$\mathbb{F}_{p}$</span> be a finite field of order <span class="math-container">$p$</span> and <span class="math-container">$H_{n}(\mathbb{F}_{p})$</span> be the subgroup of
<span class="math-container">$GL_n(\mathbb{F}_{p})$</span> of upper triangular matrices with a diagonal
of one... | hunter | 108,129 | <p>The center is isomorphic to a copy of <span class="math-container">$\mathbb{F}_p$</span> coming from the upper right corner.</p>
<p>Proof: for <span class="math-container">$j > i$</span> write <span class="math-container">$E_{ij}$</span> for the matrix which has a <span class="math-container">$1$</span> in spot ... |
1,469,695 | <p><a href="https://i.stack.imgur.com/fLWrM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLWrM.png" alt="enter image description here"></a></p>
<p>I have the following graph, and i'm trying to determine the maximum. When I use conventional methods, I end up with the ordered pair (2.35, 1.28) roug... | DanielWainfleet | 254,665 | <p>Use the technique of incorporating new information into a given equation: Let $a=da'$ and $b=db'$ where $a',b'$ are integers. The main equation is then $$d=ar+bs=da'r+db's=d(a'r+b's).$$</p>
<p>We can divide through by $d$ to get $a'r+b's=1.$ Now if $e$ is any common divisor of $r$ and $s,$ we have $$(e|r\land e|... |
479,249 | <p>Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a set of zero Lebesgue measure?</p>
<p>Thanks.</p>
| Kernel_Dirichlet | 368,019 | <p>This is relates very closely to the Luzin N property, which informally states an absolutely continuous function maps measure zero sets to measure zero sets. </p>
<p>Formally, for $f:X\rightarrow Y$ (let's consider $X,Y\subset \mathbb{R}$ for simplicity), and $\mu(X)=0$, we have $\mu(f(X))=0$. Such a function is sai... |
578,938 | <p>Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well.</p>
<p>If k is arbitrary and we assume it's correct, then how come one can't say, </p>
<p>j = (k+1)</p>
<p>and assume p(j) is true because j is arbitrary just like k and it has the same form of p(k).</p>
<p>That log... | Stephan Kulla | 32,951 | <p>The <a href="https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem" rel="nofollow">Berry–Esseen theorem</a> gives an estimate for the normal approximation of the binomial distribution:</p>
<p>$$\sup_{x\in\mathbb R} \left|P\left(\frac{B(n,p)-np}{\sqrt{np(1-p)}} \le x\right)-\Phi(x)\right| \le \frac{C(p^2+q^2)}{... |
2,584,862 | <p>I'm trying to evaluate the following integral $$\int_{0}^{\infty} \frac{(x+1)}{(1+(1+x)^2)x^{1/3}}dx$$
I'm using the branch cut from $[0, \infty)$ (positive real axis to be precise). For this cut, when I try to evaluate the residues ( they're $(+i-1)$ and $(-i-1)$ ) and I get the following values : $ \frac{1}{2(i-1... | Doug M | 317,176 | <p>A quick way to find the residuals:</p>
<p>$\lim_\limits{z\to -1+i} \frac{(z+1)}{(z+1-i)(z+1+i)z^{1/3}} (z+1-i) = \frac {i}{2i(-1+i)^{\frac 13}}$</p>
<p>And</p>
<p>$\lim_\limits{z\to -1-i} \frac{(z+1)}{(z+1-i)(z+1+i)z^{1/3}} (z+1+i) = \frac {i}{(-2i)(-1-i)^{\frac 13}}$</p>
|
932,907 | <p>So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p \rangle=e^{-ixp/\hbar}$. I can see that these are the eigenfunctions of position and momentum respectively, but I don't see how they'... | Robin Goodfellow | 176,079 | <p>If anything, $\sin^2(x)$ is the ambiguous notation. To some it might mean $\sin(\sin(x))$ (this is why $\sin^{-1}$ is sometimes used as arcsine), and to others it might mean $(\sin(x))^2$. I cannot think of a case where anyone would see $\sin(x)^2=\sin(x^2)$ (except when, say, $x=0$). However, we mathematicians avoi... |
833,814 | <p>Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so </p>
<ol>
<li>$z=-96+160i$</li>
<li>$z=96-160 i$</li>
<li>$z=160-96i$</li>
<li>$z=-160+96i$</li>
</ol>
<p>what is the right answer ? Thanks in advance.</p>
| user1337 | 62,839 | <p>Using a change of variables, it suffices to prove the equivalent statement</p>
<p>$$f(0)=\lim_{x \to 0^+} x \int_{1}^{1/x} f(1/u)du .$$</p>
<p>Step 1: The statement clearly holds for all monomials $f(x)=1,x,x^2,\dots$.</p>
<p>Step 2: Since the problem is linear, it holds for all polynomial $f(x)$.</p>
<p>Step 3:... |
232,930 | <p>It is a well-known result that all profinite groups arise as the Galois group of <em>some</em> field extension.</p>
<blockquote>
<p>What profinite groups are the absolute Galois group
$\mathrm{Gal}(\overline{K}|K)$ of some extension $K$ over
$\mathbb{Q}$?</p>
</blockquote>
<p>The answer is simple enough in t... | Community | -1 | <p>There is a paper of Jochen Königsmann which restricts the profinite groups isomorphic to absolute Galois groups. Unfortunately, I can't remember the title.</p>
<p>Edit: I found it: <a href="http://math.usask.ca/fvk/jkproduct.pdf" rel="nofollow">http://math.usask.ca/fvk/jkproduct.pdf</a> "Products of absolute Galois... |
232,930 | <p>It is a well-known result that all profinite groups arise as the Galois group of <em>some</em> field extension.</p>
<blockquote>
<p>What profinite groups are the absolute Galois group
$\mathrm{Gal}(\overline{K}|K)$ of some extension $K$ over
$\mathbb{Q}$?</p>
</blockquote>
<p>The answer is simple enough in t... | Leonid Positselski | 2,106 | <p>This is a very good question which is a big open problem. There are a number of theorems, some of them easy and some very difficult, and also a number of conjectures, restricting the class of groups which may turn out to be absolute Galois groups. But it seems that nobody has any idea about how a precise descripti... |
232,930 | <p>It is a well-known result that all profinite groups arise as the Galois group of <em>some</em> field extension.</p>
<blockquote>
<p>What profinite groups are the absolute Galois group
$\mathrm{Gal}(\overline{K}|K)$ of some extension $K$ over
$\mathbb{Q}$?</p>
</blockquote>
<p>The answer is simple enough in t... | Claudio Quadrelli | 89,259 | <p>In order to study absolute Galois groups of fields one may also use another consequence of the aforementioned Rost-Voevodsky theorem: in fact, for any field $K$, the cohomology algebra $H^*(G_K,\mathbb{F}_p)$ is a quadratic algebra over the field $\mathbb{F}_p$ also for $p$ odd; and this is still true for the algebr... |
1,182,429 | <p>Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water the... | Robert Soupe | 149,436 | <p>Because it readily bridges the gap between different measures. If you didn't know about least common denominators, you'd probably wind up rediscovering the concept and wondering why no one else had thought of it before.</p>
<p>Let's say you swap the contents of the two beakers. You find that $$\frac{2}{4} > \fra... |
3,449,808 | <p>I am having a tough time solving this problem because I cannot seem to picture this problem. I don't know how the picture in this problem is supposed to look like and therefore I don't know where to start.</p>
<p><a href="https://i.stack.imgur.com/W6LdN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | ViHdzP | 718,671 | <p>Your solution seems correct. However, it'd be much faster to simply notice that if <span class="math-container">$x,y\geq1$</span>, <span class="math-container">$$154x+24y\geq178>30.$$</span></p>
|
3,449,808 | <p>I am having a tough time solving this problem because I cannot seem to picture this problem. I don't know how the picture in this problem is supposed to look like and therefore I don't know where to start.</p>
<p><a href="https://i.stack.imgur.com/W6LdN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | John Hughes | 114,036 | <p>You're looking at
<span class="math-container">$$
77+12=15
$$</span>
right? For positive <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.
