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 |
|---|---|---|---|---|
710,518 | <p>I'm having a brainfart while trying to solve a problem for differential equations that requires me to recall some Calculus. If I have $y' = f(t, y) = 1 - t + 4y$, what is $y''$? Do I just differentiate with respect to $t$ to get $y'' = -1$?</p>
| Guy | 127,574 | <p>I am assuming $y'=\frac{dy}{dt}$ since there are no other variables here.</p>
<p>We have $\large \frac{dy}{dt} = 1 - t +4y$ </p>
<p>Differentiating with respect to $t$ (your idea is right, but the final answer isn't)</p>
<p>$$\frac{d^2y}{dt^2} = -1+4\frac{dy}{dt}$$</p>
<p>Substituting back the original value of ... |
196,173 | <p>Recently I have stumbled upon links which are closures of braids, of the form $\sigma = \tau^{n}$. Such links generalize torus links. Are there any papers studying such links? In particular I am interested in questions like which links appear in this way, what can we say about polynomial invariants of such links, ar... | Marco Golla | 13,119 | <p>If you only cared about knots, then you'd be looking at a special case of the so-called <em>periodic</em> knots, and the latter have been studied by several people.</p>
<p>There is <a href="http://www.indiana.edu/~jfdavis/papers/dliv1.pdf" rel="nofollow">this paper</a> by Davis and Livingston where some results are... |
27,509 | <p>I have a set of data of the kind {{$x_i$, $y_i$, $z_i$}, ...} at randomly chosen points $x_i, y_i$. </p>
<p>The $z_i$ are supposed to be a smooth function of the $x_i$ and $y_i$. Unfortunately they turn out not to be. A few points stick out. These are local extrema - I want to find them and remove them from the dat... | Szabolcs | 12 | <p>The idea is to construct the Delaunay triangulation and get the points that have higher values than all of their neighbours (according to this triangulation).</p>
<p>Here's a worked example:</p>
<p>Generate the sample data:</p>
<pre><code>f[x_, y_] := Sin[x] Sin[y]
points = RandomReal[{-3, 3}, {300, 2}];
zval = ... |
3,232,319 | <p>I need to show that <span class="math-container">$e^{-it}=\frac{1}{e^{it}}$</span>.
but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me.
Is the claim true even if t is not real?
Thank you</p>
| Ahmad Bazzi | 310,385 | <p>Using Eulers formula,</p>
<p><span class="math-container">$$\cos(x) + i\sin(x) = e^{ix}$$</span></p>
<p><span class="math-container">$$\cos(x) - i\sin(x) = e^{-ix}$$</span></p>
<p>Multiply the two equations and you’re done (work out the left hand side containing trigonometric quantities which should turn out to b... |
3,232,319 | <p>I need to show that <span class="math-container">$e^{-it}=\frac{1}{e^{it}}$</span>.
but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me.
Is the claim true even if t is not real?
Thank you</p>
| José Carlos Santos | 446,262 | <p>You have<span class="math-container">$$e^{it}\cdot e^{-it}=e^{it-it}=e^0=1$$</span>and therefore<span class="math-container">$$e^{-it}=\frac1{e^{it}}.$$</span></p>
|
1,671,572 | <p>$\|A\vec{x}\|\leq\|A\|\space\|\vec{x}\|$ where $A$ is a $m\times n$ matrix and $\vec{x}$ is a n-dimensional column vector. Assume that $\|A\|=\sqrt{\Sigma_{i}\Sigma_{j}a_{ij}^{2}}$</p>
| Emilio Novati | 187,568 | <p>If you are using an <a href="https://en.wikipedia.org/wiki/Matrix_norm#Induced_norm" rel="nofollow noreferrer">induced norm</a>, the inequality is guaranteed by the definition, as noted in the other answers.</p>
<p>From your comment it seems that you are using the Frobenius norm. in this case you can find a proof o... |
23,564 | <p>In the <a href="http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf" rel="noreferrer">December 2009 issue</a> of the <a href="http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X" rel="noreferrer">newsletter of the European Mathematical Society</a> there is a very interesting interview with Pierre C... | algori | 2,349 | <p>I'm not a logician at all, but since I'm using categories, I decided at some point to find out what is going on. A logician would probably give you a better answer, but in the mean time, here is my understanding.</p>
<p>Grothendieck's Universe axiom (every set is an element of a Grothendieck universe) is equivalent... |
23,564 | <p>In the <a href="http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf" rel="noreferrer">December 2009 issue</a> of the <a href="http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X" rel="noreferrer">newsletter of the European Mathematical Society</a> there is a very interesting interview with Pierre C... | Dan Doel | 5,880 | <p>With regard to what Darij said, if Haskell is viewed as a formal system, it's quite easy to derive contradictions; they correspond to infinite loops:</p>
<pre><code>loop :: forall a. a
loop = loop
</code></pre>
<p>However, when viewed in this light, the quote becomes something like:</p>
<blockquote>
<p>Nowadays... |
4,528,027 | <p>Im trying to prove that the function <span class="math-container">$$\begin{cases}f(x,y)=\dfrac{(2x^2y^4-3xy^5+x^6)}{(x^2+y^2)^2}, & (x,y)≠0\\ 0, & (x,y)=0\end{cases}$$</span> is continuous at point (0,0) using the rigorous defintion of a limit.</p>
<p>Attempting to find the upper limit of the function:
<span... | Lelouch | 991,491 | <p>You can just use :
<span class="math-container">$x \leq \sqrt{x^2+y^2} =r$</span> and <span class="math-container">$y \leq \sqrt{x^2+y^2} =r $</span></p>
<p>Thus
<span class="math-container">$$ 0 \leq | f(x,y) | \leq \frac{2r^6+3r^6+r^6}{r^4} = 6r^2 $$</span></p>
<p>Then just make <span class="math-container">$r \to... |
2,775,443 | <p>The statement $\displaystyle \sum_{n=1}^{\infty}a_n$ converges $\implies$ $\displaystyle \sum_{n=1}^{\infty}\cfrac{1}{a_n}$ looks natural but do we have this implication? I am checking alternating series as an counter-example but could not find one yet. What can we say about the implication?</p>
| Kirk Fox | 551,926 | <p>The $n$-th term divergence test states that if $\lim_{n\to\infty} a_n \ne 0$, then $\sum_{n=1}^{\infty} a_n$ diverges. The contraposition of this tells us that if $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n\to\infty} a_n = 0$. If we then apply the $n$-th term divergence test to $\frac{1}{a_n}$, we get
$$\lim_... |
502,994 | <p>If the image is f(x), why does f(|x|) look like two triangles above the x axis (basically the right side duplicated on the left)?</p>
<p><img src="https://i.imgur.com/pGIdNYZ.jpg" alt="function image"></p>
| William Ballinger | 79,615 | <p>$f(|x|)$ will equal $f(x)$ if $x$ is positive, and $f(-x)$ otherwise. Therefore, the positive part of graph of $f(|x|)$ will be identical to the positive part of the graph of $f(x)$, and the negative part will be a reflection of the positive part of the graph of $f(x)$.</p>
|
1,231,082 | <p>Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$.</p>
<ul>
<li>I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a <em>formal</em> proof?). However, I am confused about how I should calculate $\text{dim}(\text{Im(tr)})$ and $\text{dim}(\text... | lab bhattacharjee | 33,337 | <p>We don't even need Fermat's Little theorem if we use <a href="https://math.stackexchange.com/questions/641443/proof-of-anbn-divisible-by-ab-when-n-is-odd">Proof of $a^n+b^n$ divisible by a+b when n is odd</a></p>
<p>Now $105=3\cdot5\cdot7$</p>
<p>So, check for $2^3+3^3,2^5+3^5,2^7+3^7$</p>
|
1,825,076 | <p>Prove that $\sum_{n=1}^\infty \frac{x^{2n}}{(1 + x + \dots + x^{2n})^2}$ converges uniformly when $x \geq 0$.</p>
