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 |
|---|---|---|---|---|
1,748,211 | <p>I don't understand the determinant condition on SU(3) group, broadly.</p>
<p>I know that the determinant of such matrices should be equal to 1.
But what is the real intention of that 1?</p>
<p>Is it the real number 1 i.e.
$1+i 0$ ?</p>
<p>Or the determinant of such matrices is a complex number per se and the abs... | Eric Towers | 123,905 | <p>The purpose is to keep the matrices that preserve orientation and to ignore scaling.</p>
<p>Since every matrix, $M \in M^{3 \times 3}(\Bbb{R})$, has a determinant that is positive, negative, or zero This partitions the matrices into three chunks : those with negative determinant (which reverse orientation when they... |
3,116,768 | <p>The title is preliminary and should be changed if anyone has a better idea how to express this.</p>
<p>This is the series in question:
<span class="math-container">$$\sum_{n=1}^{\infty} \frac{n+4}{n^2-3n+1} := \sum a_n$$</span></p>
<p>What feels natural is a comparison to the harmonic series, since for any <span c... | Mark | 470,733 | <p>It is absolutely fine if there are some negative terms in the beginning of the sequence. The most important thing is that there is <span class="math-container">$n_0\in\mathbb{N}$</span> such that <span class="math-container">$a_n\geq 0$</span> for all <span class="math-container">$n\geq n_0$</span>. So you can use t... |
2,522,951 | <p>So we studied the triangle congruence criteria and we proved the ASA criterium: <em>If two triangles have respectively congruent two angles and the side included in them, then the two triangles are congruent</em>.</p>
<p>The proof is a proof of contradiction, and starts with: <em>Assume the triangles ABC and A'B'C'... | Stefan4024 | 67,746 | <p>This is an old International Mathematical Olympiad problem. Show that $p \mid 2^{p-2} + 3^{p-2} + 6^{p-2} - 1$ for all primes $p\ge 5$ by using Fermat's Little Theorem. Hence the only solution is $1$, as $(a_1,2) = 2$ and $(a_2,3) = 3$</p>
|
1,752,848 | <p>A strictly increasing sequence of positive integers $a_1, a_2, a_3,...$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ arithmetic. If $a_{13}=539$. Then how can I find $a_5.$</p>
<p>Any help will be app... | Jack D'Aurizio | 44,121 | <p>Just work backwards. You know that $539 a_{11} = a_{12}^2$, but $539=7^2\cdot 11$, hence $a_{11}=11 n^2$ and $a_{12}=77 n$. The sequence has to be strictly increasing, hence $n\leq 6$. What happens if we take $n=6$?
It follows that $a_{11}=11\cdot 6^2 = 396$ and $a_{12}=462$, so $a_{10}=2\cdot a_{11}-a_{12}=330$ and... |
2,159,970 | <p>In Simmons's Calculus with Analytic Geometric, 2nd, pg. 114, there is the following problem:</p>
<blockquote>
<p>$\text{47) Sketch the curve}$ $f(x)=\frac{2}{1+x^2}$ $\text{and find the points on it at which the normal passes through the origin.}$</p>
</blockquote>
<p>This is how I approached it: </p>
<p>let $P... | egreg | 62,967 | <p>The derivative is
$$
f'(x)=-\frac{4x}{(1+x^2)^2}
$$
so the normal at $(a,f(a))$ has equation
$$
y-f(a)=\frac{(1+a^2)^2}{4a}(x-a)
$$
<em>provided</em> $a\ne0$. This passes throught the origin if and only if
$$
-\frac{2}{1+a^2}=-a\frac{(1+a^2)^2}{4a}
$$
that is, $(1+a^2)^3=8$, so $1+a^2=2$, therefore $a=\pm1$.</p>
<p... |
3,897,110 | <p>If <span class="math-container">$f'(x)=\sqrt{x^3+1}$</span> for all <span class="math-container">$x>0$</span> and <span class="math-container">$f(2)=10$</span>, then <span class="math-container">$f(5) > 16$</span>.</p>
<p>I have what is below so far, but I am not sure how to show f(5) > 16.</p>
<p>Assume <s... | Vercassivelaunos | 803,179 | <p>A function <span class="math-container">$f:X\to Y$</span> consists of three pieces of information: the domain <span class="math-container">$X$</span>, the codomain <span class="math-container">$Y$</span>, and the graph <span class="math-container">$G_f\subseteq X\times Y$</span>. So formally, it makes sense to defin... |
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | user53763 | 53,763 | <p>You could argue on the contradiction by assuming your given function has a fixed point. By definition a function has a fixed point iff $f(x) = x$. If you substitute your function into the definition it would be clear you get an impossible mathematical equality, thus you have proved by contradiction that your functi... |
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | Sushant | 495,085 | <p>One way is to use the property of continuous function on compact set. Take g(x) =|x - f(x)| on [a,b]. Clearly g is continuous so image of g must be compact, hence it will achieve it's minimum value. Can you show now that minimum will be zero ?</p>
|
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Alexei Averchenko | 3,793 | <p>To prove that $S^n$ is path-connected, prove that through each two points $x, y \in S^n$ passes a unique big circle. Then prove that a circle is path-connected (represent it as the unit circle in $\mathbb C$ and use complex multiplication).</p>
|
2,972,235 | <p>Which function grows faster </p>
<p><span class="math-container">$()= 2^{^2+3}$</span> and <span class="math-container">$() = 2^{+1}$</span></p>
<p>by using the limit theorem I will first simplify </p>
<p>then I will just get <span class="math-container">$$\lim_{n \to \infty} \dfrac{2^{n^2+3n}}{2^{n+1}}=\lim_{n \... | KM101 | 596,598 | <p><strong>Before Edit:</strong>
Your idea was correct, but you didn’t simplify the limit properly.
<span class="math-container">$$\lim_{n \to \infty} \frac{2^{n^2+3n}}{2^n+1}$$</span>
It is enough to divide both the numerator and denominator by <span class="math-container">$2^n$</span>.