Let <span class="math-container">$u = x-1, v = y-1$</span>, then (1) <span class="math-container">$u$</span> and <span class="math-container"... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| Bixxli | 145,805 | <p>A very nice proof is via classifying spaces of categories. It goes as follows: Take the “walking isomorphism“ category <span class="math-container">$J$</span>, that is the unique category with two isomorphic objects and four morphisms in total. The geometric realization of its simplicial nerve is exactly the infinit... |
2,162,375 | <p>I want to prove that $\sqrt{a} + \sqrt{b} \le 2 \times \sqrt{a+b}$, I had the idea to draw it:<br>
<a href="https://i.stack.imgur.com/cyqj5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/cyqj5.png" alt="enter image description here"></a></p>
<p>Would it be enough to prove what I want to prove? If not, i... | Vincent Laberge | 418,755 | <p>Not all problems have a nice geometrical interpretation. I think it is surely better to do this one in a purely algebraic way.</p>
<p>It would also be more convenient to assume a,b $\geq$ 0 for your problem.</p>
<p>Because a,b$\geq$0, then a$\leq$a+b and b$\leq$a+b, and because f(x)=$\sqrt{x}$ is increasing on $ R... |
4,426,896 | <p>Let <span class="math-container">$R$</span> be a ring <span class="math-container">$I\subset R$</span> an ideal.
Are the following true or false conditionals?</p>
<ol>
<li>If <span class="math-container">$R$</span> is a field then so is <span class="math-container">$R/I$</span>.</li>
<li>If <span class="math-contain... | Sourav Ghosh | 977,780 | <p><strong>Theorem 1</strong> : A commutative ring with unity is a field iff it has only two trivial ideals.</p>
<p>Proof : Let, <span class="math-container">$R$</span> be a commutative ring with unity with two trivial ideals.</p>
<p>Claim : <span class="math-container">$R$</span> is a field.</p>
<p>We need to show for... |
3,384,793 | <p>I have the following sequence,
<span class="math-container">\begin{align*}
P_n=\displaystyle \cfrac{1}{n^2}{\prod_{k=1}^{n}(n^2+k^2)^\frac{1}{n}} \:\:\:\: \:\: n\geq 1
\end{align*}</span>
The sequence seems to converge toward zero. But I have a hard time proving it. My strategy is to use a the following theorem.... | zhw. | 228,045 | <p>Hint:</p>
<p><span class="math-container">$$P_n = \left (\prod_{k=1}^n(1+(k/n)^2)\right)^{1/n}.$$</span></p>
|
1,066,921 | <p>Solve the system of equations $x^2=y^3, x^y=y^x$ in positive real numbers.</p>
<p>Taking $\ln$ of the second equation, we have $\ln x/x=\ln y/y$. This function is increasing in $(0,e)$ and decreasing in $(e,\infty)$. For any value of $x\neq e$, we can find a unique value of $y$ such that $x^y=y^x$. But how can we f... | Edward Jiang | 179,276 | <p>Write $y=x^{\frac{2}{3}}$, then</p>
<p>$$x^{x^{\frac 23}}=x^{\frac{2x}{3}}$$</p>
<p>so</p>
<p>$$x^{\frac{2}{3}}=\frac{2x}{3}$$</p>
<p>Can you continue?</p>
|
1,751,764 | <p>I'm doing some studies in mathematical methods for physics and I came across something that I don't really understand. I have only been using the $\epsilon_{ijk}$ when I cross some vectors or operators etc... But what does it mean when you write something like this, with Poisson brackets we have that</p>
<p>$$
\{L_... | Hans Lundmark | 1,242 | <p>From the context, it seems that the indices $a$, $b$, $c$ can only take the values 1, 2 and 3, and then the formula means that
$\{L_1,A_1\}=0$,
$\{L_1,A_2\}=+A_3$,
$\{L_1,A_3\}=-A_2$,
and so on.</p>
<p>(The positive sign if $abc$ come in the right cyclic order, negative sign if the wrong order, or zero if there is ... |
357,102 | <p>If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$?</p>
<p>It has been established that this is $m^n$. </p>
<p>How many of these are one-to-one?
I think this means that each element of $|A|$ is fixed to a unique $|B|$.</p>
| ferson2020 | 59,689 | <p>If you consider an arbitrary function $f$, from $A$ to $B$, each element in $A$ has $m$ choices for where $f$ could take it in $B$. There are $n$ elements in $A$, hence $n$ choices to be made. Therefore, there are $m^n$ such functions.</p>
<p>If you want your function to be one-to-one, then put some ordering on $A$... |
203,836 | <p>Following the wonderful methodologies provided in <a href="https://mathematica.stackexchange.com/q/203517/52181">this post</a>, I have learned how to process microscopic images of the types shown below in order to analyze them. When it comes to analysing the size distribution of the particles, the images always have... | Alx | 35,574 | <p>First you need to find scaling coefficient from your *.jpg by using Coordinate Tools. Place cursor on first tick of scale, horizontal coordinate is 234, last tick corresponds to 391. So, scaling coefficient is <code>hscale=500./(391-234)</code>. Assumimg that area has a shape of circle, <code>Histogram</code> comman... |
203,836 | <p>Following the wonderful methodologies provided in <a href="https://mathematica.stackexchange.com/q/203517/52181">this post</a>, I have learned how to process microscopic images of the types shown below in order to analyze them. When it comes to analysing the size distribution of the particles, the images always have... | MelaGo | 63,360 | <p>You could extract the conversion factor in a semi-automated way.</p>
<pre><code>imgcrop = ImageTake[img, {-20, -15}, {-150, -1}]
</code></pre>
<p><a href="https://i.stack.imgur.com/sVQSV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sVQSV.png" alt="enter image description here"></a></p>
<pre>... |
171,038 | <p>After some computations I end up with the following expression:</p>
<pre><code>Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]]
</code></pre>
<p>where $z$ is actually complex.</p>
<p>For reasons I don't understand, <em>Mathematica</em> won't simplify this to $1$.</p>
<p>According to the <a href="http://referen... | Steffen Jaeschke | 61,643 | <pre><code>ComplexPlot3D[
Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]], {z, -2 \[Pi] -
2 \[Pi] I, 2 \[Pi] + 2 \[Pi] I}, PlotLegends -> Automatic]
</code></pre>
<p><a href="https://i.stack.imgur.com/mkrN6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mkrN6.png" alt="ComplexPlot3D" /></... |
2,463,756 | <p>I'm trying to prove that $f(x)=x^2-6x-40$ is injective for $f: [3, \infty) \rightarrow [-49, \infty)$. Note that I cannot use calculus.</p>
<p>I tried letting $f(a)=f(b)$ and I arrived at $a^2-6a=b^2-6b$. Then I tried to find a solution for $a$ in terms of $b$ and a solution for $b$ in terms of $a$ and I got $a=\fr... | J126 | 2,838 | <p>First complete the square:
$$
x^2 - 6x -40 = (x - 3)^2 - 49.
$$
Now, if $f(a) = f(b)$ you'd get
$$
(a - 3)^2 - 49 = (b - 3)^2 - 49.