| Jack D'Aurizio | 44,121 | <p>By setting $f_n(x)=\frac{x^{2n}}{(1+x+\ldots+x^{2n})^2}$ we may easily check that $f_n(x)=f_n(x^{-1})$, so it is enough to prove the series is uniformly convergent over $[1,+\infty)$ or $(0,1]$. Moreover, for any $z\in\mathbb{R}^+$ we have $z+\frac{1}{z}\geq 2$, so:
$$ 0\leq f_{n}(x) = \frac{1}{\left(x^{-n}+x^{1-n}+... |
2,714,738 | <p>If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?</p>
| rschwieb | 29,335 | <p>$R[x]\cong \bigoplus_{i=0}^\infty R$ as $R$ modules, so obviously not.</p>
|
2,714,738 | <p>If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?</p>
| Bernard | 202,857 | <p>A noetherian $R$-module has all its submodules finitely generated – in particular it has to be finitely generated as an $R$-module. If $R[x]$ were finitely generated, it would be generated by a finite number of polynomials as an $R$-module, so that polynomials would have a bounded degree.</p>
|
2,768,249 | <p>In the middle of a proof, I have had to analyze the asymptototic behavior of
$$
\mathbb{E}\left[\frac{1}{(1+X)^2}\right] = \frac{1}{2^n}\sum_{k=0}^n \binom{n}{k}\frac{1}{(1+k)^2}\tag{1}
$$
where $X$ is Binomially distributed with parameters $n$ and $1/2$. (I also had to handle $\mathbb{E}\left[\frac{1}{(1+X)^4}\righ... | Community | -1 | <p>For <a href="https://math.stackexchange.com/questions/172841/explain-why-ex-int-0-infty-1-f-x-t-dt-for-every-nonnegative-rando?noredirect=1&lq=1">non-negative random variables</a>, $X$, </p>
<p>$$ E[X] = \int_0^\infty P(X > x)dx $$</p>
<p>Since $|X| \leq M$ almost surely, $\{|X| > x\} \subseteq \{M > ... |
3,600,868 | <p>Ellipse can be
<a href="https://math.stackexchange.com/q/3594700/122782">perfectly packed with <span class="math-container">$n$</span> circles</a>
if </p>
<p><span class="math-container">\begin{align}
b&=a\,\sin\frac{\pi}{2\,n}
\quad
\text{or equivalently, }\quad
e=\cos\frac{\pi}{2\,n}
,
\end{align}</span> <... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/wwA8R.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wwA8R.png" alt="enter image description here"></a></p>
<p>This is example of the
ellipse, perfectly packed with three circles,
inscribed into the equilateral triangle <span class="math-container">$ABC$</span... |
587,162 | <p>I want to prove that the following holds, where the $+$ means Minkowski sum:</p>
<p>$$
conv(A+B)=conv(A)+conv(B)
$$</p>
<p>Let the convex hull of $A+B$ be
$$
conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k)
$$</p>
<p>I don't know how to continue from here. </p>
| Post No Bulls | 111,742 | <p>A set $C$ is convex iff for all $t\in (0,1)$ we have $tC+(1-t)C= C$.</p>
<p>It follows that the Minkowski sum of convex sets $A, B$ is convex:
$$t(A+B)+(1-t)(A+B) = (tA+(1-t)A) + (tB+(1-t)B) = A+B$$</p>
<p>Therefore, for general sets $A,B$ the sum $\operatorname{conv}(A)+\operatorname{conv}(B)$ is convex; and sin... |
3,364,263 | <p>How many 8 digit (its digits from left to right are labeled a through g) satisfy the following constraints:</p>
<ol>
<li><span class="math-container">$a_1 < a_2 < a_3 < a_4$</span></li>
<li><span class="math-container">$a_4 > a_5$</span></li>
</ol>
| Dr. Sonnhard Graubner | 175,066 | <p>Your equation is equivalent to
<span class="math-container">$$x \left( x-5 \right) \left( {x}^{2}-x+7 \right) \left( {x}^{2}-6\,x+
12 \right)
=0$$</span></p>
|
207,399 | <p><em>Let $A$ and $B$ be non-empty sets, and let $f\,:\,A\rightarrow B$ be a function.</em>
<br/></p>
<hr/>
<blockquote>
<p>$ \color{darkred}{\bf Theorem}$: The function $f$ is injective if and only if $f\circ g=f\circ h$ implies $g=h$ for all functions $g,h:\,Y\rightarrow A$ for all sets Y. ($f\,:\,A\, \rightarr... | Community | -1 | <p>assume $f(a) = f(b)$ with $a \neq b$. Then define $g,h: A \rightarrow A$ with $g(x) := b$ as $x=a$ and $g(x) := a$ as $x = b$ and $x$ else. Take for $h:= Id_A$. Then we have $f\circ g =f \circ h$ but $f \neq h$.</p>
|
4,033,453 | <p>How many roots does the equation <span class="math-container">$$\left\lfloor\frac x3\right\rfloor=\frac x2$$</span> have?</p>
<ol>
<li><span class="math-container">$1$</span></li>
<li><span class="math-container">$2$</span></li>
<li><span class="math-container">$3$</span></li>
<li>infinitely many</li>
</ol>
<p>I che... | VIVID | 752,069 | <p>First observe that
<span class="math-container">$$\color{blue}{\frac x3 } \ge \left\lfloor\frac x3\right\rfloor=\color{blue}{\frac x2} \implies x \le 0 \\
\color{blue}{\frac x3 - 1} \le \left\lfloor\frac x3\right\rfloor=\color{blue}{\frac x2} \implies x \ge -6$$</span>
Now, from the equation, notice that <span class... |
620,340 | <p>I am stuck by this question from Liu's algebraic geometry textbook on quasi-coherent modules.</p>
<p>Let X be an affine scheme $\mbox {Spec} A $. Let $\mathcal {F} $ be a quasi-coherent $\mathcal{O}_{X} $ module. Show that for any affine open subset U of X we have a canonical isomorphism</p>
<p>$\mathcal {F}(X)\ot... | enoughsaid05 | 31,754 | <p>I attempted the question but it will be too long as a comment in the preceding answer, hope someone can help me verify. (And because I am typing this on the phone :( )</p>
<p>I need to show that $\mathcal {F}(X)\otimes_{A}\mathcal {O}_{X}(U)\cong \mathcal {F}(U) $.</p>
<p>The $\mathcal{O}_{X} $-module $ \mathcal {... |
2,696,281 | <blockquote>
<p>$\triangle ABC$ is an equilateral triangle of side $1$. $ \triangle BDC$ is an isosceles triangle with $D$ opposite to $A$ and $DB = DC$ and $\angle BDC = 120^\circ$. If points $M$ and $N$ are on $AB$ and $AC$, respectively, such that $\angle MDN = 60^\circ$, find the perimeter of $\triangle AMN$.... | John Glenn | 528,976 | <p>Like Vasya and Qurultay in the comments, I too get 2.</p>
<p>To find the length of $BD$, let $K$ be a perpendicular bisector of $BC$, which also bisects $\angle BDC$, thus we have a right triangle $\triangle BKD$. Thus:
$$\sin 60^\circ=\frac{0.5}{BD}\Rightarrow BD=\frac{0.5}{\sin 60^\circ}$$</p>
<p>Note that $\an... |
1,992,009 | <p>It is known that $$\lim_{x \to 0}\frac{f(x)}{x} = -\frac12$$ </p>
<p>Solve
$$\lim_{x \to 1}\frac{f(x^3-1)}{x-1}.$$</p>
<p>Beforehand, I know that I should aim to get rid of the denominator $(x-1)$ and as such I factor the numerator to get:</p>
<p>$$\lim_{x \to 1}{f(x^2+x+1)}{}.$$</p>
<p>Now that I factored the d... | hamam_Abdallah | 369,188 | <p>We have</p>
<p>$$\frac{f(x^3-1)}{x-1}=$$</p>
<p>$$(x^2+x+1)\frac{f(x^3-1)}{x^3-1}.$$</p>
<p>and when $x\to 1\;$, your limit is $3.\frac{-1}{2}=\frac{-3}{2}$.</p>
|
335,295 | <p>How do you show that a deduction exist in the Hilbert Proof System, as used in Herbert Enderton, <em>A Mathematical Introduction to Logic</em>.</p>
<p>L is a FOL (First Order Language) which contains R, where R is a single binary predicate symbol.</p>
<p>a1, a2, a3 are defined as:</p>
<p>a1 = $∀x∀y∀z(Rxy → (Ryz →... | Alex Kruckman | 7,062 | <p>Well, there are two options.</p>
<p>Option 1: You could write out a proof using the Hilbert Proof System. The details of this proof are going to depend on the details of the proof system. Since I don't have a copy of Enderton on hand, I can't help you here.</p>
<p>Option 2: If you know the completeness theorem for... |
106,838 | <p>Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it necessarily true that $K \cong L$?</p>
<p>I'm also interested in the corresponding question for the special linear group i... | Mikhail Borovoi | 4,149 | <p>The answer is "yes", see below.</p>
<p>Dieudonné in his book "La géométrie des groupes classiques"
considers the abstract group $SL_n(K)$ for a field $K$, not necessarily commutative,
and writes $PSL_n(K)$ for $SL_n(K)$ modulo the center.
