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 |
|---|---|---|---|---|
41,123 | <p>What is a reference for the (classical and well-known) proof of Weyl's lemma that states: </p>
<blockquote>
<p>Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U \hspace{1mm}{f\phi_\bar{z}}=0\;\;\;\forall \phi \in C_c^{\infty}(U) $, then $f$ is a.e. equal to a holomorphic function.</p... | allizdog | 381,913 | <p>There's a proof in the book Riemann surfaces by H.M. Farkas and I. Kra, I believe they prove the weak version.</p>
|
73,928 | <p>Suppose I wanted to use Mathematica graphics primitives to create a gradient of colors between two circular arcs. It's easy enough to make an area between two circles a solid color, but what if I wanted to have the area between two colors blur from black to white as you go out in radius?</p>
<p>VertexColor provides... | Szabolcs | 12 | <p>It is not possible to do this <em>directly</em> circular arcs. <code>VertexColor</code> works for triangles (any other polygon will be broken up into triangles before the colours are applied). If you need a gradient fill for an arbitrary shape, it needs to be approximated using a set of triangles first, then each ... |
305,434 | <p>If $f(\frac{x}{y})=\frac{f(x)}{f(y)} \, , f(y),y \neq 0$ and $f'(1)=2$ then $f(x)=$?</p>
<p>I am not sure where to begin, any hints on starting and steps is apreciated.</p>
<p>Thank you</p>
| Ritana | 62,214 | <p>Linear functions satisfied this function-equation trivially.</p>
<p>Let $f(x)=2x$ then $f'(x)=2$ for all $x$ and $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$.</p>
<p>That's all :-)</p>
<p>R</p>
|
2,589,143 | <p>Let $(a_n)_n$ be a Cauchy sequence in $\mathbb C$. I want to show</p>
<blockquote>
<p>$(a_n)_n$ is Cauchy $\iff$ the real part $(\Re(a_n))_n$ and the imaginary part $(\Im(a_n))_n$ are both Cauchy</p>
</blockquote>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>As $ah+bk+c=0, k=-\dfrac{ah+c}b$</p>
<p>So, $P\left(h,-\dfrac{ah+c}b\right)$ will represent any point on $ax+by+c=0$</p>
<p>Now if $Q(p,q)$ is the reflection of $P,$</p>
<p>the midpoint $\left(\dfrac{p+h}2,\dfrac{bq-ah-c}{2b}\right)$ will lie on $$x+y+1=0\ \ \ \ (1)$$</p>
<p>Again $PQ$ will be perp... |
2,249,320 | <p>We have to find </p>
<blockquote>
<p>$$g(x)=\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$$</p>
</blockquote>
<p>I could not get any good idea .</p>
<p>Intialy I thought of using </p>
<p>$$\cos a+\cos b=2\cos(a+b)/2\cos (a-b)/2$$</p>
| The Dead Legend | 433,379 | <p>Let $z=\cos\theta+i\sin\theta$ i.e. $z=e^{i\theta}$</p>
<p>Your sum:$$e^{i\theta}+e^{3i\theta}+e^{5i\theta}+...e^{(2n-1)i\theta}$$</p>
<p>This is a GP with common ratio $e^{2i\theta}$</p>
<p>Therefore sum is $$\frac{a(r^n-1)}{r-1}$$
$$\frac{e^{i\theta}(e^{2ni\theta}-1)}{e^{2i\theta}-1}$$
$$\frac{(\cos \theta+i\si... |
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| Amzoti | 38,839 | <p>You might want to have a look at the following (peruse them at your favorite online book store).</p>
<p><strong>Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry</strong>, George F. Simmons</p>
<p><strong>Pre-calculus Demystified 2/E</strong>, Rhonda Huettenmueller</p>
<p>Some other food for ... |
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| saru44 | 290,028 | <p>There is a FREE AND OPEN SOURCE book titled 'Precalculus' by Carl Stitz and Jeff Zeage<em>r</em>. Its quite detailed at close to 1000 pages. It is an ebook with no official hard copy published. Quality definitely is NOT lacking just because its free and open source. Its up there with the best in the market. Hope yo... |
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| guest | 472,303 | <p>Frank Ayres, First Year College Mathematics, Shaum's Outline</p>
<p>Based on how you describe yourself, think this is very efficient thing for you to study from. In general Shaum's gets good response from students. And for someone like you who is well into other courses but needs to go back and remediate a gap, i... |
482,290 | <p>I have a tide guide that gives me four readings for the day - 2 high tides and two low tides. This means it completes two full revolutions within a day. What I'm having trouble with is taking the four measurments and making a graph and equation of the entire function for that specific day. </p>
<p>One example is...... | Andrew | 85,681 | <p>It is not possible to solve this system of equations with a function of the form given. When plotting the points, observe that any sine or cosine function that has a minimum of 11.8 cannot possibly reach a value of 1. </p>
<p>It is possible however, to write a linear combination of such functions up to a constant t... |
465,631 | <blockquote>
<p>Is it possible to embed $\Bbb{C}(x)$ (the field of rational functions over the complex numbers) in $\Bbb{C}$ ?</p>
</blockquote>
<p>Thank you!</p>
| BBBB | 36,200 | <p>Yes, it is possible; although the solution relies heavily on the axiom of choice, and is not constructive or geometric in any way. If you believe the axiom, then every pair of algebraically closed fields of the same transcendence degree over the base field and of the same characteristic are isomorphic - simply pick ... |
40,998 | <p>I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$</p>
<p>This is a problem in Calculus by Simmons. It's in the end of chapter review and it's associated with the section about the alternating series test. There's a hint: refer to an equation from a previ... | Ajat Adriansyah | 10,001 | <p>For more literate solution:
Consider $$\kappa(s)=\sum_{n=2}^{\infty} \frac{(-1)^{n+1}\ln n}{n^s}$$</p>
<p>when $s=1$ we can use abel's theorem to verified the convergence since the above series is absolutely convergent for $s\geq 1$ Now suppose that $s>1$. Define $$\psi(s)=\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}... |
2,025,246 | <p>I know for a fact that it's a trig substitution where u=tanØ (let's pretend Ø is theta) and u=x and a=1.</p>
<p>For some reason, I keep going in circles. </p>
<p><a href="https://i.stack.imgur.com/BVYWB.png" rel="nofollow noreferrer">This is the integral that I need to solve</a></p>
| Dave | 334,366 | <p>Substitute $x=\tan (\theta)$ so $dx=\sec^2 (\theta)d\theta$. Use the identity $\tan^2 (\theta)+1=\sec^2 (\theta)=\frac {1}{\cos^2 (\theta)} $.</p>
|
2,910,243 | <p>How to prove that $\sin(\sqrt{x})$ is not periodic?
THe definition of a periodic function is $f(x+P)=f(x)$. </p>
<p>So I assume that $\sin(\sqrt{x+P})=\sin(\sqrt{x})$. This is equivalent to $\sin(\sqrt{x+P})-\sin(\sqrt{x})=0$. This implies $2cos(\frac{\sqrt{x+P}+\sqrt{x}}{2})\sin(\frac{\sqrt{x+P}-\sqrt{x}}{2})$. Wh... | Ixion | 409,792 | <p>Note that the domain $D$ of a $P$-periodic function $f$ must be "invariant by translations of $P$", i.e.: $D+P=D$. In this case $\forall P>0, \ D+P$ is a proprer subset of $D$ hence $f(x)$ can not be a periodic function.</p>
|
445,651 | <p><a href="https://math.stackexchange.com/questions/327995/problem-with-infinite-product-using-iterating-of-a-function-expx-x-cdot">In a previous question</a> I arrived at the family of functions (depending on a real parameter <em>b</em> for the base of exponentiation/logarithm):
$$ f(x) = x - \log_b(x) \qquad \qquad ... | Sheldon L | 43,626 | <p>One important point in the system is $b=\sqrt{e}$, as was pointed out by Gottfried. For $b>\sqrt{e}$, there is a stable attracting fixed point for $f(x)=x-\log_b(x)$, and that fixed point is x=1. For $1.518120456732599974768513856<b<\sqrt{e}$, the fixed point in the neighborhood of one is repelling, and i... |
445,651 | <p><a href="https://math.stackexchange.com/questions/327995/problem-with-infinite-product-using-iterating-of-a-function-expx-x-cdot">In a previous question</a> I arrived at the family of functions (depending on a real parameter <em>b</em> for the base of exponentiation/logarithm):
$$ f(x) = x - \log_b(x) \qquad \qquad ... | Gottfried Helms | 1,714 | <p>This is not an answer, only additional data for clarification.