$$</p>
|
3,209,773 | <p>Is the following result true? Or Is there any known result of fractions like this?</p>
<p>Let <span class="math-container">$n$</span> be fixed.</p>
<blockquote>
<p>There are infinitely many integer solutions for <span class="math-container">$$\sum_{i=1}^n \frac{1}{x_i} = 0,$$</span> where <span class="math-conta... | Mike | 544,150 | <p>No for fixed <span class="math-container">$n$</span> and <span class="math-container">$k$</span> there is only a finite number of solutions.</p>
<p>Indeed, let us try to build a solution. Let first pick <span class="math-container">$x_1,\ldots, x_{\ell}$</span>; <span class="math-container">$x_{i}$</span> negative ... |
978,927 | <p>How would one prove the equality of the sum of squares of diagonals and twice the sum of squares of the two sides:</p>
<p>$$\left|\mathbf{p} + \mathbf{q}\right|^2 + \left|\mathbf{p} - \mathbf{q}\right|^2 = 2\left|\mathbf{p}\right|^2 + 2\left|\mathbf{q}\right|^2 $$</p>
<p>where $\mathbf{p}$ and $\mathbf{q}$ are vec... | Git Gud | 55,235 | <p>Picking up from where you left off.</p>
<p>From $\text{rank}(A-5I)=3$ and from $\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$ you get
$$\text{rank}(A+2I)+\text{rank}(A+3I)=6.$$</p>
<p>Now prove that $A+2I$ is invertible. What does that tell you about the rank of $A+2I$?</p>
<p>Infer that $\text{rank}(A... |
684,639 | <p>Prove that in the inverse function theorem, the hypothesis that $f$ is $C^1$ cannot be weakened to the hypothesis that $f$ is differentiable. I read an example of my teacher, but I can't have any analysis argument for the fact that $f$ is not one to one in any neighborhood of $0$. Here it is, </p>
<p>$$f(0)=0, \qq... | Carsten S | 90,962 | <p>Except at <span class="math-container">$0$</span>, the function is <span class="math-container">$C^1$</span>, so you can analyse it by examining its derivate on the interval <span class="math-container">$(0,\varepsilon)$</span>. If <span class="math-container">$f$</span> was injective on <span class="math-container"... |
684,639 | <p>Prove that in the inverse function theorem, the hypothesis that $f$ is $C^1$ cannot be weakened to the hypothesis that $f$ is differentiable. I read an example of my teacher, but I can't have any analysis argument for the fact that $f$ is not one to one in any neighborhood of $0$. Here it is, </p>
<p>$$f(0)=0, \qq... | Lutz Lehmann | 115,115 | <p>The upper and lower bounds for the function tell us that $0<y=f(x)$ may have solutions between the positive solutions of </p>
<p>$y=x\pm2x^2=2(x\pm\tfrac14)^2-\tfrac18$ or $x=\tfrac14(\sqrt{1+8y}\mp1)$. </p>
<p>This segment of length $\tfrac12$ at height $y$ is long enough for multiple oscillations of the $\sin... |
2,800,441 | <p>Anyone can give me the path for this, could not figure out which theory/method to use for this...
$$\lim_{n\to\infty}\left(\frac{1}{1\cdot4}+\frac{1}{4\cdot7} + \frac{1}{7\cdot10}+.....+ \frac {1}{(3n-2)\cdot(3n+1)}\right)=?$$</p>
| Przemysław Scherwentke | 72,361 | <p>HINT: It is
$$
\lim_{n\to\infty}\frac13\left(
\frac11-\frac14+\frac14-\frac17+\frac17-\frac1{10}\ldots+\frac1{3n-2}-\frac1{3n+1}
\right)
$$</p>
|
2,187,929 | <p>A speaks truth $3$ times out of $4$ and $B$ $7$ times out of $10$ . they both agree that a white ball has been drawn out from a bag containing $6$ balls of different color . find the probability that the statement is true . </p>
<p>my try .</p>
<p>probability when they say false and agree = $\left(\dfrac56\right)\... | hey | 417,906 | <p>Ans: $\dfrac{35}{36}$</p>
<p>From the problem,
P(A speaking truth) $=\dfrac{3}{4}$
P(A not speaking truth) $=1-\dfrac{3}{4}=\dfrac{1}{4}$
P(B speaking truth) $=\dfrac{7}{10}$
P(B not speaking truth) $=1-\dfrac{7}{10}=\dfrac{3}{10}$
P(drawing a white ball) $=\dfrac{1}{6}$
P(drawing a non-white ball) =$\dfrac{5}{6}$... |
2,348,293 | <p>Sorry, I feel like this should be simple, but I'm stumped and I've searched everywhere. Is it possible to calculate the length of a photographed object in the attached scenario if you know the width and assume it is a perfect rectangle?</p>
<p><img src="https://i.stack.imgur.com/FlSFm.png" alt="pic of flat rectang... | Gaëtan Brochard | 460,521 | <p>You have to know more parameters.</p>
<p>Imagine you're raytracing the rectangle, your viewplane has a certain x, y size value. In general y / x == WINY / WINX. WINY and WINX being the size of your window in pixels. </p>
<p>Image explaining the very basics of raytracing in case you don't know about it :
<img src=... |
2,348,293 | <p>Sorry, I feel like this should be simple, but I'm stumped and I've searched everywhere. Is it possible to calculate the length of a photographed object in the attached scenario if you know the width and assume it is a perfect rectangle?</p>
<p><img src="https://i.stack.imgur.com/FlSFm.png" alt="pic of flat rectang... | Community | -1 | <p>If we assume the rectangle to be horizontal (and the picture vertical) at relative height $z$ of the optical axis and distance $d$ of the viewing plane, the coordinates of the two right corners in the image are given by</p>
<p>$$\left(\frac{fw}{2d},-\frac{fz}{d}\right),\left(\frac{fw}{2(d+l)},-\frac{fz}{d+l}\right)... |
3,784,223 | <p>Let <span class="math-container">$g : \mathbb{R} \to \mathbb{R}$</span> be a differentiable function such that <span class="math-container">$g(0) = g^\prime(0) = 0$</span> and <span class="math-container">$g^{\prime\prime}(0)$</span> exists and is positive. Prove that there exists <span class="math-container">$x >... | Chris | 255,344 | <p>Try a Taylor expansion:
<span class="math-container">$$
g(x) = g(0) + g'(0) x + \frac{1}{2}g''(0)x^2 + R(x) = \frac{1}{2}g''(0)x^2 + R(x),
$$</span>
where <span class="math-container">$R(x)$</span> is the remainder term and is <span class="math-container">$o(x^2)$</span>, i.e. <span class="math-container">$\lim_{x \... |
4,027,927 | <p>I'm trying to find derivative of <span class="math-container">$\frac{\cos t-\sin t}{\cos t+\sin t}$</span> in a different way: there is a trick to find derivative of the form <span class="math-container">$\frac{ax+b}{cx+d}$</span>:
<span class="math-container">$$\left(\frac{ax+b}{cx+d}\right)'=\frac{ad-bc}{(cx+d)^2... | Arroz con Tomate | 655,827 | <p>Instead of trying to get the derivative of <span class="math-container">$f=\frac{\cos-\sin}{\cos+\sin}$</span>, try to get the derivative of <span class="math-container">$f^2$</span>. This way you have that <span class="math-container">$$f^2(t)=\dfrac{-2\sin(t)\cos(t)+1}{2\sin(t)\cos(t)+1}.$$</span> Now you can appl... |
40,063 | <p>I'm trying to answer the following question:</p>
<p>"A person collects coupons one at a time, at jump times of a Poisson process $(N_t)_{t\geq 0}$ of rate $\lambda$. There are m types of coupons, and each time a coupon of type j is obtained with probability $p_j$, independently of the previously collected coupons a... | Shai Covo | 2,810 | <p>Hint: Try showing ${\rm E}[T]=(1/\lambda){\rm E}[L]$, noting that $1/\lambda$ is the mean of the waiting time...</p>
<p>It is also interesting to note the following.</p>
<p>Let's show using the formula for ${\rm P}(T \leq t)$ that ${\rm E}[L] = \lambda {\rm E}[T]$ in the case when all the $p_j$ are equal, that is ... |
259,156 | <p>I am using a piecewise function to define the height of columns. My goal is to make a nice picture that illustrates the different heights of the columns using color. Below in the picture I have described what I have generated using code on the left and on the right I have illustrated what I am trying to achieve:</p>... | Syed | 81,355 | <pre><code>Manipulate[
Plot[f[a x], {x, 0, 2 Pi}
],
{{f, Sin}, {Sin, Cos, Tan}},
{a, 1, 10}
, ControlType -> {PopupMenu, Slider}
]
</code></pre>
|
1,180,719 | <p>I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory
, I would like to have some practical examples for this method (a system of linear equations with n>=1000, preferrably in .txt form), as well as some implementation details (maybe block GS or something, since I haven't looked... | mathlove | 78,967 | <p>Note that for $x,y,n\in\mathbb N$, we have
$$\frac{1}{x}+\frac{1}{y}=\frac{1}{n}\iff ny+nx=xy\iff (x-n)(y-n)=n^2.$$</p>
|
2,691,818 | <p>I know from integration that the answer is -4. However, I am messing something up somewhere while working through the Riemann sums. Going cross-eyed trying to find my mistake. I included the pertinent steps and skipped the details. I do have those; just didn't type them in. Can anybody help me out? </p>
<p>$$\in... | Andrew Li | 344,419 | <p>When rewriting a definite integral as a Riemann sum, it is rewritten as so:</p>
<p>$$\int_a^b f(x)\, dx = \lim_{n\to \infty} \sum_{i=1}^n f(x_i)\Delta x$$</p>
<p>Where the following are given:</p>
<p>$$\Delta x = {b - a\over n}$$
$$x_i = a + i\Delta x $$</p>
<p>You've forgotten to add $a$ in $x_i = a + i\Delta x... |
1,279,105 | <p>how to prove convergence of $\{\sin(1/n)\}$ sequence when $n$ goes infinity, using limit definition?