In Ch. IV, Section 9, he considers the question whether $PSL_n(K)$ can be iso... |
3,237,094 | <p>Find all differentiable functions <span class="math-container">$f\colon [0,\infty)\to [0,\infty)$</span> for which <span class="math-container">$f(0)=0$</span> and <span class="math-container">$f^{\prime}(x^2)=f(x)$</span> for any <span class="math-container">$x\in [0,\infty)$</span>. </p>
<p>I have tried to reduce... | Lázaro Albuquerque | 85,896 | <p>First note that <span class="math-container">$f$</span> and <span class="math-container">$f'$</span> are both nondecreasing.</p>
<p>By MVT, there is a <span class="math-container">$0 < \lambda < 1$</span> such that <span class="math-container">$f(1)=f'(\lambda)$</span>. But <span class="math-container">$f'(\l... |
3,625,886 | <blockquote>
<p>Show that if <span class="math-container">$a$</span> and <span class="math-container">$b$</span> belong to <span class="math-container">$\mathbb{Z}_+$</span> then there are divisors <span class="math-container">$c$</span> of <span class="math-container">$a$</span> and <span class="math-container">$d$<... | Calvin Lin | 54,563 | <p>It is not correct.</p>
<p>If <span class="math-container">$ a = 4, b = 2$</span>, then you have <span class="math-container">$ d = b = 2, c = a/ \gcd(a,b) = 4/2 = 2$</span>.<br>
This doesn't satisfy <span class="math-container">$ \gcd(c,d) = 1$</span>. </p>
<p>Now, figure out where your reasoning breaks down, and ... |
104,186 | <p>I'm having difficulty understanding how to express text in <code>Epilog</code> that is dynamically updated using <code>Log[b, x]</code>. <em>Mathematica</em> changes this to base $e$, but I would like it to be <code>Log[b, x]</code> in traditional format with base $b$, and I can't seem to make it work. I'm guessin... | Michael E2 | 4,999 | <p>Here's a way that let's <em>Mathematica</em> take care of the typesetting. Use <a href="http://reference.wolfram.com/language/ref/With.html" rel="nofollow noreferrer"><code>With</code></a> to inject the value of <code>b</code> into a held expression for the logarithm (see <a href="http://reference.wolfram.com/langu... |
1,357,488 | <p>Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual basis. Now we define $\omega:=e^1\wedge\cdots\wedge e^n$ as element of volume of $E$.</p>
<p>How can I prove that $\omega... | HK Lee | 37,116 | <p>Hint : (1) Let $$ U=[u_1\cdots u_n],\ V=[v_1\cdots v_n]$$</p>
<p>Then $$ (U^TV)_{ij} = u_i\cdot v_j =g(u_i,v_j) $$</p>
<p>(2) ${\rm det} (U^TV)={\rm det}\ U {\rm det}\ V$</p>
|
3,822,822 | <p>Four cards are face down on a table. You are told that two are red and two are black, and you need to guess which two are red and which two are black. You do this by pointing to the two cards you’re guessing are red (and then implicitly you’re guessing that the other two are black). Assume that all configurations ar... | fny | 28,533 | <p>Here's a general answer that should provide you with insight.</p>
<p>Say we have <span class="math-container">$2n$</span> cards where <span class="math-container">$n$</span> are red and <span class="math-container">$n$</span> are black. There are <span class="math-container">${2n \choose n}$</span> arrangements of t... |
129,993 | <p>Let $p$ and $q$ be relative primes, $n$ positive integer.</p>
<p>Given</p>
<ul>
<li>$n\bmod p$ and</li>
<li>$n\bmod q$</li>
</ul>
<p>how do I calculate $n\bmod (pq)$ ?</p>
| Bill Dubuque | 242 | <p>One may use the Bezout identity $\rm\:a\:p + b\:q = 1\:$ obtained by the extended Euclidean algorithm. But, in practice, it's often more convenient to use the form below, e.g. see my <a href="https://math.stackexchange.com/search?q=user%3A242+easy-CRT">Easy CRT posts.</a></p>
<p><strong>Theorem (Easy CRT)</strong> ... |
1,990,033 | <p>Suppose I have a point $P(x_1, y_1$) and a line $ax + by + c = 0$. I draw a perpendicular from the point $P$ to the line. The perpendicular meets the line at point $Q(x_2, y_2)$. I want to find the coordinates of the point $Q$, i.e., $x_2$ and $y_2$.</p>
<p>I searched up for similar questions where the coordinates ... | lab bhattacharjee | 33,337 | <p>Using $\sin3B=3\sin B-4\sin^3B$</p>
<p>$$32\sin^3x\cdot\cos^2x\cdot\cos2x=\sin x(3-4\sin^2x)$$</p>
<p>If $\sin x=0,x=180^\circ n$ where $n$ is any integer</p>
<p>Else using $\cos2A=2\cos^2A-1=1-2\sin^2A,$</p>
<p>$$32\cdot\dfrac{1-\cos2x}2\cdot\dfrac{1+\cos2x}2\cdot\cos2x=3-2(1-\cos2x)$$</p>
<p>$$\iff8(c-c^3)=1... |
1,482,152 | <p>Show that four non-coplanar points in $\mathbb{R}^3$ determine an unique sphere.</p>
<p>I have no idea how to solve this exercise. Thank you for your help.</p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>The equation of a sphere of center $C=(\alpha,\beta,\gamma)$ and radius $r$ is:
$$
(x-\alpha)^2+(y-\beta)^2+(z-\gamma)^2=r^2
$$
that becomes
$$
x^2+y^2+z^2+ax+by+cz+d=0
$$
with
$$
a=-2\alpha \quad b=-2\beta \quad c=-2\gamma \quad d=\alpha^2+\beta^2+\gamma^2-r^2
$$</p>
<p>so if we have four points, sub... |
1,708,420 | <p>Here's what I am trying to figure out: Find all vectors $\vec{b}$ that are in $span \{\vec{u},\vec{v},\vec{w}\}$ where $\vec{u},\vec{v},\vec{w}$ are vectors.</p>
<p>I'm given specific vectors in $\mathbb{R}^3$ for $\vec{u},\vec{v},\vec{w}$, but I really just want to ask about concepts. I can do the algebra, certain... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use $\displaystyle I=\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx$</p>
<p>$\displaystyle I+I=\int_a^bf(x)\ dx+\int_a^bf(a+b-x)\ dx=\int_a^b[f(x)+f(a+b-x)]\ dx$</p>
|
961,304 | <p>I am studying a book and I am stagnating on a what should be a straightfoward proof:</p>
<p>Show that if $X$ is compact, $V\subset X$ is open and $x\in V$, then there exists an open set $U$ in $X$ with $x\in U\subset \bar U\subset V$</p>
<p>I don't know how to find the appropriate set $U$. I am guessing you need t... | orangeskid | 168,051 | <p>That would be $y$ so that the subsets of $[a,b]$ </p>
<p>$\{x \in [a,b] \ | \ f(x) \le y\}$ and $\{x \in [a,b] \ | \ f(x) \ge y\}$ </p>
<p>have the same measure (with some care). </p>
<p>Obs: if $f$ is monotonous it will be $f(\frac{a+b}{2})$.</p>
|
3,122,989 | <p>Let <span class="math-container">$A\in\mathbb{R}^{n\times n}$</span> and define conjugation by <span class="math-container">$GL_n$</span> on <span class="math-container">$\mathbb{R}^{n\times n}$</span> in the usual way (e.g. for all <span class="math-container">$A\in\mathbb{R}^{n\times n}$</span> and <span class="ma... | Dietrich Burde | 83,966 | <p>Let <span class="math-container">$T=\begin{pmatrix} t_1 & t_2\cr t_3 & t_4\end{pmatrix}$</span> with <span class="math-container">$\det(T)=1$</span>.
