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 |
|---|---|---|---|---|
1,231,113 | <p>There are plenty of questions about the homology of the connected sum of two $n$-manifolds, but I didn't find an explicit explanation of the computation done in degree $n-1$.
Let's show some examples of such questions:</p>
<p><strong>1)</strong> In <a href="https://math.stackexchange.com/questions/453132/connected-... | Joe S | 224,892 | <p>There is a basic theorem for the case of triangulable orientable compact n-manifolds without boundary that the top dimensional Z-homology is Z and is generated by an oriented sum of the n-simplices of the triangulation. </p>
<p>The picture is that each n-simplex shares each of its (n-1)-faces with exactly one other... |
82,716 | <p>There seems to be two competing(?) formalisms for specifying theories: <a href="http://ncatlab.org/nlab/show/sketch" rel="noreferrer">sketches</a> (as developped by Ehresmann and students, and expanded upon by Barr and Wells in, for example, <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf" rel="nore... | Peter Arndt | 733 | <p>Well, as you said yourself the category of signatures is left unspecified in the general setup of institution theory - it just tells you what, given some notion of signature, can be said about the relation of syntax and semantics. It's quite amazing how many meaningful things can be said at such a level of abstracti... |
1,158,712 | <p>My question is : Given an invertible matrix $A$ ( with complex entries ) , if $A^n$ is normal,is $A$ normal?</p>
<p>This is related to the question : <a href="https://math.stackexchange.com/questions/1158600/if-a-is-an-invertible-n-times-n-complex-matrix-and-some-power-of-a-is-diag">If $A$ is an invertible $n\times... | egreg | 62,967 | <p>Consider a triangularization $A=UTU^*$ of $A$. Saying that $A^n$ is normal means that $T^n$ is normal, so diagonal. Divide $T$ in blocks,
$$
T=\begin{bmatrix}
T_1 & X \\
0 & T_2
\end{bmatrix}
$$
where $T_1$ is $2\times2$ (the assertion is trivial for $1\times1$ matrices. Then, using block multiplication, we ... |
4,646,498 | <p>I have an assignment for university and I’m a bit confused as to how I should translate the following sentence:</p>
<p>Neither Ana nor Bob can do every exercise but each can do some.</p>
<p>I've identified the atomic sentences A=Ana can do every exercise and B=Bob can do every exercise and managed to translate the f... | Angelo | 771,461 | <p>Kai, you can solve it without any change of variables, indeed :</p>
<p><span class="math-container">$\displaystyle y’=-\int4x\,\mathrm dx=-2x^2+C\;\;,$</span></p>
<p><span class="math-container">$\displaystyle y=\int\left(-2x^2+C\right)\mathrm dx=-\dfrac23x^3+Cx+D\,.$</span></p>
|
3,531,809 | <p>In how many ways can we place 7 identical red balls and 7 identical blue balls into 5 distinct urns if each urn has at least 1 ball?</p>
<p>This is how I approached the problem:</p>
<p>1) Compute the number of total combinations if there were no constraints:</p>
<p>Placing just the red balls, allowing for empty u... | Daniel S. | 362,911 | <pre><code>A=[ nchoosek(1:11,4)-ones(size(nchoosek(1:11,4))), diff(nchoosek(1:11,4),[],2) - ones(size(diff(nchoosek(1:11,4),[],2))), -nchoosek(1:11,4)+11*ones(size(nchoosek(1:11,4)))];
B=A(:, [1,5,6,7,11]);
valid=0;
for i=1:size(B,1)
for j=1:size(B,1)
C=B(i,:)+B(j,:);
if (min(C) > 0)
... |
1,572,126 | <p>I did solve, I got four solutions, but the book says there are only 3.</p>
<p>I considered the cases $| x - 3 | = 1$ or $3x^2 -10x + 3 = 0$.</p>
<p>I got for $x\leq 0$: $~2 , 3 , \frac13$</p>
<p>I got for $x > 0$: $~4$ </p>
<p>Am I wrong? Is $0^0 = 1$ or NOT?</p>
<p>Considering the fact that : $ 2^2 = 2 \cd... | Archis Welankar | 275,884 | <p>You are wrong $0^0=indeterminate$ you will get only three solutions. mod $| |$ gets opened with $\pm$ if we put it as '-'. We will get expression...=$-1$ so we can express negative number as raised to something only with a complex number here $i^2$ but we want real solutions so $...expression=1$ and you get three s... |
1,146,050 | <p>given $f(x)=\frac{x^4+x^2+1}{x^2+x+1}$.</p>
<p>Need to find the min value of $f(x)$.</p>
<p>I know it can be easily done by polynomial division but my question is if there's another way</p>
<p>(more elegant maybe) to find the min? </p>
<p><strong>About my way</strong>: $f(x)=\frac{x^4+x^2+1}{x^2+x+1}=x^2-x+1$. (... | Amirali | 794,843 | <p>Multiply the fraction by <span class="math-container">$\frac{x^2-1}{x^2-1}$</span>:<span class="math-container">$$\frac{(x^2-1)(x^4+x^2+1)}{(x+1)(x-1)(x^2+x+1)}=\frac{x^6-1}{(x+1)(x^3-1)}=\frac{x^3+1}{x+1}=x^2-x+1$$</span></p>
|
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| Michael Hardy | 11,667 | <p>First approach:</p>
<blockquote>
<p>Let <span class="math-container">$P= 2\times3\times5\times7\times11\times13.$</span> Then:</p>
<p>The next number after <span class="math-container">$P$</span> that is divisible by <span class="math-container">$2$</span> is <span class="math-container">$P+2.$</span></p>
<p>The nex... |
264,770 | <p>If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:</p>
<ol>
<li>May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,3) \implies (1,2,3)$ or does the constant always go along with the vector?</li>
<li>If yes on question 1... | Ink | 34,881 | <p>The answer is yes to both questions. In any normed vector vector space, $\|\alpha v\| = |\alpha|\|v\|$ for any vector $v$ and scalar $\alpha$.</p>
|
169,531 | <p>Let me preface this by saying that I have essentially no background in logic, an I apologize in advance if this question is unintelligent. Perhaps the correct answer to my question is "go look it up in a textbook"; the reasons I haven't done so are that I wouldn't know which textbook to look in and I wouldn't know ... | hmakholm left over Monica | 14,366 | <p>You are right that $T'=T+G$ must be consistent. If it were not, then we could prove a contradiction from $T$ together with $G$, which would amount to a proof by contradiction of $\neg G$ in $T$. But that contradicts the fact that $G$ is independent of $T$!</p>
<p>On the other hand, when you ask</p>
<blockquote>
... |
194,724 | <p>All graphs discussed are finite and simple. The <em>cycle sequence</em> of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished by the vertices they contain, not by the edges they contain. </p>
<p>For example, $C(K_{3,2})=4,4,4$ and $C... | Gordon Royle | 1,492 | <p><strong>Second Answer</strong></p>
<p>I'm adding this as another separate answer, rather than editing the first "answer" because otherwise anyone coming late to this discussion will end up doubly confused.</p>
<p>So let's try again, and say that the answer to your question is still "Yes". </p>
<p>If you type the ... |
430,629 | <p>The <strong>full linear monoid</strong> <span class="math-container">$M_N(k)$</span> of a field <span class="math-container">$k$</span> is the set of <span class="math-container">$N \times N$</span> matrices with entries in <span class="math-container">$k$</span>, made into a monoid with matrix multiplication.
