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 |
|---|---|---|---|---|
3,541,524 | <blockquote>
<p>Decide whether the following ie true or false
<span class="math-container">$$\lvert\arcsin z \rvert \le \left\lvert \frac {\pi z} {2} \right\rvert $$</span>
whenever <span class="math-container">$z\in\Bbb C$</span> . </p>
</blockquote>
<p><span class="math-container">$\arcsin z =-i \text{Log } (... | Andreas | 317,854 | <p>Note <span class="math-container">$\arcsin(z) = \sum_{n=0}^\infty \frac{1 }{2^{2n}}\binom{2n}{n} \frac{ z^{2n+1}}{2n+1}$</span>. So we have that
<span class="math-container">$\lvert\arcsin z \rvert \le \arcsin\lvert z \rvert$</span>.</p>
<p>Further, we have that <span class="math-container">$\arcsin\lvert z \rvert$... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Tyler | 2,465 | <p><strong>HINT:</strong> $(k + 1)^3 - k^3 = 3k^2 + 3k + 1$. Telescope the left hand side, solve for $k^2$.</p>
<p>If you need more of a hint I'll be glad to elaborate later. In case you'd like a reference, this is one of the first exercises in Spivak's Calculus (I don't have the latest edition, but it's in the sectio... |
1,109,759 | <p>I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$,
$$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$</p>
<p>I think we're supposed to use Cauchy-Schwarz: $\lvert \int_0^1 f(x)g(x) dx \rvert \le \left( \int_0^1 \lvert f(x) \rvert ^2 dx \right)^{1/2... | user14717 | 24,355 | <p>There's no reason to do all these calculations. The norm induced from <em>any</em> inner product obeys the triangle inequality as a consequence of the Cauchy-Schwarz inequality, so just state that your norm is induced from the inner product
\begin{align}
\left<f, g\right> := \int_{0}^{1} f(x)\overline{g(x)} \,... |
280,346 | <p>I am wondering how to tell Mathematica that a function, say <code>F[x]</code>, is a real-valued function so that, e.g., the <code>Conjugate</code> command will pass through it:</p>
<pre><code>Conjugate[E^(-i k x)F[x]] = E^(i k x)F[x]
</code></pre>
<p>I tried to make a huge calculation using the <code>Conjugate</code... | sebqas | 14,660 | <p>I'm surprised, as Bob's answer works as you wish</p>
<p><a href="https://i.stack.imgur.com/LsetX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LsetX.png" alt="enter image description here" /></a></p>
<p>However, if you don't want to use FullSimplify, you may redefine Conjugate[]</p>
<pre><code>U... |
3,586,346 | <p>Basically, I'd like to model sin x, but make it's derivative tend towards 0, so as x increases, it becomes a constant y = 0. The function begins like a typical sin x function, but slowly the fluctuation decreases until it isn't there anymore. If this works as I'm trying to have it work, I think some constant between... | Lt. Commander. Data | 632,103 | <p>Any function of the form
<span class="math-container">$$y(t)=Ae^{-\gamma t}\sin(\omega t+\phi)$$</span>
works. For example, substituting <span class="math-container">$\gamma = 0.2$</span>, <span class="math-container">$A=1$</span>, <span class="math-container">$\omega = 1$</span> and <span class="math-container">$\p... |
1,056,038 | <blockquote>
<p>Each of $n$ balls is independently placed into one of $n$ boxes, with all boxes equally likely. What is the probability that exactly one box is empty? (Introduction to Probability, Blitzstein and Nwang, p.36).</p>
</blockquote>
<ul>
<li>The number of possible permutations with replacement is $n^n$</l... | Jimmy R. | 128,037 | <p>Your approach (although nice) has a flaw in the second bullet. The problem is that there you count two different things: on the one hand ways to choose a box and on the other hand ways to choose a ball and this results to a confusion. In detail</p>
<ol>
<li>Your denominator is correct,</li>
<li>Your numerator is mi... |
4,615,947 | <p>Let <span class="math-container">$a,b\in\Bbb{N}^*$</span> such that <span class="math-container">$\gcd(a,b)=1$</span>. How to show that <span class="math-container">$\gcd(ab,a^2+b^2)=1$</span>?</p>
| chroma | 1,006,726 | <p><span class="math-container">\begin{align*}I & =\int_{-\infty}^\infty\frac{\sin x}{(x+1)^2+1}dx=\int_{-\infty}^{\infty}\frac{\sin(x-1)}{x^2+1}dx \\ & =\cos(1)\int_{-\infty}^\infty\frac{\sin x}{x^2+1}dx-\sin(1)\int_{-\infty}^\infty\frac{\cos x}{x^2+1}dx \\ &=-2\sin(1)\int_{0}^\infty\frac{\cos x}{x^2+1}dx=... |
1,038,713 | <p>Suppose I am given a circle $C$ in $\Bbb C^*$ and two points $w_1,w_2$. Given another circle $C'$ and points $z_1,z_2$, what is the procedure to find a Möbius transformation that sends $C\to C'$, $w_i\to z_i,i=1,2$? Here $z_1\in C\not\ni z_2$; $w_1\in C'\not\ni w_2$. For example, take $|z|=2$, $w_1=-2,w_2=0$. Then,... | Joonas Ilmavirta | 166,535 | <p>First, take a Möbius transform $\phi$ that takes $C$ to $\mathbb R$ and $z_1$ to $i$.
This $\phi$ always exists uniquely, provided $z_1\notin C$; it's not too hard to map a circle to the real axis, and the transforms fixing the real axis are easy to classify.
Let $\psi$ be the corresponding map for $C'$ and $w_... |
2,820,464 | <p>Find the limit of the sequence {$a_{n}$}, given by$$ a_{1}=0,a_{2}=\dfrac {1}{2},a_{n+1}=\dfrac {1}{3}(1+a_{n}+a^{3}_{n-1}), \ for \ n \ > \ 1$$</p>
<p>My try:</p>
<p>$ a_{1}=0,a_{2}=\dfrac {1}{2},a_{3}=\dfrac {1}{2},a_{4}=0.54$ that is the sequence is incresing and each term is positive. Let the limit of the s... | Boyku | 567,523 | <p>We will prove that all $a_n$ are smaller than ${2 \over 3}=0.6666...$. </p>
<p>By induction, suppose that $0, 1/2, ... a_{n-1}, a_n < 2/3$.</p>
<p>then $a_{n+1} < {1 + 2/3 + 8/27 \over 3 }= {53 \over 81} < {54 \over 81} = {2\over 3}$</p>
<p>since $a_1 =0<{2 \over 3}$, for all n , $0 \leqslant a_n <... |
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | Jack D'Aurizio | 44,121 | <p>Using only <span class="math-container">$3$</span>s and <span class="math-container">$4$</span>s, with <span class="math-container">$n$</span> of them you can make any integer number between <span class="math-container">$3n$</span> and <span class="math-container">$4n$</span>.<br>
Let <span class="math-container">$\... |
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | Calum Gilhooley | 213,690 | <p>Each term in either of these sums is equal to either <span class="math-container">$\left\lfloor\pi\right\rfloor = 3$</span> or <span class="math-container">$\left\lceil\pi\right\rceil = 4$</span>:
<span class="math-container">\begin{align*}
\pi & = \lim_{n\to\infty}\frac{\left\lfloor{n\pi}\right\rfloor}{n} =
\li... |
2,714,450 | <p>Suppose $A$ and $B$ are two square matrices so that $e^{At}=e^{Bt}$ for infinite (countable or uncountable) values of $t$ where $t$ is positive.</p>
<p>Do you think that $A$ <strong>has to be equal to</strong> $B$?</p>
<p>Thanks,
Trung Dung.</p>
<hr>
<p>Maybe I do not state clearly or correctly.</p>
<p>I mean t... | José Carlos Santos | 446,262 | <p>No. Let $A$ be the null matrix and let $B=2\pi i\operatorname{Id}$. Then$$(\forall t\in\mathbb{Z}):e^{tA}=e^{tB}.$$</p>
|
133,370 | <p>In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition,
$$\Gamma^{\gamma}_{\alpha\beta}\equiv\mathbf{a}^{\gamma}\cdot\mathbf{a}_{\alpha,\beta}=\mathbf{a}^{\gamma}\cdot\mathbf{r}_{,\alpha\beta}=\mathb... | Peter Michor | 26,935 | <p>Levi-Civita means metric compatible and torsion free. Adding a skew symmetric $\binom{1}{2}$ tensor field (= your favorite torsion) to a covariant derivative does not change metric compatibility.</p>
|
3,435,256 | <p>The following statement is given in my book under the topic <em>Tangents to an Ellipse</em>:</p>
<blockquote>
<p>The <a href="http://mathworld.wolfram.com/EccentricAngle.html" rel="nofollow noreferrer">eccentric angles</a> of the points of contact of two parallel tangents differ by <span class="math-container">$\... | mathlove | 78,967 | <blockquote>
<p>Kindly explain the reason behind this fact.</p>
</blockquote>
<p>The reason is that an ellipse can be obtained by stretching/shrinking a circle. The strech/shrink is a <a href="https://en.wikipedia.org/wiki/Linear_map" rel="nofollow noreferrer">linear map (linear transformation)</a>.</p>
<p>Let's co... |
617,275 | <p>$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$
And I have proven $f(x)=\|x\|^2$ and $\|x\|\leq \|f\|$. Prove that $\|f\|=\|x\|$.</p>
| Prahlad Vaidyanathan | 89,789 | <p>Since $f(x) = \|x\|^2$, for any $y\in E$, one has
$$
f(y) = f(x) + f(y-x) \leq \frac{1}{2}\|x\|^2 + \frac{1}{2}\|y\|^2 \leq \max\{\|x\|^2, \|y\|^2\}
$$
Hence, for $z \in X$ such that $\|z\| \leq 1$, let $y = \|x\|z$, then $\|y\| \leq \|x\|$, hence
$$
\|x\|f(z) = f(y) \leq \|x\|^2
$$
$$
\Rightarrow f(z) \leq \|x\|
$$... |
4,036,558 | <p><span class="math-container">$f(x)=e^x(x^2+x)$</span>, derive <span class="math-container">$\dfrac{d^n\,f(x)}{dx^n}$</span></p>
<p>may use Leibniz formula but i'm not sure:(</p>
| Z Ahmed | 671,540 | <p>By Newton-Lebniz formula for <span class="math-container">$D^n[u(x) v(x)]$</span>, we het
<span class="math-container">$$f(x)=(x^2+x)e^x \implies D^n[e^x(x^2+x)]= (D^n e^x) (x^2+x)+ {n\choose 1} (D^{n-1} e^x) D(x^2+x)+{n \choose 2} (D^{n-2} e^x) D^2(x^2+x)+0$$</span>
<span class="math-container">$$f^{n}(x)=e^x(x^2+x... |
2,621 | <p>Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).</p>
<p>As a converse of the above, let ... | WWright | 249 | <p>EDIT: a previous comment posted the same answer, I just noticed</p>
<p>You could use montecarlo method to approximate $\pi$.