Then <span class="math-container">$T^{-1}AT$</span> has upper right corner equal to <span class="math-container">$b$</span> if and only if
<span class="math-co... |
814,899 | <p>Suppose that $\{X_n\}$ is an independent sequence and $E[X_n]=0$. If $\sum \operatorname{Var}[X_n] < \infty$, then $\sum X_n$ converges with probability $1$. Is independence necessary condition here ? I am thinking of a counterexample. The intuition behind the other assumptions is clear. </p>
| ajotatxe | 132,456 | <p>As you hace been said, we need to assume that $A$ is connected. Then, it is possible to assume WOLOG that $V$ is connected, hence, $f(V)$ is connected.</p>
<p>Suppose that $f(V)$ is not open. Then, there exists some $y\in f(V)$ (say, $f(x)=y$) such that for every $\epsilon>0$ there exists $y_\epsilon\in(y-\epsil... |
2,717,842 | <p>I want to show that if $X$ is a locally connected topological space, $A\subseteq X$ is a subspace and $f:X \rightarrow A$ is continuous such that $f|_{A} = Id_{A}$, then $A$ must be locally connected as well.</p>
<p>My progress so far:</p>
<p>Take $U\subseteq A$ $A-$open and $x\in U$. Since $f^{-1} (U)$ is $X-$ope... | red whisker | 833,796 | <p>It is easier to instead show that components of open sets are open, which is equivalent to local connectedness. Let <span class="math-container">$r\colon X\to A$</span> be the retraction and suppose <span class="math-container">$N$</span> is an open set in <span class="math-container">$A$</span> and <span class="mat... |
2,347,995 | <p>Not sure what I'm doing wrong.</p>
<p>Here's my work:</p>
<p>Expressing the first part $A \setminus (B\setminus C)$ using logical symbols:</p>
<p>$A \land \neg(B \land \neg C)$ becomes</p>
<p>$A \land \neg B\lor C$ (De Morgan's law)</p>
<p>While the second expression $(A \setminus B) \cup (A \cap C)$ is </p>
<... | user217285 | 217,285 | <p>The inequality is obvious for $x = 0$ and the left-hand side is even in $x$ so we may assume $x \in (0,1)$. Taylor's theorem implies that
$$ \cos x = 1 - \frac{x^2}{2} + \frac{\xi^4}{24}$$
for some $\xi \in (0,x)$, so
$$ \frac{1- \cos x}{x^2} = \frac12 - \frac{\xi^4}{24x^2}.$$
Hence it is enough to show that $\xi^4/... |
966,420 | <p>How can one write $x$ is a factor of $y$ (as a constraint)? I am also not sure what else to add to meet the question quality requirements. </p>
| Traklon | 103,134 | <p>Well $|$ is used to describe divisibility so I think it meets our requirements.</p>
<p>For example, $3|6$.</p>
<p>$x$ is a factor of $y$ $\Leftrightarrow$ $x|y$. </p>
|
2,106,983 | <p><strong>Question</strong></p>
<p>how to evaluate $\tan x-\cot x=2.$</p>
<p>Given that it lies between on $\left[\frac{-\pi} 2,\frac \pi 2 \right]$.</p>
<p><em>My Steps so far</em></p>
<p>I converted cot into tan to devolve into $\frac{\tan^2 x-1}{\tan x}=2$.</p>
<p>Then I multiply $\tan{x}$ on both sides and th... | Fawad | 369,983 | <p>$\tan x - \cot x =2$</p>
<p>$\dfrac{\sin x }{\cos x} - \dfrac{\cos x}{\sin x }=2$</p>
<p>$\dfrac{\sin^2 x - \cos ^2 x}{\sin x \cos x } =2$</p>
<p>$-\cos 2x = 2\sin x \cos x$</p>
<p>$-\cos 2x=\sin2x$</p>
<p>$\tan 2x=-1$</p>
<p>$2x= n\pi -\frac{\pi}{4}$</p>
<p>Put n values to get values of x in required range .... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Graham Kemp | 135,106 | <p>Where $I$ is a sufficiently close interval to the limit point, a limit will exists if:</p>
<p>Right Sided $\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < x -2 < \delta \Rightarrow \lvert \frac 1{1-x} +1 \rvert<\varepsilon)$</p>
<p>Left Sided $\forall\varepsilon > 0\;\exists \d... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Hamed | 191,425 | <p>Define $f(x) = \frac{1}{1-x}$. We want to show that $f(x)$ is continuous at $-1$. With $\epsilon-\delta$ we need to show that for any $\epsilon>0$ there exists $\delta>0$ such that $|x-2|<\delta$ yields $|f(x)+1|<\epsilon$. This much is definition. Now
$$f(x)+1 = \frac{1}{1-x}+1 = \frac{x-2}{x-1}$$
I cla... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | liuyao | 1,189 | <p>In boundary value problems, physicists consider the infinity (in space and in time) to be part of the boundary. Mathematicians know there's a distinction between compact and non-compact spaces.</p>
|
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Gene S. Kopp | 8,410 | <p>The use of <a href="http://en.wikipedia.org/wiki/Random_matrix">random matrix theory</a> to model energy levels of heavy nuclei and other physical systems. See also the following <a href="http://www.williams.edu/go/math/sjmiller/public_html/ntrmt10/handouts/general/Hayes_spectrum_riemannium.pdf">historical piece</a... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Jamahl Peavey | 21,268 | <p>Yang-Mills Equations are experimentally proven but have no strong mathematical foundations. In the Clay Mathematics Institute the mass gap problem is worth one million dollars.</p>
|
1,197,056 | <p>I tried to evaluate the following limits but I just couldn't succeed, basically I can't use L'Hopital to solve this... </p>
<p>for the second limit I tried to transform it into $e^{\frac{2n\sqrt{n+3}ln(\frac{3n-1}{2n+3})}{(n+4)\sqrt{n+1}}}$ but still with no success...</p>
<p>$$\lim_{n \to \infty } \frac{2n^2-3}{... | Alex | 38,873 | <p>For the second one pay attention to the order of the numerator and denominator: the largest terms converge to some constant, the rest to 0, so you should get $(\frac{3}{2})^2$. </p>
|
3,795,655 | <p>Let <span class="math-container">$H(\mu|\nu)$</span> be the relative entropy (or Kullback-Leibler convergence) defined in the usual way. I am looking for a proof or reference to the following fact: <span class="math-container">$\mu,\nu$</span> two-dimensional probability measures with marginals <span class="math-con... | E-A | 499,337 | <p>The below answer is false; need to think; most likely the claim made is false for arbitrary reference measures; I know it true to be suitably chosen gaussians and uniforms.</p>
<blockquote>
<p>This is an immediate consequence of chain rule and the joint convexity of KL divergence. You can look at the proofs of the i... |
625,746 | <p>If the nth term of a series is given by $T(n)=T(n-1) + T(n-1) \times C$ where $C$ is a given constant and $T(1)=A$ and $n \ge 2$ . I need to tell whether its value will be greater than $G$ or not before its mth term.</p>
<p>EXAMPLE : Say A that is first term is $2$ and say $C$ is also $2$ . If we need to check wea... | Thanos Darkadakis | 105,049 | <p>$T(n)=T(n-1)+T(n-1)*C$</p>
<p>$T(n)=(1+C)*T(n-1)$</p>
<p>$T(n)=(1+C)^{n-1}*T(1)$</p>
<p>$T(n)=(1+C)^{n-1}*A$</p>
<p>Do you need something more?</p>
|
1,129,567 | <p>Using the substitution $x=\cosh (t)$ or otherwise, find
$$\int\frac{x^3}{\sqrt{x^2-1}}dx$$
The correct answer is apparently
$$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$
I seem to have gone very wrong somewhere; my answer is way off, can someone explain how to get this answer to me.</p>
<p>Thanks.</p>
<p>My working:
$$\int... | graydad | 166,967 | <p>If you want to do this by using the substitution $x = \cosh(t)$ you also need $dx = \sinh(t)dt$, which means $$\int \frac{x^3}{\sqrt{x^2-1}}dx = \int \frac{\cosh^3(t)\sinh(t)dt}{\sqrt{\cosh^2(t)-1}}$$ and now use $\cosh^2(t) = 1+\sinh^2(t)$ in the numerator to get $$\int \frac{\cosh^3(t)\sinh(t)dt}{\sqrt{\cosh^2(t-... |
