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 |
|---|---|---|---|---|
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | Michael Rozenberg | 190,319 | <p>$$\lim\limits_{x\rightarrow8}\frac{\frac{1}{\sqrt{x+1}}-\frac{1}{3}}{x-8}=\lim\limits_{x\rightarrow8}\frac{8-x}{3(x-8)\sqrt{x+1}\left(3+\sqrt{x+1}\right)}=-\frac{1}{3\cdot3\cdot6}=-\frac{1}{54}$$</p>
|
2,316,561 | <p>How to evaluate the integral $$\frac{1}{2 \pi i}
\int \limits_{c-i \infty}^{c+i \infty} \frac{ds}{s(1-q^{1-s})}\text{?}$$ I tried with Perron's formula but I couldn't solve it. The result of the integral is $\frac{1}{2}$. Can someone help please?!</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
4,111,835 | <p>So I was given the following prompt:</p>
<blockquote>
<p>When <span class="math-container">$x=−2$</span>, for what values of p does the series converge?
<span class="math-container">$$\sum_{n=1}^\infty\left(\frac{(-1)^{n+1}(x-3)^n}{5^n\cdot n^p}\right)$$</span></p>
</blockquote>
<p>I ended up working out this proble... | user0102 | 322,814 | <p><strong>HINT</strong></p>
<p>For <span class="math-container">$x = -2$</span>, the proposed series reduces to
<span class="math-container">\begin{align*}
\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(-5)^{n}}{5^{n}n^{p}} & = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}(-1)^{n}5^{n}}{5^{n}n^{p}} = -\sum_{n=1}^{\infty}\frac{1}{n^{p}... |
109,037 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/107336/why-doesnt-dx-n-x-n1-rightarrow-0-as-n-rightarrow-infty-imply-x-n">Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?</a> </p>
</blockquote>
<p>my questio... | André Nicolas | 6,312 | <p>Your analysis is sound: though differences between neighbours may be ultimately small, these differences can build up. However, they do not <em>need</em> to build up: after all, there <em>are</em> Cauchy sequences. So the analysis has to be used to generate specific situations in which the differences <em>do</em> b... |
3,758,635 | <p>I'm trying to prove <span class="math-container">$n+\left(-1\right)^n\ge \dfrac{n}{2}$</span> is true for all natural numbers <span class="math-container">$n \ge 2$</span> via induction. The base case is trivial as
<span class="math-container">$$2+(-1)^2 \ge \frac{1}{2}(2)$$</span>
<span class="math-container">$$3 \... | Bananach | 70,687 | <p>An overkill solution would be to use the Gelfand formula, which states that</p>
<p><span class="math-container">$$
\rho(A) =\lim_n \|A^n\|^{1/n}
$$</span>
where the spectral radius <span class="math-container">$\rho(A)$</span> is defined as the supremum of the absolute values of all <span class="math-container">$x$<... |
1,193,558 | <p>I ran into this problem in a math camp, but I can't seem to solve it via elementary techniques. </p>
<p>If $a$ and $b$ are positive integers such that $a^n+n\mid b^n + n$ for all positive integers $n$, prove that $a=b$.</p>
| Anubhav Mukherjee | 72,329 | <p>No you are wrong...</p>
<p>$S$ is not discrete... $(m,n) \in S$ ( when $p=q=0$) where $m,n \in \mathbb{Z}$, and it is a limit point of a sequence in $S$.</p>
<p>But $S$ is countbale, and so $\mathbb{R^2}-S$ is path connected... you can find a general proof here <a href="https://math.stackexchange.com/questions/240... |
672,412 | <p>I am reading an e-book called <a href="http://www.ldsinsight.org/">To Infinity and Beyond</a> by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".</p>
<p>In talking about it, he seems to claim that because half a pie can be b... | John Habert | 123,636 | <p>If you want your mind blown some more, try <a href="http://en.wikipedia.org/wiki/Banach-Tarski_paradox">this</a>.</p>
<p>As for what is going on here, an infinite set is always the same size as half of itself because we can define a bijection between the two. The problem you seem to be running into is the author tr... |
672,412 | <p>I am reading an e-book called <a href="http://www.ldsinsight.org/">To Infinity and Beyond</a> by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".</p>
<p>In talking about it, he seems to claim that because half a pie can be b... | mjqxxxx | 5,546 | <p>There are (at least) two kinds of "size" in mathematics. One is <em>cardinality</em>. In set theory, the cardinality of a set is the number of elements it contains, and two sets have the same cardinality if there is a one-to-one mapping between them. This is a coarse kind of "size", in that many different sets sha... |
1,132,922 | <p>Let $f$ and $g$ be two differentiable functions s.t $ f '(x) \le g '(x) $ for all $ x\lt 1$ and $ f '(x) \ge g'(x) $ for all $ x\gt 1$ then</p>
<ol>
<li><p>If $f(1) \ge g(1)$, then $f(x)\ge g(x)$ for all $x$</p></li>
<li><p>If $f(1) \le g(1)$, then $f(x)\le g(x)$ for all $x$</p></li>
<li><p>$f(1) \le g(1)$</p></li>... | DeepSea | 101,504 | <p>$(f-g)' \leq 0 \to (f-g)(x) \geq (f-g)(1) \geq 0$ for $x < 1$ and $(f-g)(x) \geq (f-g)(1) \geq 0$ for $x > 1$. Thus: $1)$ is true.</p>
|
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Neal | 20,569 | <p>Other answers give good counterexamples ($\mathbb{R}$ and a point, $\mathbb{S}^2$ and a point), so I'm just going to write a little expository answer about searching for converses. </p>
<p>The fundamental group is far too weak to detect homeomorphism type. In fact, knowing the entire sequence of homotopy groups is ... |
1,423,491 | <p>Suppose you are flipping a coin with probability of Heads being 0.4931 and Tails being 0.5069</p>
<p>Can someone please tell me what is the probability of hitting 6 and 7 tails in a row in 24 tries? how about 44 tries?</p>
<p>OK. I have been asked to edit my question!