A <st... | Nicholas Kuhn | 102,519 | <p>As Ben Steinberg has been suggesting in the comments, the representation theory of these have been studied for quite awhile. His book on this, <em>the Representation Theory of Finite Monoids</em>, is an excellent modern general reference for how to think about these.</p>
<p>The general theme is that the maximal sub... |
381,177 | <p>I have a problem in which I have to compute the following integral: <span class="math-container">$$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$</span>
where this notation means that I want to integrate over <span class="math-container">$... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand\R{\mathbb R}\newcommand\1{\mathbf1}$</span>When you say "I want to integrate over <span class="math-container">$\mathbb{R}^k$</span> restricted to the plane where <span class="math-container">$\sum_{i=1}^{k}y_i=x$</span>", you have to specify the measure over the p... |
2,196,539 | <p>I'm having a complete mind blank here even though i'm pretty sure the solution is relatively easy.</p>
<p>I need to make X the subject of the following equation:</p>
<p>$$AB - AX = X $$</p>
<p>All i've done so far is:
$$A(B-X) = X$$
$$B-X = A^{-1} X$$</p>
<p>Not sure if thats right?</p>
<p>Thanks in advance.</p... | Random-generator | 427,836 | <p>Continuing with your expression, $B-X = A^{-1} X$, $B = (I+A^{-1}) X$, $X = (I+A^{-1})^{-1}B$.</p>
|
4,632,865 | <p>This was on a mock test for an examination that grants admission to an undergraduate course in mathematics. So, in theory, a school-going <span class="math-container">$17$</span> year old with a bit of extra knowledge should be able to solve it. As such, I'm looking for "elementary" answers.</p>
<p>Q. Supp... | Lorago | 883,088 | <p>Take <span class="math-container">$\delta\in(0,1)$</span>, set <span class="math-container">$I_\delta=(1-\delta,1+\delta)$</span>, and set</p>
<p><span class="math-container">$$M_\delta:=\max_{x\in[0,2]\setminus I_\delta}\lvert 2x-x^2\rvert.$$</span></p>
<p>Notice that then <span class="math-container">$M_\delta<... |
4,632,865 | <p>This was on a mock test for an examination that grants admission to an undergraduate course in mathematics. So, in theory, a school-going <span class="math-container">$17$</span> year old with a bit of extra knowledge should be able to solve it. As such, I'm looking for "elementary" answers.</p>
<p>Q. Supp... | Phobo Havuz | 563,245 | <p>Here's how to solve this by forming a recurrence relation. First, complete the square and perform a trigonometric substitution <span class="math-container">$x-1=\sin\phi,\text{d}x=\cos\phi\text{ d}\phi$</span>.
<span class="math-container">$$\begin{aligned}
I_n&=\int_0^2\left(2x-x^2\right)^n\text{ d}x\\
&=\i... |
2,134,928 | <p>Let <span class="math-container">$ \ C[0,1] \ $</span> stands for the real vector space of continuous functions <span class="math-container">$ \ [0,1] \to [0,1] \ $</span> on the unit interval with the usual subspace topology from <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$$\... | Upstart | 312,594 | <p>Correct answers$=x$ ,Incorrect answers$=10-x$</p>
<p>$3.x -1.(10-x)=18$</p>
<p>$4x=28$</p>
<p>$x=7$</p>
|
4,182,343 | <p>I am trying to solve the following simple equation:</p>
<p><span class="math-container">$$\frac{df}{dx}=\sin{f}$$</span> This is the "kink" solution to the Sine-Gordon equation. To solve this, I do the following substitution:</p>
<p><span class="math-container">$$f=\tan^{-1}g$$</span></p>
<p>Then we can us... | dbrane | 521,734 | <p>The trick is to use a slightly modified substitution:</p>
<p><span class="math-container">$$f=2\tan^{-1}g$$</span> Then, the RHS becomes <span class="math-container">$$\sin{(2\tan^{-1}g)}=2\sin{(\tan^{-1}g)}\cos{(\tan^{-1}g)}$$</span> solving the problem</p>
|
15,063 | <p>Let $F:R \to S$ be an étale morphism of rings. It follows with some work that $f$ is flat. </p>
<p>However, faithful flatness is another story. It's not hard to show that faithful + flat is weaker than being faithfully flat. An equivalent condition to being faithfully flat is being surjective on spectra. </p>
... | Anton Geraschenko | 1 | <p>The definition of $f:R\to S$ being faithfully flat that I first saw is that $S\otimes_R-$ is exact and faithful (meaning that $S\otimes_R M=0$ implies $M=0$). I'm not sure exactly what your definition of "faithfully flat" is, but it looks like you're happy with "flat and surjective on spectra." You get flatness for ... |
2,485,529 | <p>The integral is
$$\int{\left[\frac{\sin^8(x) - \cos^8(x)}{1 - 2 \sin^2(x)\cos^2(x)}\right]}dx$$</p>
| Corey | 492,030 | <p>Hint: $\sin^8(x)- \cos^8(x)=(\sin^2(x)- \cos^2(x))(1-2\sin^2(x) \cos^2(x))$</p>
|
370,212 | <p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. For <span class="math-container">$\alpha\in \; ]0,1[\;$</span>, let <span class="math-container">$$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$</span> (Note that we could have written <span c... | Max Alekseyev | 7,076 | <p>No. If <span class="math-container">$\alpha$</span> is rational, set <span class="math-container">$n$</span> to the denominator of <span class="math-container">$\alpha$</span>. Otherwise set <span class="math-container">$n$</span> to the denominator of the third <a href="https://en.wikipedia.org/wiki/Continued_fract... |
186,638 | <p>$f(x)=\max(2x+1,3-4x)$, where $x \in \mathbb{R}$. what is the minimum possible value of $f(x)$.</p>
<p>when, $2x+1=3-4x$, we have $x=\frac{1}{3}$</p>
| ronno | 32,766 | <p>At $x = \frac13, f(x) = \frac53$ and this is the minimum value, since for $x>\frac13$, $2x+1>\frac53$ and for $x<\frac13$, $3-4x >\frac53$.</p>
|
1,215,537 | <p>I need to prove that
$ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$.
I have proved that $\sum_1^\infty \frac {\sin^2(n \delta)}{n^2 \delta}=\frac{\pi-\delta}{2}$ for $0<\delta<\pi$ and I'm supposed to use this identity.</p>
| Olivier Bégassat | 11,258 | <p>Here is an alternate proof. It hinges on a lemma, which isn't too difficult to prove (it requires cutting $\Bbb R_+$ in two halves $[0,A]$ and $[A,+\infty)$ for some meaningful $A$, and using Riemann sums on the segment)</p>
<blockquote>
<p>Suppose $f:\Bbb R_+\to\Bbb R$ is continuous integrable and eventually dec... |
2,541,709 | <p>For example Calculate the probability of getting exactly 50 heads and 50 tails after flipping a fair coin $100$ times. then is ${100 \choose 50}\left(\frac 12\right)^{50}\left(\frac 12\right)^{50}$
the reason that we multiply $\left(\frac 12\right)^{50}$ twice is because the first one $\left(\frac 12\right)^{50}$ is... | user1101010 | 184,176 | <p>First, note $C_0(\mathbb{R})$ is the closure of $C_c(\mathbb{R})$ in the uniform metric and $C_c(\mathbb{R})$ is dense in $L^1(\mathbb{R})$. Then choose sequences $\{f_n\} \subseteq C_c(\mathbb{R})$ and $\{g_n\} \subseteq C_c(\mathbb{R})$ such that $\| f_n - f\|_u \to 0$ and $\|g_n - g\|_1 \to 0$. Then we should be ... |
198,521 | <p>I am trying to overlap two <code>Graphics</code> objects <code>g1</code> and <code>g2</code> with <code>Show</code>. However, I found that when the coordinates of each object is defined to quite different ranges, I need to "scale" and "shift" the coordinates of one object to get the desired look. </p>
<p>For exampl... | kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/Scale.html" rel="nofollow noreferrer"><code>Scale</code></a> on <code>First @ g2</code> (which contains the graphics directives and primitives):</p>
<pre><code>Show[g1, Graphics @ Scale[First @ g2, 2, {1, 1}], AspectRatio -> 1,
Axes -> True, ... |
495,622 | <p>Well, it may seem trivial, but I cannot find it on google. Is a constant function continuously differentiable, of all orders?</p>
<p>Thank you.</p>
| BaronVT | 39,526 | <p>Let's assume this is a constant function on $\mathbb{R}$ (i.e. $f : \mathbb{R} \to \mathbb{R}$, $f(x) = c$ for some fixed $c$, for all $x \in \mathbb{R}$).</p>
<p>Fix any $x_0$. What is</p>
<p>$$
\lim_{h\to 0} \frac{f(x_0 + h) - f(x_0)}{h} \ \ ?