You basically define a square of side length 2 and inscribe a circle of radius 1 in it. Let the center of the circle be at the origin.
Use a random number generator to pick an x-coordinate be... |
1,357,922 | <p>How can I find the period of real valued function satisfying <span class="math-container">$f(x)+f(x+4)=f(x+2)+f(x+6)$</span>?</p>
<p>Note: Use of recurrence relations not allowed. Use of elementary algebraic manipulations is better!</p>
| Michael Burr | 86,421 | <p>Observe, since
$$
f(x)+f(x+4)=f(x+2)+f(x+6),
$$
we can substitute $x+2$ for $x$ to get
$$
f(x+2)+f(x+6)=f(x+4)+f(x+8).
$$</p>
<p>Equating these, we know that $f(x)=f(x+8)$.</p>
|
1,357,922 | <p>How can I find the period of real valued function satisfying <span class="math-container">$f(x)+f(x+4)=f(x+2)+f(x+6)$</span>?</p>
<p>Note: Use of recurrence relations not allowed. Use of elementary algebraic manipulations is better!</p>
| JimmyK4542 | 155,509 | <p>You are given that $f(x)+f(x+4) = f(x+2)+f(x+6)$ for all $x \in \mathbb{R}$. </p>
<p>Replace $x$ with $x+2$ to get $f(x+2)+f(x+6) = f(x+4)+f(x+8)$ for all $x \in \mathbb{R}$. </p>
<p>Thus, $f(x)+f(x+4) = f(x+2)+f(x+6) = f(x+4)+f(x+8)$ for all $x \in \mathbb{R}$. </p>
<p>Subtract $f(x+4)$ from the left and right s... |
298,791 | <blockquote>
<p>If a ring $R$ is commutative, I don't understand why if $A, B \in R^{n \times n}$, $AB=1$ means that $BA=1$, i.e., $R^{n \times n}$ is Dedekind finite.</p>
</blockquote>
<p>Arguing with determinant seems to be wrong, although $\det(AB)=\det(BA ) =1$ but it necessarily doesn't mean that $BA =1$.</p>
... | Andreas Caranti | 58,401 | <p>I believe arguing with the determinant works, as $1 = A B$ implies $1 = \det(A B) = \det(A) \det(B)$, so $\det(A) \in R$ is invertible, and $A$ is.</p>
<p><strong>PS</strong> I believe this argument is implicit in @YACP comment to the original post.</p>
|
4,634,180 | <p><span class="math-container">$$\int \frac{\sin^2(x)dx}{\sin(x)+2\cos(x)}$$</span></p>
<p>I tried to use different substitutions such as <span class="math-container">$t=\cos(x)$</span>, <span class="math-container">$t=\sin(x)$</span>, <span class="math-container">$t=\tan(x)$</span>, and after expressing <span class="... | Sofia Ibragimova | 1,146,983 | <p>After applying the method with rational fractions i’ve got three fractions in the integral and calculated it separately so that’s the answer i got in the end
-1/5Cos(x)-2/5Sin(x)-4/5^2,5ln(|2/5^0,5tg(x/2)-1/5^0,5-1/(2/5^0,5tg(x/2)-1/5^0,5+1)|+constant
I’m sorry for giving the answer in such an awkward form, but i’ve... |
764,632 | <p>The question is this :</p>
<p>$$\lim_{x\to-\infty} {\sqrt{x^2+x}+\cos x\over x+\sin x}$$</p>
<p>The solution is $-1$ and this seems to be only obtained from the change variable strategy, such as $t=-x$.</p>
<p>However, I have no idea why this isn't just solved by simply eliminating $x$ in numerator and denominato... | Karolis Juodelė | 30,701 | <p>You are thinking to do $\sqrt{x^2 + x} = x\sqrt{1+\frac 1 {x^2}}$, however, $\sqrt{x^2 + x}$ and $\sqrt{1+\frac 1 {x^2}}$ are (defined to be) positive, so how can you do this with a negative $x$?</p>
|
764,632 | <p>The question is this :</p>
<p>$$\lim_{x\to-\infty} {\sqrt{x^2+x}+\cos x\over x+\sin x}$$</p>
<p>The solution is $-1$ and this seems to be only obtained from the change variable strategy, such as $t=-x$.</p>
<p>However, I have no idea why this isn't just solved by simply eliminating $x$ in numerator and denominato... | DeepSea | 101,504 | <p>$t = -x$ gives: $L$ = $-\displaystyle \lim_{t \to \infty} \dfrac{\sqrt{t^2 - t} + \cos t}{t + \sin t} = -\displaystyle \lim_{t \to \infty} \dfrac{\sqrt{1 - \dfrac{1}{t}} + \dfrac{\cos t}{t}}{1 + \dfrac{\sin t}{t}} = -1$ because $\displaystyle \lim_{t \to \infty} \dfrac{\cos t}{t} = \displaystyle \lim_{t \to \infty}... |
20,982 | <p>Let E be an elliptic curve over a finite field k (char(k) is not 2) be given by y^2 = (x-a)(x-b)(x-c) where a,b and c are distinct and are in k. Then why is (c,0) is in [2]E(k) iff c-a and c-b is a square in k-{0}? </p>
| Pete L. Clark | 1,149 | <p>For an elliptic curve over any field $K$ of characteristic different from $2$, the Kummer sequence reads</p>
<p>$0 \rightarrow E(K)/2E(K) \stackrel{\iota}{\rightarrow} H^1(K,E[2]) \rightarrow H^1(K,E)[2] \rightarrow 0$.</p>
<p>In particular, $\iota$ is an injection. Therefore $P \in 2E(K) \iff$ the image $[P]$
o... |
20,982 | <p>Let E be an elliptic curve over a finite field k (char(k) is not 2) be given by y^2 = (x-a)(x-b)(x-c) where a,b and c are distinct and are in k. Then why is (c,0) is in [2]E(k) iff c-a and c-b is a square in k-{0}? </p>
| Robin Chapman | 4,213 | <p>Pete's is certainly the right way to look at this problem,
but in this example one can argue naively using explicit
calculations. One loses no generality by assuming $c=0$
(by replacing $x$ by $x+c$). Then using the duplication formula,
one finds that the solutions of $[2]P = (0,0)$ are $P=(uv,uv(u+v))$
where $u$ an... |
2,081,792 | <p>I learnt the derivation of the distance formula of two points in first quadrant I.e., $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where it is easy to find the legs of the hypotenuse (distance between two points) since the first has no negative coordinates and only two axes ($x$ coordinate and $y$ coordinate). while f... | Logan Luther | 347,317 | <p>Consider the following diagram:<img src="https://i.stack.imgur.com/RiihX.jpg" alt="Distance is highlighted in red."></p>
<p>Now to find the wanted distance,By the Pythagorean theorem,we need to know the size of the Edges BC and AC.</p>
<p>Suppose A and B to be:$$A(x_1,y_1),B(x_2,y_2)$$
And to find AC ,We need to s... |
3,089,493 | <p>Calculate the volume between <span class="math-container">$x^2+y^2+z^2=8$</span> and <span class="math-container">$x^2+y^2-2z=0$</span>. I don't know how to approach this but I still tried something:</p>
<p>I rewrote the second equation as: <span class="math-container">$x^2+y^2+(z-1)^2=z^2+1$</span> and then combin... | G Cab | 317,234 | <p>A geometric view of the problem will be much of help to solve it.</p>
<p>One is a sphere of radius <span class="math-container">$\sqrt{8}$</span> centered at the origin. </p>
<p>The other is a paraboloid of revolution, given by the revolution of <span class="math-container">$z=x^2/2$</span>
around the <span clas... |