Here is the story:</p>
<p>The game "Baccarat"... | M. Aykens | 403,744 | <p>I recently asked a similar question here:</p>
<p><a href="https://math.stackexchange.com/questions/2081699/coin-flipping-likely-to-hit-7-heads-or-tails-in-a-row-answered-and-betting-p/2082919#2082919">Coin flipping - likely to hit 7 heads or tails in a row (answered) and betting progression</a></p>
<p>In my questi... |
3,921,255 | <p>If a function <span class="math-container">$f:M\to\mathbb{R}$</span> is continuous at point <span class="math-container">$x_0$</span> we know that for an arbitrary <span class="math-container">$\epsilon>0$</span> there exists a <span class="math-container">$\delta>0$</span> such that for all <span class="math-... | Aphelli | 556,825 | <p>Let’s prove something stronger: for each such manifold <span class="math-container">$M$</span> (without boundary), the path-connected component of the identity in the group of homeomorphisms of <span class="math-container">$M$</span> acts transitively on <span class="math-container">$M$</span>.</p>
<p>Since it’s a c... |
3,399,276 | <p>If the problem is to write the following with simplified polynomials</p>
<p><span class="math-container">$$\frac{x^2 + 5x + 6}{x^2+1}$$</span></p>
<p>Is it possible to do this problem with synthetic division? If so, how?</p>
<p>I've tried googling, finding on youtube, even plug this in Wolfram Alpha, no helpful r... | Andrew Chin | 693,161 | <p>Synthetic division is only used when you have a polynomial divided by a linear divisor. However, we can <em>creatively</em> decompose the numerator into something that we want.</p>
<p><span class="math-container">\begin{align}
\frac{x^2+5x+6}{x^2+1}&=\frac{\color{blue}{x^2}+5x+\color{blue}{1}+5}{x^2+1}\\
&... |
4,083,512 | <p>How exactly does multiplication make sense in synthetic geometry? I'll use a theorem expressing circle inversion. Let <span class="math-container">$C$</span> be some circle with radius <span class="math-container">$r$</span> and center <span class="math-container">$O$</span>, and let <span class="math-container">$P'... | shintuku | 755,010 | <p>EDIT: Found a simpler, more intuitive proof:</p>
<p><a href="https://i.stack.imgur.com/wzPwj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wzPwj.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/3G7Od.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... |
4,083,512 | <p>How exactly does multiplication make sense in synthetic geometry? I'll use a theorem expressing circle inversion. Let <span class="math-container">$C$</span> be some circle with radius <span class="math-container">$r$</span> and center <span class="math-container">$O$</span>, and let <span class="math-container">$P'... | Micah | 30,836 | <p>It is possible to define proportionality relationships between line segments in purely geometric terms. The trick is that you force the line segments to be legs of a right triangle, which makes everything well-defined.</p>
<p>More formally, we say that <span class="math-container">$AB:AC=AD:AE$</span> if in the foll... |
154,955 | <blockquote>
<p>Let $M\in M_n(F)$ and define $\phi:M_n(F)\to M_n(F)$ by $\phi(X)=AX$
for all $X\in M_n(F)$. Prove that $\det(\phi)=\det(A)^n$.</p>
</blockquote>
<p>I can prove it by considering the matrix representation with respect to the usual basis, which turns out to be a block diagonal form consisting of $n$ ... | JLA | 30,952 | <p>The eigenvalue equation is $\phi(X)=\lambda_k X\implies AX=\lambda_k X$, so the columns of $X$ are the eigenvectors of $A$ corresponding to the eigenvalue $\lambda_k$. The operator $A$ has n eigenvalues counting multiplicity, and the multiplicity of $\lambda_k$ corresponding to $\phi(X)$ equation is $n$ times the t... |
4,553,332 | <p>What do we call the property that if <span class="math-container">$a = b$</span>, then <span class="math-container">$f(a) = f(b)$</span>?</p>
<p>Wikipedia calls it "substitution property" but is that correct?</p>
| ajotatxe | 132,456 | <p>Well, I'd say, if you want to enter into philosophy, that two objects <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> can be distinguished only if there is some function <span class="math-container">$f$</span> that <span class="math-container">$f(x_1)\neq f(x_2)$</span>. (For ... |
249,597 | <p>I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that</p>
<p>$2x^2 + 4x + 1 = 0$</p>
<p>Can have at most two solutions. In calculus this is proven by the ... | rundavidrun | 43,933 | <p>You should definitely commit the formula to memory. Here's a cool video to the tune of row-row-row-your-boat, but there are plenty more out there. <a href="http://www.youtube.com/watch?v=HRcj9slciqM" rel="nofollow">http://www.youtube.com/watch?v=HRcj9slciqM</a>. Once you get on youtube, search around as there are lo... |
3,987,552 | <p>I need to prove or refute a property about a sequence of numbers.