<span class="math-container">$$... |
1,209,546 | <p>$$F(x) = \int{f(x)}\,dx$$</p>
<p>$$G(x) = \int_0^x{g(z)}\,dz$$</p>
<p>I am confused about the exact meaning about these functions. The second function is clear to me, $G(x)$ is just the area under the graph of $g(x)$ from $0$ to some $x$. But the first function is not so clear.</p>
<p>Also, why is the following... | davidlowryduda | 9,754 | <p>These are good questions.</p>
<p>The notation $\displaystyle \int f(x) dx$ is shorthand for "an antiderivative of $f(x)$." That is, a function with the property that $F'(x) = f(x)$. Part of the depth of the fundamental theorem of calculus is that antiderivatives are also ways to calculate the area under a curve. Th... |
3,013,177 | <blockquote>
<p>Find a curve <span class="math-container">$\alpha : (−ε,ε) → \Sigma$</span> on the sphere which has <span class="math-container">$\alpha(0) = (1,0,0)$</span> and <span class="math-container">$\alpha′(0) = (0, 5, 6)$</span>.</p>
</blockquote>
<p>I'm unsure how to approach this. I know the parametariza... | Paul Frost | 349,785 | <p>Yes. </p>
<p>1) Choose an open neighborhood <span class="math-container">$W$</span> of <span class="math-container">$x_0$</span> such that <span class="math-container">$W \subset U$</span> and <span class="math-container">$\lvert f(x) - f(x_0) \rvert < \epsilon/2$</span> for <span class="math-container">$x \in ... |
44,746 | <p>Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.</p>
<p>I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perhaps as $1^{-\in... | 410 gone | 8,572 | <p>You could say that <span class="math-container">$\frac{1}{\infty} = 0$</span>, so <span class="math-container">$1-\frac{1}{\infty} = 1$</span>. But then, you're stretching the definition of division past breaking point - division as you know it isn't defined for infinity, so the answer is undefined. Otherwise, you c... |
3,214,662 | <p><strong>Q1</strong> Prove that every simple subgroup of <span class="math-container">$S_4$</span> is abelian.</p>
<p><strong>Q2</strong> Using the above result, show that if <span class="math-container">$G$</span> is a nonabelian simple group then every proper subgroup of <span class="math-container">$G$</span> has... | Dr. Mathva | 588,272 | <h2>Method 1: The Law of Cosines</h2>
<p>Define
<span class="math-container">$$a:=\sqrt{(x_c-x_b)^2+(y_c-y_b)^2}\qquad b:=\ldots \qquad c:=\ldots$$</span></p>
<p>Observe that, in any triangle, <span class="math-container">$\angle ABC$</span> is acute, if and only if <span class="math-container">$$\cos(\angle ABC)=\frac... |
3,214,662 | <p><strong>Q1</strong> Prove that every simple subgroup of <span class="math-container">$S_4$</span> is abelian.</p>
<p><strong>Q2</strong> Using the above result, show that if <span class="math-container">$G$</span> is a nonabelian simple group then every proper subgroup of <span class="math-container">$G$</span> has... | Dr. Mathva | 588,272 | <h2>Method 2: Your method (an analytical approach)</h2>
<p>Observe that the line through <span class="math-container">$B,C$</span> has the slope <span class="math-container">$$m=\frac{y_c-y_b}{x_c-x_b}$$</span> Therefore, the lines <span class="math-container">$l_1$</span> and <span class="math-container">$l_2$</span>,... |
67,171 | <p>I am sure <a href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse">all those symbols</a> are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me.</p>
<p>How could I do this on a basic calculator? or with a few lines of programmer's code which... | Bill Dubuque | 242 | <p>One need not understand congruence arithmetic to understand the <a href="http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm" rel="nofollow">extended Euclidean algorithm</a> as applied to computing modular inverses. By <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">Bezout's Iden... |
160,779 | <p>I am having a bit of difficulty trying to answer the following question:</p>
<blockquote>
<p>What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$?</p>
</blockquote>
<p>So far I have factored $X^8-1$ as </p>
<p>$$X^8-1=(X+10)(X+1)(X^2+1)(X^4+1).$$ </p>
<p>I know $X^2+1$ is irreducible over $\mathbb{F}_{11... | Mikko Korhonen | 17,384 | <p>Let $E$ be a splitting field of $x^8 - 1$ over $F_{11}$. Then $E = F_{11}(\alpha)$, where $\alpha$ is a primitive $8$th root of unity. </p>
<p>It follows that $G = \operatorname{Gal}(E/F)$ is isomorphic to a subgroup of $(\mathbb{Z}/8\mathbb{Z})^*$, which is isomorphic to the Klein four-group. Now $G$ cannot have o... |
2,855,142 | <p>I'm hosting a bar sports tournament with $10$ teams and $6$ different sports (pool, darts, table tennis, foosball, beer pong and cornhole). Trying to get the fixtures as fair as possible so that each team plays each sport twice and playing against the same team multiple times is minimised. Are there any formulas to ... | saulspatz | 235,128 | <p>I've spent a good deal of time on this, and I haven't been able to do it, so in one sense, this isn't an answer at all, more of a progress report. On the other hand, someone may be able to suggest a modification to the method that will allow for solution. I tried it with both $10$ teams and $12$ teams, without s... |
2,828,060 | <blockquote>
<p>Let $f(x)=x^3 +4x^2 +6x$ and $g(x)$ be its inverse. Then the value of $g'(-4)$ is?</p>
</blockquote>
<p>My attempt at the solution:
$$
g(x) = f^{-1}(x) \implies f(g(x))=x,
$$
<em>differentiating both sides,</em>
$$
f'(g(x)) \cdot g'(x)=1 \implies g'(x)=\frac1{f'(g(x))}.
$$
I am stuck here as we do no... | mathreadler | 213,607 | <p>Using vector notation $[x,y]^T$ :
First we need to write down the points:
$$P_0 = [0,1]^T, P_1 = [0,0]^T, P_2 = [1,0]^T$$
Now to the formula:
$$B(t) = (1-t)^2[0,1]^T + (1-t)t[0,0]^T + t^2[1,0]^T = ...\\ [t^2,(1-t)^2]^T$$</p>
<p>So it is true you get quadratic functions, but the quadratic functions are in both $x$ a... |
2,370,851 | <p>How to solve the Integral $\int{\frac{1}{\sqrt{1-\sqrt{4-x}}}}$dx with steps</p>
<p>I have tried to make the substitution $\frac{du}{dx}=4-x$ but it seems that there is no continuation road.</p>
<p>I have seen that there is a substitutions that gives the following results.</p>
<p>$\int{\frac{1}{2\sqrt{-u}} \fr... | Claude Leibovici | 82,404 | <p>Considering $$I=\int{\frac{dx}{\sqrt{1-\sqrt{4-x}}}} $$ first let $$\sqrt{4-x}=t^2\implies x=4-t^4 \implies dx=-4t^3\,dt$$ making $$I=-4\int \frac{ t^3}{\sqrt{1-t^2}}\,dt$$ Now, use $$t=\sin(y)\implies dt=\cos(y)\,dy$$ making $$I=-4\int \sin^3(y)\,dy=4\int (1-\cos^2(y))\, d(\cos(y))$$ Just finish !</p>
|
109,528 | <p>Let S be a finite set of integers, do I can check with Gap that this set be a set of character degrees of small group?</p>
| Geoff Robinson | 14,450 | <p>Dima is correct that the problem is much easier with multisets. Furthermore, taking direct products with Abelian groups, it is clear that for any given set of irreducible character degrees which actually occurs, there are infinitely many groups for which it occurs, if we allow multiplicities. I had a recollection th... |
109,528 | <p>Let S be a finite set of integers, do I can check with Gap that this set be a set of character degrees of small group?</p>
| Marty Isaacs | 9,694 | <p>Given an arbitrary set $S$ of powers of some prime $p$, subject only to the condition that $1 \in S$, there necessarily exists a $p$-group $P$ such that $S$ is exactly the set of irreducible character degrees of $P$. In fact, it is always possible to find $P$ having nilpotence class $2$. This theorem appears in my p... |
2,710 | <p>The <em>Mandelbrot set</em> is the set of points of the complex plane whos orbits do not diverge. An point $c$'s <em>orbit</em> is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.</p>
<p>The shape of this set is well known, <strong>why is it that if you zoom into parts of the filaments you will find slight... | MathematicalOrchid | 29,949 | <p>It's slightly tricky for the Mandelbrot set, because this exists in parameter space. It's easier to think about the corresponding Julia sets - although the idea is similar.</p>
<p>The answer that I like is this: A Julia set is a "hall of mirrors". When you look at one, you are seeing a reflection of a reflection of... |
1,190,320 | <p>I need to evaluate $$\lim_{x \to \infty}x\int_0^{1/x}e^t \cos(t) \space \text{d}t$$ and I'm not really sure how to start. Do I have to find the integral or is there another way to figure it out?</p>
| André Nicolas | 6,312 | <p>The simplest approach is through L'Hospital's Rule and the Fundamental Theorem. But there are more complicated ways.