I proved that $\{\sin(n)\}$ is divergent but i can not do this with that method.</p>
| ajotatxe | 132,456 | <p>No. Consider $f:\Bbb R\to(-1,1)$ with
$$f(x)=\frac1\pi\tan^{-1}(x)$$</p>
|
2,410,940 | <p>The question states that if $f (x)$ is a polynomial such that $x-1|f(x^n)$ prove that $f(x^n)$ is divisible by $x^n-1$</p>
<p>This is how I proceeded since$x-1|f(x^n)$ </p>
<p>$f(1)=0$ </p>
<p>$\frac {f (x^n)-f(1)}{x^n-1}=g(x)$ </p>
<p>since $f(1)=0$ </p>
<p>$\frac{f(x^n)}{x^n-1}=g(x)$
hence $x^n-1|f(x^n)$</p>... | farruhota | 425,072 | <p>Assume the polynomial of degree $m$ has the roots $x_1,x_2,...,x_m$:
$$f(x)=a_0x^m+a_1x^{m-1}+\cdots +a_m=a_0(x-x_1)(x-x_2)\cdots (x-x_m).$$
Then:
$$f(x^n)=a_0(x^n-x_1)(x^n-x_2)\cdots (x^n-x_m).$$
Since $f(1)=f(1^n)$, then $x-1|f(x^n) \Rightarrow x-1|f(x)$. It implies that at least one root of $f(x)$ is $1$, hence $... |
1,947,915 | <p>Can I ask what is the Laplace transform of $\sqrt[3]{t}$ using the Gamma function?
This was my initial answer. Note that there is a theorem, $\Gamma(\frac{1}{3})=3\Gamma(\frac{4}{3})$. $$\int^{\infty}_{0}\exp^{-\beta} \beta^{-\frac{2}{3}}=3\int^{\infty}_{0}\exp^{-\beta} \beta^{\frac{4}{3}}$$ And was able to get the ... | reuns | 276,986 | <p>The correct proof needs some complex analysis.</p>
<p>Define for $Re(s) > 0$ : $$\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}dt$$</p>
<p>Then for real $s \in (0,\infty)$ you have
$$\mathcal{L}[t^{1/3}1_{ t > 0}](s) = \int_0^\infty t^{1/3} e^{-st} dt \underset{x \ = \ st}= s^{-4/3}\int_0^\infty x^{1/3} e^{-x} dx ... |
1,664,385 | <blockquote>
<p>A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x axis and end point B lies on y axis. A point P(x,y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.</p>
</blockquote>
<p>I understand the definitions of... | user247327 | 247,327 | <p>As A and B move along the x and y axes, respectively, P moves over the xy-plane.</p>
|
101,805 | <p>Let $m$ be a positive integer. Let $a,b$ be integers with $0 \leq a,b < m$, $a,b$ not both zero, $\gcd(a,b,m)=1$.</p>
<p>Do there necessarily exist integers $x,y$ such that<br>
$x \equiv a \pmod{m}$<br>
$y \equiv b \pmod{m}$<br>
$(x,y)=1$?</p>
<p>Equivalently, are there integers $c,d,k,l$ such that
$$
c(a+mk)... | Gerry Myerson | 8,269 | <p>Let $t$ be the product of all the primes dividing $b$ but not $a$ (if there are no such primes then $t$ is the empty product which, by convention, is $1$). Let $x=a+tm$, $y=b$. Then clearly $x\equiv a\pmod m$, and $y\equiv b\pmod m$, so we just have to check that $\gcd(x,y)=1$. </p>
<p>Let $p$ divide $y$. If $p$ al... |
3,230,619 | <p>Hello i cant figure out how to evaluate this integral and find a volume for a triple integral which is given by <span class="math-container">$z=x^2+y^2 \text{ and } z = x+y$</span> this figure for what i found is paraboloid intersected by a straight plane. I have tried to find the intersection of the two figures but... | Disintegrating By Parts | 112,478 | <p>A pointwise result is typically needed for such a proof. For example,
<span class="math-container">$$
\lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R}^{R}e^{isx}\widehat{\chi_{[a,b]}}(s)ds = \frac{1}{2}(\chi_{[a,b]}(x)+\chi_{(a,b)}(x)) \;\;(*)
$$</span>
The form on the right is a convenient way to expre... |
340,417 | <p>How to calculate following integration?</p>
<p><span class="math-container">$$\int 5^{x+1}e^{2x-1}dx$$</span></p>
| Caran-d'Ache | 66,418 | <p>Use integration by parts. Set $2x=y$ and you'll get:
$$5e^{-1}\int5^{\frac{y}{2}}e^ydy=5e^{-1}\int5^{\frac{y}{2}}d(e^y)$$
After that just a little effort. :)
$$5e^{-1}\int5^{\frac{y}{2}}e^ydy=5e^{-1}\int5^{\frac{y}{2}}d(e^y)=5e^{-1}(5^{\frac{y}{2}}e^y-\frac{\ln(5)}{2}\int5^{\frac{y}{2}}e^ydy)$$
Combining the first a... |
2,617,576 | <p>I'm trying to check whether or not this set is linearly independent for all $n$, where $A$ is $n \times n$ and $A, A^2, \dots, A^{n^2}$ are distinct matrices and $I_n$ is the identity matrix.</p>
<p>Clearly, if we take $n = 2$, and $A = 3 I_n$ then the set $\{I, A, A^2, A^3, A^4 \}$ is not linearly independent. Is ... | Pavel Čoupek | 82,867 | <p>By the way, you can show something (much) stronger:</p>
<blockquote>
<p>Given any $n \times n$ matrix $A$, the set $\{I_n, A, A^2, \dots, A^n\}$ is linearly dependent.</p>
</blockquote>
<p>For proof, use the <a href="https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem" rel="nofollow noreferrer">Cayley-... |
2,617,576 | <p>I'm trying to check whether or not this set is linearly independent for all $n$, where $A$ is $n \times n$ and $A, A^2, \dots, A^{n^2}$ are distinct matrices and $I_n$ is the identity matrix.</p>
<p>Clearly, if we take $n = 2$, and $A = 3 I_n$ then the set $\{I, A, A^2, A^3, A^4 \}$ is not linearly independent. Is ... | Alberto Andrenucci | 370,680 | <p>Consider $V$ a vector space over a field $\mathbb{K}$. Then, if you take any matrix $M \in \mathcal{M}_n(\mathbb{K})$ you can compute its characteristic polynomial which you know has degree $n$. Then, notice this:</p>
<blockquote>
<p>Let $m_A(t)$ be the minimal polynomial of $A$. Then, if you call $k=deg(m_A(t))$... |
2,347,775 | <p>Suppose we have arbitrary predicates are $P$ and $Q$.</p>
<p>Let the statements be defined as follows:</p>
<p>$F1:$ [for all x, P(x)] is false OR [for some x, Q(x)] is true</p>
<p>$F2:$ [for some x, P(x)] is false OR [for all x, Q(x)] is true</p>
<p>Prove that $F1 \neq\implies F2$ (does not apply)</p>
<p>We hav... | Mauro ALLEGRANZA | 108,274 | <p>Consider:</p>
<blockquote>
<p>$(x=0)$ as $P(x)$ and $(x > 0)$ as $Q(x)$.</p>
</blockquote>
<p>Thus, we have:</p>
<blockquote>
<p>$\mathbb N \vDash \lnot \forall x \ (x=0) \lor \exists x \ (x > 0)$</p>
</blockquote>
<p>but:</p>
<blockquote>
<p>$\mathbb N \nvDash \lnot \exists x \ (x=0) \lor \forall x... |