Here is what is given to me:</p>
<p>Sequence (<span class="math-container">$a_1,a_2,...,a_k,a_{k+1},a_{k+2}$</span>) containing <span class="math-container">$k+2$</span> numbers. Every number <span class="math-container">$0 < a_i \leq M, i=1,...,k+... | Hagen von Eitzen | 39,174 | <p>If <span class="math-container">$a_i+a_{i+1}>M$</span> for all applicable <span class="math-container">$i$</span>, then
<span class="math-container">$$ N=\sum_{i=1}^{k+2}a_i=\frac{a_1}2+\sum_{i=1}^{k+1}\frac{a_i+a_{i+1}}2+\frac{a_{k+2}}2>\frac{k+1}2M$$</span>
and in fact
<span class="math-container">$$ N=\sum_... |
3,294,446 | <p>I'm currently working on a definite integral and am hoping to find alternative methods to evaluate. Here I will to address the integral:
<span class="math-container">\begin{equation}
I_n = \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx
\end{equation}</span>
Where <span class="math-container">$n \in \mathbb{N}$</s... | skbmoore | 321,120 | <p>Make an exponential generating function with <span class="math-container">$I_k$</span>,
<span class="math-container">$$I_k: = \frac{1}{2} (1 + (-1)^k) \int_0^\infty \frac{ \log^{k}(u) }{u^2+1} du $$</span>
Then
<span class="math-container">$$I(x)=\sum_{k=0}^\infty I_k\,\frac{x^k}{k!} = \frac{1}{2}\int_0^\infty \frac... |
4,602,464 | <p>I've been asked to provide context to my original question, so here's the context:</p>
<p>The rectangle in the problem below represents a pool table whose "pool table light" cannot be easily moved, but CAN easily be rotated. No portion of the pool table's perimeter can be too close to a wall in order for t... | rych | 73,934 | <p>The reconciling difference is in one word that is missing in the latter quote: <strong>differential</strong> 1-form. Differential 1-form is indeed a <em>smooth field</em> of (algebraic) 1-forms. This is described in @Nick's answer</p>
|
1,254,820 | <p>Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ C^{*} $-algebra? We can show that the map $ f \mapsto \overline{f} $ is an involution, but with this involution, $ {L^{1}... | Norbert | 19,538 | <p>A $C^*$-algebra $A$ is isometric to an $L_1$-space (even as Banach space!) iff it is one dimensional.</p>
<p>Assume $A$ is isometric to $L_1$ space and $\operatorname{dim}(A)>1$, then $A$ is <a href="http://en.wikipedia.org/wiki/Banach_space#Weak_convergences_of_sequences" rel="nofollow">weakly sequentially comp... |
1,595,206 | <p>For a practice question I have been given I have been told to find a spanning tree using a breadth first search for the following graph:</p>
<p><a href="https://i.stack.imgur.com/pKYwB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pKYwB.png" alt="enter image description here"></a></p>
<p>From ... | Element118 | 274,478 | <p>In general, you can use any searching method on a connected graph to generate a spanning tree, with any source vertex.</p>
<p>Consider connecting a vertex to the "parent" vertex that "found" this vertex. Then, since every vertex is visited eventually, there is a path leading back to the source vertex.</p>
<p>By pi... |
2,339,964 | <p>I wondered whether it is possible to define a sequence without relying on sets or natural numbers and tried with this definition.</p>
<ol>
<li>A symbol which is not a comma is a sequence. </li>
<li>If $S$ is sequence and $s$ a symbol which is not a comma:
<ul>
<li>$S, s$ is a sequence, where every symbol occur... | Xodarap | 2,549 | <p>Why not make it even simpler?</p>
<ol>
<li>Any symbol is a sequence</li>
<li>Sequences are closed under concatenation</li>
</ol>
<p>This is similar to the definition of the <a href="https://en.wikipedia.org/wiki/Free_monoid" rel="nofollow noreferrer">free monoid</a>.</p>
<p>In terms of addressing elements in a se... |
2,546,792 | <blockquote>
<p><strong>Definition.</strong> A metric space $M$ is connected if there is no other open disjoint $A$ and $B$ in $M$ such that $M=A\cup B$ than the empty set and the total space $M$. A subset $C\subset M$ is connected if the subspace $C$ is connected.</p>
<p><strong>Definition</strong>. A connected... | jgon | 90,543 | <p>Note that you can cut each petal in half longways, and then rearrange the half petals by translation to see that the shaded region has the same area as the part of a circle of radius the circumradius of the equilateral triangle minus an inscribed regular hexagon. Since you've computed the circumradius, $r=\sqrt{2}$,... |
2,546,792 | <blockquote>
<p><strong>Definition.</strong> A metric space $M$ is connected if there is no other open disjoint $A$ and $B$ in $M$ such that $M=A\cup B$ than the empty set and the total space $M$. A subset $C\subset M$ is connected if the subspace $C$ is connected.</p>
<p><strong>Definition</strong>. A connected... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/HKGAD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HKGAD.png" alt="enter image description here"></a></p>
<p>The center $O_a$
of the circular arc $COB$
with the radius $R_a=|O_aO|=|O_aB|=|O_aC|$
is found as $O_a=DOa\cap EO_a$,
$|DO|=|DB|$,
$|EO|=|EB|$,
$DO_a... |
99,750 | <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p>