We examine
$$\int_0^{1/x}e^t\cos t\,dt$$
for large positive $x$. Note that $e^t\cos t$ is increasing in the interval $(0,\pi/4)$ (look at the derivative). So if $x$ is large, and $0\le t\le 1/x$, we ... |
1,059,270 | <p>I was send here from stackoverflow because they thought maybe you can help me.</p>
<p>Here my original post: <a href="https://stackoverflow.com/questions/26799476/a-faster-way-then-doing-14-for-loops">https://stackoverflow.com/questions/26799476/a-faster-way-then-doing-14-for-loops</a></p>
<p>What I want:</p>
<p>... | user200365 | 200,365 | <p>what i think the writer is trying to say is:</p>
<p>is their a way to enter all of your exact cards into a simulator and have the simulator tell you the best result for your entire inventory</p>
<p>example :</p>
<p>list of 200 exact cards
vs</p>
<h1>demon</h1>
<p>simulator tells you the best 10 cards you have t... |
3,604,877 | <p>Does a parabola eventually form a sort of ellipse when stretched to infinity along its axis? I am asking because I am trying to intuitively understand the following picture and the fact that the line at infinity is a tangent of parabola: </p>
<p><a href="https://i.stack.imgur.com/J64XZ.png" rel="nofollow noreferrer... | User2020201 | 765,960 | <p>Elipse has <span class="math-container">$2$</span> focuses. What would fouces be in that case? </p>
<p>If we imagine point <span class="math-container">$F$</span> in infinity along that axis and <span class="math-container">$G$</span> is focus of parabola then for every point <span class="math-container">$P$</span>... |
4,911 | <p>$$
\frac{\sqrt{3(m-n)^2 n^2}}{2}
$$
This expression is getting correctly rendered here on meta. I copied and pasted it from math.stackexchange.com. There, the horizontal line did not extend far enough in either direction, going to just above the middle of the $n$ on the right and failing to cover the $3$ on the le... | Davide Cervone | 7,798 | <p>MathJax v2.1 should resolve this issue for Chrome users. Math.SE is now using the beta version of MathJax v2.1, so you should see the correct rendering for square roots (and fraction bars) now. Please let me know if that is not the case.</p>
<p>Note that if you keep Chrome open continually, you may need to empty ... |
97,920 | <p>Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$
for all $a>0$ and $0\lt q\lt 1$ ?</p>
| Hjalmar Rosengren | 10,846 | <p>There is a well-established name for this series; it is called a partial theta function. So if by "closed form", you mean "expression in terms of objects that are interesting enough that people have a name for them", the answer is yes. For more information on the partial theta function, <a href="http://arxiv.org/fin... |
395,994 | <p>Suppose that $f_{x,y}(x,y) = \lambda^2 e^{\displaystyle-\lambda(x+y)}, 0\leq x , 0\leq y.$ Find $\operatorname{Var(X+Y)}$. </p>
<p>I'm having trouble with this problem the way to find $\operatorname{Var(X+Y)} = \operatorname{Var(X)}+\operatorname{Var(Y)}+2\operatorname{Cov(X,Y)}$, however if $X$ and $Y$ are indepen... | GeoffDS | 8,671 | <p>If you can factor the joint density into a product that is a function of x times a function of y, then $X$ and $Y$ are independent and their marginal densities are a constant multiple of the two functions in the product. That is, if
$$f_{X, Y}(x, y) = g(x) h(y)$$
for some functions $g(x)$ and $h(y)$, then $X$ and ... |
2,421,896 | <p>I have the following integral to find:</p>
<p>$$\int 12x^2(3+2x)^5 dx$$</p>
<p>Now, I am aware of the integration by parts property - </p>
<p>$$\int \ u \frac{dv}{dx} = uv - \int v\frac{du}{dx}$$</p>
<p>Now, my question is the following - </p>
<p>When I make $u = 12x^2$, I find a different answer to when I make... | David Quinn | 187,299 | <p>You will get the same answer which ever way you do it, but setting $u=(3+2x)^5$ is going to take a long time. </p>
<p>But an easier method is just a direct substitution $u=3+2x$, much quicker than integration by parts.</p>
|
379,140 | <p>I have this irregular line and I want to split it in, for example, ten equal parts. How can I do that?</p>
<p><img src="https://i.stack.imgur.com/4AcBP.png" alt="Example"></p>
<p>Thank you!</p>
| rohit | 75,205 | <p>Consider sequence Dn = { e^i*(theta) , where 0<=theta < 2pi-1/n}
clearly each Dn is simply connected, but their union is the unit circle which is not simply connected.</p>
|
244,309 | <p>I have some data as</p>
<pre><code>data={{257.3`, 493.7`}, {43.666666666666664`,490.5`}, {111.91176470588235`,461.20588235294116`},{345.2142857142857`,460.5`}, {420.88461538461536`, 436.34615384615387`}, {318.1`,408.46`}, {277.`,400.7`}, {273.5`, 383.`}, {444.`,381.5`}, {208.28571428571428`,379.7857142857143`}, {510... | Tugrul Temel | 60,365 | <p>This is a partial answer because it does not have the flexibility of choosing a specific set of parameter values. It simply gives us sets of outputs:</p>
<pre><code> f[x_, y_, z_] := x y + z^2 + y z;
Manipulate[
outputs=Flatten[Table[{x, y, z, f[x, y, z], f[x, y, z]^2,f[x, y, z]^3},{x,1,5,1}, {y,1,5,1}, {z,.1,1,... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | Vincent | 332,815 | <p>You should not divide by $15$, but by the number of votes. In your case, the pondered sum is 90 (and not 52) and once you divide it by your 28 votes, you get an average note of about 3.2</p>
|
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | JMP | 210,189 | <p>In your first example you should calculate;</p>
<p>$$\dfrac{1\cdot5+2\cdot3+3\cdot1+4\cdot17+5\cdot2}{5+3+1+17+2}$$
$$=\dfrac{92}{28}\approx3.29$$</p>
|
502,295 | <p>Use proof by contradiction to show that every integer greater than 11 is a sum of two composite numbers</p>
<p>My Solution: </p>
<p>Statement: For all integers $x$, if $x>11$, then $x = y + z$ whereby $y$ and $z$ are any composite numbers.</p>
<p>Proof by contradiction: There exists an integer $x$ such that $x... | Community | -1 | <p>See, You want to prove that $n>11$ is a sum of two composite numbers.</p>
<p>Given hint is to check "Are $n−4$, $n−6$ and $n−8$ all prime?"</p>
<p>Suppose one of then is composite then :</p>
<p>$n=(n-4)+4$ (if $n-4$ is not prime) (Thus $n$ is sum of two composite numbers $n-4$ and $4$)</p>
<p>$n=(n-6)+6$ (if ... |
3,688,208 | <p>Let <span class="math-container">$f: ]0,1[ \to \mathbb{R}$</span> be a function. Suppose that for every sequence <span class="math-container">$(\epsilon_n)_n$</span> in <span class="math-container">$]0,1[$</span> with <span class="math-container">$\epsilon_n \searrow 0$</span> we have that <span class="math-containe... | Reinhard Meier | 407,833 | <p>Let <span class="math-container">$u=x^9$</span> and <span class="math-container">$v=y^{11}$</span>. Then
<span class="math-container">$$
\left| \frac{x^5 y^5}{|x|^9+|y|^{11}}\right|
=\frac{\left|u^{\frac{5}{9}}v^{\frac{5}{11}}\right|}{|u|+|v|}
$$</span>
If <span class="math-container">$|u| \geq |v|,$</span> then
<sp... |
1,393,423 | <p>Given the function</p>
<p>$$y=Ax + B\sqrt x$$</p>
<p>where $A$ and $B$ are real constants, $x$ is real and $x > 0$</p>
<p>I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible without approximating the square root term, right?</p>
<p>Either
$$y=B\sqrt x$$
or
$$y=Ax $$</... | abiessu | 86,846 | <p>One approach is to "complete the square" by treating $\sqrt x$ as the variable quantity. Then you have</p>
<p>$$y=Ax+B\sqrt x\\
=Au^2+Bu$$</p>
<p>By the quadratic formula, we then have</p>
<p>$$u={-B\pm\sqrt{B^2+4Ay}\over 2y}=\sqrt x$$</p>
<p>Continuing the inversion should be easy from this point.</p>
|
1,393,423 | <p>Given the function</p>
<p>$$y=Ax + B\sqrt x$$</p>
<p>where $A$ and $B$ are real constants, $x$ is real and $x > 0$</p>
<p>I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible without approximating the square root term, right?</p>
<p>Either
$$y=B\sqrt x$$
or
$$y=Ax $$</... | David K | 139,123 | <p>It depends on the values of $A$ and $B$.</p>
<p>To get the trivial objections out of the way first,
if $A = 0$ or $B = 0$ you can easily invert the function, but
if $A = B = 0$ then clearly you cannot invert the function,
as there is no inverse function for $y = 0$.</p>
<p>So assume that neither $A$ nor $B$ is zer... |
3,597,812 | <p><span class="math-container">$f: \mathbb C \to \mathbb C$</span> is entire function such that</p>
<blockquote>
<blockquote>
<p><span class="math-container">$f(1/n)=1/n^2$</span> for all <span class="math-container">$n \in \mathbb N$</span>, Then to show <span class="math-container">$f(z)=z^2$</span></p>
</b... | copper.hat | 27,978 | <p>Let <span class="math-container">$\phi(z) = f(z)-z^2$</span>. Then <span class="math-container">$\phi({1 \over n}) = 0$</span> for all <span class="math-container">$n$</span> and <span class="math-container">${ 1 \over n} \to 0$</span>.</p>
|
966,504 | <p>I had been following all the blogs, but I would like to understand, whether an attempt has been made to understand how many cycles are possible apart from the 1-4-2-1 cycle in collatz problem</p>
| G Tony Jacobs | 92,129 | <h2>Cycle Shapes</h2>
<p>A cycle under the Collatz map is completely determined by its "shape", in the following sense: A cycle contains a certain number of odd elements, and between each odd element and the next, there are a certain number of divisions-by-2 that occur.</p>
<p>For example, the famous <span class="mat... |
494,239 | <p>How would you find the 4th term in the expansion $(1+2x)^2 (1-6x)^{15}$?</p>
<p>Is there a simple way to do so?</p>
<p>Any help would be appreciated</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>The $r$ th term $T_{r+1}$ of $(a+b)^n$ is $$\binom nr a^{n-r}b^r$$ where $0\le r\le n$</p>
|
2,696,400 | <p>The Peano axioms are intended to be able to prove very general statements about arithmetic, such as "all natural numbers can be written as the sum of two primes".</p>
<p>However, how can we use the peano axioms to mathematically derive all the rules that are being taught to primary school children, about how to add... | David C. Ullrich | 248,223 | <p>Computing $5\times 4$ would take a little space. Instead here's a proof from the axioms that $2\times 2=4$. </p>
<p>Note first that the <em>definitions</em> of $2$ and $4$ are $$2=SS0,\quad 4=SSSS0.$$</p>
<p>So $$2\times 2=(SS0)(SS0)=(S0)(SS0)+SS0=SS0+SS0=SSS0+S0=SSSS0.$$</p>
|
4,019,754 | <p>I have no idea how to approach this?</p>
<p><a href="https://i.stack.imgur.com/XHH84.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XHH84.png" alt="enter image description here" /></a></p>
| JLMF | 551,373 | <p>This doesn't seem true to me without more restrictions.