131,051 | <p>So we want to find an $u$ such that $\mathbb{Q}(u)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. I obtained that if $u$ is of the following form: $$u=\sqrt[6]{2^a5^b}$$Where $a\equiv 1\pmod{2}$, and $a\equiv 0\pmod{3}$, and $b\equiv 0\pmod{2}$ and $ b\equiv 1\pmod{3}$. This works since $$u^3=\sqrt{2^a5^b}=2^{\frac{a-1}{2}}5^{\... | Gerry Myerson | 8,269 | <p>The field has degree 6 over the rationals. Any element $w$ of degree 6 will generate the field. </p>
<p>Now, every element of the field has degree 1, 2, 3, or 6. The only elements of degree 1 are the rationals. The only elements of degree 2 are those of the form $a+b\sqrt2$ (although it takes some work to check thi... |
139,385 | <p>Can anyone help me prove if $n \in \mathbb{N}$ and is $p$ is prime such that $p|(n!)^2+1$ then $(p-1)/2$ is even?</p>
<p>I'm attempting to use Fermats little theorem, so far I have only shown $p$ is odd.</p>
<p>I want to show that $p \equiv 1 \pmod 4$</p>
| marlu | 26,204 | <p>If $p$ divides $(n!)^2+1$, then $(n!)^2 \equiv -1 \pmod p$, so $n!$ has order $4$ in $\mathbb F_p^\times$. By Lagrange's theorem, 4 divides the order of $\mathbb F_p^\times$ which is $p-1$, hence $p \equiv 1 \pmod 4$.</p>
|
1,499,949 | <p>Prove that for all event $A,B$</p>
<p>$P(A\cap B)+P(A\cap \bar B)=P(A)$</p>
<p><strong>My attempt:</strong></p>
<p>Formula: $\color{blue}{P(A\cap B)=P(A)+P(B)-P(A\cup B)}$</p>
<p>$=\overbrace {P(A)+P(B)-P(A\cup B)}^{=P(A\cap B)}+\overbrace {P(A)+P(\bar B)-P(A\cup \bar B}^{=P(A\cap \bar B)})$</p>
<p>$=2P(A)+\un... | Elekko | 101,668 | <p>They are the same, it's the exponent representation of "root operator".
Example $4^{\frac{1}{2}}=\sqrt[2]{4}=2$</p>
|
83,512 | <p>Question: (From an Introduction to Convex Polytopes)</p>
<p>Let $(x_{1},...,x_{n})$ be an $n$-family of points from $\mathbb{R}^d$, where $x_{i} = (\alpha_{1i},...,\alpha_{di})$, and $\bar{x_{i}} =(1,\alpha_{1i},...,\alpha_{di})$, where $i=1,...,n$. Show that the $n$-family $(x_{1},...,x_{n})$ is affinely independe... | Agustí Roig | 664 | <p>So, we want to prove that these two statements are equivalent:</p>
<ul>
<li><p>(a) The <em>points</em> $x_1, \dots , x_n \in \mathbb{R}^d$ are <em>affinely</em> independent.</p></li>
<li><p>(b) The <em>vectors</em> $\overline{x}_1, \dots , \overline{x}_n \in \mathbb{R}^{d+1}$ are <em>linearly</em> independent.</p><... |
380,452 | <p>A relation R is defined on ordered pairs of integers as follows :</p>
<p>$(x,y) R(u,v)$ if $x<u$ and $y>v.$ </p>
<p>Then R is </p>
<ol>
<li><p>Neither a Partial Order nor an Equivalence relation</p></li>
<li><p>A Partial Order but not a Total Order</p></li>
<li><p>A Total Order </p></li>
<li><p>An Equivalen... | rschwieb | 29,335 | <p>I think the touchstone for understanding direct limits is understanding <a href="http://en.wikipedia.org/wiki/Directed_union#Examples" rel="nofollow">directed unions</a>.</p>
<p>A collection $C$ of sets is directed if for every $X,Y\in C$, there exists $Z\in C$ containing both $X$ and $Y$. This becomes a direct sys... |
596,005 | <p>Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation
$$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$
for all points $(x,y) \in \mathbb{R}^2$ if and only if for all $(x,y)\in \mathbb{R}^2$ and $t \in \mathbb{R}$ we have:
$$f(x, y + 2t) + f(x, y) = f(x + t,y + t) ... | Brian Rushton | 51,970 | <p>Try setting $u=x+y,v=x-y$, and notice that the inverse equations are $x=(u+v)/2,y=(u-v)/2$. This changes your equation to $f_{uv}=0$, which makes the problem much easier.</p>
<p><em>Edit</em>: Apparently we have some critics. Do what I said to do; the solution of this equation is any function of the form $f=g(u)+h(... |
2,249,020 | <p><a href="https://i.stack.imgur.com/L7PXf.jpg" rel="nofollow noreferrer">The Math Problem</a></p>
<p>I have issues with finding the Local Max and Min, and Abs Max and Min, after I find the Critical Point. How do I do this problem in its entirety? </p>
| Chappers | 221,811 | <p>The scalar product of vectors $a=(a_1,a_2)$ and $b=(b_1,b_2)$ is given by the two formulae (provable equivalent using the cosine rule, see <a href="https://math.stackexchange.com/a/2227712/221811">here</a>)
$$ a \cdot b = a_1b_1+a_2b_2 = \sqrt{a_1^2+a_2^2}\sqrt{b_1^2+b_2^2} \cos{\theta}, $$
where $\theta$ is the ang... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | Ty. | 760,219 | <p>Consider the following for large n and finite x:
<span class="math-container">$$e^{\frac{x^2}{n^2}} \approx 1+\frac{x^2}{n^2}$$</span>
Therefore, rewrite the limit as:
<span class="math-container">$$\lim_{n \to \infty} {\left(e^{\frac{x^2}{n^2}}\right)}^{\frac{n}{2}}$$</span>
<span class="math-container">$$=\lim_{n... |
44,391 | <p>The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF. </p>
<p>Is there a similar equation for getting the foci or directrix for a general ellipse, parabola, hyperbola from ABCDEF? Please assume that a non-degenera... | ccorn | 75,794 | <p>The appropriate setting for this is the <em>complex projective plane</em>.
While declaring some symbols, I will talk a tiny bit about that,
but do not mistake this as a proper introduction to the subject.</p>
<p>In the projective plane,
we use triples $(X:Y:Z)$ of homogenous coordinates for points,
not all equal to... |
537,228 | <p>I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems quite likely to me that something like that can be constructed. I am an amateur however and my scope is small. I have... | Did | 6,179 | <p>This has been asked before:</p>
<ul>
<li><a href="https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical">Is there an introduction to probability theory from a structuralist/categorical perspective?</a></li>
</ul>
<p>And for the notion of product:</... |
2,998,189 | <p>I'm looking at a matrix operator in which <span class="math-container">$T \in \mathcal{L}(\mathbb{R}^2)$</span> by <span class="math-container">$T(x,y) = (x, -y)$</span>. So its basis is <span class="math-container">$ \mathcal{M}(T) = \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$</span>. </p>
<p>How do I show ... | Fred | 380,717 | <p>We have <span class="math-container">$\mathcal{M}(T)=\mathcal{M}(T)^t$</span>, hence <span class="math-container">$\mathcal{M}(T)$</span> is symmetric, thus <span class="math-container">$T$</span> is self-adjoint.</p>
|
3,856,567 | <p>I’m new to number theory and I’m solving questions in the textbook one by one.