2,182,994 | <p>Consider $\text{SL}(2,\mathbb{R})$ with the left-invariant metric obtained by translating the standard Frobenius product at $T_I\text{SL}(2,\mathbb{R})$. (i.e $g_I(A,B)=\operatorname{tr}(A^TB)$ for $A,B \in T_I\text{SL}(2,\mathbb{R})$).</p>
<p>One can show that the geodesics starting at $I$ are of the form of</p>
... | Robert Bryant | 84,371 | <p>The geodesic leaving $I_2\in\mathrm{SL}(2,\mathbb{R})$ with velocity
$$
v = \begin{pmatrix} v_1 & v_2+v_3\\ v_2-v_3 & -v_1\end{pmatrix}
\in {\frak{sl}}(2,\mathbb{R})\simeq\mathbb{R}^3
$$
is given by $\gamma_v(t) = e^{t\,v^T}e^{t\,(v{-}v^T)}$. Thus, the <em>geodesic</em> exponential mapping for this metric... |
2,615,825 | <p>What I did was:
I tested for $\lim_\limits{n\to\infty}u_n$ by taking log</p>
<p>$$\lim_\limits{n\to\infty} \frac{\ln\ \left(4 - \frac{1}{n}\right)} {\frac{n}{(-1)^n}}$$</p>
<p>Applying L'hopital's rule,</p>
<p>$$\lim_\limits{n\to\infty} \frac{\left(\frac{1} {4-\frac{1}{n}}\right)\left(\frac{1}{n^2}\right)}{(-1)^n... | Christian Blatter | 1,303 | <p>The set $E\subset{\mathbb R}^2$ defined by the given equation is symmetric with respect to the lines $y=\pm x$, since
$$(x,y)\in E\Leftrightarrow (y,x)\in E,\qquad (x,y)\in E\Leftrightarrow(-y,-x)\in E\ .$$
It follows that these two lines are the principal axes of $E$. Putting $y=x$ in the equation gives $34x^2=7$, ... |
2,615,825 | <p>What I did was:
I tested for $\lim_\limits{n\to\infty}u_n$ by taking log</p>
<p>$$\lim_\limits{n\to\infty} \frac{\ln\ \left(4 - \frac{1}{n}\right)} {\frac{n}{(-1)^n}}$$</p>
<p>Applying L'hopital's rule,</p>
<p>$$\lim_\limits{n\to\infty} \frac{\left(\frac{1} {4-\frac{1}{n}}\right)\left(\frac{1}{n^2}\right)}{(-1)^n... | mathreadler | 213,607 | <p>Yet another method is with matrices and linear algebra. If we stuff $(x,y,1)^T$ into a vector, then the matrix</p>
<p>$${\bf M}=\left[\begin{array}{ccc}10&7&0\\7&10&0\\0&0&-7\end{array}\right]$$</p>
<p>Can be used to express the ellipse as a scalar product with matrix multiplication: </p>
... |
2,131,709 | <p><span class="math-container">$$F=\left(y\cos \left(xy\right)+e^{x+y}\right)i+\left(x\cos \left(xy\right)+e^{x+y}\right)j$$</span></p>
<p>also show that <span class="math-container">$$∫_cF ⋅ dr = e^2-e^{-2}$$</span>
where c is the straight line from <span class="math-container">$\left(-1,-1\right)$</span> to <span c... | Our | 279,869 | <p>In order to find the potential function of a conservative force, you choose some point, lets say (-1,-1), than take the line integral of the force from (-1,-1) to (x,y), which will give you the potential function of the force.</p>
<p>$$\int_{(-1,-1)}^{(x,y)} \vec{F} \cdot d \vec{r} = U(x,y)$$</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Ilya Nikokoshev | 65 | <p>When you consider how polynomial $f$ of degree $n$ acts on a big circle $R$, it gives rise to a map $S^1 \to S^1$ of degree $n$. Such a map cannot be continued to a map of a disk $D \to D-\{0\}$, thus $f(D)$ contains point 0.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | skupers | 798 | <p>There is a proof using clutching functions over the sphere and the first Chern class. It is quite similar to the fundamental group proof of FTA. The trick is a polynomial without zeroes allows one to construct an isomorphism between a vector bundle with first Chern class $\deg d$ and a vector bundle with first Chern... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Terry Tao | 766 | <p>This is not a serious answer, but one can "prove" the fundamental theorem of algebra by applying the spectral theorem to the matrix</p>
<p>$$ A := \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\\ 0 & 0 & 1 & \ldots & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ -a_0... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Pablo Lessa | 7,631 | <p>A proof using dynamical systems:</p>
<p>If $p$ is a non-constant polynomial without roots then $f = \text{Re}(\frac{1}{p})$ is a bounded harmonic function which goes to zero at infinity. Consider the gradient flow for $f$. This flow is area preserving because $f$ is harmonic. Also, the value of $f$ is strictly i... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | J. M. Almira | 22,915 | <p>Perhaps it could be of interest for you to know that there exists purely geometric proofs of this result. Concretely, it can be shown that, if the FTA fails then there exists a plane Riemannian metric over the Sphere S^2. Of course, this produces a contradiction since the sphere is not flat. This proof can be locate... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Community | -1 | <p>There is this "Math Help Forum" which discusses about high school level and undergraduate level mathematics. You can find lots and lots of things. Here is one problem website : <a href="http://amc.maa.org/e-exams/e8-usamo/usamo.shtml" rel="nofollow">http://amc.maa.org/e-exams/e8-usamo/usamo.shtml</a></p>
<p>Calculu... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | J. M. ain't a mathematician | 498 | <p>If you haven't been using <a href="http://rads.stackoverflow.com/amzn/click/0471789011" rel="nofollow">"The Art and Craft of Problem Solving" by Zeitz</a>, you should. Like, right at this moment. :)</p>
<p>For neat problems... you might want to look into <a href="http://rads.stackoverflow.com/amzn/click/0883853256"... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Isaac | 72 | <p><a href="http://www.keypress.com/x5488.xml" rel="nofollow">Problem Solving Strategies: Crossing the River with Dogs and Other Mathematical Adventures</a></p>
|
2,494,160 | <p>I've been having trouble using the definition of a limit to prove limits, and at the moment I am trying to prove that
$$\lim_{n\to\infty} \frac{n^x}{n!}=0$$</p>
<p>for all $x$ which are elements of natural numbers.