I kept the base constant (here I used: $b = \exp(0.3)$) and was able to find cycles of lengthes <em>2</em> to <em>16</em> with the help of Newton-Raphson. The resp. start-values simply follow one from the other, I started initially with $x_0=0.3$ for cy... |
4,188,257 | <p>I am looking to solve:</p>
<p>Show that <span class="math-container">$$\sum_{n \ge 0} \sum_{k \ge 0} {n\choose k} {2k \choose k} y^k x^n = \frac{1}{\sqrt{(1-x)(1-x(1+4y))}}$$</span></p>
<p>and then use that to show that</p>
<p><span class="math-container">$$\sum_{k \ge 0} {n \choose k} {2k \choose k} (-2)^{-k} = \be... | grand_chat | 215,011 | <p>For the first part: To prove the identity
<span class="math-container">$$\sum_{n\ge0}\sum_{k\ge0}{n\choose k}{2k\choose k}y^kx^n=\frac1{\sqrt{(1-x)(1-x(1+4y))}}\tag1
$$</span>
you should try swapping the order of summation. After swapping, the inner sum over <span class="math-container">$n$</span> will be
<span clas... |
3,753,563 | <p>In the proof of the Stone-Weierstrass theorem (7.26), Rudin claims <span class="math-container">$Q_n \to 0$</span> uniformly. Can someone explain why this is the case? I don't see how that immediately follows from the bound.</p>
<p><a href="https://i.stack.imgur.com/vTCvf.png" rel="nofollow noreferrer"><img src="ht... | Greg Nisbet | 128,599 | <p>Here's one way to do it.</p>
<p>First, a word on notation.</p>
<p>Let <span class="math-container">$\mathbb{N}$</span> refer to the positive integers.</p>
<p>Let <span class="math-container">$\mathbb{N}_>$</span> refer to the upward closed sets of <span class="math-container">$\mathbb{N}$</span>.</p>
<p><span cla... |
2,618,524 | <p>I managed to prove this using a direct proof but my prof suggested I try proving it using the contrapositive. Here's what I have so far:</p>
<p>Contrapositive: $(x \le 0) \lor (x \ge 1) \Rightarrow x^4 + 2x^2 - 2x \ge 0$</p>
<p>Splitting this into two, ($P_1 \Rightarrow Q)\land(P_2 \Rightarrow Q)$:</p>
<p>$$x \ge... | Ted Shifrin | 71,348 | <p>So, as you go around the little rectangle, the flux is (approximately) the sum of the $\mathbf V\cdot\mathbf n \Delta s$ contributions. Starting with the bottom edge and proceeding counterclockwise, we have
\begin{align*}
\text{FLUX} &\approx\mathbf V\cdot (-\mathbf j) \Delta x + \mathbf V\cdot\mathbf i \Delta y... |
1,224,268 | <p>I was given this problem (I have to prove) and not sure if I use fermat's theorem
$24^{31} ≡ 23^{32} (mod 19)$</p>
<p>If I do use fermat's is this right:</p>
<p>I would do the LHS first:</p>
<p>$24^{18}·24^{13} ≡ 1·24·24^{12}$</p>
<p>RHS:</p>
<p>$23^{18}·23^{14} ≡ 1·23^{14}$</p>
<p>I am not sure where to from... | Rolf Hoyer | 228,612 | <p>So far, so good.
A couple of simplifications might help make further progress.</p>
<ul>
<li>Rewrite 23 and 24 as 4 and 5 mod 19, respectively.</li>
<li>Now you can rewrite $4 = 2^2$, so that $4^{14} = 2^{28} = 2^{10} \mod 19$.</li>
<li>A similar simplification might be $5^{12} = 25^6 = 6^6 \mod 19$.</li>
</ul>
<p>... |
1,224,268 | <p>I was given this problem (I have to prove) and not sure if I use fermat's theorem
$24^{31} ≡ 23^{32} (mod 19)$</p>
<p>If I do use fermat's is this right:</p>
<p>I would do the LHS first:</p>
<p>$24^{18}·24^{13} ≡ 1·24·24^{12}$</p>
<p>RHS:</p>
<p>$23^{18}·23^{14} ≡ 1·23^{14}$</p>
<p>I am not sure where to from... | Bernard | 202,857 | <p>Begin with reducing $24$ and $23$ mod. 19. Little Fermat asserts that non zero elements have order a divisor of $18$, i.e., it can be $1, 2, 3, 6, 9$ or $18$. </p>
<p>Let's do the computations:
$$24^{31}=5^{31\bmod 18}=5^{13}, \qquad 23^{32}=4^{32}=2^{64\bmod 18}=2^{10}.$$</p>
<p>Now, modulo $19$:
$$5^{2}\equiv ... |
138,050 | <p>How is <code>a~b</code> parsed and evaluated? I would expect an error, but it goes through without any complaints, yielding no apparent output. At first I thought it evaluated to <code>Null</code>, but then I realized every expression I've tried containing <code>a~b</code> (either directly or through something lik... | Carl Woll | 45,431 | <p>I only have a partial answer. The input <code>a~b</code> is not syntactically correct, and so the input is not evaluated. One way to see that it is not syntactically correct is to use:</p>
<pre><code>ToExpression[RowBox[{"a","~","b"}],StandardForm]
</code></pre>
<blockquote>
<p>ToExpression::esntx: Could not par... |
910,870 | <p>Can somebody explain me how to calculate this integral?</p>
<p>$$\int \frac{\left(x^2+4x\right)}{\sqrt{x^2+2x+2}}dx$$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $\displaystyle\frac{d(x^2+2x+2)}{dx}=2x+2,$</p>
<p>write $$x^2+4x=(x^2+2x+2)+(2x+2)-4$$</p>
<p>Use $\#1,\#8$ of <a href="http://www.sosmath.com/tables/integral/integ11/integ11.html" rel="nofollow">this</a></p>
|
910,870 | <p>Can somebody explain me how to calculate this integral?</p>
<p>$$\int \frac{\left(x^2+4x\right)}{\sqrt{x^2+2x+2}}dx$$</p>
| user84413 | 84,413 | <p>Let $x+1=\tan\theta$, so $x=\tan\theta-1$, $dx=\sec^{2}\theta d\theta$, and $\sqrt{x^2+2x+2}=\sqrt{(x+1)^2+1}=\sec\theta$.</p>
<p>Then $\displaystyle\int\frac{x^2+4x}{\sqrt{x^2+2x+2}}dx=\int\frac{(\tan\theta-1)^2+4(\tan\theta-1)}{\sec\theta}\sec^{2}\theta d\theta$</p>
<p>$\displaystyle=\int(\tan^{2}\theta+2\tan\th... |
49,336 | <p>Fix a hyperkähler manifold $X$ and an identification of $S^2$ with the hyperkähler sphere of $X$. Now consider the twistor space $T := S^2\times X$ equipped with the tautological complex structure. For each $x\in X$, we have a holomorphic map $u_x:S^2\to T$ defined by $u_x(\theta):=(\theta,x)$.</p>
<p><strong>Que... | Sebastian | 4,572 | <p>As explained by Sascha, the answer to this question is no. But there is a reformulation of the question as follows: The twistor space has a natural anti-holomorphic involution given by $\rho(I,x)=(-I,x)$ for $(I,x)\in S^2\times X.$ Then you can ask whether all holomorphic sections which are real, i.e. $s(-I)=\rho(s(... |
4,093,732 | <p>The equation of a curve is <span class="math-container">$$ y=8\sqrt x -2x $$</span>
We have to find the values of <span class="math-container">$x$</span> at which the line <span class="math-container">$y = 6$</span> meets the curve</p>
<p>I tried equating them and doing using the quadratic formula like this:
<span c... | Shinrin-Yoku | 789,929 | <p><span class="math-container">$(a-b)^2=a^2+b^2-2ab$</span> not <span class="math-container">$a^2+b^2$</span> as you have done.</p>
|
10,047 | <p>Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map? </p>
| Robin Chapman | 226 | <p>This raises another question. If the restriction maps from $S(X)$ to $S(U)$
are surjective for all open sets $U$ (a sheaf with this
property is called <em>flasque</em> or <em>flabby</em>) then certainly the maps
from $S(X)$ to each stalk $S_x$ are surjective. So are there sheaves
$S$ with the property that each map... |
47,636 | <h3>Background</h3>
<p>Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot when one can be made into the other by a sequence of <strong><a href="http://en.wikipedia.org/wiki/Reidermei... | Ryan Budney | 1,465 | <p>In some strict sense I think the answer to your question is <em>no</em>, there are likely no finite collection of 2-cells doing what you want. If you were to ask the more natural question where you're looking for a 2-complex whose inclusion into $Emb(S^1,\mathbb R^3)$ is an isomorphism on $\pi_1$ (component-by-comp... |