Here is one :
If <span class="math-container">$m\geq 1$</span> and <span class="math-container">$n\geq2$</span> , which both of them are natural numbers , prove this statement:</p>
<p><span class="math-container">$$(n-1)^2 | (n^m-1) \iff ... | Alessio K | 702,692 | <p>You want to show that <span class="math-container">$$(n-1)\mid(n^{m-1}+n^{m-2}+\ldots+n+1)\qquad\iff\qquad (n-1)|m$$</span></p>
<p>But <span class="math-container">$n-1\equiv 0 \pmod {n-1}$</span>, so <span class="math-container">$n\equiv 1 \pmod {n-1}.$</span> Now use this to evaluate<span class="math-container">$(... |
3,856,567 | <p>I’m new to number theory and I’m solving questions in the textbook one by one.
Here is one :
If <span class="math-container">$m\geq 1$</span> and <span class="math-container">$n\geq2$</span> , which both of them are natural numbers , prove this statement:</p>
<p><span class="math-container">$$(n-1)^2 | (n^m-1) \iff ... | sirous | 346,566 | <p>Answer in more details:</p>
<p>Conder <span class="math-container">$n^{m-1}+n^{m-2} + . . . +n+1$</span> which has m terms and must also be divisible by <span class="math-container">$n-1$</span>. Now add <span class="math-container">$m-1$</span> times (-1) and (m-1) times (+1) you get:</p>
<p><span class="math-conta... |
41,174 | <p>I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a refere... | Richard Wentworth | 9,867 | <p>With regard to $PGL(n,{\mathbb C})$ vs $SL(n,{\mathbb C})$, you're right that the $n=2$ case generalizes. On a Riemann surface, the moduli of rank $n$ degree $d$ semistable Higgs bundles with fixed determinant is a $n^{2g}$ cover of a component of the moduli space of $PGL(n,{\mathbb C})$ representations of the fu... |
3,125,093 | <p>Let us remember, the conditions to apply L'Hôpital's Rule:</p>
<p>Let suppose:</p>
<p><span class="math-container">$f(x)$</span> and <span class="math-container">$g(x)$</span> are real and differentiable for all <span class="math-container">$x\in (a,b)$</span> </p>
<p>1-) <span class="math-container">$ \lim_{x\t... | jmerry | 619,637 | <p>Yes, we can do it.</p>
<p>The cancellation laws immediately get us <span class="math-container">$i=jk$</span> and <span class="math-container">$k=ij$</span>. Multiply the first by <span class="math-container">$i$</span> on the right, and <span class="math-container">$i^2=jki$</span>, leading to <span class="math-co... |
3,975,832 | <p>I think the following claim is clearly correct, but I cannot prove it.</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be sets. If <span class="math-container">$f:A \times B \to \mathbb{R}$</span> satisfies <span class="math-container">$f(a, b) \leq C_a$</s... | Hank Igoe | 806,514 | <p>Let <span class="math-container">$f(a,b)=min(a,b).$</span> Then <span class="math-container">$f$</span> will be bounded by <span class="math-container">$min(C_a, C_b)$</span>, but <span class="math-container">$f(a,a)$</span> will diverge to <span class="math-container">$\infty$</span>.</p>
|
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Thomas Kalinowski | 12,674 | <p>In <a href="https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=0198534450" rel="nofollow noreferrer" title="Alexandre Borovik and Ali Nesin: Groups of Finite Morley Rank (Oxford Logic Guides, 26)">groups of finite Morley rank</a> there is a Sylow theory for the prime <span class="math-containe... |
1,613,863 | <p>How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$? </p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $\binom{n}{2}=\dfrac{n(n-1)}{2}, \binom{4}{2} = 6, \binom{n+2}{4} = \dfrac{(n+2)!}{4!\cdot (n-2)!}= \dfrac{(n-1)n(n+1)(n+2)}{24}$</p>
|
840,700 | <p>I have to show that the following function $f:[0,1]\rightarrow\mathbb{R}$ is Riemann Integrable:</p>
<p>$$f(x) =
\left\{
\begin{array}{ll}
1 & \mbox{if } x = \frac{1}{n} \\
0 & \mbox{otherwise}
\end{array}
\right.$$</p>
<p>For the upper and lower Riemann sum I am using the following definitions:</p>... | YTS | 126,222 | <p>Try the following: </p>
<p>The set $F=\{x\in [0,1]: f(x)>\epsilon \}$ is finite for every $\epsilon>0$. Then you can form a partition such that if an interval contains some $x\in F$ then it have no other. Finally you can choose the partition such that the sum of interval who contains some $x\in F$ is $<\ep... |
2,450,007 | <p>Show that if $x\in Q^p$, then there exists $-x\in Q^p$ where $$Q^p=\{a_{-l}p^{-l}+a_{-l+1}p^{-l+1}+...|l\in Z,a_i\in\{0,1,...,p-1\}\}$$ and p is a prime number.</p>
<p>Actually I don't quite understand p-adic numbers and how addition and multiplication work in this number system. For this question, I think I need ... | Alex Ravsky | 71,850 | <p>For instance, the discrete topology on the set $\Bbb Z$ or $\Bbb R$ can be generated by a discrete uniformity with a base $\{\Delta\}$ or by a uniformity with a base $\{U_n:n\in\Bbb N\}$, where $U_n=\{(x,y): x=y$ or $x,y\ge n\}$.</p>
|
1,327,644 | <p>Using EM summation formula estimate
$$
\sum_{k=1}^n \sqrt k
$$</p>
<p>up to the term involving $\frac{1}{\sqrt n}$</p>
<p>My attempt is
$$
\sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 2 (\sqrt n -1)+ \frac{1}{24} (\frac{1}{\sqrt n} -1) + \int_1^n P_{2k+1}(x)f^{(2k+1)}(x)dx
$$
I am not s... | Mark Viola | 218,419 | <p>The correct expansion is given by</p>
<p>$$\begin{align}
\sum_{i=1}^n f(i) &= \int_1^n f(x)dx + B_1 [f(n)+f(1)]\\\\
& + \sum_{k=1}^m \frac{B_{2k}}{(2k)!} \left(f^{(2k-1)}(n)-f^{(2k-1)}(1)\right)\\\\
& +\frac{1}{(2m+1)!}\int_1^n P_{2m+1}(x)f^{(2m+1)}(x)
\end{align}$$</p>
<p>For $f(x)=x^{1/2}$ and $m=1$,... |
1,991,238 | <p>How can I integrate this? $\int_{0}^{1}\frac{\ln(x)}{x+1} dx $</p>
<p>I've seen <a href="https://math.stackexchange.com/questions/108248/prove-int-01-frac-ln-x-x-1-d-x-sum-1-infty-frac1n2">this</a> but I failed to apply it on my problem.</p>
<p>Could you give some hint?</p>
<p>EDIT : From hint of @H.H.Rugh, I'v... | Antonio Vargas | 5,531 | <p>In terms of the <a href="https://en.wikipedia.org/wiki/Harmonic_number">harmonic numbers</a> $H_n$, your sequence is</p>
<p>$$
s_n = H_{F_{n+1}} - H_{F_n-1}
$$</p>
<p>As $n \to \infty$ it's known that $H_n = \log n + \gamma + o(1)$, so</p>
<p>$$
\begin{align}
s_n &= \log F_{n+1} + \gamma + o(1) - \log(F_n-1) ... |
4,292,618 | <p>I have the following function <span class="math-container">$$\frac{1}{1+2x}-\frac{1-x}{1+x} $$</span>
How to find equivalent way to compute it but when <span class="math-container">$x$</span> is much smaller than 1? I assume the problem here is with <span class="math-container">$1+x$</span> since it probably would b... | Vasile | 959,234 | <p>If the therms in <span class="math-container">$x^2$</span> can be neglegted:</p>
<p><span class="math-container">$$\frac{1}{1+2x}\approx\frac{1-4x^2}{1+2x}=1-2x$$</span>
<span class="math-container">$$\frac{1-x}{1+x}\approx\frac{(1-x)(1-x^2)}{1+x}=(1-x)^2\rightarrow$$</span>
<span class="math-container">$$f(x)\appro... |