2,309,527 | <p>I was sincerely hoping someone could explain to me how for the function found below I would determine its standard matrix and whether or not the function is 1-to-1 and onto. </p>
<blockquote>
<p>The linear function $T:\mathbb{R}^2 \to \mathbb{R}^3$ is given by $$ T(x,y) = \begin{pmatrix} x-y \\ 5x+3y \\ 2x+4y \en... | Martin Argerami | 22,857 | <p>All the operators you list are finite-rank, so compact. </p>
<p>More generally, if you define
$$
Tx=(a_1x_1,a_2x_2,\ldots)
$$
then $T$ is compact if and only if $\lim_{n\to\infty}a_n=0$. </p>
|
3,975,162 | <p>Let <span class="math-container">$(X, A, µ)$</span> be a positive metric space. If <span class="math-container">$\mu(X) < \infty$</span> and <span class="math-container">$(A_n)_{(n \in N^*)},A \in X$</span> <br />
show that if <span class="math-container">$\mu(A\bigtriangleup A_n)\rightarrow 0$</span> then <span ... | Stinking Bishop | 700,480 | <p>If <span class="math-container">$y$</span> divides <span class="math-container">$(y+1)^2=y^2+2y+1$</span>, knowing that <span class="math-container">$y$</span> already divides the terms <span class="math-container">$y^2$</span> and <span class="math-container">$2y$</span>, we conclude that <span class="math-containe... |
3,975,162 | <p>Let <span class="math-container">$(X, A, µ)$</span> be a positive metric space. If <span class="math-container">$\mu(X) < \infty$</span> and <span class="math-container">$(A_n)_{(n \in N^*)},A \in X$</span> <br />
show that if <span class="math-container">$\mu(A\bigtriangleup A_n)\rightarrow 0$</span> then <span ... | Bill Dubuque | 242 | <p>Since you tagged it "polynomials" we highlight their key role here.</p>
<p>Note that <span class="math-container">$\ y\mid y^2\!+\!1\iff y\mid (\overbrace{y^2\!+\!1\bmod y}^{\large \color{#c00}1})\!\iff y\mid \color{#c00}1$</span></p>
<p><strong>Generally</strong> <span class="math-container">$\ y\mid f(y... |
137,006 | <p><code>FunctionDomain[(x^2-x-2)/(x^2+x-6),x]</code> </p>
<p>gives </p>
<blockquote>
<p><code>x < -3 || -3 < x < 2 || x > 2</code>.</p>
</blockquote>
<p>However, when I factor the numerator and denominator the result is different:</p>
<p><code>FunctionDomain[((x - 2) (x + 1))/((x - 2) (x + 3)),x]</co... | Szabolcs | 12 | <p>Evaluate <code>((x - 2) (x + 1))/((x - 2) (x + 3))</code> and see what it gives. It <em>automatically</em> simplifies to <code>(1 + x)/(3 + x)</code>. The second input you show is effectively</p>
<pre><code>FunctionDomain[(1 + x)/(3 + x), x]
</code></pre>
<p>The different results you get are not due to <code>Func... |
1,606,709 | <p>I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The cotangent sheaf is later defined in terms of the diagonal map. Why is this geometrically interesting? I tried to write a s... | Christian Blatter | 1,303 | <p>Begin by investigating the sequence
$$x_0:=0,\quad x_{k+1}:=\sqrt{{1\over2}+x_k}\quad(k\geq0)\ .$$
It has a certain limit $\xi$. Knowing $\xi$ draw conclusions about the limit
$$\lim_{n\to\infty}\>\prod_{k=1}^n x_k\ .$$ </p>
|
2,804,074 | <p>Hello I am self teaching foundational math, I am thinking about union of a set. It definition assumes the set to have only elements that are also sets, else it would break. But I started thinking is the empty set infact an element of any mathematical object, like say the integer n? In that case taking the union of t... | Ishan Rai | 458,220 | <p>I really think the answer depends on <strong>n</strong> more than <strong>x</strong>. As it involves <strong>frac(x)</strong> the value of that doesn't change even if <strong>x->infinity</strong>.</p>
<p>Before giving an answer take a look at this, maybe it gives you a hint. Sometimes graphing can actually help a l... |
3,247,841 | <p>I'm working on an algorithm to colour a map drawn in an editor using 4 colours, as a visual demonstration of the four colour theorem. However, my (imperfect) algorithm was able to colour all maps except this one, which after giving it a go myself I struggled to do. I was also unable to collapse it into an 'untangled... | Sudix | 470,072 | <p>One possible algorithm would be the following:</p>
<ol>
<li>Take the map, and turn it into a graph.</li>
<li>Delete all nodes with degree <span class="math-container">$<4$</span>.</li>
<li>Turn the graph into a clause set (a special form of a logical formula) by doing the following: <br>
a. For every edge <span ... |
4,301,673 | <p>Suppose we have chosen <span class="math-container">$n$</span> random points in a line segment <span class="math-container">$[0, t]$</span>, <span class="math-container">$n\leq t+1$</span>. What is the probability that distance between each pair of adjacent points is > 1? Or more formally, let <span class="math-c... | MXXZ | 966,405 | <p>Answer: <span class="math-container">$\mathbb{P}(\bigcap_{i=1}^{n-1} U_{(i+1)} - U_{(i)} > 1) = \frac{(t-(n-1))^n}{t^n}$</span>.</p>
<p><strong>1. <span class="math-container">$\mathbb{P}(\bigcap_{i=1}^{n-1} U_{(i+1)} - U_{(i)} > 1) = n! \cdot \mathbb{P} (\forall 1 \leq i \leq n-1 \colon U_i + 1 < U_{i+1})$... |
121,431 | <p>I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they are defined). This topic seems pretty standard to me, but when I needed a basic reference on this (both to check my mem... | Carlos De la Mora | 40,832 | <p>In my humble opinion the best reference is Dixmier $C^*$-algebras. The first half of the book has a very complete explanation of what you need to know about $C^*$-algebras. In chapter 8 he goes over what is the decomposition of a trace for $C^*$-algebras. Then from Chapter 13 on he goes into the theory for a locally... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | Jean Marie | 305,862 | <p>There is a very easy-to-check necessary and sufficient condition :</p>
<p>You will have the first figure (triangle) if and only if there exists a linear combination of the LHS of your system of equations (1),(2),(3) making <span class="math-container">$0$</span> <strong>without the RHS being so</strong> with the sam... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | Ingix | 393,096 | <p>I guess the reason you "immediately knew" that the system</p>
<p><span class="math-container">$$ x_1 -3x_2 +2x_3 = -2 \tag1 \label{eq1}$$</span>
<span class="math-container">$$ x_1 +3x_2 -2x_3 = 5 \tag2 \label{eq2}$$</span>
<span class="math-container">$$-6x_2 + 4x_3 = 3 \tag3 \label{eq3}$$</span></p>
<p>b... |
3,243,328 | <p>X is a random variable with values from <span class="math-container">$\Bbb N\setminus{0}$</span></p>
<p>I am trying to show that <span class="math-container">$E[X^2]$</span> = <span class="math-container">$\sum_{n=1}^\infty (2n-1) P(X\ge n)$</span> iff <span class="math-container">$E[X^2]$</span> < <span class="... | Oliver Díaz | 121,671 | <p>Just as you did it:
<span class="math-container">\begin{align}
\sum^\infty_{n=1}(2n-1)P(X\geq n) &=\sum^\infty_{n=1}(2n-1)\Big(\sum^\infty_{j=n}P(X=j)\Big)\\
&=\sum^\infty_{j=1}\sum^j_{n=1}P(X=j)(2n-1)\\
&=\sum^\infty_{j=1}P(X=j)\Big(\sum^j_{n=1}(2n-1)\Big)\\
&=\sum^\infty_{j=1}P(X=j)j^2
\end{align}<... |
4,577,925 | <p>Which expression is larger,
<span class="math-container">$$
99^{50}+100^{50}\quad\textrm{ or }\quad 101^{50}?