<p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd ... | Dima Pasechnik | 11,100 | <p>If you start from basics, then J.Tits' "Local approach to buildings" [1] would certainly win, as you won't even need a definition of a group to describe the natural geometries for the exceptional Lie groups.</p>
<p>[1] Tits, J. "A local approach to buildings", The geometric vein: The Coxeter Festschrift, Springer-V... |
1,336,209 | <p>I need to understand very good how the properties of this formula</p>
<p>$\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $</p>
<p>Taken from the paper "Radian Reduction for Trigonometric Function" (Hanek Payne Algorithm)</p>
<p>Some remarkable... | Community | -1 | <p>Let
$$S(n):=1^4+2^4+3^4+4^4+\ldots+n^4.$$
The first order backward difference verifies
$$\nabla S(n):=S(n)-S(n-1)=n^4,$$
so that $S(n)$ must be a polynomial of the fifth degree in $n$, and the ratio $$\frac{S(n)}{n^5}$$ has a finite limit.</p>
<hr>
<p>More precisely, if the leading term of $S(n)$ is $an^5$, the le... |
3,752,676 | <p>what is <span class="math-container">$P(P(P(333^{333})))$</span>, where P is sum of digit of a number. for an example <span class="math-container">$P(35)=3+5=8$</span></p>
<p>a)18</p>
<p>b)9</p>
<p>c)33</p>
<p>d)333</p>
<p>f)5</p>
<p>I tried to find this but I couldn't. I started to find a pattern for an example the... | Batominovski | 72,152 | <p>Note that <span class="math-container">$G$</span> is generated by <span class="math-container">$a:=(1,0)$</span> and <span class="math-container">$b:=(0,1)$</span>. Let <span class="math-container">$\pi:G\to (G/H)$</span> be the canonical projection. Then, <span class="math-container">$\pi(a)$</span> and <span cla... |
308,251 | <p>I asked this question on Mathematics Stackexchange (<a href="https://math.stackexchange.com/q/2863312/660">link</a>), but got no answer.</p>
<p>Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$. </p>
<p>Recall that $A$ can be defined as the set of expressi... | Duchamp Gérard H. E. | 25,256 | <p>A side remark (but strongly connected). Let <span class="math-container">$X$</span> be the set of variables (in the question, <span class="math-container">$X=\{x_1,x_2,\dots \}$</span>) and <span class="math-container">$K[[X]]$</span> be the corresponding ring (<span class="math-container">$K$</span>-algebra) of ser... |
2,815 | <p>This is a final exam question in my algorithms class:</p>
<p>$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.</p>
<p>I don't know if the problem had a typo or not, because $O(n^3)$ seems more reas... | Miguel Velilla | 240,780 | <p>There is a faster algorithm to check if a given integer is a sum (or difference) of two cubes $n=a^3+b^3$</p>
<p>I don´t know if this algorithm is already known (probably yes, but I can´t find it on books or internet). I discovered and use it to compute integers to $n < 10^18$ </p>
<p>This process uses a single... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| OR. | 26,489 | <p>The operations that are relatively easy to compute by hand are addition, multiplication, and their inverses, subtraction, and division. With these operations we can compute all rational functions, e.g. $\frac{2x^2-1}{x^3+x-1}$.</p>
<p>We know that $$\ln(x)=\sum_{k=1}^{\infty}(-1)^k\frac{(x-1)^k}{k}$$</p>
<p>for va... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Jaume Oliver Lafont | 134,791 | <p>We can represent the logarithm of positive rational numbers as follows.</p>
<p>First, consider the following null conditionally convergent series (cancelled harmonic series):</p>
<p><span class="math-container">$$0=(1-1)+\left(\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{3}\right)+\left(\frac{1}{4}-\f... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Community | -1 | <p>We have the CORDIC method, which can be quite effective for by-hand computation as it requires additions/subtractions only (an one multiply by a small integer).</p>
<p>There are two limitations though:</p>
<ul>
<li><p>it is better performed in base $2$, so a preliminary change of base is needed for the input argum... |
2,567,989 | <p>I need to prove that if $\gamma_1,\gamma_2:[0,1] \to \Bbb C - \{0\}$ are closed paths and have the same index ( or winding number) around $0$ , then they are homotopic.</p>
<p>So, I think I have an idea on how to show it:</p>
<p>we have $g_1,g_2:[0,1] \to \Bbb C$ continuous logarithms for $\gamma_1 , \gamma_2$ , t... | eranreches | 208,983 | <p>Observe that for every $s$, the function $H\left(s,\cdot\right):\left[0,1\right]\rightarrow\mathbb{C}$ should also map under $\exp$ to a closed path with index $k$. Ergo, exactly as with the $g$'s</p>
<p>$$\frac{H\left(s,1\right)-H\left(s,0\right)}{2\pi i}=k$$</p>
<p>is the winding number of the path $\exp\Big(H\l... |
66,480 | <p>I have the following problem:</p>
<blockquote>
<p>Given $P(A)=0.2$, $P(B)=0.4$, $P(C)=0.8$, $P(D)=0.5$, find $P(A\cup B\cup C\cup D)$</p>
</blockquote>
<p>And the final answer should be 0.952</p>
<p>I know how to find the union of two and three elements (for 2, its: $A+B-AB$), but the formula becomes clumsy aft... | Sasha | 11,069 | <p>Use the following identity:</p>
<p>$$\mathbb{P}( A \cup B \cup C \cup D) = 1 - \mathbb{P}( (A \cup B \cup C \cup D)^c ) =
1 - \mathbb{P}( A^c \cap B^c \cap C^c \cap D^c )$$</p>