For example, take <span class="math-container">$X$</span> the zero vector space in <span class="math-container">$H_1$</span>, then <span class="math-container">$X$</span> is not dense in <span class="math-container">$Y$</span>.</p>
<p>Edit: What if we consider ... |
3,073,640 | <p>I think that I know how to do this task but I need know that I have right. This is my idea: I create matrix with vectors from lin and after elementary matrix I have: <span class="math-container">$$\begin{bmatrix}
0 & 0 & 1 & 0 \\
1 & -0,5 & 0 & -1,5
\end{bmatrix}$$</span> Then I think that th... | José Carlos Santos | 446,262 | <p>No, <span class="math-container">$A$</span> doesn't have to be open. Let <span class="math-container">$\{q_n\,|\,n\in\mathbb{N}\}$</span> be an enumeration of <span class="math-container">$\mathbb Q$</span>. For each <span class="math-container">$n\in\mathbb N$</span>, let <span class="math-container">$V_n=\mathbb{R... |
2,163,306 | <p>Find the following limit $$I = \lim_{n \to\infty} \int_{n}^{e^n} xe^{-x^{2016}} dx$$</p>
<p>My attempt </p>
<p>Assumption: as $n \to \infty$ we can assume and interval on the positive real axis $[n,e^n]$</p>
<p>Here the function $e^{-x^{2016}}$ is a decreasing function, using this fact we use the sandwich lemma t... | robjohn | 13,854 | <p>Use <a href="https://en.wikipedia.org/wiki/Partial_fraction_decomposition" rel="nofollow noreferrer">Partial Fractions</a> and <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hôpital</a>:
$$
\frac{P(x)}{Q(x)}=\sum_{k=1}^n\frac{A_k}{x-a_k}\\
$$
where $A_k=\lim\limits_... |
2,205,087 | <p>Could you please give me an intuitive explanation why the dot product is defined this way?</p>
| Community | -1 | <p>A function associates to every value in the domain exactly one value in the codomain.</p>
<p>Addition, subtraction, multiplication and division are functions (of two arguments).</p>
<p>The composition of two functions is a function because the uniqueness of the values carries over.</p>
|
51,026 | <p>I just wasted the last hour on google looking in vain for an excerpt of Weil's writings describing the process of discovering mathematics. I believe he once beautifully described the feeling of loss that accompanies the realization that the discovery you made seems, in retrospect, trivial. Am I misremembering or j... | t.b. | 5,363 | <p>You might have the following passage in mind:</p>
<p><img src="https://i.stack.imgur.com/UkDCL.png" alt="De la métaphysique aux mathématiques"></p>
<p>It appears in A. Weil, <em>De la métaphysique aux mathématiques</em>, Science <strong>60</strong>, p. 52–56 (see also Collected P... |
1,703,491 | <p>I have the following problem:
$$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$
I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify this?</p>
| DonAntonio | 31,254 | <p>$$\int\frac{x+2}{x^2+x+1}dx=\frac12\int\frac{2x+1}{x^2+x+1}dx+\frac32\int\frac{dx}{\frac34+\left(x+\frac12\right)^2}=$$</p>
<p>$$=\frac12\log(x^2+x+1)+2\int\frac{dx}{1+\left(\frac{2x+1}{\sqrt3}\right)^2}=$$</p>
<p>$$=\log\sqrt{x^2+x+1}+\sqrt3\,\arctan\frac{2x+1}{\sqrt3}+K$$</p>
|
500,678 | <blockquote>
<p>Let $a$ and $b$ belong to a group $G$. Find an $x$ in $G$ such that $xabx^{-1}= ba$.</p>
</blockquote>
<p>This is what I have done so far, but I am stuck and not sure if I am in the right direction:</p>
<p>$xabx^{-1} = ba$</p>
<p>Multiply both sides on the right by $x$.</p>
<p>$xabx^{-1}x = bax$.<... | amWhy | 9,003 | <p>Try letting $x = a^{-1}$, so that $x^{-1} = (a^{-1})^{-1} = a$.</p>
<p>Or, if you prefer, let $x = b$, so $x^{-1} = b^{-1}$.</p>
<p>(Since $a, b \in G$, so are $a^{-1}, b^{-1} \in G$, since $G$ is a group and a group is closed under taking inverses.)</p>
|
2,720,539 | <p>Prove that for every $x \in(0,\frac{\pi}{2})$, the following inequality:</p>
<p>$\frac{2\ln(\cos{x})}{x^2}\lt \frac{x^2}{12}-1$</p>
<p>holds</p>
<p>I don't see room to use derivatives, since it seems a little messy to calculate the $\lim_{x\to 0}$ of $\frac{2\ln(\cos{x})}{x^2}$
(which, I think, is necessary in o... | orangeskid | 168,051 | <p>The Maclaurin series of the tangent function has all coefficients positive (see <a href="https://en.wikipedia.org/wiki/Taylor_series#Trigonometric_functions" rel="nofollow noreferrer">formula</a>). Integrating we get that the Maclaurin series of the function $-\log(\cos x)$ has all coefficients positive. Up to order... |
3,991,914 | <p>I've been thinking about this one for a while, at first I thought it only to be valid if <span class="math-container">$U,W$</span> were open (the union of open sets in <span class="math-container">$\mathbb{R}^{n}$</span> is open and <span class="math-container">$\mathbb{R}^{n}$</span> is an open set), then I remembe... | Marko Karbevski | 45,470 | <p>The way that it is stated, it is not true (see the answer by Duncan Ramage and the comments)</p>
<p>Judging by the discussion though, this nice question will lead you to the notion of <a href="https://en.wikipedia.org/wiki/Connected_space" rel="nofollow noreferrer"><em>connectedness</em></a>. For more info on the to... |
1,858,002 | <p>Is there any standard solution or way to solve the following integration</p>
<p>$$I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$$</p>
<p>where, $n=0, 1, 2,\dots$ and $p=0, 1, 2,\dots$ and $p> n$</p>
| Community | -1 | <p><strong>Hint</strong>:</p>
<p>Using the complex representation with $z=e^{i\theta}$,</p>
<p>$$I=-i\int_1^{-1}\left(\frac{z+z^{-1}}2\right)^n\frac{z^p+z^{-p}}2\frac{dz}z\
=-\frac i{2^{n+1}}\int_1^{-1}\sum_{k=0}^n\binom nk\left(z^{2k-n+p-1}+z^{2k-n-p-1}\right)dz.$$</p>
<p>Then for $m\ne-1$,</p>
<p>$$\int_0^\pi z^m... |
72,536 | <p>In this question I will use the following definition of a Dedekind domain:</p>
<p>An integral domain $A$ is a Dedekind Domain if:</p>
<p>1) $A$ is a Noetherian Ring.</p>
<p>2) $A$ is integrally closed.</p>
<p>3) Every non-zero prime ideal of $A$ is maximal.</p>
<p>I also know that every non-zero ideal $I \subse... | jspecter | 11,844 | <p>Yes. Factor $I = \displaystyle\prod_{i =1}^n \mathfrak{p}_i^{e^i}.$ Then by the <a href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem#Statement_for_general_rings" rel="noreferrer">Chinese Remainder Theorem</a> $A/I \cong \displaystyle\bigoplus_{i =1}^n A/\mathfrak{p}_i^{e^i}.$ So it is enough to show each... |
72,536 | <p>In this question I will use the following definition of a Dedekind domain:</p>
<p>An integral domain $A$ is a Dedekind Domain if:</p>
<p>1) $A$ is a Noetherian Ring.</p>
<p>2) $A$ is integrally closed.</p>