$$</p>
<p>In particular, are there any points where this limit fails... |
2,586,618 | <p>I'm trying to study for myself a little of Convex Geometry and I have some doubts with respect the proof of the Theorem 1.8.5 of the book Convex Bodies: The Brunn-Minkowski Theory. Before I presented the proof and my doubts, I will put the definitions used in the theorem below.</p>
<p><span class="math-container">$\... | Community | -1 | <p>You can use Green's formulas:</p>
<p>$\int_{S}1=\frac{1}{2}\oint_{\partial S}xdy-ydx$</p>
<p>Your vector field is exactly the one described in the RHS, so, defining $S:
\frac{x^2}{4}+\frac{y^2}{9}=1; y\geq 0$:</p>
<p>$\oint_{L}A=-2\cdot Area (S)=-2\cdot\frac{1}{2}\pi\cdot2\cdot3=-6\pi$</p>
<p>(You get the minus ... |
2,993,551 | <p>I'm studying for a first year Discrete Mathematics course, I found this question on a previous paper and am lost on how to solve:</p>
<blockquote>
<p>Let <span class="math-container">$n$</span> be a fixed arbitrary integer, prove that there are infinitely
many integers <span class="math-container">$m$</span> s.... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Recall that in general <span class="math-container">$n$</span> linearly independent vectors <span class="math-container">$\in \mathbb{R^n}$</span> are a basis and then span <span class="math-container">$\mathbb{R^n}$</span>, to check that we can consider the matrix <span class="math-con... |
4,216,740 | <p>Let <span class="math-container">$f:\mathbb{R}^k\rightarrow \mathbb{R}$</span> be a vanishing at infinity function, also infinitely differentiable, i.e. <span class="math-container">$f\in C_0^\infty(\mathbb{R}^k,\mathbb{R})$</span>. Is it true that I can always approximate <span class="math-container">$f$</span> wit... | Mark Saving | 798,694 | <p>Yes and no.</p>
<p>If you restrain <span class="math-container">$f$</span> to take a compact domain <span class="math-container">$C \subseteq \mathbb{R}^k$</span>, then the answer is an unambiguous yes. For the Stone-Weirstrass theorem only requires an algebra of functions which can discriminate between points, and ... |
4,216,740 | <p>Let <span class="math-container">$f:\mathbb{R}^k\rightarrow \mathbb{R}$</span> be a vanishing at infinity function, also infinitely differentiable, i.e. <span class="math-container">$f\in C_0^\infty(\mathbb{R}^k,\mathbb{R})$</span>. Is it true that I can always approximate <span class="math-container">$f$</span> wit... | zhw. | 228,045 | <p>If you're interested in uniform convergence of polynomials on all of <span class="math-container">$\mathbb R^k,$</span> there's not much good news: The only <span class="math-container">$f\in C_0^\infty$</span> that can be uniformly approximated by polynomials is the zero function. That's because if <span class="mat... |
98,361 | <p>I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems trying to get the second chapter right. I have been able to get the definitions and work out some problems, but I am sti... | Community | -1 | <p>This approach may not suit you, but I definitely found that it helped me. My suggestion while studying Rudin chapter two is to look at Munkres' <em>Topology</em> chapters 2 and 3. What I found was though general topological spaces are in a more abstract setting, they it gave me a lot of motivation. For example if yo... |
98,361 | <p>I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems trying to get the second chapter right. I have been able to get the definitions and work out some problems, but I am sti... | Yesid Fonseca V. | 252,386 | <p>I feel very identified with you, I am also an engineer, electronic engineer and I have a background similar to yours. My motivation to start studying topology is to understand Hilbert spaces and someday understand functional analysis.</p>
<p>Initially, also thought I understood the concepts of Chapter 2 until Secti... |
2,147,571 | <p>Both $A$ and $B$ are a random number from the $\left [ 0;1 \right ]$ interval.</p>
<p>I don't know how to calculate it, so i've made an estimation with excel and 1 million test, and i've got $0.214633$. But i would need the exact number.</p>
| spaceisdarkgreen | 397,125 | <p>You can use the fact that the distribution of the ratio of two independent uniform $[0,1]$'s is $$ f_Z(z) = \left\{\begin{array}{ll}1/2& 0 < z < 1 \\\frac{1}{2z^2} & z >1 \end{array}\right.$$</p>
<p>Then you can calculate the probability that the closest integer is $i$: $$\int_{i-1/2}^{i+1/2} p_Z(... |
379,893 | <p>Define the sequence <span class="math-container">$b_1=1$</span> and
<span class="math-container">$$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$</span></p>
<p>By now, there is enough in the literature that <span class="math-container">$C_n$</span> is odd iff <span class="math-container">$n=2^k-1$</sp... | Giorgio Mossa | 14,969 | <blockquote>
<p>Is there a way to recapture the additional understanding imparted by multicategories using higher categories?</p>
</blockquote>
<p>Well I would say so. Multicategories are basically categories whose morphisms have multiple sources instead of only one. Bicategories generalize categories by adding 2-morph... |
995,489 | <p>This is taken from Trefethen and Bau, 13.3.</p>
<p>Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression:</p>
<p>$$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - 4032x^4 + 5376x^3 - 4608x^2 + 2304x - 512 $$</p>
<p>Where exactly is the problem?</p>
<p>Thanks.</p>
| Adam | 44,654 | <p>I have tried to plot this 2 expresions near x=2.0. Look at the image. The behaviour ( shape) seems very different . </p>
<p><img src="https://i.stack.imgur.com/ohKfY.png" alt="enter image description here"></p>
<p>Here is the Maxima CAS code :</p>
<pre><code>draw2d(
file_name = "d",
color = blue,
key = "sec... |
3,020,988 | <p>Here's my attempt at an integral I found on this site.
<span class="math-container">$$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$</span>
<strong>I'm not asking for a proof, I just want to know where I messed up</strong></p>
<p>Recall that, for all <span class="math-container">$x$</span>,
<span class="ma... | Seewoo Lee | 350,772 | <p>To solve the integral, you may consider <span class="math-container">$$\int_{C} \frac{e^z}{z}dz$$</span>
where <span class="math-container">$C$</span> is a unit circle, and see its real part.</p>
|
3,020,988 | <p>Here's my attempt at an integral I found on this site.