3,096,526 | <p>The book "basic category theory" states that in the category <strong>Mon</strong>, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?</p>
| vadim123 | 73,324 | <p>Quoting <a href="https://en.wikipedia.org/wiki/Epimorphism" rel="noreferrer">Wikipedia</a>, we see:</p>
<p>In the category of monoids, Mon, the inclusion map <span class="math-container">$\mathbb{N}\to\mathbb{Z}$</span> is a non-surjective epimorphism. To see this, suppose that <span class="math-container">$g_1$</s... |
157,420 | <p>Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent $F:\mathbb{S}^{2}\to \mathbb{S}^{2}$ in the form $F=(f,g,h)$. then as a consequence of the Gauss Bonnet theorem we have \be... | Liviu Nicolaescu | 20,302 | <p>$\newcommand{\bR}{\mathbb{R}}$ If $\Sigma\subset \bR^3$ is a cooriented surface then its Gauss map $\Gamma:\Sigma\to S^2$ has a symplectic nature. Its graph, viewed as a submanifold of $\bR^3\times S^2$ is a Legendrian submanifold with respect to the canonical contact structure on $\bR^3\times S^2$. </p>
<p>T... |
2,103,984 | <p>Imagine we are given three intersecting circles centered on the vertices of an equilateral triangle of side length $1$, with $(0,0)$ arbitrarily placed at the bottom left corner. The circles have radii $r$, $r+a$, and $r+b$ respectively (going clockwise again arbitrarily), where $r$ is unknown but $a$ and $b$ are g... | Carl Schildkraut | 253,966 | <p>Let the coordinates of the triangle be $(0,0),(1,0),\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. Then, we can write three equations of the form</p>
<p>$$(x-x_0)^2+(y-y_0)^2=r_0^2$$</p>
<p>for various $x_0,y_0,r_0$. What happens if you assume a solution to each of these concurrently and then solve for $r$?</p>
|
2,103,984 | <p>Imagine we are given three intersecting circles centered on the vertices of an equilateral triangle of side length $1$, with $(0,0)$ arbitrarily placed at the bottom left corner. The circles have radii $r$, $r+a$, and $r+b$ respectively (going clockwise again arbitrarily), where $r$ is unknown but $a$ and $b$ are g... | David K | 139,123 | <p>Consider a general point $(x,y)$ inside the triangle,
and let the distance of $(x,y)$ from the vertices $(0,0),$
$\left(\frac12,\frac{\sqrt3}2\right),$ and $(1,0)$ be
$r_1$, $r_2,$ and $r_3$ respectively.
(That is, each of $r_1$, $r_2,$ and $r_3$ is a function of $(x,y)$.)</p>
<p>We want to find a point $(x,y)$ suc... |
2,103,984 | <p>Imagine we are given three intersecting circles centered on the vertices of an equilateral triangle of side length $1$, with $(0,0)$ arbitrarily placed at the bottom left corner. The circles have radii $r$, $r+a$, and $r+b$ respectively (going clockwise again arbitrarily), where $r$ is unknown but $a$ and $b$ are g... | rays | 408,255 | <p>First, note that (x,y) is determined by the intersection of the circles with radii r and r+b, Given the "safe" values assumed, so we should be able to express both x and y in terms of r and b only. </p>
<p>Safe values values are those such that:<br>
* 2r + a > 1<br>
* 2r + b > 1<br>
* r + a < 1<br>
* r +... |
1,837,924 | <p>The primality test of Fermat with base $2$ seems to be as secure as the computer hardware for testing numbers <em>big enough</em>. However, I think there are an infinite numbers of false primes using this method, while there are other, slower methods without known exceptions.</p>
<blockquote>
<p>My question is, g... | DanaJ | 117,584 | <p>For many people, using a good method such as BPSW (not a single Fermat test), is good enough. More importantly, the code for such a test is much simpler than good proof software, so in many cases it has more trust.</p>
<p>A few example times with your number to give some idea. Your times will vary depending on so... |
4,096,806 | <p>I'm given by symmetry matrix <span class="math-container">$A$</span> and <span class="math-container">$3$</span> eigenvalues of it: <span class="math-container">$1,2,3$</span>. And I know two of it's eigenvectors: <span class="math-container">$(3,2,1)$</span> and <span class="math-container">$(2,-1,-4)$</span>, whic... | Community | -1 | <p>You could take the cross-product: <span class="math-container">$(3,2,1)\times(2,-1,-4)=\begin{vmatrix}i&j&k\\3&2&1\\2&-1&-4\end{vmatrix}=-7i+14j-7k$</span>. This <em>must</em> be an eigenvector, since it is the only orthogonal direction left.</p>
|
1,426,155 | <p>The question is :</p>
<p>$$\lim_{x\rightarrow 1} \frac{\sqrt{x+3}-2}{x-1}$$</p>
<p>I know I probably have to do some sort of factorisation of the numerator in order to cancel the denominator, but the surd has me stumped I'm afraid.</p>
| MathAdam | 266,049 | <p>You end up with $\frac{0}{0}$ if you simply substitute $1$ for $x$. So, I would try L'Hôpital's rule. Take the derivative of the numerator and divide by the derivative of the denominator. See what happens after that. </p>
<p>$$\lim_{x\rightarrow 1} \frac{\sqrt{x+3}-2}{x-1}=\lim_{x\rightarrow 1} \frac{\frac{d}{dx}\... |
4,415,279 | <p>This question is from assignment 1 of my algebraic geometry course. For theory, I have been following my class notes.</p>
<blockquote>
<p>Question: Let K be an arbitrary field, S be the set of all polynomials in <span class="math-container">$K[X_1, ..., X_n]$</span> which have no zeroes in <span class="math-containe... | drhab | 75,923 | <p>In this situation:</p>
<p><span class="math-container">$$Y:=X_{(n)}-X_{(1)}
\text{ has Beta-distribution with parameters }n-1\text{ and }2$$</span></p>
<p><span class="math-container">$$\mathsf{Cov}\left(X_{(n)},X_{(1)}\right)=\frac1{(n+1)^2(n+2)}$$</span></p>
<p><span class="math-container">$$\mathsf{Var}\left(X_{(... |
455,060 | <p>I always see questions on here that deal with this modular stuff, and I have no idea what any of it means, so I figured I would ask here.</p>
<p>So lets say we have
$$a \equiv b\pmod n$$
The example on wiki is $$38\equiv 14\mod 12$$
This is because 38-14 = 24, which has a factor of 12. Why is it 12 instead of 24,... | AJMansfield | 50,951 | <p>If you do programming, you should be familiar with the usage of <code>%</code> as an operator that (depending on your language) is either the <em>remainder after division</em> or <em>modulus</em> operator (they are slightly different in how they handle negative numbers, but they are essentially the same idea). If y... |
19,119 | <p>I need to estimate $\pi$ using the following integration:</p>
<p>$$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$</p>
<p>using monte carlo </p>
<p>Any help would be greatly appreciated, please note that I'm a student trying to learn this stuff so if you can please please be more indulging and try to explain in depth..</p>
| Narcis | 6,284 | <p>C# implementation. Thanks to all who contributed!! Ross, Esteban, Shai</p>
<pre><code>class Program
{
static void Main(string[] args)
{
double x = 0;
var rd = new Random();
double sum = 0;
for (int i = 0; i < 1000000; i++) // the higher the more precise
{
... |
1,524,879 | <p>Suppose that we have some function $f: \mathbb R \to \mathbb R$ such that $f$ is integrable (Riemann or Lebesgue, choose one, or some other maybe more general type of integration, if there is such) on some inteval $(a,b)$. </p>