<p>3) Every non-zero prime ideal of $A$ is maximal.</p>
<p>I also know that every non-zero ideal $I \subse... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\:$ By the approximation theorem, Dedekind domain ideals $\rm\:J\!\ne\! 0\:$ are strongly two generated, i.e. for each $\rm\:0\ne i\in J\:$ there exists $\rm\:j\in J\:$ such that $\rm\:J = (i,j)\:.\:$ Therefore every ideal $\rm\: J\supset I\:$ can be presented in the form $\rm\:J = (i,j)\:$ fo... |
3,804,347 | <p>I am told that the directional derivative is defined as
<span class="math-container">$$
D_vf(x) = \lim_{h \rightarrow 0} \frac{f(x+hv)-f(x)}{h}
$$</span>
So my way of deriving this kind of stuff has always been the Taylor expansion (<span class="math-container">$v^j$</span> and <span class="math-container">$x^j$</sp... | mathcounterexamples.net | 187,663 | <p>What you say is correct. However, you also have to use the coordinates of <span class="math-container">$v$</span> in the other coordinate system, polar for example.</p>
<p>Let say that <span class="math-container">$A$</span> is the matrix to move from the cartesian to the polar coordinates.</p>
<p>You have
<span cla... |
3,804,347 | <p>I am told that the directional derivative is defined as
<span class="math-container">$$
D_vf(x) = \lim_{h \rightarrow 0} \frac{f(x+hv)-f(x)}{h}
$$</span>
So my way of deriving this kind of stuff has always been the Taylor expansion (<span class="math-container">$v^j$</span> and <span class="math-container">$x^j$</sp... | peek-a-boo | 568,204 | <p>(1) and (2) are both right, but it's just that the <span class="math-container">$v^{\phi}$</span> in your two formulas mean different things, and you've unknowingly abused notation by calling them both <span class="math-container">$v^{\phi}$</span>. This issue boils down to the distinction between the tangent vector... |
250,919 | <p><em>We have a point $A(6,0)$ and a line $k:y=2$. Show that the equation of the parabola with a locus $A$ and a directrix $k$ has the formula: $\dfrac{1}{4}x^2-3x+8$</em>.</p>
<p>I had a test on analytic geometry today and this was one of the questions. I was 100% sure that the question was wrong, however, all my cl... | matovitch | 26,304 | <p>I think the sign of c depend of the position of the focus compared to the directrix (question of orientation) because if you set it at -1 you will get the answer I compute (-1/4 for 1/4).</p>
<p>By the way, you forgot to divide by four the affine term.</p>
<p>Please excuse the deplorable english of a french (thoug... |
250,919 | <p><em>We have a point $A(6,0)$ and a line $k:y=2$. Show that the equation of the parabola with a locus $A$ and a directrix $k$ has the formula: $\dfrac{1}{4}x^2-3x+8$</em>.</p>
<p>I had a test on analytic geometry today and this was one of the questions. I was 100% sure that the question was wrong, however, all my cl... | David Mitra | 18,986 | <p>If the directrix is the line $y=2$:</p>
<p>Your value of $c$ should be negative, since the focus lies below the directrix. So, your initial equation should be $$\tag{1} y-1={-1\over 4}(x-6)^2.$$
Apart from $c=-1$, and not $c=1$, what you had there is correct.</p>
<p>But, it seems you made some algebraic errors w... |
3,546,184 | <p>Take three numbers <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, and <span class="math-container">$x_3$</span> and form the successive running averages <span class="math-container">$x_n = (x_{n-3} + x_{n-2} + x_{n-1})/3$</span> starting from <span class="math-container">$x_4$<... | Arthur | 15,500 | <p>The naive, immediate approach would be this: any term in the sequence is a linear combination of <span class="math-container">$x_1,x_2$</span> and <span class="math-container">$x_3$</span>. A state would be the coefficients of this linear combination, for the most recent three terms.</p>
<p>So, the initial state is... |
1,212,433 | <p>Problem statement:</p>
<p>Let $ f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ be defined as $ f(m, n) = (3m + 7n, 2m + 5n) $. Is $f$ a bijection, i.e., one- to-one and onto? If yes then give a formal proof, based on the definitions of one-to-one and onto, and derive a formula for $ f^-1 $. If ... | simonzack | 112,423 | <p>You can try to find an inverse to see that it is a bijection.</p>
<p>Suppose $f(m, n) = (x, y)$. Then $3m+7n=x, 2m+5n=y$. Now this is just a linear equation over $\mathbb{Z}$, with determinant $3\times 5 - 7\times 2=1$, so $x, y$ will be a linear combination of $m, n$ with integer coefficients, which is in the doma... |
2,881,488 | <p>Let $M$ be a metric space, $x_n\in M$ a sequence which converges to $x\in M$</p>
<p>Prove: $F=\{x_n\}\cup \{x\}$ is a closed set</p>
<p>So we have $x_n\to x$ such that $x_n\in F$ and $x\in F$ and we know that a set is closed if it contains all of its accumulation points, so $F$ is closed</p>
<p>Or must I look at ... | Tsemo Aristide | 280,301 | <p>You can also show that $F^c$ is open, let $y\in F^c$, suppose that for every $n>0$, there exists $x_{n_p}$in the open ball $B(y,{1\over n})$, this implies that $lim_nx_{n_p}=y=x$ contradiction. Thus there exists $n_0$ such that $B(y,{1\over n_0})$ does not contain any $x_n$, it does not contain $x$ also, we deduc... |
2,881,488 | <p>Let $M$ be a metric space, $x_n\in M$ a sequence which converges to $x\in M$</p>
<p>Prove: $F=\{x_n\}\cup \{x\}$ is a closed set</p>
<p>So we have $x_n\to x$ such that $x_n\in F$ and $x\in F$ and we know that a set is closed if it contains all of its accumulation points, so $F$ is closed</p>
<p>Or must I look at ... | user247327 | 247,327 | <p>How you prove a set is "closed" depends strongly on what definition of "closed" you are using. One commonly used definition is that a set is closed if and only if it contains all of its limits points. Show that the only limit point of this set is x.</p>
|
2,632,527 | <p>Suppose the product $z_1z_2$ of two complex numbers is a nonzero real constant. Show that $z_2$ = $k\overline{z}_1$, where $k$ is a real number.</p>
<p>Hint: $z_2$=$k$ conjugate of $z_1$</p>
| Fred | 380,717 | <p>If $0 \ne z_1z_2=c \in \mathbb R$, then $z_2= \frac{c}{z_1}=\frac{c \overline{z_1}}{z_1\overline{z_1}}=\frac{c \overline{z_1}}{|z_1|^2}=k \overline{z_1}$, where $k= \frac{c }{|z_1|^2}$.</p>
|