<p>Here $A^c$ means complement of set $A$. </p>
<p>Given independence of events $\mathbb{P}( A^c \cap B^c \cap C^c \cap D^c ) = \math... |
63,064 | <p>If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on linear algebra/functional analysis norms are studied ? </p>
<p>Aren't there also other functions that always give r... | Matthew Daws | 1,438 | <p>If a metric $d$ on a vector space $V$ is translation invariant, and compatible with scalar multiplication, in your sense, then define $$\|x\| = d(x,0)$$I claim that this is a norm:</p>
<ul>
<li>$\|x\|=0$ iff $d(x,0)=0$ iff $x=0$</li>
<li>$\|\lambda x\| = d(\lambda x,0) = d(\lambda x,\lambda 0) = |\lambda| d(x,0) = ... |
3,693,814 | <p>For <span class="math-container">$x,y,z>0.$</span> Prove<span class="math-container">$:$</span> <span class="math-container">$$P={x}^{4}y+{x}^{4}z+3\,{x}^{3}{y}^{2}-11\,{x}^{3}yz+3\,{x}^{3}{z}^{2}+3
\,{x}^{2}{y}^{3}+3\,{x}^{2}{y}^{2}z+3\,{x}^{2}y{z}^{2}+3\,{x}^{2}{z}^{
3}+x{y}^{4}-11\,x{y}^{3}z+3\,x{y}^{2}{z}^{2}... | Michael Rozenberg | 190,319 | <p>Another way by hand.</p>
<p>By AM-GM twice we obtain:
<span class="math-container">$$\sum_{cyc}(x^4y+x^4z+3x^3y^2+3x^3z^2-11x^3yz+3x^2y^2z)=$$</span>
<span class="math-container">$$=\sum_{cyc}(x^4y+x^4z-x^3y^2-x^3z^2+4x^3y^2+4x^3z^2-8x^3yz-3x^3yz+3x^2y^2z)=$$</span>
<span class="math-container">$$=\sum_{cyc}(x-y)^2... |
4,270,857 | <p>Solve by separation of variables
<span class="math-container">$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}$$</span>
given intitial conditions:</p>
<p><span class="math-container">$$\frac{\partial u}{\partial x}=0 \text{ at } x=a \text{ and } x=-a, \forall t\geq 0;$$</span>
<span class="math-con... | sankhya | 871,803 | <p>Are the coordinates of <span class="math-container">$X$</span> chosen independently and uniformly at random from <span class="math-container">$\{1,2,\ldots,n\}$</span>? Then, <span class="math-container">$E[x_ix_j]=E[x_i]E[x_j]$</span> and you can also calculate <span class="math-container">$E[x_i^2]$</span>.</p>
|
1,545,019 | <p>I'm sure this is probably an extremely simple problem but I'm stuck figuring this out.<br>
For example: </p>
<p>$(\frac{1}{5})^{x} + (\frac{7}{10})^{x} = 1$</p>
<p>What would be the steps to solve for x?</p>
| John | 7,163 | <p>Both terms on the left side are decreasing functions in $x$. Note also that the left hand side is $2$ when $x=0$ and $0.9$ when $x=1$.</p>
<p>A binary search between $x=0$ and $x=1$ works. I'll show you how to do this semi-manually with Excel or some similar program.</p>
<p>Enter the following (column A is blank... |
1,545,019 | <p>I'm sure this is probably an extremely simple problem but I'm stuck figuring this out.<br>
For example: </p>
<p>$(\frac{1}{5})^{x} + (\frac{7}{10})^{x} = 1$</p>
<p>What would be the steps to solve for x?</p>
| Claude Leibovici | 82,404 | <p>As Brevan Ellefsen answered, Newton method as described in the link given is a very good candidate for solving this kind of equations.</p>
<p>Just for your curiosity, let us apply it to your case $$f(x)=(\frac{1}{5})^{x} + (\frac{7}{10})^{x} - 1$$ using $x_0=1$ since $f(0)=1$ and $f(1)=-\frac 1 {10}$. The successiv... |
1,704,707 | <p>The function $f: ℝ → ℝ$ defined by $f(x) = x^{3}$ is onto because for any real number $r$, we have that $\sqrt[3]r$ is a real number and $f(\sqrt[3]r)=r$. Consider the same function defined on the integers $g: ℤ → ℤ$ by $g(n) = n^3.$ Explain why $g$ is not onto $ℤ$ and give one integer that $g$ cannot output. </p>
... | Kernel_Dirichlet | 368,019 | <p>As others have said, since each of the $f_k$'s are Lipschitz, then they are equicontinuous, they are of course also uniformly bounded since $||f_k||_\infty$ $<1$. This is enough to conclude that $D$ is compact. You do not need to prove closure because compact subsets of a Hausdorff space (all metric spaces, inclu... |
1,771,961 | <p>Suppose $U = \{(x,x,y,y) \in \mathbb{F}^4 : x, y \in \mathbb{F}\}$ </p>
<p>Find a subspace W of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U\oplus W$</p>
<p>Attempt: Now from what I understand I would think that an element of $\mathbb{F}^4$ would look like $$(w,x,y,z) \quad\mbox{such that}\quad
w,x,y,z \in \mat... | Ege Erdil | 326,053 | <p>$ U $ is a subspace spanned by the linearly independent set $ S = \{ (1, 1, 0, 0), (0, 0, 1, 1) \} $. Therefore, it suffices to pick a subspace $ W $ which is spanned by two vectors such that their adjoinment to $ S $ would not disturb its linear independence. In other words, we need to extend $ S $ to a basis of $ ... |
3,769,222 | <p>Evaluate: <span class="math-container">$\lim_{n\to \infty } \int_0^n (1-\frac{x}{n})^n \cos(\frac{x}{n})dx$</span></p>
<p>For fixed <span class="math-container">$n$</span>, I first rewrite my integral as
<span class="math-container">$$\int_0^n \left(1-\frac{x}{n}\right)^n \cos\left(\frac{x}{n}\right)dx = \int_0^\inf... | Oliver Díaz | 121,671 | <p><span class="math-container">$$
0\leq 1-\frac{x}{n}\leq e^{-x/n}\qquad 0<x\leq n$$</span></p>
<p>and so <span class="math-container">$0\leq \Big(1-\frac{x}{n}\Big)^n\mathbb{1}_{(0,n]}(x)\leq e^{-x}$</span>. The function <span class="math-container">$x\mapsto\cos(x/n)$</span> is controlled easily since <span class... |