$$</span></p>
<p>Idea is to use the Binomial Theorem:</p>
<p>The right hand side then becomes
<span class="math-container">$$
101^{50}=(100+1)^{50}=\sum_{k=0}^{50}\binom{50}{k}1^{50-k}100^k=100^{50}+\sum_{k=... | Claude Leibovici | 82,404 | <p>It could be amazing to compare
<span class="math-container">$$(2x)^x +(2 x-1)^x \qquad \text{and} \qquad (2 x+1)^x$$</span> or, better, their logarithms.</p>
<p>So, consider that you look for the zero of function
<span class="math-container">$$f(x)=\log \left((2x)^x +(2 x-1)^x\right)-\log \left((2 x+1)^x\right)$$</s... |
1,344,690 | <p>I was wondering how to find the vertices of an equilateral triangle given its center point?</p>
<p>Such as:</p>
<pre><code> A
/\
/ \
/ \
/ M \
B /________\ C
</code></pre>
<p>Provided that <code>AB, AC, BC = x</code> and <code>M = (50,50)</code> and <code>M</code> is the ... | Hagen von Eitzen | 39,174 | <p>Draw the circle of radius $\frac {\sqrt 3}3x$ around $M$. Pick an arbitrary point $A$ on this circle. Then intersect the circle of radius $X$ around $A$ with the first circle to determine $B,C$ as intersection points. </p>
<p>Note that $A$ could be picked anywhere on the circle, hence the result is not unique. </p>... |
3,056,031 | <p>From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):</p>
<blockquote>
<p>To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
<span class="math-container">\begin{equation}
x → x_1,
\textbf{ }
y → x_2
\end{e... | Renzo Vizarreta | 761,926 | <p>When you choose to represent the system of equations as a sum, it looks like this: </p>
<p><span class="math-container">$$ x^{'}_i = \sum_{j=1}^N a_{ij}x_j $$</span></p>
<p>For example, in the two-dimensional case (N=2), it follows that:</p>
<p><span class="math-container">$$ x^{'}_1 = a_{11}x_1 + a_{12}x_2 $$</s... |
3,123,681 | <p>let us consider following problem taken from book</p>
<p><em>An appliance store purchases electric ranges from
two companies. From company A, 500 ranges are
purchased and 2% are defective. From company B,
850 ranges are purchased and 2% are defective.
Given that a range is defective, find the probability
that it... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,273,477 | <blockquote>
<p>What is the unit normal vector of the curve $y + x^2 = 1$, $-1 \leq x \leq 1$? </p>
</blockquote>
<p>I need this to calculate the flux integral of a vector field over that curve.</p>
| marwalix | 441 | <p>The curve is given by $F(x,y)=x^2+y-1=0$. A normal vector is $\operatorname{grad}{F}=(F_x,F_y)=(2x,1)$. We now normalize to get</p>
<p>$$n=\left(\frac{2x}{\sqrt{4x^2+1}},\frac{1}{\sqrt{4x^2+1}}\right)$$</p>
|
1,273,477 | <blockquote>
<p>What is the unit normal vector of the curve $y + x^2 = 1$, $-1 \leq x \leq 1$? </p>
</blockquote>
<p>I need this to calculate the flux integral of a vector field over that curve.</p>
| E.H.E | 187,799 | <p>this is another way used when you want to find the unit normal vector in plane
$$y=1-x^2$$
$$y'=-2x$$
$$v=2xi+j$$
$$|v|=\sqrt{(2x)^2+1^2}=\sqrt{4x^2+1}$$
$$n=\frac{v}{|v|}=\frac{2xi}{\sqrt{4x^2+1}}+\frac{j}{\sqrt{4x^2+1}}$$</p>
|
519,224 | <p>Using the Squeeze Theorem, how do I find: $$\lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)$$
I thought I knew the Squeeze Theorem, but I haven't encountered anything like this yet, so I honestly have no idea how and where to start.</p>
<p>I would appreciate any help I get because I really want to be able ... | Community | -1 | <p>By these inequalities</p>
<p>$$0\leq\lim_{x\to 3} |(x^2-2x-3)^2\cos\left(\pi \over x-3\right)|\leq\lim_{x\to 3} |(x^2-2x-3)^2|=0$$
we have
$$\lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)=0$$</p>
|
4,036,903 | <p>Suppose <span class="math-container">$\sigma_1:\Delta^k \rightarrow X$</span> is a singular <span class="math-container">$k$</span>-simplex and <span class="math-container">$\sigma_2:\Delta^l \rightarrow X$</span> is a singular <span class="math-container">$l$</span>-simplex. Is there a singular <span class="math-co... | Sam Freedman | 245,133 | <p>Were there a common extension <span class="math-container">$\sigma : \Delta^2 \to X$</span> with <span class="math-container">$\sigma|_{[v_0, v_1]} = \sigma_1$</span> and <span class="math-container">$\sigma|_{[v_1, v_2]} = \sigma_2$</span> where 1-simplices <span class="math-container">$\sigma_1$</span> and <span c... |
3,383,672 | <p>I wanted to prove <span class="math-container">$(Ma)\times(Mb)=(\det M)(M^{-1})^T(a\times b)$</span> where the formula could be found here <a href="https://en.wikipedia.org/wiki/Cross_product#Algebraic_properties" rel="nofollow noreferrer">wiki's cross product page</a> However, I had a hard time counting the index, ... | user1551 | 1,551 | <p>They are equal because
<span class="math-container">\begin{aligned}
\left[(\det M)(M^{-1})^T(a\times b)\right]\cdot Mc
&=(\det M)(a\times b)\cdot M^{-1}Mc\\
&=\det(M)(a\times b)\cdot c\\
&=\det(M)\det[a,b,c]\\
&=\det[Ma,Mb,Mc]\\
&=(Ma\times Mb)\cdot Mc
\end{aligned}</span>
for every vector <span ... |
3,383,672 | <p>I wanted to prove <span class="math-container">$(Ma)\times(Mb)=(\det M)(M^{-1})^T(a\times b)$</span> where the formula could be found here <a href="https://en.wikipedia.org/wiki/Cross_product#Algebraic_properties" rel="nofollow noreferrer">wiki's cross product page</a> However, I had a hard time counting the index, ... | J.G. | 56,861 | <p>By the definition of the determinant, <span class="math-container">$\epsilon_{ijk}M_{il}M_{jm}M_{kn}=(\det M)\epsilon_{lmn}$</span>. Multiplying both sides by <span class="math-container">$a_lb_m$</span>, <span class="math-container">$[M^T(Ma\times Mb)]_n=(\det M)(a\times b)_n$</span>, i.e. <span class="math-contain... |
1,210,194 | <p>for example: I have a series <img src="https://i.stack.imgur.com/hV9QG.jpg" alt="enter image description here"></p>
<p>is there numerical computation method to find it ? thanks</p>
| Claude Leibovici | 82,404 | <p>Yoy can write
<span class="math-container">$$(1 - 4 n^2)^2 (36 n^2 + 1)=(2 n-1)^2 (2 n+1)^2 (6 n-i) (6 n+i)$$</span> Using partiai fractions, the summand is
<span class="math-container">$$\frac{7}{100 (2 n+1)}-\frac{7}{100 (2 n-1)}+\frac{81 i}{200 (6
n+i)}-\frac{81 i}{200 (6 n-i)}+$$</span> <span class="math-cont... |
2,440,785 | <p>if $f: X \to X$ is continuous where $X$ is a topological space with a cofinite topology, then:</p>
<p>$$(i) \ f^{-1}(x) \text{ is finite for all $x$} \\ \text{or} \\ (ii) \ f \text{ is constant}$$</p>
<p>My approach:</p>
<p>I couldn't build up a proper approach here to be honest. I believe we need to use the fac... | Community | -1 | <p>Hint: $f^{-1}(X\setminus \{ x\}) = X\setminus f^{-1}(\{x\})$ must be open.</p>
|
2,440,785 | <p>if $f: X \to X$ is continuous where $X$ is a topological space with a cofinite topology, then:</p>
<p>$$(i) \ f^{-1}(x) \text{ is finite for all $x$} \\ \text{or} \\ (ii) \ f \text{ is constant}$$</p>
<p>My approach:</p>
<p>I couldn't build up a proper approach here to be honest. I believe we need to use the fac... | Ice sea | 37,787 | <p>First, every point set $\{x\}$ is a closed set, the continuity of $f$ implies that $f^{-1}(x)$ is a closed set in $X$. Hence, $(f^{-1}(x))^c$ is an open set in $X$. So it has to be a finite set. </p>
|
475,151 | <blockquote>
<blockquote>
<p>Determine the volume of
$$
M:=\left\{(x,y,z)\in\mathbb{R}^3: z\in [0,2\pi],(x-\cos(z))^2+(y-\sin(z))^2\leq\frac{1}{4}\right\}
$$</p>
</blockquote>
</blockquote>
<p>My idea is to use the principle of Cavalieri, i.e. to set
$$
M_z:=\left\{(x,y)\in\mathbb{R}^2: (x,y,z)\in M\right... | Pratyush Sarkar | 64,618 | <p>You are making a mistake in concluding the following statement:
$$
(x-\cos(z))^2+(y-\sin(z))^2\leq\frac{1}{4} \iff \\
\cos(z)-\frac{1}{2}\leq x\leq\cos(z)+\frac{1}{2} \,\,\,\,\, \text{and} \,\,\,\,\, \sin(z)-\frac{1}{2}\leq y\leq\sin(z)+\frac{1}{2}
$$
And so you conclude the set $M_z$ is the rectangle $R_z = \left[\... |
1,957,166 | <p>For a given set $A$,
An element such that $a \in A $ exists. </p>
<p>If $A$ is a set of all natural numbers, then:</p>
<p>$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$</p>
<p>Would maths normally be written like this, if it is correct? </p>
| Bargabbiati | 352,078 | <p>A couple of things, if $A$ is the empty set doesn't exists any $a\in A$. If $A$ is the set of all natural numbers than you have $A= \mathbb N$. The inclusions are ok.</p>
|
1,957,166 | <p>For a given set $A$,
An element such that $a \in A $ exists. </p>
<p>If $A$ is a set of all natural numbers, then:</p>
<p>$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$</p>
<p>Would maths normally be written like this, if it is correct? </p>
| ForgotALot | 295,090 | <p>You have written:</p>
<p>$$a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$</p>
<p>and told us to assume $a\in A$ and $A=\mathbb{N}$. Under that assumption, the inclusion $A \in \mathbb{N}$ is incorrect; the set of all natural numbers is not a natural number (sorry I don't have a reference handy for... |
1,545,092 | <p>The following sum represents the number of relevant kinds of lines in an N-dimensional tic-tac-toe game, which is why I am interested in finding a closed form, but it also is the sum of all possible combinations of N unique elements when any number of the elements from 1 to N can be chosen, which is also cool, and s... | Aguila | 288,634 | <p>$S = 2^{N}-1$.