25,263 | <p>Can you efficiently parallelize this? The parallel versions are much slower than the sequential version, and I'm not sure why. Does <code>SetSharedVariable</code> allow simultaneous reads for different kernels? It appears that it doesn't even though the documentation says you should use <code>CriticalSection</code> ... | Szabolcs | 12 | <p>To answer your actual question, <code>SetSharedVariable</code> does not allow simultaneous reads. It forces the variable to be evaluated on the main kernel, effectively disabling the parallelization.</p>
<hr />
<p>A more interesting question for me is: why is <code>ParallelMap</code> so slow when not using <code>Set... |
849,191 | <p>I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I thought, it's a small group of kids that have strong math skills and seem to have an endless desire for knowledge. I... | gadfly1974 | 160,416 | <p>When I work with a particularly motivated group of students, I don't feel it's my responsibility to do all of the heavy lifting.</p>
<p>Encourage this group of kids to take what you've shown them and allow them to create their own fallacies and evaluate one another's work.</p>
<p>Have them sign up for their own St... |
754,057 | <p>I have the problem: $\sin(2x)=\tan(x)$</p>
<p>I used the double angle formula to get $2\sin(x)\cos(x)=\tan(x)$.</p>
<p>But after that step, I do not know whether or not to subtract $\tan(x)$ or to set $2\sin(x)\cos(x)$ to $u$ and solve for $U$.</p>
| Community | -1 | <p>$$2\sin x\cos x=\dfrac{\sin x}{\cos x}\\
\implies \cos x=\dfrac{1}{\sqrt{2}}\text{ and/or }\sin x=0\\
\implies x=\dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{5\pi}{4},\dfrac{7\pi}{4},0,\pi,2\pi$$</p>
|
1,181,405 | <p>If $b_n>0$ and $\sum b_n\,$ converges, prove $\sum {b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\,$ converges for all $\alpha>{1\over2}$.</p>
<p>I know
${b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\leq{b_n}^{1\over2}\dot\,{1\over{n^{1\over2}}}$</p>
<p>Since $b_n$ converges to $0$, I cannot say ${b_n}^{1\over2}\leq... | robjohn | 13,854 | <p>Without knowledge of Cauchy-Schwarz, we can use that
$$
0\le\left(b_n^{1/2}-\frac1{n^\alpha}\right)^2=b_n-2b_n^{1/2}\frac1{n^\alpha}+\frac1{n^{2\alpha}}
$$
implies
$$
b_n^{1/2}\frac1{n^\alpha}\le\frac12\left(b_n+\frac1{n^{2\alpha}}\right)
$$
Sum both sides to get that
$$
\sum_{n=1}^\infty b_n^{1/2}\frac1{n^\alpha}\l... |
4,143,585 | <p>Recently, I came across <a href="https://www.youtube.com/watch?v=joewDkmpvxo" rel="noreferrer">this</a> video, the method shown seemed good for a few logarithms. Then I tried to plot the equation <span class="math-container">$$ \frac{x^{\frac{1}{2^{15}}}-1}{0.000070271} $$</span> and it looks <em>exactly</em> like t... | Joe Lamond | 922,532 | <p>Try differentiating <span class="math-container">$a^x$</span> using the definition of the derivative. You should find that
<span class="math-container">$$
\frac{d}{dx}(a^x)=a^x\cdot\lim_{h\to0}\frac{a^h-1}{h} \, . \\
$$</span>
But we also have
<span class="math-container">$$
\frac{d}{dx}(a^x)=\frac{d}{dx}(e^{x\ln a}... |
3,670,178 | <p>Simplification: <span class="math-container">$x(t)=\exp{(-m\cdot t)}\cdot (a\cdot cos(k\cdot t) + b\cdot sin(k\cdot t)) = A\cdot \exp{(-m\cdot t)}\cdot cos(k\cdot t + c)$</span>.</p>
<p>There is no further explanation in the script, but I found this theorem: <a href="https://mathworld.wolfram.com/HarmonicAdditionTh... | Vishu | 751,311 | <p>To hold the equality, it is required that <span class="math-container">$$\tan c =-\frac ba \implies c =\tan^{-1} \left(- \frac ba\right) + n\pi$$</span> Since <span class="math-container">$-\frac{\pi}{2} \le \tan^{-1} \left(-\frac ba\right) \le \frac{\pi}{2}$</span>, we have that <span class="math-container">$$-\fra... |
134,606 | <p>As part of a larger investigation, I am required to be able to calculate the distance between any two points on a unit circle. I have tried to use cosine law but I can't determine any specific manner in which I can calculate theta if the angle between the two points and the positive axis is always given.</p>
<p>Is ... | Community | -1 | <p><strong>Hint</strong></p>
<ul>
<li><p>Points on the unit circle centered at $(0,0)$ on the argand plane are of the form $(\cos \theta, \sin \theta)$, with $0 \leq \theta \lt 2\pi$. </p></li>
<li><p>Can you use <strong>distance formula</strong> now to calculate the requires to distance?</p></li>
</ul>
<hr>
<p>With... |
225,813 | <p>Let M be a finite dimensional von Neumann algebras with a normal faithful trace. Let e and f be two projections with rank 1. I want to know if e and f have identical traces. (This is obviously true if M is a factor.)</p>
<p>I guess it is false, while i have no counterexample.</p>
| H1ghfiv3 | 78,554 | <p>Although this question has already been answered, I would like to shortly explain the idea on how to prove that $f$ is $\pi_1$-surjective, as it has been established in the comments below my original question. The result then follows from the classification of Haken manifolds (See Hempel's "3 Manifolds", for example... |
3,560,742 | <p><span class="math-container">$X_1, \ldots , X_n$</span>, <span class="math-container">$n \ge 4$</span> are independent random variables with exponential distribution: <span class="math-container">$f\left(x\right) = \mathrm{e}^{-x}, \ x\ge 0$</span>. We define <span class="math-container">$$R= \max \left( X_1, \ldots... | drhab | 75,923 | <p>You can go for calculating another integral:</p>
<p><span class="math-container">$$\begin{aligned}\mathbb{E}\max\left(X_{1},\dots,X_{n}\right) & =\int_{0}^{\infty}P\left(\max\left(X_{1},\dots,X_{n}\right)>x\right)dx\\
& =\int_{0}^{\infty}1-P\left(\max\left(X_{1},\dots,X_{n}\right)\leq x\right)dx\\
&... |
1,231,156 | <p>I am supposed to give a 30-minute class presentation on any PDE subjects as end-of-semester project. Do you have any pet subject you would love to suggest? </p>
<p>I have very little applied maths in my background, therefore I am not very advanced in PDE; I took this class as elective only. Hence I prefer a subject... | Mister Benjamin Dover | 196,215 | <p>I would like to suggest presenting the Malgrange-Ehrenpreis theorem and its applications, using e.g. Stein's functional analysis or Hörmander's PDE book.</p>
|
2,246,161 | <p>The parameter for the normal trigonometric functions represents the length of the opposite and adjacent sides of a triangle in a unit circle. The parameter is the angle of the triangle that is located at the radius. The vertex that touches the circle has the coordinates of $(\cos{\theta},\sin{\theta})$.</p>
<p>From... | zoli | 203,663 | <p>The meaning of the parameter $a$ in $\sinh(a)$ and in the case $\cosh(a)$ is the same as in the case of $\sin(a)$ and $\cos(a)$. Take a look at the figure below.