<p>Now suppose that there exist point $x_0 \in (a,b)$ and sequence $\varepsilon_n$ such ... | Hosein Rahnama | 267,844 | <p>Simple short answers. The following two are logically equivalent</p>
<p>$$(a \implies b) \iff (\neg b \implies \neg a)$$ </p>
<p>and also the negation of an implication is <strong>not</strong> an implication</p>
<p>$$\neg (a \implies b) \iff (a \wedge \neg b)$$</p>
<p>If you want to prove $(a \implies b)$ is t... |
39,973 | <p>I want to numerically solve the heat equation. To do so I wrote a program in C++ - It is a BVP $(\alpha T +\kappa \frac{\partial T}{\partial r}=f(t))$, where $f(t)$ is an arbitrary function, whose address I pass to the constructor of the struct <code>BoundaryCondition</code></p>
<pre class="lang-c prettyprint-overr... | Szabolcs | 12 | <p>Since Mathematica 10.0, you can use <a href="http://reference.wolfram.com/language/LibraryLink/tutorial/InteractionWithMathematica.html#894022691" rel="nofollow noreferrer">library callback functions</a> in LibraryLink, which are designed precisely for this purpose.</p>
|
3,340,941 | <blockquote>
<p>Find The image(or reflection) of the point <span class="math-container">$(4,-13)$</span> in the line <span class="math-container">$5x+y+6=0$</span></p>
</blockquote>
<p><strong>Method 1</strong>
<span class="math-container">$$
y+13=\frac{1}{5}(x-4)\implies x-5y-69=0\quad\&\quad 5x+y+6=0\implies (... | Toby Mak | 285,313 | <p>As ganeshie8 suggested, your matrix formula is not working because the line does not pass through the origin.</p>
<p>When you translate everything up by <span class="math-container">$6$</span> units, the line now passes through the origin and you can continue as follows:</p>
<p><span class="math-container">$$\begi... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Robert Israel | 13,650 | <p><a href="https://en.wikipedia.org/wiki/Positive-definite_matrix" rel="noreferrer">Positive definite matrix</a> (and related terms). Most authors require these to be hermitian (or symmetric in the real case), but not all.</p>
<p>EDIT: also, "positive" can be ambiguous even for numbers: most authors, especially in E... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Aryeh Kontorovich | 12,518 | <p>I would say the word "kernel" is probably among the most overloaded terms in mathematics. You've got kernels of linear operators, convolutional kernels, distribution kernels, Markov kernels, and Reproducing Hilbert Space kernels. All of these notions are related but are, strictly speaking, distinct objects. Thus, th... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Joel David Hamkins | 1,946 | <p>A <em>tree</em> can be a very different thing in different parts of mathematics. It might be a certain kind of acyclic graph; or a partial order such that the predecessors of every node are linearly ordered; or a partial order where the predecessors are well-ordered; or either of these, except with successors instea... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Michael Greinecker | 35,357 | <p>Non-cooperative game theory has the odd property that essentially all authors have their own notions of what a game in <a href="https://en.wikipedia.org/wiki/Extensive-form_game" rel="noreferrer">extensive form</a> is and thus prove results about principally different mathematical objects.The used notions tend to be... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Stefan Perko | 78,650 | <p>How about "algebra"? Usually an algebra over a field is assumed to be associative by default, but sometimes it is not. </p>
<p>Not to mention the various category-theory uses of "algebra" (over a monad, over an operad, for a Lawvere theory, for an endofunctor...).</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Per Alexandersson | 1,056 | <p>The term 'Macdonald polynomial' might refer to:</p>
<ul>
<li>The symmetric function $P_\lambda(x;q,t)$ with coefficients being rational in $q,t$. </li>
<li>The symmetric function $J_\lambda(x;q,t)$, being $P_\lambda(x;q,t)$ multiplied by a normalization factor.</li>
<li>The one of the non-symmetric polynomial $E_\a... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Sabrina Gemsa | 75,338 | <p>The notion of a category: </p>
<p>It sometimes happens that some people define a category as a locally small category without stating it, or probably without knowing that the general definition is where a category consists of a class of morphisms, instead of that morphisms between any two of its objects has to be a... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | user123824 | 123,824 | <p>I'm surprised nobody has mentioned "range" which can by synonymous with "co-domain" or "image" depending on personal preference.</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Michael Hardy | 6,316 | <p>Let <span class="math-container">$X_1,X_2,X_3,\ldots$</span> be independent random variables.</p>
<p>Sometimes it is said that a <b>stopping time</b> for this random process is a random variable <span class="math-container">$T$</span> for which the truth value of <span class="math-container">$T=n$</span> (for <span... |
1,687,387 | <p>I've recently been learning factorials in school. If there is an equation (in $\mathbb N$) with $(n-5)!$, I have to ensure that $n$ is not 1, 2, 3 or 4. I've been told that I should write domain:</p>
<p>$D = \mathbb N \setminus \{1; 2; 3; 4\}$</p>
<p>My question: Is it possible to use an interval? Can I write</p>
... | user247327 | 247,327 | <p>I would not use "interval notation" here. [1, 4] would normally be interpreted as the set of all <strong>real numbers</strong> between 1 and 4 which is not what you intend. Instead, use {1, 2, 3, 4}. For a more general situation, such as "all integers between 1 and 50" or "all integers between 1 and n", use {1, 2... |
1,104,525 | <p>i have this complex number</p>
<p>$\sqrt x/2 + \sqrt x/2 i$</p>
<p>i am trying to convert it to polar form. I know that $r = \sqrt (x^2 + y^2)$ but what are the x and y, $1/2$ ?</p>
| Empy2 | 81,790 | <p>The $x$ is $\sqrt{x}/2$, and the $y$ is $\sqrt{x}/2$ as well.</p>
|
1,104,525 | <p>i have this complex number</p>
<p>$\sqrt x/2 + \sqrt x/2 i$</p>
<p>i am trying to convert it to polar form. I know that $r = \sqrt (x^2 + y^2)$ but what are the x and y, $1/2$ ?</p>
| graydad | 166,967 | <p>First, $$\frac{\sqrt x}{2} + \frac{i\sqrt x}{2} = \frac{\sqrt x}{\sqrt{2}}\left(\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2} \right)$$Now recall that the cartesian coordinate $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ is the polar coordinate $(1,\pi/4)$ and hence $$\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2}=e^{i\pi... |
37,207 | <p><strong>Note</strong>: As pointed out in a comment by Peter and echoed by Andrew in his answer, the question as stated does not make sense because "being bimorphic to" is not an equivalence relation. Nevertheless, I will leave the question as stated in case it continues to receive interesting answers.</p>
<h3>Jus... | David Roberts | 4,177 | <p>If the category in question is concrete with underlying set functor $C \to Set$, and bimorphisms are given by one-to-one and onto functions, then the more interesting question is what happens <em>inside</em> a bimorphism class. Namely, what sort of $C$-objects are there for a given underlying set? In other words, wh... |