You can get this by noticing that $\sum\limits_{i=0}^{N}\binom{N}{i} = (1+1)^{N} = 2^{N}$ by the binomial theorem.</p>
|
306,732 | <p>Let $R$ and $S$ be rings and $h$ and $g$ be homomorphisms from $R$ into $S$. </p>
<p>Let $T=\{r| r\in R$ and $h(r)=g(r)\}$</p>
<p>Prove that $T$ is a subring of $R$. </p>
<p>I understand what the question is asking but I am a little confused on how to get it started. I know that I have to prove $T\neq \emptyset$;... | Julien | 38,053 | <p>You're right. Just get started. </p>
<p>First, $h(0)=0=g(0)$ so $0\in T$ and so $T\neq\emptyset$. </p>
<p>Next take $r,t\in T$, i.e. $h(r)=g(r)$ and $h(t)=g(t)$.</p>
<p>Then $h(rt)=h(r)h(t)=g(r)g(t)=g(rt)$.
So $rt\in T$.</p>
<p>Can you continue?</p>
|
1,057,050 | <p>I am currently doing a math problem and have come across an unfamiliar notation.
A mini circle between <span class="math-container">$f$</span> and <span class="math-container">$h(x)$</span></p>
<p>The question ask me to find for 'the functions <span class="math-container">$f(x)=2x-1$</span> and <span class="math-con... | Eric Stucky | 31,888 | <p>This notation means that you take the output of $h$ and use it as the input of $f$. When we are working with a specific $x$ value, we can suggestively write $f(h(x))$ instead.</p>
<p>For instance if $f(z)=1/z$ and $h(x)=2+3x$ then
$$(f\circ h)(x) = f\big(h(x)\big) = f(2+3x) = \frac{1}{2+3x}.$$</p>
<p>(Note: I onl... |
2,571,746 | <p>$$|x+4| -4 =x $$</p>
<p>I've two questions about this equation. </p>
<ul>
<li><p>Why do we need to build an inequality?</p></li>
<li><p>If we build an inequality, in what cases do we need to analyse?</p></li>
</ul>
<p>Also I'm trying to find the negative values that $x$ can take.</p>
| Community | -1 | <p>First of all, note that this isn’t an inequality, but an equation. There are two methods to solve this:</p>
<ul>
<li><p>Case by case analysis: Note that the equation can be written as: $$|x+4|= x +4 $$ When $x\leq -4$, we know that $(x+4)\leq 0$, hence, $|x+4|=-4-x$. Thus, it boils down to solving: $$-4-x=x+4 \impl... |
1,772,675 | <blockquote>
<p>Writer is writing a book and he is doing 2 mistakes per page</p>
</blockquote>
<p>What is the probability that the 2nd mistake of the writer is in page 3?</p>
<p>What I tried to do is as follows:<br>
$X$~$Poi(2)$ -> 2 mistakes per page hence<br>
$P(X=1)=2e^{-2}$ this is the chance to do 1 mistake i... | Paul | 202,111 | <p>You don't need two mistakes on the third page; you just need the second mistake on the third page.</p>
<p>For this problem it is best to consider TWO different Poisson processes:</p>
<p>(1) One Poisson process for the number of mistakes on the first two pages, with a rate of 4 for every two pages, and</p>
<p>(2) ... |
1,772,675 | <blockquote>
<p>Writer is writing a book and he is doing 2 mistakes per page</p>
</blockquote>
<p>What is the probability that the 2nd mistake of the writer is in page 3?</p>
<p>What I tried to do is as follows:<br>
$X$~$Poi(2)$ -> 2 mistakes per page hence<br>
$P(X=1)=2e^{-2}$ this is the chance to do 1 mistake i... | Med | 261,160 | <p>$3$ possible scenarios might happen:</p>
<p>1- one error in the first page, no error in the second page and at least one error in the third page. the probability would be:</p>
<p>$2*e^{-2}*e^{-2}*\sum_{i=1}^\mathbb{\infty}\frac{2^{i}*e^{-2}}{i!}$</p>
<p>2- No error in the first page, one error in the second page ... |
545,380 | <p>Let $F$ be a field such that $|F|=3^{2n+1}$ and $r=3^{n+1}$. I want to find the number of $x\in F$ that satisfies the equation $x^{r+1}=1$.</p>
| DonAntonio | 31,254 | <p>Well, since the multiplicative field of $\;\Bbb F\;$ , namely $\;\Bbb F^*:=\Bbb F-\{0\}\;$ is cyclic of order $\;3^{2n+1}-1\;$, we get that forall </p>
<p>$$x\in\Bbb F^*\;,\;\;x^{3^{2n+1}-1}=1\;$$</p>
<p>You though want some $\;x\in\Bbb F^*\;$ s.t.</p>
<p>$$x^{3^{n+1}+1}=1\iff \left(3^{n+1}+1\right)\mid\left(3^{2... |
231,549 | <p>Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent matrix and an identity matrix? I am not looking so much for name suggestions, but rather for a generally accepted term... | Dmitry Vaintrob | 7,108 | <p>I would call it projectipotent :)</p>
|
1,371,075 | <p>$$3^x = 3 - x$$</p>
<p>I have to prove that only one solution exists, and then find that one solution.</p>
<p>My approach has been the following:</p>
<p>$$\log 3^x = \log (3 - x)$$</p>
<p>$$x\log 3 = \log (3 - x)$$</p>
<p>$$\log 3 = \frac{\log (3 - x)}{x}$$</p>
<p>And this is where I get stuck. Any help will b... | Zain Patel | 161,779 | <p>Unfortunately, finding the solution explicitly is not possible in terms of elementary functions. You'll need to use the Lambert W function.</p>
<p>You have several methods of doing so, one is to simply sketch the graphs and show that they only intersect once as done below:</p>
<p><a href="https://i.stack.imgur.com... |
6,534 | <p>I apologize if this isn't the right place to ask this question.</p>
<p>Two features of stackexchange would be very useful for a personal math blog -- Latex works great, and comments and replies can be voted upon.</p>
<p>Is there any way to use the stackexchange functionality in a personal math blog?</p>
| GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Thanks to <a href="https://github.blog/2022-05-19-math-support-in-markdown/" rel="nofollow noreferrer">GitHub's recent announcement of MathJax support in GitHub Markdown</a>, you may setup a GitHub repo to write Markdown and LaTeX code to be stored in text files inside the repo. Others can give feedback through Git... |
9,304 | <p>I have the following question: I have a file that has structure:</p>
<pre><code>x1 y1 z1 f1
x2 y2 z2 f2
...