I'm able to start the usual setup, namely let $0<\epsilon$ and attempt to obtain $\left\lvert\dfr... | zwim | 399,263 | <p>$n!=\overbrace{1\times 2\times 3\times 4}^{>2^4}\times\overbrace{5}^{>2}\times\overbrace{6}^{>2}\times\cdots\times\overbrace{n}^{>2}\ge2^4\times2^{n-4}\ge2^n$</p>
<p>Or you can prove it by induction: <a href="https://math.stackexchange.com/questions/76946/prove-the-inequality-n-geq-2n-by-induction?rq=1"... |
1,066,311 | <p>Given a vector $\vec a\in\mathbb R^n$ and another $\alpha=(\|\vec a\|,0,\dots,0)$, how could I define an orthogonal matrix $M$ such that $M\vec a=\alpha$ and $M^{-1}=M^t$? For $\mathbb R^2$ I tried to use a generic matrix $\; Q=\left(\begin{array}{cc}a & c\\b & d\end{array}\right)$ and $M=Orthogonalize(Q)$ u... | abel | 9,252 | <p>use reflection transformation $R = 2uu^T/u^Tu - I$ or $I - 2vv^T/v^Tv$ where $u = a + |a|e_1$ and $v = a - |a|e_1.$</p>
<p>these reflection matrices are sometimes called householder transformations.</p>
|
121,678 | <p>While teaching number theory this quarter, I have come across a phenomenon which was already addressed <a href="https://mathoverflow.net/questions/16141">in another MO posting</a>, but I have new questions. Let $p$ be a prime congruent to 3 mod 4. Then an elementary refinement of Wilson's theorem says that $\frac{... | John Blythe Dobson | 23,807 | <p>As suggested in the comment by Greg Martin, the determination of the parity of $(h + 1)/2$ can be performed more efficiently than by computing the factorial expression in Lagrange’s refinement of Wilson’s theorem.
Dirichlet, “Question d’analyse indéterminée,” <em>Journal für die Re... |
3,033,601 | <p>I am interested in the following question:</p>
<blockquote>
<p>Does there exist a continuous function <span class="math-container">$f:S^2\to S^2$</span> such that, for any <span class="math-container">$p\in S^2$</span>, <span class="math-container">$|f^{-1}(\{p\})|=2$</span>?</p>
</blockquote>
<p>I suspect the a... | FMB | 480,187 | <p><strong>EDIT</strong>: the argument is incomplete. As pointed in the comments by yoyo, the separation of the pre-images is not guaranteed by the compactness.</p>
<p>There is no such map.</p>
<p>For is there is, I prove below that it has to be a covering map and you proved that it is not possible
(or simply remark ... |
3,695,971 | <p>The way I understand it currently, saying "<em>only if <span class="math-container">$P$</span>, then <span class="math-container">$Q$</span></em>" is like saying that "<em>only if <span class="math-container">$P$</span> happens, <span class="math-container">$Q$</span> happens.</em>" To me, it seems to say the same t... | Brian M. Scott | 12,042 | <p>No. <em>Only if</em> <span class="math-container">$P$</span>, <em>then</em> <span class="math-container">$Q$</span> means that <span class="math-container">$Q$</span> cannot happen without <span class="math-container">$P$</span>: if <span class="math-container">$Q$</span> happens, <span class="math-container">$P$</... |
1,660,402 | <p>Suppose I transform an integral $$I=\int f(x,y) \, dx \, dy$$ using polar coordinates, setting $x=r\cos\theta$ and $y=r\sin\theta$. We get
$$
\begin{split}
dx &= \cos\theta \, dr - r\sin\theta \, d\theta\\
dy &= dr\,\sin\theta + r \cos \theta \, d\theta.
\end{split}
$$</p>
<p>Now the volume element $dx \, d... | gt6989b | 16,192 | <p>Multivariable transformations don't quite work out so easily in multi-d to a simple 1d change. You have to generalize the 1D change of variables carefully (see <a href="http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/change/change.html" rel="nofollow">here</a> for example).</p>
<p... |
1,432,003 | <p>A is a $n\times k $ matrix.</p>
<p>I have to show that $\|A\|_2\leq \sqrt{\|A\|_1\cdot \|A\|_\infty}$. </p>
<p>I know that $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| $ for every $\| \cdot \|$ submultiplicative matrix norm, but I don't know how to conclude.</p>
<p>Any idea? </p>
| Svetoslav | 254,733 | <p>As you noted that you now the inequality $\|A\|_2^2=\rho(A^HA)$ then it is assumed that you work with the standard induced matrix norms.</p>
<p>The <strong>definition</strong> is as follows:</p>
<p>Let $B\in \mathbb R^{n\times k}$ and let $\|.\|$ is some vector norm in $\mathbb R^k$. Then the induced matrix norm, ... |
1,432,003 | <p>A is a $n\times k $ matrix.</p>
<p>I have to show that $\|A\|_2\leq \sqrt{\|A\|_1\cdot \|A\|_\infty}$. </p>
<p>I know that $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| $ for every $\| \cdot \|$ submultiplicative matrix norm, but I don't know how to conclude.</p>
<p>Any idea? </p>
| R.N | 253,742 | <p>$\|A\|_{2}^2\leq trac(A^H\cdot A)\leq{\Vert A^H\cdot A\Vert_1\leq \| A\|_1\|A\| _\infty}$. </p>
<p>for more information see $D.46$ and $D.52$ of abstract harmonic analysis hewitt&ross pages 706 and 709</p>
|
798,606 | <p>One can block-diagonally embed a copy $H$ of the unitary group $U(n-1)$ into $U(n)$ by
$$A \mapsto \begin{bmatrix}\det(A)^{-1}&0\\0& A\end{bmatrix}.$$</p>
<p>According to a remark in the example section of Paul Baum's dissertation, the coset space $$U(n)/H \approx S^1 \times \mathbb{C}\mathrm{P}^{n-1}.$$
... | Matt E | 221 | <p>Consider $SU(n)$ acting on $\mathbb C P^{n-1}$ (as a subgroup of $\mathrm{GL}_n(\mathbb C)$. Check that it acts transitively, and that the
stabilizer of a point is $U(n-1)$ (embedded as in the OP).</p>
<p>This shows that $SU(n)/U(n-1) = \mathbb C P^{n-1}$. Now we may embed
$U(1) = S^1$ into $U(n)$ as $z \mapsto \... |
2,479,483 | <p>given the matrix equation $Ax=b$ (shown as $[A|b]$ )</p>
<p>$$\left[\begin{array}{ccc|c} 2 & 2 & 0 &2 \\ 0 &k &1 &1 \\ 1 &2 &k&2 \end{array}\right]$$</p>
<p>I've used Gaussian elimination to find that this equals:</p>
<p>$$\left[\begin{array}{ccc|c} 1&1 &0 &1 \\ 0 &... | Robert Israel | 8,508 | <p>By Zorn's lemma, your set has a maximal linearly independent subset. Show that its span is the span of the original set.</p>
|
948,819 | <blockquote>
<p>Find the domain of the function $$g(x)=\log_3(x^2-1)$$</p>
</blockquote>
<p>This is what I got so far:</p>
<p>$$\{ x\mid x^2-1>0\} =$$
$$\{ x\mid x^2>1\} =$$
$$\{ x\mid x>\sqrt { 1 } \}= $$</p>
<p>I don't know where to go from here to arrive at the correct answer... I would like a nudge in... | qwr | 122,489 | <p>$A_c = A_t = A_a + 0.04$ is not correct. That would only apply if the children and teens have the same average. Also, $\left(\frac{A_{c}+A_{t}}{2}\right)=A_{a}+4$ is not correct either. That is the average of the average teen and average child score, which is not the same as the average of 200 teens and 100 children... |
1,838 | <p>I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction... | BWW | 638 | <p>You seem to have a very idealised idea of what is likely to happen. In the vicinity of a pole the magnetic field lines are nearly vertical. Any compass that needs gimbals is not going to be reliable over a wide area.</p>
|
3,295,098 | <p>I have a problem with finding the center coordinates of each side in an equilateral triangle. I've linked an image below that shows exactly which coordinates I'm after.</p>