37,207 | <p><strong>Note</strong>: As pointed out in a comment by Peter and echoed by Andrew in his answer, the question as stated does not make sense because "being bimorphic to" is not an equivalence relation. Nevertheless, I will leave the question as stated in case it continues to receive interesting answers.</p>
<h3>Jus... | Andrew Stacey | 45 | <p>(Disqualifier: I don't have too much expertise) I would say that, at least at the beginning of one's categorical quest, the most common use of the concept of "bimorphism" is to find conditions where "bimorphism => isomorphism". This isn't quite the same as finding those categories which are balanced as there may b... |
1,855,961 | <p>How can I prove that the sequence</p>
<p>$f(n) = n(n+3)/2 = 0, 2, 5, 9, 14, 20, 27, 35, 44, ...$</p>
<p>does not contain primes except $2$ and $5$?</p>
| John Hughes | 114,036 | <p>Write $(a^k)^n = (a^{dr})^n = (a^d)^{rn}$ to see that every power of $a^k$ is also a power of $a^d$, hence $\langle a^k\rangle \subset \langle a^d\rangle$.</p>
|
23,547 | <p>The motivation for this question comes from the novel <em>Contact</em> by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at the end) says that if humans compute enough digits of $\pi$, they will discover that after some point there is... | Gerry Myerson | 3,684 | <p>Summing up what others have written, it is widely believed (but not proved) that every finite string of digits occurs in the decimal expansion of pi, and furthermore occurs, in the long run, "as often as it should," and furthermore that the analogous statement is true for expansion in base b for b = 2, 3, .... On th... |
23,547 | <p>The motivation for this question comes from the novel <em>Contact</em> by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at the end) says that if humans compute enough digits of $\pi$, they will discover that after some point there is... | Spice the Bird | 14,167 | <p>I do not know what kind of effect this has on your question, but it might be a good thing to look at in this context. In 1995, a formula for the $n$-th digit of $\pi$ written in hexadecimal was found. See the wikipedia article, <a href="http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula" rel... |
255,810 | <p>Let $A$ be an $n\times n$ complex matrix, and write $A=X+iY$, where $X$ and $Y$ are real $n\times n$ matricies. Suppose that for every square submatrix $S$ of $A$, $|\mathrm{det}(S)|\leq 1$ (i.e., all minors of $A$ are complex numbers with modulus $\leq 1$). This includes the assumption that $|\mathrm{det}(A)|\leq... | Suvrit | 8,430 | <p>As a complement to Fedor's answer, let me note the following special case, where the desired claim does hold.</p>
<blockquote>
<p><strong>Prop.</strong> Let $A$ be a complex matrix and $A=X+iY$ be its <em>Cartesian decomposition</em>, i.e., $X=\frac12(A+A^*)$ and $Y=\frac{1}{2i}(A-A^*)$. If $X>0$ (i.e., it is ... |
3,840,282 | <p>Given two lines</p>
<p><span class="math-container">\begin{cases}
ax+by+c=0\\
ax+by=0
\end{cases}</span></p>
<p>and a point <span class="math-container">$V(x_p, y_p)$</span>, the point-line distance formula is:</p>
<p><span class="math-container">$$\frac{\lvert ax_p+by_p+c\rvert}{\sqrt{a^2+b^2}}$$</span></p>
<p>I un... | Rezha Adrian Tanuharja | 751,970 | <p>Hint: <span class="math-container">$c=-ax_{l}-by_{l}$</span> where <span class="math-container">$(x_{l},y_{l})$</span> lies on line <span class="math-container">$ax+by+c=0$</span>. Inner product again?</p>
|
308,750 | <p>In the Art Gallery Problem, we have given
a polygon $P$ on $n$ vertices and a number $k$ and we
want to know if there exists $k$ guards
such that every point inside the polygon
is seen by at least one of the guards.
We say a point $p$ sees a point $q$
if the entire segment $pq$ is contained inside $P$.
Let us ... | Till | 104,681 | <p>Thanks to Joseph O'Rourke we can do the following algorithm:</p>
<p>Compute $k$ visibility polygons in $O(n)$ time per visibility polygon.
According to [1] a visibility region in a simple polygon can be
computed in linear time and thus also has at most a linear number of
edges and vertices.
For polygons with $h$ ... |
704,127 | <p>I cannot prove any conclusion of this problem. Can anyone please help me?
Let $f:M\to N$ be an immersion of $M$ into $N$ and dim $M=\dim N$.Prove or disprove that $f(M)$ is a submanifold.
Thanks for any help.</p>
| Gil Bor | 118,580 | <p>$f(M)$ is an open subset of $N$ and thus a submanifold. Proof by unwrapping of definitions: say $y=f(x)\in f(M)$. Since $f$ is an immersion $df(x)$ is injective, and since $\dim(M)=\dim(N)$ $df(x)$ is in fact an isomorphism. By the inverse function theorem $f$ restricted to some neighborhood of $x$ is a diffeomorp... |
509,349 | <p>The problem I am having is figuring out the way show the following sequence is monotone:</p>
<p>let $x_1 = \frac{3}{2}$ and $x_{n+1} = {x_n}^2-2x_n+2$, show that the sequence $x_n$ is monotone and bounded and find the limit.</p>
<p>I have found the first three terms, and found that the sequence is decreasing, I ha... | J. W. Perry | 93,144 | <p>My turn... request check for typos:</p>
<p>We first rewrite the sequence as (Abel's bright idea: complete the square),
$$x_{n+1}={x_n}^2-2x_n+2=(x_n-1)^2+1.$$</p>
<p>$\textbf{Boundedness:}$ We prove that $x_n$ is bounded on the half open interval $1 \leq x_n<2$ using proof by induction on $n$.</p>
<p>For a bas... |
505,059 | <p>I have a question:</p>
<p>If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is:</p>
<p>$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$</p>
<p>But why can't I say:</p>
<p>$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}... | Avitus | 80,800 | <p>What you describe is a quite common situation which pops up when dealing with commutators of <em>operators</em>. On an appropriate space of functions $\mathcal D$ (like an $L^2$-space or the Schwartz space etc...), the operators $x$ and $p$ are given by</p>
<p>$$x(f)(x):=xf(x), $$
$$p(f)(x):=\frac{h}{i}\frac{df}{dx... |
505,059 | <p>I have a question:</p>
<p>If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is:</p>
<p>$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$</p>
<p>But why can't I say:</p>
<p>$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}... | Adnan Najeeb | 322,252 | <p>You can say </p>
<p>$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}x$</p>
<p>But that will not be equal to zero, since (1): $x h^2{d^2 \over dx^2}$ is not equal to (2): $h^2 {d^2 \over dx^2}x$.</p>
<p>It's clear that in (1), the differential operator is not operating on anything. But in (2)... |
1,440,559 | <p>Is zero considered a scalar?</p>
<p>In other words, is $\begin{bmatrix}0\\0\\\end{bmatrix}$ a scalar multiple of $\begin{bmatrix}a\\b\\\end{bmatrix}$ where $a$ and $b$ are real numbers?</p>
| MathAdam | 266,049 | <p>Yes, $0$ is indeed a scalar. But, to be clear, $\begin{bmatrix}0\\0\\\end{bmatrix}$ is <em>not</em> a scalar; it's a vector, the result of multiplying the scalar, $0$ by the vector, $\begin{bmatrix}a\\b\\\end{bmatrix}$, where $a$ and $b$ are real numbers.</p>
|
1,440,559 | <p>Is zero considered a scalar?</p>