xn yn zn fn
</code></pre>
<p>I can easily visualize it with <em>Mathematica</em> using <code>ListContourPlot3D</code>. But could you please tell me how I can plot contour plot for this surface? I mean with t... | Mr.Wizard | 121 | <p>I'm not claiming this is a good method, I'm just getting some ink the page:</p>
<pre><code>data = Table[{x, y, z, x^2 + y^2 - z^2}, {x, -2, 2, 0.2}, {y, -2, 2, 0.2}, {z, -2, 2, 0.2}] ~Flatten~ 2;
ListContourPlot3D[data, Contours -> {0.5, 2}, Mesh -> None]
</code></pre>
<p><img src="https://i.stack.imgur.com... |
9,304 | <p>I have the following question: I have a file that has structure:</p>
<pre><code>x1 y1 z1 f1
x2 y2 z2 f2
...
xn yn zn fn
</code></pre>
<p>I can easily visualize it with <em>Mathematica</em> using <code>ListContourPlot3D</code>. But could you please tell me how I can plot contour plot for this surface? I mean with t... | Heike | 46 | <p>You can use the options <code>MeshFunctions</code> in combination with <code>Mesh</code> for this. </p>
<p>I'm borrowing Mr.Wizard's data here for a moment:</p>
<pre><code>data = Flatten[Table[{x, y, z, x^2 + y^2 - z^2}, {x, -2, 2, 0.2},
{y, -2, 2, 0.2}, {z, -2, 2, 0.2}], 2];
</code></pre>
<p>Suppose you... |
42,411 | <p>I have some products that I want to increase in value such that a 20% discount gives their current value. It's been ~25 years since college algebra and so I'm a bit rusty on setting up the equation.</p>
<p>I've been trying to figure out how to solve for X being the percentage increase needed in order that 20% off ... | Jonas Meyer | 1,424 | <p>Increasing an amount $A$ by $X$ percent means adding $A\cdot \frac{X}{100}$ to $A$, resulting in $A+A\cdot\frac{X}{100}=A\left(1+\frac{X}{100}\right)$. Decreasing an amount $B$ by $Y$ percent means subtracting $B\cdot\frac{Y}{100}$, resulting in $B-B\cdot\frac{Y}{100}=B\left(1-\frac{Y}{100}\right)$. To have an $X$... |
3,109,036 | <p>I must prove this tautology using logical equivalences but I can't quite figure it out. I know it has something to do with the fact that not p and p have opposite truth values at all times. Any help would be appreciated.</p>
| J.G. | 56,861 | <p>Rewrite <span class="math-container">$p\to\neg q$</span> as <span class="math-container">$q\to\neg p$</span> (both are equivalent to <span class="math-container">$\neg p\lor\neg q$</span>).</p>
|
2,523,000 | <p>Three players are playing a game and have a fair six-sided die. This is an arbitrary game conditioned on the following rule:</p>
<p>Player 1 rolls first, Player 2 roles until he has a different number to 1, Player 3 rolls until he has a different number to players 1 and 2. </p>
<p>$\underline{Question}$: How do I... | Christian Blatter | 1,303 | <p>Since the numerical values of the rolls are of no relevance there could as well be six different animals on the faces of the die. </p>
<p>This should make it clear that the expected value is $3.5$ for each of the three players, by symmetry.</p>
|
1,560,192 | <ol>
<li><p>Compute the determinant of \begin{pmatrix}
1 & 2 & 3 & ...& n\\
-1 & 0 & 3 & ...& n\\
-1 & -2 & 0 & ...& n\\
...& ...& ...& ...& \\
-1 & -2 & -3 & ...& n
\end{pmatrix}
after some elementary raw operation, one can reach:
\... | Tim Raczkowski | 192,581 | <p>Yes, there is a relationship. Note that
$$(n+1)^2=n^2+2n+1.$$ </p>
<p>So, the difference between a perfect square, $n^2$, and the next perfect square, $(n+1)^2$, is $2n+1$ which is always an odd number.</p>
|
1,560,192 | <ol>
<li><p>Compute the determinant of \begin{pmatrix}
1 & 2 & 3 & ...& n\\
-1 & 0 & 3 & ...& n\\
-1 & -2 & 0 & ...& n\\
...& ...& ...& ...& \\
-1 & -2 & -3 & ...& n
\end{pmatrix}
after some elementary raw operation, one can reach:
\... | costrom | 271,075 | <p>This result does not need calculus to be shown, just knowing how to expand $(a+b)^n$ for $n = 2$:</p>
<p>$(x+1) = x^2+2x+1$</p>
<p>Then subtract off $x^2$ (since that is the preceding square) and you get:</p>
<p>$x^2+2x+1-x^2 = 2x+1$</p>
<p>so each successive difference will increase by two (as it depends on $x$... |
25,784 | <p>As many Americans know, the “traditional” high school sequence is:</p>
<p>Algebra 1</p>
<p>Geometry</p>
<p>Algebra 2</p>
<p>PreCalculus</p>
<p>Calculus</p>
<p>For those who take developmental education at the community college level, it consists of something like:</p>
<p>Developmental Algebra</p>
<p>Intermediate Alg... | Argyll | 20,601 | <p>Typically, a geometry class in high school teaches Euclidean geometry. Depending on how much time is spent and the exact class, Euclidean geometry as rendered today explores properties of triangle, parallelogram, and circle, while following an axiomatic system, much of which was initially laid out thousands of years... |
1,550,934 | <p>I want to calculate the nth derivative of <span class="math-container">$\arcsin x$</span>. I know
<span class="math-container">$$
\frac{d}{dx}\arcsin x=\frac1{\sqrt{1-x^2}}
$$</span>
And
<span class="math-container">$$
\frac{d^n}{dx^n} \frac1{\sqrt{1-x^2}} = \frac{d}{dx} \left(P_{n-1}(x) \frac1{\sqrt{1-x^2}}\right)... | Lucian | 93,448 | <p>Let $~P_n(x)~=~\dfrac{2^n}{n!}~\Big(\sqrt{1-x^2}\Big)^{2n+1}~\bigg(\dfrac1{\sqrt{1-x^2}}\bigg)^{(n)}.~$ Then its coefficients form the sequence described <a href="http://oeis.org/A051288" rel="nofollow">here</a>.</p>
|
849,486 | <p>Find all <em>distinct</em> integers $x$ and $y$ that satisfy the following equation.</p>
<p>$$
x\log y=y\log x.
$$</p>
<p>Obviously, if $x=y$, the equation is satisfied.
I found $x=2$, and $y=4$. I think we cannot find all solutions (if there are many).</p>
<p>P.S. The base of the $\log(\cdot)$ is not important.<... | lab bhattacharjee | 33,337 | <p>For positive real $x,y$ $$x\log y=y\log x\iff \frac1y\log(y)=\frac1x\log(x)\implies y^{\dfrac1y}=x^{\dfrac1x}$$</p>
<p>Now, show that $f(z)z=z^{\dfrac1z}$ is not constant </p>
|
4,127,468 | <p>Suppose we have two functions <span class="math-container">$f,g:\Bbb R\rightarrow \Bbb R$</span>. The chain rule states the following about the derivative of the composition of these functions, namely that
<span class="math-container">$$
(f \circ g)'(x) = f′(g(x))\cdot g′(x).
$$</span>
However, the equivalent expre... | user21820 | 21,820 | <p>While the other answers deal with the modern definition of derivatives, it is <strong>not</strong> actually impossible to make the original Leibniz notation completely rigorous as I sketched <a href="http://math.stackexchange.com/a/2118909/21820">here (see "Notes")</a>. In fact, doing so yields a generaliz... |
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