<p>I understand that I can work out the height of then triangle by using pythagoras theorem. I can split the triangle down the middle and calcu... | Vasili | 469,083 | <p>Assuming one of the vertices is on axis <span class="math-container">$y$</span>, it's coordinates will be A<span class="math-container">$(0,\sqrt{3})$</span> (use Pythagoras and <span class="math-container">$30-60-90$</span> triangle). The bottom side will lie on the line <span class="math-container">$y=-0.5\sqrt{3}... |
3,295,098 | <p>I have a problem with finding the center coordinates of each side in an equilateral triangle. I've linked an image below that shows exactly which coordinates I'm after.</p>
<p>I understand that I can work out the height of then triangle by using pythagoras theorem. I can split the triangle down the middle and calcu... | Allawonder | 145,126 | <p><em>Hint:</em> Draw a line parallel to the base through the centre. Then drop a perpendicular from one of the <span class="math-container">$?$</span>-points to meet this line. You have a right triangle whose legs are the coordinates of that point.</p>
<p>To calculate these legs, use parallelisms to determine that t... |
3,295,098 | <p>I have a problem with finding the center coordinates of each side in an equilateral triangle. I've linked an image below that shows exactly which coordinates I'm after.</p>
<p>I understand that I can work out the height of then triangle by using pythagoras theorem. I can split the triangle down the middle and calcu... | MarianD | 393,259 | <p>The height of the equilateral triangle is <span class="math-container">$$h={\sqrt 3 \over 2}a$$</span> </p>
<p>(you have <span class="math-container">$a=3$</span>), and coordinates of its vertices are</p>
<p><span class="math-container">$$A = \left(-\frac a 2, -\frac h 3\right),\quad B = \left(\frac a 2, -\frac h ... |
2,170,833 | <p>A proposition in a lecture script on measure and integration theory reads:</p>
<blockquote>
<p><strong>Proposition 1.1.4</strong> Let $$\mathcal{I} = \{ (a,b]: -\infty \leq a < b < \infty \}\cup\{ (b,\infty) : -\infty < b < \infty \}\cup\{ \emptyset \}\cup\mathbb{R}.$$ Then $$
\epsilon = \left\{ \big... | zipirovich | 127,842 | <p>You're misinterpreting what $\mathcal{I}$ is. $\mathcal{I}$ is a collection (set) of subsets (more specifically, certain types of intervals) of $\mathbb{R}$. In other words, $\mathcal{I}$ is a subset of $\operatorname{Pow}(\mathbb{R})$, but <strong>NOT</strong> a subset of $\mathbb{R}$. That's why $\mathcal{I}$ can'... |
59,429 | <p>I am trying to learn more about the Berlin Airlift transport problem. Two links I could find are here:</p>
<p><a href="http://drmohdzamani.com/notes/file/Simplex%20Method.pdf" rel="nofollow noreferrer">http://drmohdzamani.com/notes/file/Simplex%20Method.pdf</a></p>
<p><a href="http://www.cabrillo.edu/%7Emladdon/math... | joriki | 6,622 | <p>A cost function usually specifies a quantity to be minimized. In the present case, you're trying to <em>maximize</em> the cargo capacity. A more general term for a function to be optimized (minimized or maximized) is "objective function". If you want to use an optimization algorithm or package that expects a cost fu... |
2,998,735 | <p>I am trying to solved this inequality for <span class="math-container">$k$</span>.</p>
<p><span class="math-container">$x^{2k}<\varepsilon\cdot k^k$</span></p>
<p>Here <span class="math-container">$k\in\mathbb{N}$</span> and <span class="math-container">$x,\varepsilon$</span> are fixed such that <span class="ma... | Bhavesh Singhal | 470,492 | <p>Choosing <span class="math-container">$k=\max \left( \lceil x^2\rceil+1,\ \lceil\frac{x^2}{\varepsilon}\rceil \right)$</span> works. Thanks to everyone for their suggestions.</p>
|
2,998,735 | <p>I am trying to solved this inequality for <span class="math-container">$k$</span>.</p>
<p><span class="math-container">$x^{2k}<\varepsilon\cdot k^k$</span></p>
<p>Here <span class="math-container">$k\in\mathbb{N}$</span> and <span class="math-container">$x,\varepsilon$</span> are fixed such that <span class="ma... | fleablood | 280,126 | <p>Suppose <span class="math-container">$0 < a < c$</span> and <span class="math-container">$b = 1$</span>. The <span class="math-container">$b^k = 1$</span> for all <span class="math-container">$k$</span> and defining <span class="math-container">$\log_1$</span> makes no sense. So you can't use logarithms base ... |
244,207 | <p>Prove that $a^2b + b^2c + c^2a \ge ab + bc+ ac$ for positive real numbers $a,b,c$ such that $a+b+c=3$.</p>
| Pot | 50,617 | <p>The theorem, as stated, is false, as the following counterexample shows.</p>
<p>First note that both sides of the inequality and the restriction $a + b + c = 3$ are continuous functions of $a$, $b$, and $c$, so we can consider non-negative values rather than positive. Then if $a = 0$ we have
$$
b^2 c \geq bc
$$
f... |
239,622 | <p>For what values of $1\le p \le \infty$ does $f(x,y)=\frac{1}{1+|x|+|y|}$ with $(x,y) \in \mathbb{R}^2$ belong to $L^p(\mathbb{R}^2)$?</p>
<p>Using Wolfram Alpha I've found that the answer should be $p > 2$, but I don't know how to begin proving it. I've thought about doing a change of variable, after considering... | Christian Blatter | 1,303 | <p>Because of the $1$ in the denominator we only have to bother about the complement of the unit disk. As $r:=\sqrt{x^2+y^2}\leq|x|+|y|\leq 2r$ we have
$$r\leq 1+|x|+|y|\leq 3r\qquad(r\geq 1)\ .$$
It follows that the integral
$$\int_{{\mathbb R}^2}{1\over (1+|x|+|y|)^p}\ {\rm d}(x,y)$$
is finite iff the integral
$$2\pi... |
234,137 | <p>I have a long list consists of float numbers with some of them duplicated as shown below.</p>
<p><a href="https://i.stack.imgur.com/Qwz8I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qwz8I.png" alt="list output" /></a></p>
<p>I want to divide this list into bins;</p>
<ol>
<li>with equal bin wid... | kosif | 75,572 | <p>Regarding the first goal item, modifying the answer belongs to @azerbajdzan, I came up with a function that generates binning with the arguments; list itself and requested bin amount.</p>
<pre><code> binner[list_, binamount_] :=
Module[{min, max, maxwidth, binnedlist, binamountcheck, result},
min = Min[list];... |
1,526,442 | <p>I have problem with proving following equation:</p>
<p>$$
\binom{n}{0}0^2+\binom{n}{1}1^2+\binom{n}{2}2^2+...\binom{n}{n}n^2=n(1+n) \cdot 2^{2n-2}
$$</p>
<p>Thanks for any help!</p>
| Alex Fish | 149,127 | <p>Let $f(x)=(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k$. Derive and multiply by $x$ to get $xf'(x)=\sum_{k=1}^n \binom{n}{k}kx^k$. Derive again and get $(xf'(x))'=\sum_{k=1}^n \binom{n}{k}k^2x^{k-1}\mid_{x=1}=\sum_{k=1}^n \binom{n}{k}k^2$. $$(xf'(x))'=f'(x)+xf''(x)=n(1+x)^{n-1}+xn(n-1)(1+x)^{n-2}\mid_{x=1}=n2^{n-1}+n(n-1)2^... |
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| TooTone | 77,151 | <p>Because it has some very useful properties. For example</p>
<p>$$ \frac{d}{dx} e^x = e^x $$</p>
<p>but with another base you would need a constant.</p>
<p>An analogy is using radians instead of degrees once you get more advanced at trigonometry. Here again</p>
<p>$$ \frac{d}{dx} \sin x = \cos x $$</p>
<p>when $... |
465,555 | <p>Let $f , g : X \rightarrow Y$ be continuous where $Y$ is Hausdorff. Prove that $A = \{x : f(x) = g(x)\}$ is closed in $X$.