<span class="math-container">$$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$</span>
<strong>I'm not asking for a proof, I just want to know where I messed up</strong></p>
<p>Recall that, for all <span class="math-container">$x$</span>,
<span class="ma... | Henry Lee | 541,220 | <p>you could try this:
<span class="math-container">$$I=\int_0^{2\pi}e^{\cos(2x)}\cos\left[\sin(2x)\right]dx$$</span>
<span class="math-container">$$=\Re\left(\int_0^{2\pi}e^{\cos(2x)}\cos\left[\sin(2x)\right]dx+i\int_0^{2\pi}e^{\cos(2x)}\sin\left[\sin(2x)\right]dx\right)$$</span>
<span class="math-container">$$=\Re\le... |
956,110 | <p>I am struggling with thinking about this. Any help would be great!!</p>
<p>A medical research survey categorizes adults as follows:</p>
<ul>
<li>by gender (male or female)</li>
<li>by age group (age groups are 18-25, 26-35, 36-50, 51+)</li>
<li>by income (less than 30k/year, 30k-60k/year, more than 60k/year)</li>
... | mar10 | 177,164 | <p>Based on this</p>
<p><img src="https://i.stack.imgur.com/MQ1xa.png" alt="PowerPoint Lecture Video for Discrete Math"></p>
<p>I believe you would have (choices)</p>
<ul>
<li><p>(5)</p>
<p>Woman : Yes</p>
<p>Woman : No</p>
<p>Male : Frequently</p>
<p>Male : Rarely</p>
<p>Male : Never</p></li>
<li><p>(4) by ... |
322,134 | <p>$$2e^{-x}+e^{5x}$$</p>
<p>Here is what I have tried: $$2e^{-x}+e^{5x}$$
$$\frac{2}{e^x}+e^{5x}$$
$$\left(\frac{2}{e^x}\right)'+(e^{5x})'$$</p>
<p>$$\left(\frac{2}{e^x}\right)' = \frac{-2e^x}{e^{2x}}$$
$$(e^{5x})'=5xe^{5x}$$</p>
<p>So the answer I got was $$\frac{-2e^x}{e^{2x}}+5xe^{5x}$$</p>
<p>I checked my answ... | Zev Chonoles | 264 | <p><strong>Hint:</strong>
$$\frac{d}{dx}(e^{ax})=ae^{ax}$$
(this works for negative values of $a$ too, so no need to make your life more complicated with the quotient rule)</p>
|
3,218,525 | <p>Let <span class="math-container">$f:[0,1] \to [0, \infty)$</span> is a non-negative continuous function so that <span class="math-container">$f(0)=0$</span> and for all <span class="math-container">$x \in [0,1]$</span> we have <span class="math-container">$$f(x) \leq \int_{0}^{x} f(y)^2 dy$$</span><br>
Now consider ... | ajotatxe | 132,456 | <p>Just a comment that is too long to post it as a comment.</p>
<p><span class="math-container">$$f(x)\le\int_0^xf(y)^2dy\le x\sup_{0\le t \le 1}f(t)^2$$</span></p>
<p>Since the supremum of <span class="math-container">$f$</span> is reached at some point <span class="math-container">$x=x_M$</span> (Weierstrass' theor... |
4,019,561 | <p>If <span class="math-container">$M_{n\times n}$</span> is the set of invertible matrices with real entries. Find two matrices <span class="math-container">$A,B\in M_{n \times n}$</span> with the propriety that there not exists such a continuous function</p>
<p><span class="math-container">$$f:[0,1]\to M, \quad f(0)... | Tommaso Rossi | 678,717 | <p>Let A be any matrix with positive determinant, B any matrix with negative determinant. If there is such a <span class="math-container">$f$</span> , then <span class="math-container">$det(f(t))$</span> is a continuous function of <span class="math-container">$t\in[0,1]$</span> (it is a polinomial), with <span class="... |
142,993 | <p>I'm challenging myself to figure out the mathematical expression of the number of possible combinations for certain parameters, and frankly I have no idea how.</p>
<p>The rules are these:</p>
<p>Take numbers 1...n. Given m places, and with <em>no repeated digits</em>, how many combinations of those numbers can be ... | Wonder | 27,958 | <p>First one can be placed in $8^2$ ways, given it second one can be placed in $7^2$ ways, given the first two third one can be placed in $6^2$ ways. Divide by $3!$ to account for the fact that the points could have been picked in any order. So total number of ways = $\frac{8^2.7^2.6^2}{3!} = 18816$</p>
<p>EDIT: Sorry... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | Michael E2 | 4,999 | <p>For problems with exact coordinates, one could code up the definition of eigenvector. The function <code>eigV</code> finds the eigenvalue for a given vector in the form <code>L == value</code> or returns <code>False</code> if there is none; the function <code>eigQ</code> returns <code>True</code> if there exists a... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | Αλέξανδρος Ζεγγ | 12,924 | <p>How about this:</p>
<pre><code>eigenVectorQ[mat_, vec_] := Abs[Dot[#1\[Conjugate], #2]] == Norm[#1] Norm[#2] &[mat.vec, vec]
</code></pre>
<p>Then <code>eigenVectorQ[h, y]</code> returns <code>False</code>.</p>
|
1,885,751 | <p>An urn contains 15 Balls (5 white, 10 Black). Let's say we pick them one after the other without returning them. How many white balls are expected to have been drawn after 7 turns?</p>
<p>I can calculate it by hand with a tree model but is there a formula for this?</p>
| barak manos | 131,263 | <p>Split it into disjoint events, and then add up their probabilities:</p>
<ul>
<li>The probability of getting exactly $\color\red2$ ones is $\binom{5}{\color\red2}\cdot\left(\frac16\right)^{\color\red2}\cdot\left(1-\frac16\right)^{5-\color\red2}$</li>
<li>The probability of getting exactly $\color\red3$ ones is $\bin... |
1,564,729 | <p>Find:
$$
L = \lim_{x\to0}\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}
$$</p>
<p>My approach:</p>
<p>Because of the fact that the above limit is evaluated as $\frac{0}{0}$, we might want to try the De L' Hospital rule, but that would lead to a more complex limit which is also of the form $\frac{0}{0}$. </p>
<p... | Community | -1 | <p><strong>By L' Hospital anyway</strong>:</p>
<p>$$\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}$$ yields</p>
<p>$$\cos\left(1-\frac{\sin(x)}x\right)\frac{\sin(x)-x\cos(x)}{2x^3}.$$</p>
<p>The first factor has limit $1$ and can be ignored.</p>
<p>Then with L'Hospital again:</p>
<p>$$\frac{x\sin(x)}{6x^2},$$</p... |
3,282,206 | <p>I'm familiar with Fermat's Little Theorem and Euler's Totient, but I'm wondering whether the fact that the only shared factor of <span class="math-container">$(a,N)=1$</span> has something to do with the fact that, given the prior constraints there exists at least one <span class="math-container">$x$</span> (with <s... | TonyK | 1,508 | <p>Consider all powers <span class="math-container">$a^r\bmod N$</span> for <span class="math-container">$r=1,2,3,\ldots$</span> There are an infinite number of them, but they can only take values <span class="math-container">$0\le a^r\bmod N\le N-1$</span>. So sooner or later we must encounter a repeated value; say <s... |
2,648,492 | <p>I am having trouble with this problem. When they say spot I think they are essentially saying the sum, so its the probability that the sum of dice is $11$ or less.</p>
<p>I understand that there are $6^5$ combinations.</p>
<p>I found 6 ways that it can equal to $11$ $(2,3,2,2,2)(3,3,1,1,3),(4,4,1,1,1),(5,2,2,1,1),... | Siong Thye Goh | 306,553 | <p>We can solve the problem using generating function.
\begin{align}\left(\frac16\sum_{i=1}^6x^6\right)^5=\frac{1}{6^5}\left( \frac{x(1-x^6)}{1-x}\right)^5 \end{align}</p>
<p>Our goal is to find the sum of coefficients of $x^5$ to $x^{11}$.</p>
<p>$$(1-x^6)^5 =1-5x^6+\text{higher order terms} $$</p>
<p>By <a href="... |
2,648,492 | <p>I am having trouble with this problem. When they say spot I think they are essentially saying the sum, so its the probability that the sum of dice is $11$ or less.</p>
<p>I understand that there are $6^5$ combinations.</p>
<p>I found 6 ways that it can equal to $11$ $(2,3,2,2,2)(3,3,1,1,3),(4,4,1,1,1),(5,2,2,1,1),... | P.Diddy | 513,123 | <p>You can solve an equivalent question which is slightly easier. Consider the following: </p>
<p>You have five different bins to which you need to distribute up to eleven balls. First you will put one ball at each bin, as Matti P suggested, since it is impossible to have a score lower than 1 on a dice. Now you have u... |
1,293,207 | <p>A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is achieved at the least possible time when Snell's law: $$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}$$ holds. </p... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>Use <a href="http://en.wikipedia.org/wiki/Fermat's_principle" rel="nofollow">Fermat's principle</a>.</p>
<blockquote>
<p>the path taken between two points by a ray of light is the path that
can be traversed in the least time.</p>
</blockquote>
<p>And you don't need Lagrange multipliers.</p>
|
1,293,207 | <p>A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is achieved at the least possible time when Snell's law: $$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}$$ holds. </p... | Matematleta | 138,929 | <p>Let the total horizontal distance travelled be $a$. The light travels in the first medium a distance $d_{1}$ at an angle $\theta _{1}$ to the vertical. Then if the horizontal distance traveled in this step is $x$ and the vertical distance traveled is $h$, the time it takes to reach the border is </p>
<p>$t_{1}=\fra... |
216,031 | <p>Using image analysis, I have found the positions of a circular ring and imported them as <code>xx</code> and <code>yy</code> coordinates. I am using <code>ListInterpolation</code> to interpolate the data:</p>
<pre><code>xi = ListInterpolation[xx, {0, 1}, InterpolationOrder -> 4, PeriodicInterpolation -> True,... | Steffen Jaeschke | 61,643 | <p>Again I tend to duplicate, despite this is an interpolation and a real 2D curve under consideration.</p>