1,125 | <p>I know volume preserving diffeomorphisms of a $sphere^2$ make a grou'p sdiff($S_2$). I would to know if it is a Lie group, which I assume if it is that makes interpolation easier (like with rotations). </p>
<p>So that is one question, is it a Lie group?</p>
<p>Also is the group path connected? If so, how can I int... | Community | -1 | <p>The term "volume preserving" sounds a bit ambiguous to me: do you mean that your map preserves the total volume or do you mean that its differential at every point preserves volume (i.e. has determinant 1)? The former is weaker than the latter, and gives you more room for interpolation.</p>
<p>In any case, there is... |
2,895,343 | <blockquote>
<p>Differentiate $\tan^3(x^2)$</p>
</blockquote>
<p>I first applied the chain rule and made $u=x^2$ and $g=\tan^3u$. I then calculated the derivative of $u$, which is $$u'=2x$$ and the derivative of $g$, which is
$$g'=3\tan^2u$$</p>
<p>I then applied the chain rule and multiplied them together, which ... | Lady | 586,433 | <p>$$u'=2x$$
$$g'=3\tan^2u \cdot sec^2u$$</p>
<p>$$f'(x)=2x \cdot 3\tan^2(x^2)\sec^2(x^2) = 6x\tan^2(x^2)\sec^2(x^2)$$</p>
|
74,290 | <p>I have this function $f(x)$ which is continuous and differentiable on $\mathbb R$. Is the following true - without the assumption that $f$ being absolutely continuous!</p>
<p>$$ \int_{a}^{b} f'(x) dx=f(b)-f(a)$$</p>
<p>Edit: $f$ is infinitely differentiable on $\mathbb R$. I think this may change everything!</... | Evgeny Savinov | 17,357 | <p>After edit, it became true, because of continuously $f'(x)$. And now you can use <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow">Newton-Leibniz formula</a></p>
|
185,097 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/1/different-kinds-of-infinities">Different kinds of infinities?</a> </p>
</blockquote>
<p>Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be in... | user17785 | 38,251 | <p>Cantor proved that there are "infinities" "bigger" than others.
For instance, $\mathbb{R}$ is strictly bigger than $\mathbb{N}$. What is meant by this can be stated the following way :
There is not enough natural numbers to number every real number.
In other words, if you have associated a real number to each natura... |
132,875 | <p>I was told that I could obtain an analytic solution to a particle falling under the influence of Newtonian gravity by using <code>DSolveValue</code>.</p>
<p><strong>What I am given</strong></p>
<ul>
<li>$G = M = m = 1$</li>
<li>$M$ is a point mass at $z=0$</li>
<li>the particle falls along the $z$ axis</li>
<li>$z... | Kagaratsch | 5,517 | <p>It seems that the differential equation solver gets hung up on the boundary conditions for some reason. Let us therefore do some intermediate steps by hand. First, consider the general solution of the differential equation:</p>
<pre><code>DSolve[{z''[t] + 1/z[t]^2 == 0}, z[t], t]
</code></pre>
<blockquote>
<p><c... |
132,875 | <p>I was told that I could obtain an analytic solution to a particle falling under the influence of Newtonian gravity by using <code>DSolveValue</code>.</p>
<p><strong>What I am given</strong></p>
<ul>
<li>$G = M = m = 1$</li>
<li>$M$ is a point mass at $z=0$</li>
<li>the particle falls along the $z$ axis</li>
<li>$z... | Jens | 245 | <p>Here is a very short answer, using <strong>energy conservation</strong>:</p>
<pre><code>energy = 1/2 z'[t]^2 - 1/z[t];
zSolution[t_] =
z[t] /. Simplify@First[DSolve[energy == 0 && z[0] == 0, z[t], t]]
(* ==> (3^(2/3) (-t)^(2/3))/2^(1/3) *)
</code></pre>
<p>Energy conservation reduces the order of th... |
1,715,324 | <p>I am wondering what is the difference between algebraic sets and algebraic varieties in complex projective space.</p>
<p>It seems that both are zero sets of polynomials, so what is the difference?</p>
| Georges Elencwajg | 3,217 | <p>If we give $\mathbb A^n$ (resp. $ \mathbb P^n$) their Zariski topology, an algebraic set is just a closed subset $V\subset \mathbb A^n$ (resp. $V\subset \mathbb P^n$).<br>
An algebraic variety is a vastly more general general concept, the basic object in classical algebraic geometry (even if since Grothendieck we... |
102,966 | <p>Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.</p>
<blockquote>
<p>Does every (abstract group) embedding $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ extend to ... | Florian Eisele | 17,498 | <p>No (at least if you really mean <em>abstract</em> group embeddings). Choose $m=n=2$. Note that
$$UT(2,\mathbb R) \cong (\mathbb R,+)\quad \textrm{ and }\quad T^*(2,\mathbb R)\cong UT(2,\mathbb R)\rtimes (\mathbb R_+,\cdot)^2 $$ </p>
<p>Note secondly that the image of the conjugation action of $T^*(2,\mathbb R)$ on ... |
2,273,477 | <p><strong>Bessel's Inequality</strong></p>
<p>Let $(X, \langle\cdot,\cdot\rangle )$ be an inner product space and $(e_k)$ an orthonormal sequence in $X$. Then for every $x \in X$ : $$ \sum_{1}^{\infty} |\langle x,e_k\rangle |^2 \le ||x||^2$$
where $\| \cdot\|$ is of course the norm induced by the inner product. </p>
... | Martin Argerami | 22,857 | <p>What you are missing is that the statement says <em>orthonormal sequence</em> and not <em>orthonormal basis</em>. When the sequence is a basis, you get Parseval's equality. But the inequality holds for "partial sums". </p>