3,352,834 | <p><span class="math-container">$A^2 + A - 6I = 0$</span></p>
<p>A= <span class="math-container">$\begin{bmatrix}a & b\\c & d\end{bmatrix}$</span></p>
<p>I was asked to find
<span class="math-container">$a + d$</span>, and <span class="math-container">$ad - bc$</span></p>
<p><span class="math-container">$a+d... | Dietrich Burde | 83,966 | <p>"<em>I was asked to find <span class="math-container">$a + d$</span>, and <span class="math-container">$ad - bc$</span></em>". </p>
<p>Note that these are the trace of <span class="math-container">$A$</span> and the determinant of <span class="math-container">$A$</span>. Here we have several possibilities under the... |
3,260,911 | <p>I am currently struggling with the following exercise:</p>
<blockquote>
<p>Let <span class="math-container">$B$</span> be a Banach space and <span class="math-container">$C, D \subset B$</span> closed subspaces of <span class="math-container">$B$</span>.<br>
There is a <span class="math-container">$M \in ]0, \i... | Jonathan Hole | 661,524 | <p>Consider the quotient maps <span class="math-container">$p: B\rightarrow B/C$</span> and <span class="math-container">$q: D \rightarrow D/D\cap C$</span>. We claim that <span class="math-container">$p(D)$</span> is closed in <span class="math-container">$B/C$</span>. Indeed if <span class="math-container">$\{p(d_n)\... |
1,492,660 | <p>I'm teaching a course on discrete math and came across <a href="http://ac.els-cdn.com/0097316573900204/1-s2.0-0097316573900204-main.pdf?_tid=700c69c2-78e3-11e5-9825-00000aacb35e&acdnat=1445535573_08d35be15f0f7d7d939fc2800d9be60b" rel="nofollow">a paper related to the Hadwiger-Nelson problem</a>. The question ask... | Cheerful Parsnip | 2,941 | <p>It does not require choice. It is true that you have to pick an isometry from each equivalence class to the equivalence class containing the origin. This can be done by translating some chosen point in the equivalence class to the origin. You can do this constructively by well-ordering $\mathbb Q^2$ and choosing the... |
749,714 | <p>Does anyone know how to show this preferable <strong>without</strong> using modular</p>
<p>For any prime $p>3$ show that 3 divides $2p^2+1$ </p>
| Bill Dubuque | 242 | <p>Without using mod: note <span class="math-container">$\, 2p^2+1 = 2(p^2-1)+3\,$</span> so it suffices to show that <span class="math-container">$\,3\mid p^2-1.\,$</span> One of the <span class="math-container">$\,3\,$</span> consecutive integers <span class="math-container">$\,p-1,\,p,\,p+1$</span> <a href="https://... |
1,980,606 | <p>Let $f:[0, 1] \to \mathbb{R}$ differentiable in $[0, 1]$ and $|f'(x)| \leq\frac{1}{2}$ for all $x \in [0, 1]$. If $a_n = f(\frac{1}{n})$, show that $\lim_{n \to \infty} a_n$ exist (Hint: Cauchy).</p>
<p>Can you help me? Thanks.</p>
| Jacky Chong | 369,395 | <p>Hint: Observe by the mean value theorem we have
\begin{align}
\left|f\left(\frac{1}{n}\right)-f\left(\frac{1}{m}\right)\right| = |f'(\xi)|\left|\frac{1}{n}-\frac{1}{m}\right| \leq \frac{1}{2}\left|\frac{1}{n}-\frac{1}{m}\right|
\end{align}
where $\xi \in [1/n, 1/m] \subset [0, 1]$.</p>
|
3,601,865 | <p>What does the correspondence theorem (or 4th isomorphism theorem for rings) for rings mean and how is it used? That is, why do we care about it?</p>
<hr>
<p>Edit:</p>
<p>My version of the correspondence theorem:</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a ring and <span class="math-con... | QuantumSpace | 661,543 | <p>The correspondence says that if <span class="math-container">$R$</span> is a ring and <span class="math-container">$I$</span> is a two-sided ideal of <span class="math-container">$R$</span>, that the following map is a bijection:</p>
<p><span class="math-container">$$\{\mathrm{\ ideals \ of \ R \ containing \ I}\}\... |
3,717,506 | <p>I am reading some text about even functions and found this snippet:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be an integrable even function. Then,</p>
<p><span class="math-container">$$\int_{-a}^0f(x)dx = \int_0^af(x)dx, \forall a \in \mathbb{R}$$</span></p>
<p>and therefore,</p>
<p><span c... | A. Kriegman | 649,089 | <p>You can model this as a Markov chain and there are known techniques for how to solve these problems. I'll explain how we can solve this example.</p>
<p>Let <span class="math-container">$p_n$</span> be the probability of having a total of <span class="math-container">$n$</span> after a large number of rolls. If we've... |
2,305,689 | <blockquote>
<p>If $x^6-12x^5+ax^4+bx^3+cx^2+dx+64=0$ has positive roots then find $a,b,c,d$.</p>
</blockquote>
<p>I did something but that don't deserve to be added here, but what I thought before doing that is following:</p>
<ol>
<li>For us, Product and Sum of roots are given.</li>
<li>Roots are positive.</li>
<l... | Michael Rozenberg | 190,319 | <p>Let $x_i$ be our roots.</p>
<p>Now, by AM-GM $$2=\frac{\sum\limits_{i=1}^6x_i}{6}\geq\sqrt[6]{\prod_{i=1}^6x_i}=2,$$
which says that all $x_i=2$.</p>
<p>Thus, $$a=2^2\cdot\frac{6(6-1)}{2}=60,$$ $$b=-2^3\cdot\frac{6(6-1)(6-2)}{6}=-160,$$
$$c=2^4\cdot\frac{6(6-1)(6-2)(6-3)}{24}=240$$ and
$$d=-2^5\cdot\frac{6(6-1)(6-... |
2,247,968 | <blockquote>
<p>$a,b$ are elements in a group $G$. Let $o(a)=m$ which means that $a^m=e$, $\gcd(m,n)=1$ and $(a^n)*b=b*(a^n)$. Prove that $a*b=b*a$.</p>
</blockquote>
<p><em>Hint: try to solve for $m=5,n=3$.</em></p>
<p>I am stuck in this question and can't find an answer to it, can anyone give me some hints?</p>... | Community | -1 | <p>If the minimum of $f$ is at $0$, then there's nothing to prove. So let's suppose that the minimum of $f$ is attained at some $a \in (0,2]$. Suppose that $f(a) < 0$. By continuity, there is $\epsilon > 0$ such that $f(x) < 0$ for all $x \in (-\epsilon + a, a)$. We have $f'(x) > 0$ for all $x \in (-\epsilo... |
4,043,787 | <p>I have <span class="math-container">$$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\
0 & |x|>n .
\end{cases}$$</span></p>
<p>Why cannot be dominated by an integrable function <span class="math-container">$g$</span> by the Dominated Convergence Theorem? I am also wondering what exactly it... | WoolierThanThou | 686,397 | <p>Well, say <span class="math-container">$g(x)\geq |f_n(x)|$</span> for every <span class="math-container">$x$</span> and <span class="math-container">$n$</span>. Then,
<span class="math-container">$$
g(x)\geq \sup_n|f_n(x)|=\begin{cases} \frac{1}{n} & |x|\in (n-1,n] \\ 0 & x=0 \end{cases}
$$</span>
and thus,
... |
2,076,908 | <blockquote>
<p><strong>Question:</strong> Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$.</p>
</blockquote>
<p>Generally we proceed by setting up the equation
$$a_1e^x + a_2xe^x+a_3x^2e^x=0_f,$$
which simplifies to $$e^x(a_1+a_2x+a_3x^2)=0_f,$$ and furthermore to
$$a_1+a_2x+a_3x^2=... | copper.hat | 27,978 | <p>Suppose $a_1e^x + a_2xe^x+a_3x^2e^x=0 $ for all $x$.</p>
<p>Setting $x=0$ shows that $a_1 = 0$.