<a href="https://i.stack.imgur.com/BuTtc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BuTtc.png" alt="enter image description here"><... |
813,209 | <p>Can Someone help me solve this</p>
<p>$$
\int\frac{19\tan^{-1}x}{x^{2}}\,dx
$$</p>
<p>We have been told to use $\ln|u|$ and $C$.</p>
<p>Thanks!</p>
| Jeb | 136,806 | <p>Using integration by parts we see:</p>
<p>$$ \int \frac{ \arctan(x) }{x^2} dx = -\frac{\arctan(x) }{x} + \int \frac{dx}{ (x^2+1) x} $$</p>
<p>Now if we use a partial fraction decomposition we obtain:</p>
<p>$$\int \frac{dx}{ (x^2+1) x} = \int \left ( \frac{1}{x} - \frac{x}{x^2 +1} \right ) dx $$</p>
<p>From her... |
2,623,519 | <p>Problem statement - $A \setminus B= A$ st $A \subset B$</p>
<p>I think this statement is wrong as by definition of difference of sets $A \setminus B$ should contain all the elements of set $A$ <strong>which are not</strong> in $B$.But if $A$ is a subset of $B$ then all the elements of set $A$ are in set $B$ by defa... | user | 505,767 | <p>Since $A\subset B$ we have that $$x\in A \implies x\in B$$</p>
<p>but $$\forall y\in A \setminus B\implies y\in A \quad\land \quad y\not\in B$$</p>
<p>than $A=\emptyset$</p>
|
540,649 | <p>Show that the Dirichlet problem</p>
<p>$$ \left\{ \begin{array}{l} u_{xx}+ u_{yy}=u^3 \ \text{in} \ x^2+y^2 \lt 1 \\ u=0 \ \text{on} \ x^2+y^2 = 1 \end{array} \right.$$</p>
<p>where $u=u(x,y)$, has only the trivial solution $u \equiv 0$</p>
<p>Thanks</p>
| Julián Aguirre | 4,791 | <p>It is not a theorem, it is part of the definition. The Riemann integral is defined for bounded functions on a bounded domain. If the function, the domain or both are unbounded, then the integral may exist as an improper integral.</p>
|
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Joel David Hamkins | 1,946 | <p>The situation isn't that any given theorem off the shelf might have a forcing proof, but rather that it is a fascinating situation when one can use forcing to prove a theorem purely about the ground model. For such proofs, one makes a conclusion about the set-theoretic universe $V$ by first going to another univers... |
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Jason Zesheng Chen | 147,031 | <p>One use of forcing as a tool to prove theorems that has not been mentioned in the answers is the method of generic ultrapowers, where we take an ideal <span class="math-container">$I$</span> on an uncountable regular cardinal <span class="math-container">$\kappa$</span> (in the sense of <span class="math-container">... |
1,969,748 | <p>I noticed that the most simple numerical approximation of a higher order-differential equation has the same form as the numerical approximation of a delayed first-order differential equation. This leads me to the following hypothesis:</p>
<blockquote>
<p><strong>Hypothesis:</strong> Delayed first-order differenti... | Chrystomath | 84,081 | <p>A delay can be written as a differential <span class="math-container">$e^Dx(t)=x(t+1)$</span>. So it might make sense to approximate <span class="math-container">$$\dot{x}(t+1)=f(x(t))$$</span> by cutting off the following series at some <span class="math-container">$n$</span>, <span class="math-container">$$e^DDx(t... |
75,355 | <p>A groupoid is a category in which all morphisms are invertible.(*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits both left and right (weak) adjoints. So you can localize (or <em>complete</em>) a category to a groupoid. If E denote... | Todd Trimble | 2,926 | <p>I think the answer is negative. For this I'm going to adapt a passage from Categories for the Working Mathematician (bottom paragraph of page 164 in the 2nd edition). The counterexample involves a skeleton of the category of countably infinite sets, which I'll denote by $C$; let $\mathbb{N}$ be the unique object. </... |
3,791,405 | <p>Find the last <span class="math-container">$2$</span> digits of <span class="math-container">$9^{100}$</span>.</p>
<hr />
<p>Well, I know that <span class="math-container">$9^{100}$</span> mod <span class="math-container">$4$</span> is <span class="math-container">$1$</span>,but I do not know how to find <span class... | Albus Dumbledore | 769,226 | <p>Hint:observe that <span class="math-container">$9^{100}={(10-1)}^{100}$</span></p>
<p>using binomial theorem this can be written in the form of <span class="math-container">$100k+1$</span></p>
|
4,174,111 | <p>In the following is Theorem 13.6 from Bruckner's Real Analysis which I don't understand some claims on it :</p>
<p>Question <em>in Blue:</em> <span class="math-container">$\mu (|f_j(x)| > \|f_j\|_∞)=0$</span> and <span class="math-container">$\mu (|f_k(x)| > \|f_k\|_∞)=0$</span>. But how that implies <span cla... | Danny Pak-Keung Chan | 374,270 | <p>An easier way to prove that the norm of a normed space is complete
is to invoke the following result:</p>
<p>Let <span class="math-container">$(X,||\cdot||)$</span> be a norm space. Then the norm is complete if
and only if for every sequence <span class="math-container">$(x_{n})$</span> in <span class="math-containe... |
928,047 | <p>In a round-robin tournament, each team plays every other team exactly once. Show that if no games end in ties, then no matter what the outcomes of the games, there will be some way to number the teams so that team 1 beat team 2, and team 2 beat team 3, and team 3 beat team 4,
and so on.</p>
<p>I have the base case ... | Adriano | 76,987 | <p>Consider a round-robin tournament with $n + 1$ teams. Arbitrarily pick a particular team, say team $x$. Then by the definition of a round-robin tournament with no ties, observe that we can partition the remaining $n$ teams into two sets $W$ and $L$ of size $a$ and $b$ respectively, where $W$ is the set of all winner... |
124,530 | <p>I want to have a function value of an expression where some variables are solutions to some set of equations, with some values of parameters. I had an idea to use pure functions for that. </p>
<p>However, since both the expression and the equations are lengthy I'd like to place them in a separate expressions define... | m_goldberg | 3,066 | <p>It can be done, but I suspect you will find it a bit more tricky than you expected. Here is one way to do it. Note that I don't introduce a variable named δ. It is easier to work with <a href="http://reference.wolfram.com/language/ref/Slot.html" rel="nofollow noreferrer"><code>Slot</code></a> ( # ).</p>
<pre><code>... |
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| hjpotter92 | 27,741 | <h2>Hint</h2>
<p>Take the sphere to be formed by the rotation of semicircle(<strong>why?</strong>) about x-axis $ x = r \cos \theta $, $ y = r \sin \theta $ where $ \theta \in [0, \pi] $.</p>
|
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| Christian Blatter | 1,303 | <p>I won't go into a proof that the surface of the sphere $S^2_r$ of radius $r$ is $4\pi r^2$.</p>
<p>Your last question asks for the area of a unit square on $S^2_r$. The notion "unit square" can mean two things: A "square" $Q\subset S^2_r$ of unit area, or a "square" $Q\subset S^2_r$ of unit side length. I guess yo... |
2,343,026 | <p>Let $a$ and $b$ be greater than $0$ and let</p>
<p>$$I=\int_0^{\infty}\frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$$</p>
<p>I know I can first manipulate by adding and subtracting $1$ in the numerator. Thus</p>
<p>$$\int_0^{\infty}\frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx=\int_0^{\infty}\frac{1+e^{ax}-1-e^{bx... | Jack D'Aurizio | 44,121 | <p>Your approach is correct but it can shortened. Since for any $c>0$