<p>What I did is using this <a href="https://mathematica.stackexchange.com/questions/84502/movingaverage-to-include-the-average-of-the-first-and-last-elements-in-a-list/84504#84504">solution</a>. I like to meet ... |
3,208,822 | <p>I'm looking at a STEP question and I'm a little confused by the logic of the method, and i'm really hoping someone could clarify what is going on for me. I have a good knowledge (At least I thought), as some STEP II and III questions are accessible but this one , I just can't wrap my head around - there must be a ga... | Henry | 6,460 | <p>Hint:</p>
<p><span class="math-container">$$n\, f(x) -1 \lt \lfloor n\, f(x)\rfloor \le n\, f(x)$$</span></p>
<p>so <span class="math-container">$$\dfrac{n\, f(x) -1}{n} \lt \dfrac{\lfloor n\, f(x)\rfloor}{n} \le \dfrac{n\, f(x)}{n}$$</span></p>
<p>i.e. <span class="math-container">$$f(x) -\dfrac{1}{n} \lt \dfrac... |
2,194,376 | <p>I'm having some trouble understanding the answers to the following questions:</p>
<p><a href="https://i.stack.imgur.com/nnuu8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nnuu8.png" alt="enter image description here"></a></p>
<p>(a)</p>
<p>Why would it make sense for Eve to test out the $gcd... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may observe that
$$
\left|\frac{\sin\frac{x}{n}\sin 2nx}{x^2+4n}\right|\le\frac{\left|\sin\frac{x}{n}\right|}{4n}\le \frac{\frac{\left|x\right|}n}{4n}\le\frac{\left|A\right|}{4n^2},\quad x \in [-A,A] \,\,(A>0),
$$ by using <a href="https://en.wikipedia.org/wiki/Weierstrass_M-test" rel="... |
3,116,693 | <blockquote>
<p>Let <span class="math-container">$C([0,1])$</span> be the space of all real valued continuous functions <span class="math-container">$f:[0,1]\to \mathbb{R}$</span>. Take the norm
<span class="math-container">$$||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}$$</span>
and the subspace
<span class="ma... | kimchi lover | 457,779 | <p>The closure of <span class="math-container">$C$</span> in <span class="math-container">$V=(C([0,1]),\|\cdot\|_2)$</span> is all of <span class="math-container">$V$</span>. For given <span class="math-container">$f\in V$</span>, let <span class="math-container">$f_n\in C$</span> be obtained from <span class="math-con... |
3,116,693 | <blockquote>
<p>Let <span class="math-container">$C([0,1])$</span> be the space of all real valued continuous functions <span class="math-container">$f:[0,1]\to \mathbb{R}$</span>. Take the norm
<span class="math-container">$$||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}$$</span>
and the subspace
<span class="ma... | mechanodroid | 144,766 | <p><strong>Lemma.</strong></p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a normed space and <span class="math-container">$f : X \to \mathbb{C}$</span> an unbounded functional on <span class="math-container">$X$</span>. Then <span class="math-container">$\ker f$</span> is dense in <span class="ma... |
456,824 | <p>It's quite easy to give the complete <em>rational</em> solution to,</p>
<p>$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$</p>
<p>One can express it in the form,</p>
<p>$$(p+q)^k+(r+s)^k+(t+u)^k=(p-q)^k+(r-s)^k+(t-u)^k\tag{2}$$</p>
<p>and impose 2 linear conditions on $p,q,r,s,t,u$. Howe... | coffeemath | 30,316 | <p>With different notation for the main variables, the system is $u+v+w=x+y+z$ and the squared equation $u^2+v^2+w^2=x^2+y^2+z^2.$ Your substitutions are
$$ad+e=u,\\ bc+e=v,\\ac+bd+e=w,\\ ac+e=x,\\ bd+e=y,\\ ad+bc+e=z.$$
From this two expressions for $e$ are $u+v-z$ and $x+y-w.$ These are compatible because of the line... |
456,824 | <p>It's quite easy to give the complete <em>rational</em> solution to,</p>
<p>$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$</p>
<p>One can express it in the form,</p>
<p>$$(p+q)^k+(r+s)^k+(t+u)^k=(p-q)^k+(r-s)^k+(t-u)^k\tag{2}$$</p>
<p>and impose 2 linear conditions on $p,q,r,s,t,u$. Howe... | Tito Piezas III | 4,781 | <p>Complementing the proof by ckrao in the <a href="http://ckrao.wordpress.com/2012/08/28/23-multigrade-equations/" rel="nofollow">link</a> given by Myerson, here is mine. The general formula for,</p>
<p>$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$</p>
<p>where $x_1 \ne y_1$ can be given a... |
3,968,905 | <p>I am trying to prove this:</p>
<p><span class="math-container">$\bullet$</span> Prove that <span class="math-container">$\Delta(\varrho_\epsilon \star u) = \varrho_\epsilon \star f $</span> in the sense of distributions, if <span class="math-container">$\Delta u = f$</span> in the sense of distributions, <span class... | cr001 | 254,175 | <p>Expansion:</p>
<p><span class="math-container">$$x^2y^2+x^2+y^2+1+2x-2y+2xy^2-2x^2y-4xy=4$$</span></p>
<p>Re-factoring:</p>
<p><span class="math-container">$$(x^2+2x+1)(y^2-2y+1)=4$$</span></p>
<p>Simplify</p>
<p><span class="math-container">$$(x+1)^2(y-1)^2=4$$</span></p>
<p>I will leave you to check the cases.</p>... |
176,893 | <p>Suppose I have a polynomial
$$
p(x)=\sum_{i=0}^n p_ix^i.
$$
For simplicity furthermore assume $p_n=1$. </p>
<p>As it is well known we may use Gershgorin circles to give an upper bound for the absolute values of the roots of $p(x)$. The theorem states that all roots are contained within a circle with radius
$$
r=\ma... | Joe Silverman | 11,926 | <p>There are actually stronger estimates that deal with all of the roots. Let $r_1,\ldots,r_n$ be the roots of your polynomial (with multiplicities as appropriate). Then
$$
\max_{0\le i\le n} |p_i| \ge \frac{1}{4^n} \prod_{j=1}^n \max\bigl\{|r_j|,1\bigr\}
$$
and
$$
\max_{0\le i\le n} |p_i| \le 4^n \prod_{j=1}^... |
1,225,655 | <p>Find the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form.</p>
<p>I don't know how can I get the tangent line, without a given equation!!, this is part of cal1 classes.</p>
| mvw | 86,776 | <p>So far we can infer
$$
T(x) = m x + n
$$
with $T(0) = \sqrt{\pi/2}$. Thus
$$
T(x) = m x + \sqrt{\pi/2}
$$
To determine the slope $m$ we need more information about the given curve.</p>
|
2,023,130 | <p>Suppose I have directed graph $G=(V,E)$ s.t at least one of the following statements is always true:</p>
<ol>
<li>for every $v$ in $V$, v doesn't have any incoming edges.</li>
<li>for every $v$ in $V$, v doesn't have any outgoing edges.</li>
</ol>
<p>How can I prove that G is bipartite? I tried to think of a proof... | dtldarek | 26,306 | <p>Your current statement is:</p>
<blockquote>
<p>Suppose I have directed graph $G=(V,E)$ s.t. at least one of the following statements is always true:</p>
<ol>
<li>for every $v$ in $V$, $v$ doesn't have any incoming edges.</li>
<li>for every $v$ in $V$, $v$ doesn't have any outgoing edges.</li>
</ol>
</b... |
2,120,763 | <p>I've been given this problem:</p>
<p>Prove that a subordinate matrix norm is a matrix norm, i.e. </p>
<p>if $\left \|. \right \|$ is a vector norm on $\mathbb{R}^{n}$, then $\left \| A \right \|=\max_{\left \| x \right \|=1}\left \| Ax \right \|$ is a matrix norm</p>
<p>I don't even understand the question, and ... | Community | -1 | <p><em><a href="https://math.stackexchange.com/questions/319297/gcd-to-lcm-of-multiple-numbers/319690#319690">From here</a>:</em></p>
<p>We have that </p>
<p><strong>Theorem:</strong> $\rm\ \ lcm(a,b,c)\, =\, \dfrac{abc}{(bc,ca,ab)}$</p>
<p><strong>Proof:</strong> $\!\begin{align}\qquad\qquad\rm\ a,b,c&\mid\rm\ ... |
1,630,733 | <p>I seem to be having a lot of difficulty with proofs and wondered if someone can walk me through this. The question out of my textbook states:</p>
<blockquote>
<p>Use a direct proof to show that if two integers have the same parity, then their sum is even.</p>
</blockquote>
<p>A very similar example from my notes... | fleablood | 280,126 | <p>"Suppose a is an even integer and b is an even integer. Then by our definitions of even numbers, we know that integers m and n exist so that a=2m and b=2m???"</p>
<p>Since a and b are different numbers they should be different m and n.</p>
<p>"Suppose a is an even integer and b is an even integer. Then by our defi... |
25,285 | <p>I am just looking for a basic introduction to the Podles sphere and its topology. All I know is that it's a q-deformation of $S^2$. </p>
| Nicola Ciccoli | 6,032 | <p>First of all some terminology. One usually talks about Podles spheres, since they are a one parameter family. If you say <strong>the</strong> Podles sphere you probably mean the one that is often referred to as the <strong>standard</strong> one. My indications will refer to the whole family.</p>
<p>You do not clari... |
1,177,988 | <p>Comparing the equation
$$x^4+3x+20=0$$<br>
With the equation
$$(x^2+\lambda)^2-(mx+n)^2=0$$
we get </p>
<p>$m^2=2\lambda,$</p>
<p>$-2mn=3,$<br>
$n^2=\lambda^2-20$ </p>
<p>Now, $4m^2n^2=9\Rightarrow 4(2\lambda)(\lambda^2-20)=9\Rightarrow 8\lambda^3-160\lambda-9=0$. </p>
<p>How can I find easily the value... | Mathlover | 22,430 | <p>HINT: Use binom expansion
$$(x+y)^3=x^3+3x^2y+3xy^2+y^3$$
Reorder the equation as:
$$(x+y)^3-3xy(x+y)-(x^3+y^3)=0$$</p>
<p>$$8\lambda^3-160\lambda-9=0$$<br>
$$\lambda^3-20\lambda-\frac{9}{8}=0$$ </p>
<p>Define:
$x+y=\lambda$,
then solve the equation with 2 unknowns
$$3xy=20$$</p>
<p>$$x^3+y^3=\frac{9}{8}$$</... |
3,678,033 | <p>let Y be an ordered set in the order topology.let X be a topological space and let <span class="math-container">$f,g:{X\to Y}$</span> be continuous function.</p>
<p>show that the set <span class="math-container">$A=\{x\in X\mid f(x)\le g(x)\}$</span> is closed in X.