Now note that $a_2xe^x+a_3x^2e^x=0 $ for all $x$ and hence
$a_2e^x+a_3xe^x=0 $ for all $x \neq 0$. Taking limits as $x \to 0$ shows
that $a_2 = 0$, and setting $x=1$ shows that $a_3 = 0$.</p>
|
1,665,064 | <p>I have a vector quadratic equation of the form
$\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$<br>
where $\boldsymbol{A}$ is symmetric and for my particular case, $\boldsymbol{x} \in \mathbb{R}^{2}$. I know that the solution for this system (if it exists) can be found ... | hardmath | 3,111 | <p>Since $A$ is (real) symmetric, it has real eigenvalues $\lambda_1$ and $\lambda_2$ which can be found analytically (quadratic formula). Therefore a real change of variables (change of basis to eigenvector components) can be made to put the equation into the form of a standard conic section:</p>
<p>$$ \lambda_1 u^2... |
248,706 | <p>Let $X$ be a compact connected manifold. Since $\mathbb T^1$ is an Eilenberg-MacLane space $K(\mathbb Z,1)$, it follows that for every morphism $\varphi\colon\pi_1(X)\to\pi_1(\mathbb T^1)$ there is a continuous map $f\colon X\to\mathbb T^1$ such that $\varphi$ coincides with the induced morphism $f_*$.</p>
<p>Now a... | William of Baskerville | 50,457 | <p>Finally I managed to work out a solution for compact Lie groups $G$.</p>
<p>By the structure theory of compact groups, $G$ can be expressed as a topological semi-direct product $G=\mathbb T^n\ltimes K$ of a torus $\mathbb T^n$ and a semi-simple Lie group $K$. Let $p\colon G\to\mathbb T^n$ be the projection morphism... |
1,549,138 | <p>I have a problem with this exercise:</p>
<p>Proove that if $R$ is a reflexive and transitive relation then $R^n=R$ for each $n \ge 1$ (where $R^n \equiv \underbrace {R \times R \times R \times \cdots \times R} _{n \ \text{times}}$).</p>
<p>This exercise comes from my logic excercise book. The problem is that I've ... | Bartłomiej Sługocki | 293,948 | <p>Sorry, that was my mistake. I haven't read my book carefully. There's small paragraph saying: for relations we define $R^1\equiv R$ and $R^{n+1} \equiv R^n\circ R$.</p>
<p>Thanks for help. Next time I will read all the definitions carefully before I post a problem here :)</p>
|
3,325,977 | <p>The function definition is as follows:</p>
<p><span class="math-container">$ f(z) = $</span> the unique v such that <span class="math-container">$|v-z|=|v-u|=|v|$</span> for some fixed <span class="math-container">$u=a+ib$</span>.</p>
<p>For this question, I'm able to understand the basic geometry of the question ... | nonuser | 463,553 | <p>Hint: </p>
<p>Remember that <span class="math-container">$$|z-\alpha| = |z-\beta|$$</span> is a set of <span class="math-container">$z$</span> that are equaly apart from <span class="math-container">$\alpha $</span> and <span class="math-container">$\beta$</span> i.e. perpendicular bisector of segment between <span... |
3,325,977 | <p>The function definition is as follows:</p>
<p><span class="math-container">$ f(z) = $</span> the unique v such that <span class="math-container">$|v-z|=|v-u|=|v|$</span> for some fixed <span class="math-container">$u=a+ib$</span>.</p>
<p>For this question, I'm able to understand the basic geometry of the question ... | Andrei | 331,661 | <p>Rewrite your condition as <span class="math-container">$$|v-z|=|v-u|=|v-0|$$</span>This means that <span class="math-container">$v$</span> is the point at equal distances from <span class="math-container">$z$</span>, <span class="math-container">$u$</span>, and <span class="math-container">$0$</span>. If <span class... |
2,668,447 | <p>Let <span class="math-container">$F$</span> be a subfield of a field <span class="math-container">$K$</span> and let <span class="math-container">$n$</span> be a positive integer. Show that a nonempty linearly-independent subset <span class="math-container">$D$</span> of <span class="math-container">$F^n$</span> rem... | Joshua Mundinger | 106,317 | <p>This result does not require determinants. Instead, we just need that <span class="math-container">$K$</span> is a vector space over <span class="math-container">$F$</span>, so that it has some basis <span class="math-container">$B$</span> over <span class="math-container">$F$</span>. Let <span class="math-container... |
2,668,447 | <p>Let <span class="math-container">$F$</span> be a subfield of a field <span class="math-container">$K$</span> and let <span class="math-container">$n$</span> be a positive integer. Show that a nonempty linearly-independent subset <span class="math-container">$D$</span> of <span class="math-container">$F^n$</span> rem... | Qiaochu Yuan | 232 | <p>This is a comment. There is something interesting to consider here in the generalization of this question to the case of modules over commutative rings. That is, suppose <span class="math-container">$f : R \to S$</span> is a morphism of commutative rings, and suppose we are given a collection of linearly independent... |
3,063,742 | <p>Consider the closed interval <span class="math-container">$[0, 1]$</span> in the real line <span class="math-container">$\mathbb{R}$</span> and the product space <span class="math-container">$([0, 1]^{\mathbb{N}}, τ ),$</span></p>
<p>where <span class="math-container">$τ$</span> is a topology on <span class="math-c... | Adam Higgins | 454,507 | <p>Hint: It might be easier to show that both <span class="math-container">$A,B$</span> are closed in <span class="math-container">$X$</span>, and then since <span class="math-container">$X = A \cup B$</span>, we immediately have that <span class="math-container">$A,B$</span> are both open in <span class="math-containe... |
383,037 | <p>I was going through "Convergence of Probability Measures" by Patrick Billingsley. In Section 1: I encountered the following problem:</p>
<p><strong>Show that inequivalent metrics can give rise to the same class of Borel sets.</strong></p>
<p>My idea is that the 2 metrics generate different topologies but the Sigma... | user642796 | 8,348 | <p><strong>Theorem.</strong> Given a separable completely metrisable (Polish) space $( X , \mathcal{T} )$, and any Borel $B \subseteq X$ you can define a new Polish topology $\mathcal{T}_B$ on $X$ which is finer than, and has the same Borel subsets as, the original topology, and in which $B$ is a clopen set. </p>
<p>... |
785,188 | <p>I found a very simple algorithm that draws values from a Poisson distribution from <a href="http://www.akira.ruc.dk/~keld/research/javasimulation/javasimulation-1.1/docs/report.pdf" rel="nofollow">this project.</a></p>
<p>The algorithm's code in Java is:</p>
<pre><code>public final int poisson(double a) {
... | craig tovey | 460,645 | <p>Yes, the previous answer is correct. This is simply a version of Knuth's algorithm. It gets slower the larger the parameter lambda. </p>
|
265,537 | <p>I have a set of inequalities</p>
<pre><code>Cos[a]Cos[b]>=Cos[t-a]Cos[b]&&Cos[a]Cos[b]>=Cos[t/2]&&Cos[a]Cos[b]>=Sin[t/2]&&a<=t<=Pi
</code></pre>
<p>How to solve this to get a range of values for <code>a,b,t</code>?</p>
| user64494 | 7,152 | <p>Reducing trigonometry to algebra by <code>Reduce[ca*cb >= ((ct^2 - st^2)*ca + 2*ct*st*sa)*cb && cb*ca >= ct && cb*ca >= st && ca^2 + sa^2 == 1 && cb^2 + sb^2 == 1 && ct^2 + st^2 == 1, {ct, st}, Reals]</code>, one obtains a huge and useless output. In order to o... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| ajotatxe | 132,456 | <p>Let $u=\sqrt[3]{38+17\sqrt{5}}$, $v= \sqrt[3]{38-17\sqrt{5}}$</p>
<p>From the equation
$$u+v=n$$
and cubing,
we obtain
$$u^3+v^3+3uv(u+v)=n^3$$
that is
$$76-3n=n^3$$
A root of this equation is $4$. In fact, it has no more real roots.</p>
<p>So we know that $u+v=4$, but we need $u-v$. We also know that $uv=-1$. The... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| Community | -1 | <p>The expression strangely reminds of Cardano's formula for the cubic equation</p>
<p>$$x=\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}4+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}4+\frac{p^3}{27}}}.$$</p>
<p>Identifying, we have </p>
<p>$$q=-76,p=3,$$</p>
<p>and $x$ is a root of $$x^3+3x-76=0.$$
By inspection (try... |
126,739 | <p><strong>I changed the title and added revisions and left the original untouched</strong> </p>
<p>For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the factorials in the denominator of the binomial theorem;
$$\sum _{k=1}^{\infty } \frac{1}{k!} \... | Community | -1 | <p>A way to get this, and also to understand the behavior for other values would be like so (though I do not know if this is not overly indirect):</p>
<p>Recall that
$$
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}
$$
so
$$
x^{m-1}e^x = \sum_{k=0}^{\infty} \frac{x^{k+m-1}}{k!}
$$
Now 'integerate', then
$$
F(x) = \sum_{k... |
1,231,772 | <p>Motivated by Baby Rudin Exercise 6.9</p>
<p>I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges.</p>
<p>My attempt: </p>
<p>$\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty \frac{\sin^2 x}{1+x} \, dx = \int_0^\infty \frac{1}{1... | Jack D'Aurizio | 44,121 | <p>The integrand function is positive and for every $x\in\mathbb{R}^+$ close to an element of $\pi\mathbb{Z}$ we have that $|\cos x\,|$ is close to one. For instance, if the distance between $x$ and $\pi\mathbb{Z}$ is $\leq\frac{\pi}{3}$ we have $|\cos x\,|\geq\frac{1}{2}$. That gives:
$$\int_{0}^{n\pi}\frac{\left|\cos... |
1,231,772 | <p>Motivated by Baby Rudin Exercise 6.9</p>
<p>I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges.</p>
<p>My attempt: </p>
<p>$\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty \frac{\sin^2 x}{1+x} \, dx = \int_0^\infty \frac{1}{1... | Zarrax | 3,035 | <p>You're on the right track. Since $\cos^2(x + {\pi \over 2}) = \sin^2 x$,
$$\int_{0}^{\infty} {\cos^2 x \over 1 + x}\,dx = \int_{{\pi \over 2}}^{\infty} {\cos^2 (x - {\pi \over 2}) \over 1 + (x - {\pi \over 2})}\,dx $$
$$= \int_{{\pi \over 2}}^{\infty} {\sin^2 x \over 1 + (x - {\pi \over 2})}\,dx $$
$$> \int_... |
446,456 | <p>Educators and Professors: when you teach first year calculus students that infinity isn't a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?</p>
| Asaf Karagila | 622 | <p>The symbols $+\infty,-\infty$ (and $\infty$) simply denote a formal symbol which means "larger/small than any real number".</p>
|
3,193,288 | <p>i have the following question. Let <span class="math-container">$\phi_1$</span> and <span class="math-container">$\phi_2$</span> fundamental system solutions on an interval <span class="math-container">$I$</span> for the second order equation
<span class="math-container">$$
y''+a(x)y= 0.