<p>In other words, is $\begin{bmatrix}0\\0\\\end{bmatrix}$ a scalar multiple of $\begin{bmatrix}a\\b\\\end{bmatrix}$ where $a$ and $b$ are real numbers?</p>
| celtschk | 34,930 | <p>Yes. In the context of vector spaces, a scalar is a member of the underlying field. That especially includes the neutral element of field addition, known as $0$.</p>
|
543,920 | <p>Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:</p>
<p>Let $\mathcal{A}$ be a class of subsets of $\mathbb{R}$ which generates $\mathcal{B}$. Show that $X$ is an RV on $\Omega\;$ if and only if $X^{-1}(A)$ $\in \mathbb{R}$ for all $A\in \mathcal{A}$.<... | Gautam Shenoy | 35,983 | <p>Hint: Pre-images preserve arbitrary unions. i.e. for $\{A_\alpha\}$ where $\alpha \in I$ where $I$ is an index set, then $X^{-1}(\bigcup_{\alpha \in I}A_{\alpha}) = \bigcup_{\alpha \in I}X^{-1}(A_{\alpha})$</p>
<p>Let me know if you need more help.</p>
|
3,292,629 | <p>My answer of total ways is <span class="math-container">$2^4\dfrac{9\times10}{2}=720$</span> but <span class="math-container">$0$</span> should not be in the first place, number of their ways is <span class="math-container">$9\times2^3=72$</span>. Required answer is <span class="math-container">$720-72=648$</span>. ... | peek-a-boo | 568,204 | <p>From the equation
<span class="math-container">\begin{align}
du = \dfrac{dx}{3x^{2/3}},
\end{align}</span>
we get
<span class="math-container">\begin{align}
dx = 3x^{2/3} \cdot du = 3u^2\, du
\end{align}</span>
(because <span class="math-container">$x^{1/3} = u$</span>)
Hence,
<span class="math-container">\begin{ali... |
3,579,587 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> be non-negative real numbers such that
<span class=... | fzen | 759,218 | <p>Thank @Michael Rozenberg, I have a proof when <span class="math-container">$k = \frac12,$</span> for weaker conditons
<span class="math-container">$$a+b+c = x+y+z,$$</span>
<span class="math-container">$$\min(x, y, z) \leqslant \min(a, b, c),$$</span>
<span class="math-container">$$\max(a, b, c) \leqslant \max(x, y... |
349,653 | <p>I'm trying to evaluate the following limit:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\ln(2x)}{\ln(x)}$</p>
<p>Using L'Hospital's rule, I end up with:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\frac{1}{x}}{\frac{2}{x}}$</p>
<p>= $\displaystyle\lim_{x \to \infty} x/2x$</p>
<p>... | Fly by Night | 38,495 | <p>You answer is incorrect, and your should have checked this by experimentation. Putting $x=10^{10}$ gives $\ln(2x)/\ln x \approx 1.03$ while putting $x=10^{50}$ gives $\ln(2x)/\ln x \approx 1.01$. It would seem that $\ln(2x)/\ln x \to 1$ as $x \to \infty$. <em>However</em>, we need to prove this rigorously. </p>
<p>... |
2,914,976 | <p>Can someone explain to me why $$ \lim\limits_{x \to \infty} x\bigg(\frac{1}{2}\bigg)^x = 0$$
Is it because the $\big(\frac{1}{2} \big)^x$ goes towards zero as $ x $ approaches $\infty$, and anything multiplied by $0 $ included $\infty$ is $0$ ?</p>
<p>Or does this kind of question require using l'hopital's rule be... | mengdie1982 | 560,634 | <p>You'd better write the process as $x \to +\infty$. Thus$$\lim_{x \to +\infty} x \left(\frac{1}{2}\right)^x=\lim_{x \to +\infty} \frac{x}{2^x}=\lim_{x \to +\infty} \frac{1}{\ln 2\cdot2^x}=\frac{1}{+\infty}=0. $$</p>
|
4,387,544 | <blockquote>
<p>If <span class="math-container">$ab+ac+bc=2$</span>, find minimum value of <span class="math-container">$10a^2+10b^2+c^2$</span></p>
<p><span class="math-container">$1)3\qquad\qquad2)4\qquad\qquad3)8\qquad\qquad4)10$</span></p>
</blockquote>
<p>I used AM-GM inequality for three variables:
<span class="m... | LHF | 744,207 | <p>Hint: <span class="math-container">$$10a^2+10b^2+c^2-4(ab+bc+ca)=\frac{2}{5}(5a-b-c)^2+\frac{3}{5}(4b-c)^2$$</span></p>
|
75,781 | <p>This is my first question on the forum. I'm wondering if the following proof is valid.</p>
<hr>
<p><strong>Proof:</strong>
Let $\{A_\lambda\}_{\lambda \in L}$ be an arbitrary collection of disjoint non-empty open subsets of $\mathbb{R}$. Since every non-empty open subset of $\mathbb{R}$ can be written uniquely as ... | Arturo Magidin | 742 | <p>It seems to me that:</p>
<blockquote>
<p>"...every non-empty open subset of $\mathbb{R}$ can be written uniquely as a countable union of disjoint open intervals..."</p>
</blockquote>
<p>is <em>essentially</em> what you are trying to prove (in fact, it's <em>stronger</em> than what you are trying to prove; it inc... |
75,781 | <p>This is my first question on the forum. I'm wondering if the following proof is valid.</p>
<hr>
<p><strong>Proof:</strong>
Let $\{A_\lambda\}_{\lambda \in L}$ be an arbitrary collection of disjoint non-empty open subsets of $\mathbb{R}$. Since every non-empty open subset of $\mathbb{R}$ can be written uniquely as ... | Pete L. Clark | 299 | <p>Your argument seems valid to me, though I think that in assuming that every open subset of $\mathbb{R}$ has a unique representation as a countable disjoint union of open intervals, you're swatting a fly with a sledgehammer.</p>
<p>How about this? In a disjoint collection $\{U_i\}_{i \in I}$ of nonempty open subset... |
3,952,488 | <p>Let <span class="math-container">$V=\begin{cases}\begin{bmatrix}x\\y\end{bmatrix}:x\in \mathbb{R}^+, y\in \mathbb{R}\end{cases}\bigg\}.$</span>
Then it can be proved that under the operations <span class="math-container">$$\alpha \cdot \begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}x^{\alpha}\\ \alpha y\end{bmat... | J.G. | 56,861 | <p>You mean find <em>a</em> basis, as it's not unique.</p>
<p>We seek <span class="math-container">$\begin{bmatrix}a\\b\end{bmatrix},\begin{bmatrix}c\\d\end{bmatrix}$</span> such that <span class="math-container">$\begin{bmatrix}a^\alpha c^\beta\\\alpha b+\beta d\end{bmatrix}$</span> can be any element of <span class="... |
3,767,261 | <p>The text <em>Topology and Groupoids</em> by Brown states the following on page 8 in Chapter 1:</p>
<blockquote>
<p>1.2.4 (Inverse rule.) Let <span class="math-container">$f \colon A \to \mathbb{R}$</span> be a real function which is injective, so that <span class="math-container">$f$</span> has an inverse <span clas... | tkf | 117,974 | <p>If the function <span class="math-container">$f$</span> is injective, and continuous then it is monotone, by the intermediate value theorem. Then you need only note that the open sets on both <span class="math-container">$A$</span> and <span class="math-container">$f(A)$</span> are induced by their orderings. Note... |
2,222,572 | <p>Does there exist a continuous function $f:[0,1]\rightarrow[0,\infty)$ such that $$\int_0^1 \! x^{n}f(x) \, \mathrm{d}x=1$$ for all $n\geq1$?</p>
| BindersFull | 363,871 | <p>How about $f(x) = \sum_{n = 0}^\infty a_n x^n$, where $a_n = \frac{n + 2}{2^{n + 1}}$? Partial sums $S_N(x) = \sum_{n = 0}^N a_nx^n$ are pointwise increasing and bounded above by $S_{\infty}(1)< \infty$ so $S_N(x)$ converges pointwise on $[0,1]$ and the bounded convergence theorem gives
\begin{eqnarray*}
\int_0^... |
1,298,938 | <p>Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of
$$
H^*(\Sigma X;R)$$
trivial? How to prove? Where can I find the result?</p>
| Lilalas | 408,870 | <p>The answer depends on which homology theory you are using.
The statement fails for <em>singular cohomology</em>.