I have done the followings.</p>
<p>$f(X)$ and $g(X)$ are two subspaces of $Y$.</p>
<p>As Y is Hausdorff, $f(X), g(X)$ and $f(X) \times g(X)$ are also.</p>
<p>$L = \{(f(X),g(X)) : f(X) = g(X)\... | Prahlad Vaidyanathan | 89,789 | <p>I don't completely understand what you are doing, but this is how I would do this problem : To show that $C = \{x\in X : f(x) = g(x)\}$ is closed, you can show that $C^c$ is open. Choose $x \in C^c$, then $f(x) \neq g(x)$. Since $Y$ is Hausdorff, there exist open sets $U, V \subset Y$ such that
$$
f(x) \in U, g(x) \... |
3,988,517 | <p>Consider
<span class="math-container">$$A=a^2+2ab^2+b^4-4bc-4b^3,$$</span>
where <span class="math-container">$a,b,c\in\mathbb{Z}$</span> and <span class="math-container">$b\neq0$</span> such that <span class="math-container">$b|a$</span> and <span class="math-container">$b|c$</span>, so <span class="math-container"... | Math777 | 874,516 | <p>In general, A is obviously not a perfect square. A counterexample is when a = 3, b=1 and c=1 in which case A = 8. However, there are many values of a, b, and c where A is a perfect square. One special case is when b = 1. If b = 1, and you let a + 1 be the hypotenuse of a right triangle whose sides are natural number... |
105,378 | <p>I want to ask you if can it be so simple to prove that $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ by divide the numerator and denominator with $x^2$ and that's it? </p>
<p>If it this simple indeed you can write a comment and I'll delete the question after I'll read it... | Ragib Zaman | 14,657 | <p>$$ \sum \left( \frac{1}{n^2} - \frac{x^2}{1+n^2x^2} \right) = \sum \frac{ 1}{n^2(1+n^2 x^2) } \leq \frac{1}{x^2} \sum \frac{1}{n^4} \to 0.$$</p>
|
4,625,085 | <p>Let <span class="math-container">$D$</span> be a digraph as follows:
<a href="https://i.stack.imgur.com/sN32Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sN32Z.png" alt="enter image description here" /></a></p>
<p>I want to compute a longest simple path of it.</p>
<p>For an acyclic digraph, <a... | Erik Satie | 698,573 | <p>I show that the limit is finite :</p>
<p>We have :</p>
<p><span class="math-container">$$f\left(x\right)=\int_{0}^{\lfloor x\rfloor}\prod_{n=1}^{\lfloor x\rfloor}\frac{\left(y+2n\right)\ln\left(y+2n-1\right)}{\left(y+2n-1\right)\ln\left(y+2n\right)}dy<\int_{0}^{\lfloor x\rfloor}\prod_{n=1}^{\lfloor x\rfloor}\frac... |
3,103,160 | <p>I am dealing with some expressions containing combinatoric numbers. Does anybody know a formula for this?</p>
<p><span class="math-container">$$\displaystyle\sum_{k=0}^{\left\lfloor \dfrac{n}{2} \right\rfloor} \binom{n}{k}\binom{m}{k}$$</span></p>
| J. W. Tanner | 615,567 | <p><span class="math-container">$$4x^2 + 2((y-1/2)^2-1/4) = 0 $$</span></p>
<p>means</p>
<p><span class="math-container">$$4x^2 + 2 (y-1/2)^2 - 1/2 = 0 $$</span></p>
<p>or</p>
<p><span class="math-container">$$8x^2 + 4 (y-1/2)^2 - 1 = 0$$</span></p>
<p>or</p>
<p><span class="math-container">$$8x^2 + (2y-1)^2 = 1... |
3,502,280 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x,y,z$</span> are positive integers such that <span class="math-container">$x+y+z=200$</span>. Find maximal value and minimal value of <span class="math-container">$M = x! + y! + z!$</span></p>
</blockquote>
<p>Could you give me some suggest... | Bartek | 671,751 | <p><span class="math-container">$n!-n$</span> is an non-decreasing sequence on <span class="math-container">$\mathbb{Z}_+$</span>. That means that to maximize the given sum under the given conditions one should make the biggest value as big as possible and to minimize one should make it as small as possible. Thus the m... |
3,502,280 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x,y,z$</span> are positive integers such that <span class="math-container">$x+y+z=200$</span>. Find maximal value and minimal value of <span class="math-container">$M = x! + y! + z!$</span></p>
</blockquote>
<p>Could you give me some suggest... | Community | -1 | <p>Suppose <span class="math-container">$x\ge y$</span>. Then the effect on <span class="math-container">$x!+y!$</span> of increasing <span class="math-container">$x$</span> by <span class="math-container">$1$</span> and decreasing <span class="math-container">$y$</span> by <span class="math-container">$1$</span> is to... |
3,856,233 | <p>Let us start from probability space <span class="math-container">$(\Omega,\mathcal{C},\mathbb{P})$</span> and a sequence of events <span class="math-container">$\{C_n\}$</span>. I know that:
<span class="math-container">$$\mathbb{P}\left(\bigcup_{m\ge n}C_m\right)\ge\mathbb{P}\left(C_m\right),\text{ for each $m\ge n... | Henno Brandsma | 4,280 | <p>The Cantor set is the intersection <span class="math-container">$$\bigcap_{n=1}^\infty C_n$$</span></p>
<p>where <span class="math-container">$C_0 = [0, 1]$</span> and <span class="math-container">$C_{n+1} = \frac{1}{3}C_n \cup \frac{2}{3}C_n$</span> for each <span class="math-container">$n\ge 0$</span>. Every <spa... |
252,810 | <p>I have generated a real antisymmetric matrix of order 6 as follows.</p>
<pre><code>k0 = {{0, 1, 0, 0, 0, 0}, {-1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0,
0, -1, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, -1, 0}}
</code></pre>
<p>Any way to generate such a matrix of order 36 X 36 without writing each terms as abo... | LouisB | 22,158 | <p>This example returns a <code>SparseArray</code>. Use <code>Normal</code> to make it a nested list, if you prefer.</p>
<pre><code>mat = Block[{b, n = 6},
b = Riffle[ConstantArray[1, n/2], 0];
SparseArray[{Band[{1, 2}] -> b, Band[{2, 1}] -> -b}, {n, n}]
];
MatrixForm @ mat
</code></pre>
<p><span class=... |
2,575,149 | <p>I have been given the following problem: </p>
<p>A spherical balloon is expanding at the rate of 60 pie in^3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches? </p>
<p>I don't understand how the way I set the problem up is not giving me the correct answer. I hav... | actinidia | 467,139 | <p>We have that $$\frac{\mathrm dV}{\mathrm dt} = 4 \pi r^2\cdot \frac{\mathrm dr}{\mathrm dt} =60$$</p>
<p>which implies $$\frac{\mathrm dr}{\mathrm dt} =\frac{60}{4 \pi r^2}$$</p>
<p>We know that $A = 4 \pi r^2$, and so we also have</p>
<p>$$\frac{\mathrm dA}{\mathrm dt} = 8\pi r\cdot \frac{\mathrm dr}{\mathrm dt}... |
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