I used the complement A and Hasdorf, but I didn't... | Thomas | 128,832 | <p>note that in <span class="math-container">$Y\times Y$</span> with the product topology, <span class="math-container">$\{(y_1, y_2)| y_1 \le y_2\}$</span> is closed, and <span class="math-container">$(f,g):X\rightarrow Y\times Y$</span> is continuous, since each factor is.</p>
|
3,678,033 | <p>let Y be an ordered set in the order topology.let X be a topological space and let <span class="math-container">$f,g:{X\to Y}$</span> be continuous function.</p>
<p>show that the set <span class="math-container">$A=\{x\in X\mid f(x)\le g(x)\}$</span> is closed in X.
I used the complement A and Hasdorf, but I didn't... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$p \notin A$</span>, so that <span class="math-container">$f(p) > g(p)$</span> in <span class="math-container">$Y$</span>. </p>
<p>Case 1: if there exists a <span class="math-container">$y_0 \in Y$</span> with <span class="math-container">$f(p) > y_0 > g(p)$</span> then che... |
183,243 | <p>Gödel's incompleteness theorem states that: "<em>if a system is consistent, it is not complete.</em>" And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc.</p>
<p>However, why does this mean that ZF is consistent? What does "<em>relatively consistent</em>" actually mean?</p>
| Cameron Buie | 28,900 | <p>Relatively consistent means that if some other system is consistent, then so is the given system. For example, ZF is relatively consistent with ZF-Foundation (and vice versa), and relatively consistent with ZFC (and vice versa). For a unidirectional example, the axioms of Peano arithmetic are relatively consistent w... |
481,173 | <p>The most common way to find inverse matrix is $M^{-1}=\frac1{\det(M)}\mathrm{adj}(M)$. However it is very trouble to find when the matrix is large.</p>
<p>I found a very interesting way to get inverse matrix and I want to know why it can be done like this. For example if you want to find the inverse of $$M=\begin{b... | littleO | 40,119 | <p>This is probably the standard way to compute the inverse of a matrix.</p>
<p>Suppose $AB = I$. Looking at this equation column by column, we see that $Ab_1 = e_1,\ldots, A b_n = e_n$, where $b_1,\ldots,b_n$ are the columns of $B$ and $\{e_1,\ldots,e_n\}$ is the standard basis of $\mathbb R^n$.</p>
<p>Thus, to com... |
1,943,478 | <p>So the prompt is merely an existence proof--just find a $u$ and $v$ that work. Well, I'm unfortunately a little stuck on getting started.</p>
<p>I know that $Q \in SO_4(\mathbb R) \implies QQ^T = I \text{ and } \det(Q) = 1$.</p>
<p>I tried to solve $Qx = uxv$ for $u,v$ but I was not able to do so successfully. Thi... | John Hughes | 114,036 | <p>Here's a proof that you may find completely unsatisfactory, depending on whether you know about bundles and things like that. </p>
<p>Consider the map $p : SO(4) \to \mathbb S^3 : [q_1, q_2, q_3, q_3] \mapsto q_1$ that selects from a matrix its first column. </p>
<p>This is a fibration, and its fiber ($p^{-1}(q_1)... |
2,668,826 | <p>I am stuck on this result, which the professor wrote as "trivial", but I don't find a way out.</p>
<p>I have the function </p>
<p>$$f_{\alpha}(t) = \frac{1}{2\pi} \sum_{k = 1}^{+\infty} \frac{1}{k}\int_0^{\pi} (\alpha(p))^k \sin^{2k}(\epsilon(p) t)\ dp$$</p>
<p>and he told use that for $t\to +\infty$ we have:</p>... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
2,707,514 | <p>I'm reading <em>A First Course in Modular Forms</em> by Diamond and Shurman and am confused on a small point in Chapter 2. Let <span class="math-container">$\Gamma$</span> be a congruence subgroup of <span class="math-container">$\operatorname{SL}_2(\mathbb Z)$</span>. <span class="math-container">$\gamma \in \ma... | Oven | 546,601 | <p>Let $\tau$ and $U$ be as in the Corollary. If $\tau' \in U$ is an elliptic point different from $\tau$, with nontrivial stabilizer $\gamma \in \text{PSL}_2(\mathbb{Z})$, then $\gamma$ also fixes $\tau$ by the first part of the Corollary. It is thus sufficient to show that if $\gamma \in \text{PSL}_2(\mathbb{R})$ fi... |
4,025,279 | <p>I have been reading material on uniformly continuous functions. And going through the problems where we have to prove that a function is not uniformly continuous or otherwise.</p>
<p>A function defined on an interval I is said to be uniformly continuous on I if to each <span class="math-container">$\epsilon$</span> ... | Ethan Bolker | 72,858 | <p>Intuitively, A continuous function is uniformly continuous when it's never too steep.</p>
<p>If you want to draw one that's not, the domain should be an open interval (bounded or unbounded) since a continuous function on a closed interval is uniformly continuous.</p>
<p>My favorite example is
<span class="math-conta... |
640,554 | <p>For the system
$$
\left\{
\begin{array}{rcrcrcr}
x &+ &3y &- &z &= &-4 \\
4x &- &y &+ &2z &= &3 \\
2x &- &y &- &3z &= &1
\end{array}
\right.