However there is a fairly easy way to show that the cup product is trivial on <em>reduced cohomology</em> <span class="math-container">$\tilde{H}{\,}^\bullet(\Sigma X)=H^\bullet(\Sigma X,\text{pt})$</span... |
384,568 | <p>We know, the following vectors form a basis of $\mathbb R ^4$ </p>
<p>$$
B:=\{(1,2,3,4)^T, (2,0,1,-1)^T, (-1,0,0,1)^T, (0,2,3,0)^T\}
$$</p>
<p>It's easy to proof that, we only need to show these vectors are linear independent and show that these 4 vectors generate $\mathbb R^4$</p>
<p>But I don't know how to exte... | Somaye | 83,530 | <p>for linear independence it is good that you consider first and second coordinate Zero in each two vector for example A=(0,0,x_1,x_2) and B=(0,0,y_1,y_2);this will help you that in linear combination $c_1(0,4,5,9)^T+c_2(3,3,3,3)^T+c_3 A^{T}+c_4 B^{T}=0$ you catch $C_1 $ and $C_2$ equal zero.it will be remain that you... |
1,944,929 | <p>I have done some digging and I cannot find any posts addressing limits with exponentials and <em>without</em> L'Hôpital's rule.</p>
<p>I have one of these questions for my assignment, but for ethical reasons I have made up a similar function: </p>
<blockquote>
<p>Find the following limit without L'Hôpital's rule... | Djura Marinkov | 361,183 | <p>You can transform them to MacLaurins series f(x)=f(0)+f'(0)x/1!+f"(0)x<sup>2</sup>/2!+...</p>
<p>so $2^x=1+xln2+x^2ln^22/2+...$ and $7^x=1+xln7+x^2ln^27/2+...$</p>
<p>you have lim equal to (ln2-ln7+o(x))/2</p>
<p>(ln means "natural logarithm").</p>
|
3,093,651 | <p>Let <span class="math-container">$U:C([0,1])\to C([0,1])$</span> defined as <span class="math-container">$U(f):=f^3$</span>. I want to show it's not continuous in the supremum norm (i.e. <span class="math-container">$L^\infty$</span> norm) nor in the <span class="math-container">$L^1$</span> norm.</p>
<p>I can show... | Yanko | 426,577 | <p><strong>It is continuous</strong> with respect to the supremum norm.</p>
<p>If <span class="math-container">$f_n\rightarrow f$</span> in the sup norm, let <span class="math-container">$M$</span> be such that <span class="math-container">$\|f_n\|_\infty,\|f\|_\infty<M$</span>. For any two functions <span class="m... |
262,660 | <p>I used this code to solve an equation and now I would like to plot the solution with its inverse. I can't get some figures?!!</p>
<pre><code>Clear["Global`*"]
eqns = {Derivative[1][y][
x] + (3/2)*(a - b)*(y[x]/x) - ((3/2)*a - 1)*(1/(y[x]*x^3)) == 0}
sol = FullSimplify[DSolve[eqns, y[x], x]]
y2[x_] = y[x... | JimB | 19,758 | <p>@DanielHuber 's answer is the way to go if you want to use what's explicitly sanctioned in the documentation. But if you're willing to try a workaround, here's one approach.</p>
<p>The issue is that when you obtain the PDF as a pure function, the "discreteness" of the random variable is not included in th... |
3,126,162 | <p>The goal is to show that <span class="math-container">$$\left(\frac{1}{3}\right)^kn=1 \Rightarrow k = \log_3 n\,.$$</span></p>
<p>So I started with <span class="math-container">$\left(\frac{1}{3}\right)^kn=1 \Leftrightarrow \left(\frac{1}{3}\right)^k=\frac{1}{n}$</span> in order to use the identity <span class="mat... | Siong Thye Goh | 306,553 | <p>Alternatively, </p>
<p><span class="math-container">$$\left( \frac{1}{3^k}\right)n=1$$</span></p>
<p>Multiplying <span class="math-container">$3^k$</span> on both sides,
<span class="math-container">$$n=3^k$$</span></p>
<p>Hence <span class="math-container">$$k = \log_3 n$$</span></p>
|
1,665,513 | <p>Consider sequences of numbers 0, 1, 2 with length n. There are $3^n$ such sequences.</p>
<p>I want to know how many sequences there are that contain a k-run of 1's followed by 2. As a regular expression: </p>
<blockquote>
<p>(^|.*[^1])[1]{k}[2].*</p>
</blockquote>
<p>Even better would be to know the number of s... | Giovanni Resta | 312,312 | <p>This is a just a partial answer, but there was not enough space in a comment.</p>
<p>The problem seems not so easy. It is easy to see that the number of sequences of length $n$ that do contain "$12$" are
$$
3^n - F_{2n+2}\,
$$
where $F_k$ denotes the $k$-th Fibonacci number.</p>
<p>The resulting sequence $0, 1,... |
1,665,513 | <p>Consider sequences of numbers 0, 1, 2 with length n. There are $3^n$ such sequences.</p>
<p>I want to know how many sequences there are that contain a k-run of 1's followed by 2. As a regular expression: </p>
<blockquote>
<p>(^|.*[^1])[1]{k}[2].*</p>
</blockquote>
<p>Even better would be to know the number of s... | Rus May | 17,853 | <p>As an answer to questions of the type "how many words of a particular length avoid/contain such-and-such patterns", there is a general method called the <a href="http://www.math.rutgers.edu/~zeilberg/gj.html" rel="nofollow">Goulden-Jackson cluster method</a> that efficiently produces a generating function for the a... |
1,727,144 | <blockquote>
<p>Suppose you roll a fair dice $12$ times in a row. What is the probability of the event "exactly $k$ of the rolls are a $5$ or a $6$" ?</p>
</blockquote>
<p>I'm just asking for some verification of my counting. Let $X$ be the random variable that counts the number of $5$ and $6$ rolled.</p>
<p>$$\beg... | drhab | 75,923 | <p>Solving this in your line (in order to expose your mistakes): </p>
<p>If $Y$ denotes the number of times that a $5$ is rolled and $Z$
denotes the number of times that a $6$ is rolled then:</p>
<p>$$P\left(Y+Z=k\right)=\sum_{i=0}^{k}P\left(Y=i\wedge Z=k-i\right)=\sum_{i=0}^{k}\frac{12!}{i!\left(k-i\right)!\left(12-... |
1,956,331 | <p>Solve for $x,y\ \mbox{and}\ z$:
$$
\left\{\begin{array}{rcr}
x + y + z & = & 2
\\[1mm]
\left(x + y\right)\left(y + z\right) + \left(y+z\right)\left(z+x\right) +
\left(z + x\right)\left(x + y\right) & = & 1
\\[1mm]
x^{2}\left(y + z\right) + y^{2}\left(x + z\right) + z^{2}\left(x + y\right)
&... | Oleg567 | 47,993 | <p>Way:</p>
<p>$1)$ express LHS-s via elementary symmetric polynomials; <br>
$2)$ consider related cubic equation.</p>
<hr>
<p>$1)$ Elementary symmetric polynomials for this problem are: <br>
$e_1(x,y,z) = x+y+z$;<br>
$e_2(x,y,z) = xy+xz+yz$;<br>
$e_3(x,y,z) = xyz$.</p>
<p>So,
$$x+y+z= e_1;$$
$$(x+y)(y+z) + (y+z)(z... |
923,373 | <p>Consider the matrix $$A=\begin{pmatrix}-1 & 3& 3& 3\\ 3& 1& -1& 5\\ 3& -1& 7& -1\\ 3&5& -1&1\end{pmatrix}.$$</p>
<p>How do I show that $1$ is not an eigenvalue for $A$, by showing that there are no eigenvectors for $\lambda = 1$?</p>
<p>This what I've done so ... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> You can use <a href="https://proofwiki.org/wiki/Dirichlet%27s_Test_for_Uniform_Convergence" rel="nofollow">Dirichlet's test</a>.</p>
<p><strong>Added:</strong> You need to bound partial sums </p>
<blockquote>
<p>$$ \sum_{i}^{n} (-1)^i\sin(i\theta) = {\frac { \left( -1 \right) ^{n}\sin \le... |
2,278,462 | <p>I think the problem is clear from the title. It included a hint which suggested reducing it to the case $A = 1$, but so far I've come up with little.</p>