$$J(c)=\int_{0}^{+\infty}\frac{dx}{1+e^{cx}}\stackrel{x\mapsto\frac{z}{c}}{=} \frac{1}{c}\int_{0}^{+\infty}\frac{dz}{1+e^{z}}\stackrel{z\mapsto-\log t}{=}\frac{1}{c}\int_{0}^{1}\frac{dt}{1+t}=\frac{\log 2}{c}$$
by partial fraction decomposition ou... |
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | Cesareo | 397,348 | <p>Making <span class="math-container">$y = \lambda x$</span> and substituting we have</p>
<p><span class="math-container">$$
\mbox{Variation for}\ \ \ x^2(3-2\lambda)\ \ \ \mbox{s. t. }\ \ x^2(1+\lambda^2) = 1
$$</span></p>
<p>or variation for</p>
<p><span class="math-container">$$
f(\lambda) = \frac{3-2\lambda}{1... |
2,917,858 | <p>Consider the vector space of all functions $f: \mathbb{R} \rightarrow \mathbb{C}$ over $\mathbb{C}$. If $W$ is a subspace spanned by $\beta$ = $\{1, e^{ix}, e^{-ix}\}$, show that $\beta$ is a basis for $W$.</p>
<p>I think I am very confused - I know I just have to show that $\beta$ is linearly independent, which me... | Sellerie | 574,887 | <p>While most of the answers given here are true, they don't really go into enough detail on what you understood wrong. When talking about the vectorspace of all functions, we need to consider any function $f\colon \mathbb{R}\to\mathbb{C}$ to be a vector, i.e. the given $\beta$ would be denoted much more appropriately ... |
27,307 | <p>Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between them. Now suppose we have $S\subset\mathbb{N}$. If $\lim_{n\rightarrow\infty} \frac{|S\cap A_n|}{|A_n|}=0$, does i... | Brian | 4,266 | <p>You could simplify it some this way:
$$\left \langle \sum_{n=1}^\infty a_{n}b_{n},\sum_{m=1}^\infty a_{m}b_{m}\right \rangle=\lim_{N \to \infty} \lim_{M\to \infty} \left \langle \sum_{n=1}^N a_{n}b_{n},\sum_{m=1}^M a_{m}b_{m}\right \rangle = \lim_{N \to \infty} \lim_{M\to \infty} \sum_{n=1}^N \sum_{m=1}^M \left \la... |
710,040 | <p>Hello I have been blasting at this inequality proof and it is just not doing what I want it to do:</p>
<blockquote>
<p>Prove that $2^{n-1}(a_1a_2\ldots a_n + 1) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ assuming that $a_1,a_2,\dots, a_n \geq 1$.$\:\:\:$</p>
</blockquote>
<p>So the base case is pretty trivial, which lead... | TTY | 19,412 | <p>The inequality is equivalent to:
$$
\left(\frac{1+a_1}{2}\right)\cdots \left(\frac{1+a_n}{2}\right) \le \left(\frac{1+a_1\cdots a_n}{2}\right)
$$
Assuming the induction hypothesis we want to show:
$$
\left(\frac{1+a_1}{2}\right)\cdots \left(\frac{1+a_{n+1}}{2}\right) \le \left(\frac{1+a_1\cdots a_{n+1}}{2}\right)
$... |
974,270 | <p>Here's a homework problem I'm having some trouble with:</p>
<blockquote>
<p>Show that $$ \int_{|z|=3} \frac{1}{z^2-1} dz = 0$$</p>
</blockquote>
<p>So far, I've shown using Cauchy's Integral Formula that $$ \int_{|z-1|=1} \frac{1}{z^2-1} dz = \pi i$$ and $$\int_{|z+1|=1} \frac{1}{z^2-1} dz = - \pi i$$ where $|z-... | DanielV | 97,045 | <p>Yes, split the path $p$ of $|z| = 3$ into two semicircles as you said to create left semicircle $p_1$ and right semicircle $p_2$. Since </p>
<p>$$\int_{-3i}^{3i} f(z)~dz + \int_{3i}^{-3i} f(z) ~ dz = 0$$</p>
<p>you will get that (in the case that $p_1$ and $p_2$ are both defined to be counter clockwise paths):</... |
3,435,209 | <p><a href="https://i.stack.imgur.com/47sW7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/47sW7.png" alt="enter image description here"></a></p>
<p>When I attempt to compute <span class="math-container">$f_{y}(0,0)$</span>, I first set <span class="math-container">$x = 0$</span> such that <span cl... | Hans Lundmark | 1,242 | <p>The function
<span class="math-container">$$
g(y)=
f(0,y) = \begin{cases}
1, & y\neq 0,\\
0, & y=0
\end{cases}
$$</span>
isn't continuous at <span class="math-container">$y=0$</span>, hence not differentiable there either, so <span class="math-container">$g'(0)=f_y(0,0)$</span> doesn't exist.</p>
|
4,344,014 | <p>Please, give advise or reference for solving first order ODE: <span class="math-container">$a(y)y^\prime + y = b(t)$</span>, where <span class="math-container">$a$</span>, <span class="math-container">$b$</span> are known function. It would be better to find just one solution.</p>
| Robert Israel | 8,508 | <p>It is very unlikely that there is a general method for finding closed-form solutions. Even in a simple case like</p>
<p><span class="math-container">$$ y^2 y' + y = t $$</span>
Maple finds no closed-form solution.</p>
|
220,335 | <p>Let's say I have a weakly connected graph like:</p>
<pre><code>vertices = Range[8];
g = Graph[vertices, { 2 -> 1, 3 -> 2, 4 -> 3, 5 ->6 , 7 -> 6, 6-> 2, 2 -> 8}, VertexLabels -> "Name"]
</code></pre>
<p><a href="https://i.stack.imgur.com/4h239.png" rel="nofollow noreferrer"><img sr... | David G. Stork | 9,735 | <p><strong>A start</strong>:</p>
<p>Here is how you color the edges:</p>
<pre><code>Graph[Range[8],
{2 -> 1, 3 -> 2, 4 -> 3, 5 -> 6, 7 -> 6, 6 -> 2, 2 -> 8},
EdgeStyle -> {
(2 -> 1) -> Red,
(3 -> 2) -> Blue,
(4 -> 3) -> Green,
(5 -> 6) -> Orange,
(7... |
220,335 | <p>Let's say I have a weakly connected graph like:</p>
<pre><code>vertices = Range[8];
g = Graph[vertices, { 2 -> 1, 3 -> 2, 4 -> 3, 5 ->6 , 7 -> 6, 6-> 2, 2 -> 8}, VertexLabels -> "Name"]
</code></pre>
<p><a href="https://i.stack.imgur.com/4h239.png" rel="nofollow noreferrer"><img sr... | Shb | 8,988 | <p>Here is my attempt at the solution: </p>
<pre><code>Destinations[g_? GraphQ]:= Flatten[Position[AdjacencyMatrix[g] //Normal, ConstantArray[0,Length[VertexList[g]]]]];
Origins[g_? GraphQ]:= Flatten[Position[AdjacencyMatrix[g] // Transpose //Normal, ConstantArray[0,Length[VertexList[g]]]]];
FindCausalPath[graph_?Gra... |
4,928 | <p>Consider the following list of countries which I would like to highlight on a world map:</p>
<pre><code>MyCountries={"Germany","Hungary","Mexico","Austria","Bosnia","Turkey","SouthKorea","China"};
</code></pre>
<p>From the documentation center <a href="http://reference.wolfram.com/mathematica/ref/CountryData.html"... | C. E. | 731 | <p><em>Mathematica 10</em> introduced new ways to highlight countries on the world map. Examples follow below. There are lots of more examples in the documentation which showcase alternative stylings.</p>
<pre><code>myCountries = Map[
Entity["Country", #] &,
{"Germany", "Hungary", "Mexico", "Austria", "Turke... |
1,191,237 | <p>I have the list $l = (6, 6, 5, 4)$ and want to how to calculate the possible number of permutations.</p>
<p>By using <em>brute force</em> I know that there are <strong>12</strong> possible permutations:</p>
<p>$$\{(6, 5, 6, 4),
(6, 6, 5, 4),
(5, 6, 6, 4),
(6, 4, 5, 6),
(6, 5, 4, 6),
(4, 6, 6, 5),
(4, 5, 6, 6... | Reveillark | 122,262 | <p>For a list of $n$ distinct elements the number of permutations if $n!$. If you have $4$ distinct elements, the number of permutations is $24$.</p>
<p>However in this case we have the number $6$ appearing twice. For every ordering of the four numbers $6,6',5,4$ there is another ordering switching the $6$ and the $6'... |
679,807 | <p>I am trying to find the value of $2^{-1-i}$. </p>
<p>I rewrite it like this, $2^{-1-i}=e^{\ln(2)(-1-i)}={1\over{e^{\ln2}e^{i\ln2}}}=1/2$</p>
<p>Since $e^{i\ln2}=e^{Re(i\ln2)}=e^0=1$. </p>
<p>This looks way nicer than it should be, I think, can anyone tell me where I go wrong, and maybe a good way to do this probl... | MPW | 113,214 | <p>This is a multi-valued function. For each $k\in\mathbb{Z}$, you can write
$$2^{-1-i} = e^{(-1-i)(\ln 2 + 2\pi ki)} = e^{-\ln 2 + 2\pi k + i(-\ln2 - 2\pi k)}
= \boxed{\frac12 e^{2\pi k}\cos\ln 2 - i\frac12 e^{2\pi k}\sin\ln 2}$$</p>
|
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