<p>If you already have $\sum_k a_ke_k=x$, then of course you get an equality. </p>
|
349,212 | <p><strong>Context:</strong> I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts like Noether's theorem etc. in a mathematically rigorous way since the standard physics literature ... | Tobias Diez | 17,047 | <p>This is meant as a long comment to the very good answer by Pedro Riberio.</p>
<p>There is a nice analog of the variational bicomplex in the functional framework. Namely, the space of differential forms on <span class="math-container">$M \times \Gamma(E)$</span> (where <span class="math-container">$E$</span> is a fi... |
2,367,788 | <p>Need to find the total numbers out of all 6 digit numbers where a digit is repeated exactly 4 times in the number. </p>
<p>Eg. 111122, 111123 is valid
But 111121 is not valid.</p>
| true blue anil | 22,388 | <p>We can avoid cases by recognizing that $\frac 9{10}$ of a string of random digits won't have a leading zero </p>
<p>Thus, [Choose and place quadruple]$\times$[Place different digit(s) in remaining two slots]$\times\frac9{10}$</p>
<p>$10\times \binom64 \times 9^2 \times \frac9{10} = 10,935$ </p>
|
3,902,836 | <p>There're similar questions already, but I somewhat struggled to apply their reasoning to this particular statement.</p>
<p>I wanted to ask if my proof is correct (and clarify several things which I've seemingly figured out when writing it here). Here's the proof:</p>
<p><span class="math-container">$$\mbox{(1) }\lim... | Ben | 754,927 | <p>WA Don nailed it, but I feel compelled to add an answer.</p>
<p>I'm going through Spivak myself and when first encountering problems like this, I tried approaches very similar to super.t's.</p>
<p>It took a while to absorb what was going on. Initially I found it a bit confusing trying to keep straight which conditio... |
3,006,595 | <p>How to prove</p>
<blockquote>
<p><span class="math-container">$$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$</span>
where <span class="math-container">$G$</span> is the Catalan's constant.</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>For the first sum,
... | Jack D'Aurizio | 44,121 | <p>The series involving <span class="math-container">$H_k$</span> and <span class="math-container">$H_{2k}$</span> can be studied in a similar way: since
<span class="math-container">$$ \frac{-\log(1-x)}{1-x} = \sum_{n\geq 1} H_n x^{n} $$</span>
we have <span class="math-container">$ \frac{-\log(1+x^2)}{1+x^2} = \sum_{... |
3,006,595 | <p>How to prove</p>
<blockquote>
<p><span class="math-container">$$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$</span>
where <span class="math-container">$G$</span> is the Catalan's constant.</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>For the first sum,
... | Ali Shadhar | 432,085 | <p><span class="math-container">\begin{align}
S&=2\sum_{n=0}^\infty\frac{(-1)^nH_{2n}}{(2n+1)^2}+\sum_{n=0}^\infty\frac{(-1)^nH_{n}}{(2n+1)^2}\\
&=2\sum_{n=0}^\infty\frac{(-1)^nH_{2n+1}}{(2n+1)^2}+\sum_{n=0}^\infty\frac{(-1)^nH_{n}}{(2n+1)^2}-2\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^3}\\
&=2\Im\sum_{n=1}^\inf... |
1,163,001 | <p>I need help to prove this congruence: </p>
<p>$$ 3^n -4(2^n) + 6(1^n) + (-1)^n \equiv 0 \pmod {24} $$</p>
<p>I have tried to used Euler's Theorem on the powers of 2 and 3 individually but now I'm stuck.</p>
| Bill Dubuque | 242 | <p>${\rm mod}\ 3\!:\ f(n) \equiv 2^n - 2^n\equiv 0\ $ for $\,n\ge 1$</p>
<p>${\rm mod}\ 8\!:\ f(n)\equiv 3^n+(-1)^n+6,\,$ for $\,n\ge 1.\ $ $\,f(1)\equiv 0\equiv f(2),\,$ and $\,f(n+2)\equiv f(n)\,$ by $\,3^2\equiv 1\equiv (-1)^2,\,$ so $\,f(n)\equiv 0\,$ for all $\,n\ge 1\,$ by induction.</p>
<p>Therefore $\ 3,8\mi... |
386,298 | <p>Kahler spaces are just certain singular spaces equipped with a Kahler metric in appropriate sense. I first came across it Demaily-Paun's classical paper <a href="https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p05.pdf" rel="nofollow noreferrer">Numercical Characterization of the Kahler cone of a ... | HYL | 14,037 | <p>For a reference in English, you could take a look at this <a href="https://eudml.org/doc/164492" rel="nofollow noreferrer">paper</a> of Varouchas, which contains a definition of Kähler spaces and relatied concepts (e.g. Kähler morphisms) and some of their fundamental properties.</p>
|
386,298 | <p>Kahler spaces are just certain singular spaces equipped with a Kahler metric in appropriate sense. I first came across it Demaily-Paun's classical paper <a href="https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p05.pdf" rel="nofollow noreferrer">Numercical Characterization of the Kahler cone of a ... | YangMills | 13,168 | <p>There is a discussion of this topic in the recent book <a href="https://smf.emath.fr/publications/cycles-analytiques-complexes-ii-lespace-des-cycles" rel="nofollow noreferrer">Cycles analytiques complexes II : l'espace des cycles</a> by Barlet and Magnússon, chapter XII.3. An English translation (to be published by ... |
1,631,366 | <p>I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$
It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.</p>
| marty cohen | 13,079 | <p>You want,
for $x > 10^{10}$,
$(\log(x))^{10}
< x
$.