$$</span>
Prove that there ... | Lutz Lehmann | 115,115 | <p>You should have found out that the Wronskian is constant as the coefficient of the first derivative term is zero.</p>
<p>After that, it is just a matter of re-scaling one or both of the solutions to get the Wronski-determinant to have the value 1 at one and thus every point.</p>
<hr>
<p>(<em>Add</em>) Interpretin... |
218,479 | <p>I am trying to evaluate the following integral with Mathematica:</p>
<p><span class="math-container">\begin{align}
I = \int_{0}^{\infty} da \, \frac{e^{-\frac{a ^2}{4s^2}} }{a^2} \mbox{sinc}\left(\tfrac{w}{2} a \right) \delta' \left( \frac{D^2}{a}- a \right),
\end{align}</span>
where the prime on the delta ... | Ulrich Neumann | 53,677 | <p>Here my attempt to solve the integral <code>Integrate[f[a] Derivative[1][DiracDelta][d^2/a - a],{a,0,Infinity}]</code>:</p>
<pre><code>f[a_] := Exp[-a^2/(4 s^2)]/a^2 Sinc[w a/2]
</code></pre>
<p>Substitution <code>u[a]=d^2/a-a</code> (integrationlimits change to u[0]=Infinity],u[Infinity]=-Infinity)</p>
<pre><co... |
218,479 | <p>I am trying to evaluate the following integral with Mathematica:</p>
<p><span class="math-container">\begin{align}
I = \int_{0}^{\infty} da \, \frac{e^{-\frac{a ^2}{4s^2}} }{a^2} \mbox{sinc}\left(\tfrac{w}{2} a \right) \delta' \left( \frac{D^2}{a}- a \right),
\end{align}</span>
where the prime on the delta ... | AestheticAnalyst | 71,610 | <p>Let's talk about the Dirac <span class="math-container">$\delta$</span>-"function". Strictly speaking, it's a linear functional
<span class="math-container">$$\delta:C^\infty(\mathbb R)\to\mathbb R\qquad\qquad\delta(f)=f(0).$$</span>
However, we usually use the notation
<span class="math-container">$$\int_{-\infty}^... |
218,479 | <p>I am trying to evaluate the following integral with Mathematica:</p>
<p><span class="math-container">\begin{align}
I = \int_{0}^{\infty} da \, \frac{e^{-\frac{a ^2}{4s^2}} }{a^2} \mbox{sinc}\left(\tfrac{w}{2} a \right) \delta' \left( \frac{D^2}{a}- a \right),
\end{align}</span>
where the prime on the delta ... | SolutionExists | 70,331 | <p>My previous answer and comments were wrong. I didn't notice the argument of the δ function was not linear in the integration variable (and I wasn't even drunk).</p>
<p>In the <a href="https://en.wikipedia.org/wiki/Dirac_delta_function" rel="nofollow noreferrer">Wikipedia page</a>, there is this paragraph</p>
<bloc... |
3,942,512 | <p>If X and Y are independent binomial random variables with identical parameters n and p, calculate the conditional expected value of X given X+Y = m.</p>
<p>The conditional pmf turned out to be a hypergeometric pmf, but I'm a but unclear on how to relate that back into finding E[X|X+Y=m]</p>
| Aphelli | 556,825 | <p>Another argument (with quite sophisticated machinery) is that if <span class="math-container">$\mathbb{Z}[C_n]$</span> has finite projective dimension, so has <span class="math-container">$R=\mathbb{F}_p[C_n]$</span> for <span class="math-container">$p|n$</span> (a finite exact sequence of free <span class="math-con... |
1,190,759 | <p>I was trying to show the following
$\int_{-\infty}^{\infty} x^{2n}e^{-x^2}dx = (2n)!{\sqrt{\pi}}/4^nn!$ by using $\int_{-\infty}^{\infty} e^{-tx^2}dx = \sqrt{\pi/t}$
thus</p>
<p>I differentiated this exponential integral n times to get the following. </p>
<p>$\int_{-\infty}^{\infty} \frac{d^ne^{-tx^2}}{dt^n}dx ... | kobe | 190,421 | <p>The $n$th derivative of $\sqrt{\pi/t} = t^{-1/2}\sqrt{\pi}$ is </p>
<p>$$(-1)^n \left(\frac{1}{2}\right)\left(\frac{3}{2}\right)\cdots \cdot \left(\frac{2n-1}{2}\right)t^{-1/2 - n}\sqrt{\pi},$$</p>
<p>which can be written</p>
<p>$$\frac{(-1)^n(1)(3)(5)\cdots (2n-1) t^{-1/2 - n}\sqrt{\pi}}{2^n}.$$</p>
<p>This is ... |
889,155 | <blockquote>
<p>There are $2n-1$ slots/boxes in all and two objects say A and B; total number of A's are $n$ and total number of B's are $n-1$. (All A's are identical and all B's are identical.) In how many ways can we arrange A's and B's in $2n-1$ slots.</p>
</blockquote>
<p>My approach: there are $2n-1$ boxes in... | user140591 | 140,591 | <p>The problem can be reduced to this: How many ways can we arrange the n A's into 2n-1 positions?
This is because we can say that once all A's have been placed, the rest must be B's. In a given arrangement of A's, this is the only arrangement of that type since all B's are identical.</p>
<p>Hence the answer is (2n-1)... |
889,155 | <blockquote>
<p>There are $2n-1$ slots/boxes in all and two objects say A and B; total number of A's are $n$ and total number of B's are $n-1$. (All A's are identical and all B's are identical.) In how many ways can we arrange A's and B's in $2n-1$ slots.</p>
</blockquote>
<p>My approach: there are $2n-1$ boxes in... | Adi Dani | 12,848 | <p>$$\frac{(n+(n-1))!}{n!(n-1)!}=\frac{(2n-1)!}{n!((2n-1)-n)!}=\binom{2n-1}{n}$$</p>
|
794,875 | <p>Let $\{v_1, v_2,....,v_n\}$ be the standard basis for $\mathbb R^n$.Prove for any two $m\times n$ matrices that their linear transformations are equal if and only if the two matrices are equal. I know what two linear transformations need to be equal (same basis, domain and codomain), but how do I show that?</p>
| Belgi | 21,335 | <p><strong>Hint:</strong> $T(v_{i})$ is encoded into the matrices</p>
|
4,200,602 | <p>Let <span class="math-container">$\alpha$</span> be a class <span class="math-container">$\mathcal{K}$</span> function defined on <span class="math-container">$[0,a)$</span>. Then
<span class="math-container">\begin{equation}
\alpha(r_1+r_2) \leq \alpha(2r_1) + \alpha(2r_2), \quad \forall r_1,\,r_2 \in [0,\,a/2).