$$
what is the condition to determine if there is no solution or unique solution or infinite solut... | Cameron Buie | 28,900 | <p>Probably the most straightforward method (to fully distinguish between the various possibilities) that I've seen is transforming the corresponding augmented matrix into row-reduced echelon form. In this case, you would start with: <span class="math-container">$$\left[\begin{array}{ccc|c}1 & 3 & -1 & -4\\... |
36,477 | <p>The length, width and height of a rectangular box are measured to be 3cm, 4cm and 5cm respectively, with a maximum error of 0.05cm in each measurement. Use differentials to approximate the maximum error in the calculated volume.</p>
<p><br><br>
Please help</p>
| Américo Tavares | 752 | <p>Let $V=xyz$ be the actual volume of the box, where $x,y,z$ are respectively
its actual length, width and height, and let $V_{0}=x_{0}y_{0}z_{0}=3\cdot
4\cdot 5=60$ cm$^{3}$ be the measured volume.</p>
<p>The maximum error of 0.05 cm in each measurement means that $\left\vert
x-3\right\vert \leq 0.05$, $\left\vert... |
1,605,281 | <p><a href="https://i.stack.imgur.com/43uoh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/43uoh.png" alt="The question about finding the exact value of the sine of the angle between (PQ) and the plane"></a></p>
<p>I have done part (a). For part (b), I know the principle of how to do it, I tried to... | Jendrik Stelzner | 300,783 | <p>First off, $[-a,-b) = \{x \mid -a \leq x < -b\}$ and $(-b,-a] = \{x \mid -b < x \leq -a]$ are different things.</p>
<p>You also want to assume $a < b$ to ensure $[a,b)$ is non-empty, for otherwise your counterexample does not work.</p>
<p>You claim that $(-b,-a] \notin \mathcal{T}_l$, which you have to ju... |
129,295 | <p>$$\int{\sqrt{x^2 - 2x}}$$</p>
<p>I think I should be doing trig substitution, but which? I completed the square giving </p>
<p>$$\int{\sqrt{(x-1)^2 -1}}$$</p>
<p>But the closest I found is for</p>
<p>$$\frac{1}{\sqrt{a^2 - (x+b)^2}}$$ </p>
<p>So I must add a $-$, but how? </p>
| N3buchadnezzar | 18,908 | <p>The standard way to solve these problems is indeed with trigonometric substitutions.
But it is not the <em>only</em> way to solve these.
Another way that was used more before is called <em>Euler substitutions</em></p>
<blockquote>
<p>if we are to integrate $\sqrt{ax^2+bx+c}$ that has real roots $\alpha$ and
$... |
33,330 | <p>Let $p$ be a complex number. Let $ z_0 = p $ and, for $ n \geq 1 $, define $z_{n+1} = \frac{1}{2} ( z_n - \frac{1}{z_n}) $ if $z_n \neq 0 $. Prove the following:</p>
<p>i) If $ \{ z_n \} $ converges to a limit $a$, then $a^2 + 1 = 0 $</p>
<p>ii) If $ p $ is real, then $ \{ z_n \} $, if defined, does not converge</... | Adrián Barquero | 900 | <p>I'm not sure how much help this could provide but anyway. Look that what part i) tells you is that if the sequence $(z_n)$ converges then its limit must be $\pm i \in \mathbb{C}$. So in part ii) you show that if the first term $z_0 = p$ is real then the sequence does not converge because all terms are real so there'... |
2,995,408 | <blockquote>
<p><span class="math-container">$$
\lim_{x\to 2^-}\frac{x(x-2)}{|(x+1)(x-2)|}=
\lim_{x\to 2^-}\left(\frac{x}{|x+1|}\cdot \frac{x-2}{|x-2|}\right)
$$</span></p>
</blockquote>
<p>So as the title says, is it okay to separate function under absolute value like this (i.e In form of Products) as shown in the... | user | 505,767 | <p>Yes of course we can di that since the two expressions are equivalent, moreover in that case we can also go beyond that indeed</p>
<p><span class="math-container">$$\lim_{x\to 2^-}\frac{x(x-2)}{|(x+1)(x-2)|}=
\lim_{x\to 2^-}\left(\frac{x}{|x+1|}\cdot \frac{x-2}{|x-2|}\right)=\lim_{x\to 2^-}\left(\frac{x}{|x+1|}\rig... |
2,334,381 | <p>is strong topology on a metric space the topology that is induced by metric? Is a open set in weak topology also open in strong topology?</p>
| Henno Brandsma | 4,280 | <p>Yes, in a linear space $X$ that is normed, e.g. we have a metric $d$ from the norm, and the strong topology is the one induced by $d$ (so generated by the $d$-balls). The weak topology looks at all continuous linear functions from $(X,d)$ to the underlying field (often the reals). This set is called $X^\ast$, the du... |
3,400,123 | <blockquote>
<p>Proof that <span class="math-container">$$\sum_{l=1}^{\infty} \frac{\sin((2l-1)x)}{2l-1}
=\frac{\pi}{4}$$</span> when <span class="math-container">$0<x<\pi$</span></p>
</blockquote>
<p>The chapter we are working on is about Fourier series, so I guess I'd need to use that some how.</p>
<p>My id... | J.G. | 56,861 | <p>Define <span class="math-container">$f(x)$</span> to be <span class="math-container">$\frac{\pi}{4}$</span> on <span class="math-container">$(0,\,\pi)$</span>, <span class="math-container">$0$</span> at multiples of <span class="math-container">$\pi$</span> and <span class="math-container">$-\frac{\pi}{4}$</span> on... |
2,559,260 | <p>There exists a function $f$ such that $\lim_{x \rightarrow \infty} \frac{f(x)}{x^2} = 25$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = 5$</p>
<p>I am confused, I do not whether it is true or not</p>
<p>I have a counter-example, but I think thre might be such function</p>
| user | 505,767 | <p>It can't exist since :</p>
<p>$$\lim_{x \rightarrow \infty} \frac{f(x)}{x^2} =\lim_{x \rightarrow \infty} \frac{1}{x}\frac{f(x)}{x}=0\cdot5=0$$</p>
|
2,214,236 | <p>The question:</p>
<blockquote>
<p>An object is dropped from a cliff. How far does the object fall in the 3rd second?"</p>
</blockquote>
<p>I calculated that a ball dropped from rest from a cliff will fall $45\text{ m}$ in $3 \text{ s}$, assuming $g$ is $10\text{ m/s}^2$.</p>
<p>$$s = (0 \times 3) + \frac{1}{2}\... | hamam_Abdallah | 369,188 | <p>We have</p>
<p>$$h=\frac {1}{2}gt^2+v_0t+h_0$$</p>
<p>$$=\frac {1}{2}gt^2$$</p>
<p>$$=\frac {1}{2}.10. (3)^2=45 m $$.</p>
|
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Yiorgos S. Smyrlis | 57,021 | <p>Using <a href="http://en.wikipedia.org/wiki/Integration_by_parts" rel="nofollow">integration by parts</a>, we obtain
\begin{align}
\int_0^1 \frac{nx^{n-1}}{1+x}dx &=\left.\frac{x^n}{1+x}\right|_0^1+\int_0^1\frac{x^n}
{(1+x)^2}dx
=\frac{1}{2}+r_n,
\end{align}
where clearly
$$
0<r_n\le \int_0^1 x^n\,dx=\frac{1}... |
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Mercy King | 23,304 | <p>We have
$$
a_n=\int_0^1\frac{nx^{n-1}}{1+x}\,dx=\frac{x^n}{1+x}\Big|_0^1+\int_0^1\frac{x^n}{(1+x)^2}\,dx=\frac12+\int_0^1\frac{x^n}{(1+x)^2}\,dx \quad \forall n \ge 1.