<p>It is quite direct from the statement that $A, B, C$ are $n$th roots of unity. I also know that the sum of $n$th roots of unity must be zero, but haven't found... | Michael Rozenberg | 190,319 | <p>Let $A=\cos\frac{2\pi m}{n}+i\sin\frac{2\pi m}{n}$, $B=\cos\frac{2\pi k}{n}+i\sin\frac{2\pi k}{n}$ and $C=\cos\frac{2\pi l}{n}+i\sin\frac{2\pi l}{n}$,</p>
<p>where $n\geq m\geq k\geq l\geq0$.</p>
<p>Thus,
$$\cos\frac{2\pi m}{n}+\cos\frac{2\pi k}{n}+\cos\frac{2\pi l}{n}=0$$ and
$$\sin\frac{2\pi m}{n}+\sin\frac{2\pi... |
2,326,638 | <p>Is this true: </p>
<p>$$ P({+}c\mid {+}s) = P({+}c\mid{+}s,{+}r) P({+}r) + P({+}c\mid{+}s,{-}r)P({-}r) $$</p>
<p>, where ${+}x$ means $x = $true and $-x$ means $x= $false ($s,c,r$ are all boolean variables) and the commas$(,)$ mean "AND". To me, it looks like the Law of total probability: $p(a) = p(a\mid x)\cdot p... | José Carlos Santos | 446,262 | <p>No, it is not a sum. Usually, the meaning of $\int_\Gamma$ with $\Gamma=\{z\in\mathbb{C}\,:\,|z|=r\}$ means that we are integrating along the loop $\Gamma\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\Gamma(t)=re^{it}$.</p>
<p>In this specific case, the value of the integral will be $2\pi i$ times the sum of ... |
1,459,424 | <p>$$y'+y\cos x=3\cos x$$
When I find the integration factor it is $e^{\sin x}$, but as far as I know that has no solution when I try to complete this by integration by parts.</p>
| Kwin van der Veen | 76,466 | <p>This integration factor also makes the differential equation <a href="http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html" rel="nofollow">exact</a>. The implicit solution you can then find is</p>
<p>$$
y\; e^{\sin(x)} - 3\; e^{\sin(x)} = c,
$$</p>
<p>where $c$ is a constant that can be f... |
3,517,795 | <p>I need to find <span class="math-container">$A^n$</span> of the matrix <span class="math-container">$A=\begin{pmatrix}
2&0 & 2\\
0& 2 & 1\\
0& 0 & 3
\end{pmatrix}$</span> using Cayley-Hamilton theorem.</p>
<p>I found the characteristic polynomial <span class="math-container">$P(A)=(2-A... | amd | 265,466 | <p>A consequence of the Cayley-Hamilton theorem is that <span class="math-container">$$A^n=aI+bA+cA^2\tag 1$$</span> for some scalar coefficients <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$c$</span>. The above equation also holds for the eigenvalue... |
2,908,791 | <p>Given integers $x_{0},n$ with $x_{0}^{2}\equiv -1$ (mod $n$) then there are integers $y,b$ with $(y,b)=1,0<b\le\sqrt{n}$ and </p>
<p>$$\left|-\frac{x_{0}}{n}-\frac{y}{b}\right|<\frac{1}{b\sqrt{n}}$$</p>
<p>I tried to solve $|bx_{0}+ny|<\sqrt{n}$ for some integers $y,b$, but it seems do nothing.</p>
<p>Is... | TheSimpliFire | 471,884 | <p><strong>Without using calculus:</strong> Substituting <span class="math-container">$c=4-2b-a$</span>, we get <span class="math-container">$$ab+bc+ca=ab+(a+b)(4-2b-a)=(4(a+b)-(a+b)^2)-b^2$$</span> and since <span class="math-container">$f(x)=4x-x^2=4-(x-2)^2$</span> has maximum at <span class="math-container">$(2,4)$... |
2,206,575 | <p>If connected subsets have non-empty intersections pairwisely, how can I show that their union is connected? Formally, let <span class="math-container">$E_\alpha$</span> be connected for every <span class="math-container">$\alpha \in I$</span>, and suppose that <span class="math-container">$E_\alpha \cap E_{\beta} \n... | Henno Brandsma | 4,280 | <p>Define <span class="math-container">$S := \cup_{\alpha \in I} E_\alpha$</span>, and let <span class="math-container">$C$</span> be a <span class="math-container">$S$</span> clopen subset of <span class="math-container">$S$</span>.</p>
<p>Then for each <span class="math-container">$\alpha \in I$</span>: <span class="... |
1,891,398 | <blockquote>
<p>Let $a$ belong to a group and $|a|=m$. If $n$ is relatively prime to $m$ show that $a$ can be written as the $n^{th}$ power of some element in the group.</p>
</blockquote>
<p>We need to show that if $a\in G$ and $a^m=e\implies \exists \ b\in G $ such that $b^n=a$ i.e.$b^{nm}=e=({b^m})^n$ i.e. $\exis... | Zestylemonzi | 270,448 | <p>Hint: Euclid's algorithm - If $n$ and $m$ are coprime we can find $p,q \in \mathbb{Z}$ such that $pn + qm = 1.$ Then, $1=a^{qm} = a^{1-pn}.....$ can you continue from here?</p>
|
1,891,398 | <blockquote>
<p>Let $a$ belong to a group and $|a|=m$. If $n$ is relatively prime to $m$ show that $a$ can be written as the $n^{th}$ power of some element in the group.</p>
</blockquote>
<p>We need to show that if $a\in G$ and $a^m=e\implies \exists \ b\in G $ such that $b^n=a$ i.e.$b^{nm}=e=({b^m})^n$ i.e. $\exis... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Simply write $\ a = \left(a^{\Large\frac{1}n}\right )^n,\,$ valid since $\ \frac{1}n\,$ exists mod $\,m.\,$ More explicitly:</p>
<p>By Bezout $\,\gcd(n,m)= 1\,\Rightarrow\, \color{#c00}{n'n\equiv 1}\pmod{\! m}\ $ for some $\ n'\, (\equiv n^{-1}\equiv \frac{1}n)$ </p>
<p>therefore we conc... |
4,005,776 | <p>I'd like to find all 3rd roots of this number z = i - 1. Now I've found formulas on how to do it; First we transform the complex number into this form</p>
<p><span class="math-container">$$ \sqrt[n]{r} * e^{i\frac {\phi + 2k\pi}{n}} $$</span> Where n should be 3 (because of 3rd root) and k should be k = n - 1 (Inclu... | user | 293,846 | <p>Hint: In the formula you used <span class="math-container">$r$</span> and <span class="math-container">$\phi$</span> are the absolute value and the argument of the complex number <span class="math-container">$z$</span>.</p>
<p>The absolute value and argument of a complex number <span class="math-container">$z=x+iy... |
1,454,471 | <p>Show that the only matrices that commute with all other matrices in $GL_{n}(\mathbb{R})$ will be multiples of the identity matrix. Or:</p>
<p>$Z(GL_{n}(\mathbb{R}))=\{\lambda I_{n}:\lambda \in \mathbb{R}\}$</p>
<hr>
<p>One direction is clearly obvious. However, to show that every element $g$ such that $gh = hg$ f... | Groups | 28,017 | <p>Two simple, elementary observations make many things easy here.</p>
<ul>
<li><p>Order of an element is equal to the order of cyclic (sub)group generated by that element.</p></li>
<li><p>If $k$ is relatively prime to order of $x$, then $x$ and $x^k$ generate same cyclic (sub)group, i.e. $<x>=<x^k>$</p></... |
1,613,426 | <blockquote>
<p>Prove that $$|\frac{a-b}{1-\bar ab}|=1$$ if $|a|=1$ or $|b|=1$</p>
</blockquote>
<p>I assumed $|a|=1$. Then tried to show that our statement holds.</p>
<p>I wrote $a=a_1+ia_2$ and $b=b_1+ib_2$ and $\bar a=a_1-ia_2$</p>
<p>Also $$|a|=|\bar a|=a_1^2+a_2^2=1$$</p>
<p>However, after multiplying it all... | Intelligenti pauca | 255,730 | <p>If $|a|=1$ then
$$\left|\frac{a-b}{1-\bar ab}\right|=
\left|\frac{a(1-b/a)}{1-(\bar a a)(b/a)}\right|=|a|\left|\frac{1-b/a}{1-b/a}\right|=1.
$$</p>
|
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