\e... | Martin R | 42,969 | <p>Without loss of generality assume that <span class="math-container">$0 \le r_1 \le r_2$</span>. Then
<span class="math-container">$$
\begin{align}
0 \le 2 r_1 &\implies 0 = \alpha(0) \le \alpha(2r_1) \\
r_1 + r_2 \le 2 r_2 &\implies \alpha(r_1 +r_2) \le \alpha(2 r_2)
\end{align}
$$</span>
because <span cl... |
200,876 | <p>Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?</p>
<blockquote>
<blockquote>
<p>A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) = y$.</p>
</blockquote>
</bloc... | Goldstern | 14,915 | <p>No such space $C$ can exist. </p>
<p>We will derive a contradiction from the assumption that $C,c_0,c_1$ as desired exists. </p>
<p>Let κ be any cardinal greater than $|C|$. View $\kappa$ as an ordinal. For each β in κ add a copy of the unit interval between β and β+1, and add a point ∞ at the end. The resulting ... |
72,537 | <blockquote>
<p>Let $A\in M_{n}$ have Jordan canonical form $J_{n_1}(\lambda_{1})\oplus\cdots\oplus J_{n_k}(\lambda_{k})$. If $A$ is non-singular ($\lambda_i\neq 0$), what is the Jordan canonical form of $A^{2}$?</p>
</blockquote>
<p>I can prove that if the eigenvalues of $A$ are $\sigma(A)=\{\lambda_{1},\dots, \lam... | Edison | 11,857 | <p>Thank you @Mariano. Intuitively I believe this makes sense, but I just want to go through some details.</p>
<p>Given the Jordan canonical form of a matrix $A$, I want to show that an arbitrary Jordan block of $A$ corresponding to the eigenvalue $\lambda$, $J_{k}(\lambda)$, gives rise to precisely one Jordan block $... |
4,350,699 | <blockquote>
<blockquote>
<p><span class="math-container">$r:$</span>All prime numbers are either even or odd, Is it a true statement?</p>
</blockquote>
</blockquote>
<p>I was studying Mathematical Logic then i came across above question.
Since here connecting word is "OR"
so if i separate two statement then ... | ryang | 21,813 | <p><span class="math-container">$✔\quad$</span> All numbers are either even or odd.</p>
<p><span class="math-container">$✗\quad$</span> Either all numbers are even, or all numbers are odd.</p>
<p>In general, <span class="math-container">$$∀x \;\Big(A(x)\text{ or }B(x)\Big)\quad\text{does not imply}\quad∀x A(x)\;\text{ ... |
82,254 | <p>Consider the standard form polyhedron, and assume that the rows of the matrix A are linearly independent.</p>
<p>$$ \left \{ x | Ax = b, x \geq 0 \right \} $$</p>
<p>(a) Suppose that two different bases lead to the same basic solution. Show that the basic solution is degenerate (has less than m non-zero entries).<... | Mike Spivey | 2,370 | <p>(Most of this was written before the recent addendum. It addresses the OP's original question, not the addendum.)</p>
<p>(a) Suppose we have distinct bases $B_1$ and $B_2$ that each yield the same basic solution ${\bf x}$. Now, suppose (we're looking for a contradiction) that ${\bf x}$ is nondegenerate; i.e., eve... |
2,166,897 | <blockquote>
<p>Let X be a complex Banach space. Let <span class="math-container">$T\in B(X)$</span> be a bounded linear operator on <span class="math-container">$X$</span>. Let <span class="math-container">$T^*\in B(X^*)$</span> be the adjoint of <span class="math-container">$T$</span>.</p>
<p>Prove: If <span class="m... | Guy Fsone | 385,707 | <p>Obviously we have,
\begin{split}
\|x\| &=& \sup\{|\langle x,y\rangle| ; \|y\|=1\}\\
&=& \sup\{|\langle T^{-1}Tx,y\rangle| ; \|y\|=1\}\\
&=& \sup\{|\langle T^{-1}Tx,(T^{-1})^*y\rangle| ; \|y\|=1\}\\
&\le& \|Tx\|\sup\{\|T^{-1})^*y\| ; \|y\|=1\}\\
& = &\|Tx\|\|(T^{-1})^*\|
\end{... |
1,708,996 | <p>If $x = a( \theta +\sin \theta)$ and $y = a(1-\cos \theta)$ then $\frac{dy}{dx}$ will be equal to : </p>
<p>$a) \sin \frac{\theta}{2}$</p>
<p>$b) \cos \frac{\theta}{2}$</p>
<p>$c) \tan \frac{\theta}{2}$</p>
<p>$d) \cot \frac{\theta}{2}$</p>
<p>I have solved till : $\frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \t... | Rayees Ahmad | 249,254 | <p>$$\dfrac{\sin a}{1+\cos a}=\frac{\sin\frac{a}{2}cos\frac{a}{2}}{{\cos^2\frac{a}{2}}} $$</p>
<p>$=\tan\frac{a}{2} $ is the solution</p>
<p>( Rotation of cycloid?)</p>
|
2,553,175 | <p>How can I verify that
$$1-2\sin^2x=2\cos^2x-1$$
Is true for all $x$?</p>
<p>It can be proved through a couple of messy steps using the fact that $\sin^2x+\cos^2x=1$, solving for one of the trigonemtric functions and then substituting, but the way I did it gets very messy very quickly and you end up with a bunch of ... | D.R. | 405,572 | <p>Given $$\sin^2(x)+\cos^2(x)=1$$
we rearrange to get
$$\sin^2(x)=1-\cos^2(x)$$
Substituting:
$$1-2\sin^2(x)=1-2(1-\cos^2(x))=2\cos^2(x)-2+1=2\cos^2(x)-1$$
Perhaps not as messy as you imagined.</p>
|
2,381,406 | <p>Somewhere I saw that </p>
<blockquote>
<p>To show that $x^2-y^3$ is irreducible in $k[x,y]$ it suffices to show that $x^2-y^3$ is irreducible in $k(y)[x]$.</p>
</blockquote>
<p>My question is what is the relation between $k[x,y]$ and $k(y)[x]$ ?</p>
<p>Also there is a confusion that if $k(y)$ is the smallest fi... | Noah Schweber | 28,111 | <p>A space is first-countable if for each point $x$ there is a <em>single</em> sequence of neighborhoods such that <em>every</em> neighborhood of $x$ contains in some neighborhood in the sequence. Although for each specific neighborhood $V$ we can easily find a sequence with some element contained in $V$, we can't nece... |
3,816,041 | <blockquote>
<p>How many ways <span class="math-container">$5$</span> identical green balls and <span class="math-container">$6$</span> identical red balls can be arranged into <span class="math-container">$3$</span> distinct boxes such that no box is empty?</p>
</blockquote>
<p>My attempt :</p>
<p>Finding coefficient... | nguyen quang do | 300,700 | <p>Your problem can be nicely generalized as follows:
Suppose that <em>p, q</em> two distinct primes such that the class <em>p</em> mod <em>q</em> is a generator of the multiplicative group <span class="math-container">$\mathbf F_q^{\times}$</span> ; in other words, for any <span class="math-container">$d < q-1,q$</... |
1,968 | <p>We're evaluating the feasibility of <strong>sponsoring a member of the math community to speak at a conference in 2011</strong>.</p>
<p>Speaking is a relatively big "ask", so this needs to be planned many months in advance. Let's get started! </p>
<p>We'd like the community to establish <strong>where</strong> ...<... | Willie Wong | 1,543 | <p>Well, the obvious suggestion for visibility is <a href="http://jointmathematicsmeetings.org/jmm" rel="nofollow">the annual joint maths meetings</a> of the American Mathematical Society, the Mathematical Association of America, and the Society of Industrial and Applied Mathematics; it is held every year in early Janu... |
3,301,696 | <p>Prove that if <span class="math-container">$R$</span> is a non-commutative ring with <span class="math-container">$1$</span> and if <span class="math-container">$a,b \in R$</span> and if <span class="math-container">$ab =1 $</span> but <span class="math-container">$ba \neq 1$</span> then <span class="math-container... | Jens Hemelaer | 81,217 | <p>Hint: show that if <span class="math-container">$R$</span> is finite, then <span class="math-container">$a^n = 1$</span> for some natural number <span class="math-container">$n$</span>. Now compute <span class="math-container">$b$</span>.</p>
|
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Allan Henriques | 666,324 | <p><a href="https://i.stack.imgur.com/1YbEs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1YbEs.png" alt="enter image description here" /></a></p>
<p>Hopefully you can figure it out using this sketch.</p>
|
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