$$
Since
$$
\int_0^1\frac{x^n}{(1+x)^2}\,dx\le \int_0^1x^n\,dx=\frac{1}{n+1} \quad \forall n\ge 1,
$$
it follows that
$$
\lim_n\int_0^1\frac{x^n}{(1+... |
834,508 | <p>Show that for any natural number $n$, between $n^2$ and $(n+1)^2$ one can find three distinct natural numbers $a,b,c$ such that $a^2+b^2$ is divisible by $c$.</p>
<p>A friend and I found a general case that always work with a computer problem, I would like to see a different solution, or a solution that tells the m... | vonbrand | 43,946 | <p>The Frobenius problem in general is <em>hard</em> to solve (the 2 number case is easy, see e.g. <a href="http://www.cut-the-knot.org/blue/Sylvester2.shtml" rel="nofollow">here</a> for an instructive technique). I believe the (to date) definitive summary of what is known to be Ramírez-Alfonsín's "The Diophantine Frob... |
2,595,612 | <p>Is true that every limit can only converge, diverge or(exclusive) not exist?</p>
<p>Can I demonstrate that it doesn't exist after I proved it doesn't converge neither diverge?</p>
<p>I've never seen this, but it makes some sort of sense to me. If a real isn't positive nor negative, it must be zero... But with limi... | zwim | 399,263 | <p>In the following, I'll use $n$ for $n\to +\infty$ and $x$ for $x\to 0$.</p>
<p>Converge means it has a limit and this limit is finite.</p>
<ul>
<li>For instance $\frac 1n$ converges to $0$ </li>
</ul>
<p>Diverge means does not converge.</p>
<p>There are multiple forms of divergence</p>
<ul>
<li><p>The limit is ... |
236,933 | <p>When I use ListPlot, I want to show the labels, such as</p>
<pre><code>ListPlot[Callout[#, #, Above] & /@ Range[10], Joined -> True, Mesh -> All]
</code></pre>
<p>Now all positions are <strong>Above</strong>, but sometimes the labels will over other text, so I want to set some of them <strong>Below</strong... | kglr | 125 | <pre><code>ListPlot[Callout[#, #, # /. {2 | 3 | 8 -> Below, _ -> Above}] & /@ Range[10],
Joined -> True, Mesh -> All]
</code></pre>
<p><a href="https://i.stack.imgur.com/KDloV.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KDloV.png" alt="enter image description here" /></a></p>
|
377,152 | <p>Let n be a fixed natural number. Show that:
$$\sum_{r=0}^m \binom {n+r-1}r = \binom {n+m}{m}$$</p>
<p>(A): using a combinatorial argument and (B): by induction on $m$?</p>
| Arthur | 15,500 | <p>For (A), you're supposed to find something to count that can be counted in two ways. One should be naturally representable as $\sum_{r=0}^m \binom {n+r-1}r$, and the other as $\binom{n+m}{m}$. Since it is the same thing you counted the two must be equal.</p>
<p>For the other you need to check that it is actually tr... |
377,152 | <p>Let n be a fixed natural number. Show that:
$$\sum_{r=0}^m \binom {n+r-1}r = \binom {n+m}{m}$$</p>
<p>(A): using a combinatorial argument and (B): by induction on $m$?</p>
| Marko Riedel | 44,883 | <p>For the sake of completeness I add option (C): using generating functions.
We have
$$\sum_{r=0}^m \binom{n+r-1}{r} =
\sum_{r=0}^m [z^n] \frac{z}{(1-z)^{r+1}} =
[z^{n-1}] \sum_{r=0}^m \left(\frac{1}{1-z}\right)^{r+1} \\ =
[z^{n-1}] \frac{1}{1-z} \frac{1-1/(1-z)^{m+1}}{1-1/(1-z)} =
[z^{n-1}] \frac{1-1/(1-z)^{m+1}}{1-... |
2,912,759 | <p>I am trying to use the characteristic function of the uniform distribution defined on (0,1) to compute the mean. I have calculated the characteristic function (correctly) and used Euler's identity to convert it to the following form:</p>
<p>$$\phi_Y(t)=\frac{\sin(t)}{t} + i \frac{1-\cos(t)}{t}$$</p>
<p>I should be... | Kavi Rama Murthy | 142,385 | <p>You have to use limiting values. Even in your formula for $\phi_Y (t)$ there is $t$ in the denominator. It doesn't mean $\phi_Y (0)$ is not defined. The value is $1$ which is also $\lim_{t \to 0} \phi_Y (t)$. Similarly, you can find $EY$ by computing $\lim _{t \to 0} \frac {\partial} {\partial t} \phi_Y (t)$. By exp... |
1,456,411 | <p>When we're introduced to $\mathbb{R}^3$ in multivariable calculus, we first think of it as a collection of points. Then we're taught that you can have these things called <em>vectors</em>, which are (equivalence classes of) arrows that start at one point and end up at another.</p>
<p>At this point $\mathbb{R}^3$ is... | hardmath | 3,111 | <p>Note that $\mathbb{R}^3$ explicitly introduces <em>coordinates</em> for three dimensional space, and coordinates are not necessary to have three dimensional affine space. You are right in observing that once we identify points by coordinates, there is an ambiguity in whether we mean a point or a vector when coordin... |
1,117,458 | <p>How do I integrate an expression of the form
$$
\frac{f'(x)}{[f(x)]^n}
$$
with respect to $x$?</p>
<p>Could I use some kind of recognition method, thus avoiding partial fractions?</p>
<p>For example:
$$
\frac{(2x+1)}{(x^2+x-1)^2}
$$</p>
| RE60K | 67,609 | <p>Simple, substitute $t=f(x)$:
$$\int\frac{f'(x)}{f^n(x)}=\int\frac{dt}{t^n}=??$$</p>
|
3,960,282 | <p>I am trying to find <span class="math-container">$z$</span> such that
<span class="math-container">$$\dot{z} = -1 + e^{-iz^*},$$</span>
where <span class="math-container">$*$</span> denotes complex conjugate and the dots represent derivatives with respect to time. The time dependence of the dependent variables is su... | user577215664 | 475,762 | <p><span class="math-container">$$\dot{x} = -1 - e^{-y}\cos x, \quad \dot{y}= e^{-y}\sin x,$$</span>
<span class="math-container">$$\sin x \dfrac {dx}{dy}=-e^y- \cos x$$</span>
<span class="math-container">$$(\cos x)'-\cos x=e^y$$</span>
<span class="math-container">$$(e^{-y}\cos x)'=1$$</span>
That you can integrate... |
308,565 | <p>Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not canonical? If so, how bad can they be? </p>
| inkspot | 8,726 | <p>The cone over a plane curve of degree $d$ deforms to a smooth surface in $\mathbb P^3$ of degree $d$. Take $d\ge 5$ to see that things can be arbitrarily bad.</p>
|
141,423 | <p>Let $V \subset H \subset V'$ be a Hilbert triple.</p>
<p>We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\varphi(t)$$
for all $\varphi \in C_c^\infty(0,T)$.</p>
<p>Then we define the space $W = \{u \in L^2(0,T;V) : u' \i... | Guido Kanschat | 37,813 | <p>The derivative you define is the distributional derivative, which can be defined on $C^\infty_c$ or $C^1_c$. But then, you restrict to weak derivatives $u' \in L^2(I;V')$, which means, you allow test functions even from from $H^1(I;V)$, which is the class for which the left hand side of the definition of your deriva... |
62,581 | <p>I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?</p>
| davidlowryduda | 9,754 | <p>Sure. Let's approach this in a very elementary way: with matrix algebra.</p>
<p>Suppose that our two new 'basis vectors' are given by $ (\alpha _1, \alpha _2)$ and $(\beta _1, \beta_2)$, e.g. $(1,1)$ and $(1,0)$. </p>
<p>Then our goal is to find a linear combination of them such that we can express some given vect... |
1,856,530 | <blockquote>
<p>Prove that the product of five consecutive positive integers cannot be the square of an integer.</p>
</blockquote>
<p>I don't understand the book's argument below for why $24r-1$ and $24r+5$ can't be one of the five consecutive numbers. Are they saying that since $24-1$ and $24+5$ aren't perfect squa... | André Nicolas | 6,312 | <p>The numbers $24r-1$ and $24r+5$ are also divisible neither by $2$ nor by $3$, so by the previous argument, if they are part of the list they must be perfect squares. However, they are odd and respectively congruent to $-1$ and $5$ modulo $8$. But any odd perfect square must be congruent to $1$ modulo $8$.</p>
|
899,249 | <p>I got this problem from my friend. I have been doing it for hours.</p>
<p>$a_1 = 2$</p>
<p>$a_{n+1} = 2a^2_n+1$</p>
<p>$a_n = ?$</p>
<p>Could you please tell me how to solve this? Thanks!</p>
<p>BTW: I failed to solve it by using Mathematica <code>RSolve[{a[1] == 2, a[n + 1] == 2 a[n]^2 + 1}, a[n], n]</code></p... | Will Jagy | 10,400 | <p>Take $b_n = 2 a_n,$ this becomes
$$ b_{n+1} = b_n^2 + 2, \; \; b_1 = 4. $$
This is a well-known problem, the general heading is Lucas-Lehmer sequences. The best that can be done is that there is a real number $\theta > 1$ such that
$$ b_n \approx \theta^{\left( 2^n \right)},$$ where the really bad news is that ... |
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