Setting
$x = y^{10}$,
this is
$(\log(y^{10}))^{10}
< y^{10}
$
for
$y > 10$
or
$\log(y^{10}) < y$
or
$y^{10} < 10^y$.</p>
<p>Since
$x^{1/x}$ is decreasing
for $x > e$,
if $y > 10$,
$10^{1/10} > y^{1/y}$
or
$10^y > y^{10}$.</p>
|
237,464 | <p>Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.</p>
<p>Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty f_k(a;b)$$
QUESTIONs: </p>
<p>(1) Does $f(a;b)=1$ have any solution with $a\neq 0$?</p>
<p>(2) If yes: Sing... | Gerald Edgar | 454 | <p>Numerically, I get this ($500$ factors)...<br>
red is ${}> 1$, yellow is ${} < 1$.</p>
<p><a href="https://i.stack.imgur.com/nSN9s.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nSN9s.jpg" alt="graph"></a></p>
<p>the jagged parts are probably just from numerical approximation. I'm guessi... |
2,466,022 | <p>In Real and Complex Analysis, 3rd Edition, Walter Rudin advances the following:</p>
<p><a href="https://i.stack.imgur.com/XEVwK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XEVwK.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/dTwbg.png" rel="nofollow noreferrer... | Dr. Sonnhard Graubner | 175,066 | <p>if $e^z$ assumed to be $0$ then it can not be $$e^z\cdot e^{-z}=1$$ (it would be zero)</p>
|
3,203,346 | <p>I know that <span class="math-container">$f(0)=2$</span>, <span class="math-container">$f'(0)=3$</span> and <span class="math-container">$g=f^{-1}$</span>.</p>
<p>But how can I find the value of <span class="math-container">$g'(2)$</span>?</p>
| Tojra | 655,621 | <p>You have:
<span class="math-container">$g(f(x))=x$</span></p>
<p>Differentiate both sides:
<span class="math-container">$g' (f(x)) f'(x)=1$</span><br>
Put <span class="math-container">$x=0$</span>
<span class="math-container">$g' (f(0)) f'(0) =1$</span><br>
Therefore,<span class="math-container">$ g'(2)= 1/3$</span... |
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Koto | 355,087 | <p>I think I've answered this question before, but note that $e^x=\sum_{n=0}^{\infty} \frac{x^n}{n!}$, so for every fixed $x\in \mathbb{R}$ the series converges, thus the sequence of the terms $\{\frac{x^n}{n!}\}_{n\in \mathbb{N}}$ must converge to zero as $n$ goes to infinity, otherwise, the sum would not converge.</p... |
264,740 | <p>On Hilbert spaces, the following is true:</p>
<p>Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.</p>
<p>The obvious proof I see to show this uses explicitly the Hilbert space structure which is why I would like to ask:</p>
<p>Is the same result true for operators on ... | Matthew Daws | 406 | <p>(This is really a very long comment...)</p>
<p>I think maybe the actual question comes about because some of the terminology in this area is hazy. Let $T:X\supseteq D(T)\rightarrow X$ be a linear operator on a Banach space $X$. For example, <a href="https://en.wikipedia.org/wiki/Resolvent_set" rel="nofollow noref... |
4,029,725 | <p>In the book by Artificial Intelligence by Norvig and Russel, I came across following problem:</p>
<blockquote>
<p>Prove if correct: <span class="math-container">$(A ∧ B) \models (A ⇔ B).$</span></p>
</blockquote>
<p>I quickly interpreted <span class="math-container">$\models$</span> as <span class="math-container">$... | Graham Kemp | 135,106 | <p><span class="math-container">$\mathcal A \vDash \varphi$</span> says: "statement <span class="math-container">$\varphi$</span> is a semantic consequence of the list of statements <span class="math-container">$\mathcal A$</span> (often called the <em>premises</em>, or <em>assumptions</em>)" and a semantic c... |
4,449,164 | <p>My question stems from page 2 of <a href="https://www.semanticscholar.org/paper/Stability-and-positive-supermartingales-Bucy/31bbbfc842b5717ead8cd7e997a5117fe7a66373" rel="nofollow noreferrer">this paper by Bucy</a>, which states:</p>
<blockquote>
<p>[A random variable <span class="math-container">$x$</span>] is alm... | Mason | 752,243 | <ol>
<li><p>It means that there exists <span class="math-container">$c \in \mathbb{R}$</span> such that <span class="math-container">$X = c$</span> a.s.. It's essentially what you wrote, except <span class="math-container">$g$</span> is measurable (if <span class="math-container">$g$</span> is allowed to be arbitrary, ... |
13,829 | <p>I was trying to understand the notion of a connection. I have heard in seminars that a connection is more or less a differential equation. I read the definition of Kozsul connection and I am trying to assimilate it. So far I cannot see why a connection is a differential equation. Please help me with some clarificati... | Ross Millikan | 1,827 | <p>It diverges. You should be able to prove that $|\cos(x^2)|>0.1$ for most x. If you let x=$\sqrt{\pi}u$ it is easier to assess the range of u where the cosine is close to zero.</p>
|
73,675 | <p>Dear all,</p>
<p>I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf" rel="noreferrer">http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf</a>). More particularly, Theorem 6 of this lecture s... | John Palmieri | 4,194 | <p>If you're looking for a reference in print, it's in Ravenel's book <em>Complex Cobordism and Stable Homotopy Groups of Spheres</em>. See the comments after the proof of Theorem A2.2.18. (This book is <a href="http://www.math.rochester.edu/people/faculty/doug/mu.html#repub" rel="noreferrer">available online</a>, an... |
2,385,152 | <p>Given a matrix A, e.g.
$$
A=\begin{bmatrix}
a_{11}& a_{12} & a_{13} \\
a_{21}& a_{22} & a_{23} \\
a_{31}& a_{32} & a_{33} \\
\end{bmatrix}
$$
eliminating the row and the column corresponding $a_{21}$ results in a smaller matrix
$$
B=\begin{bmatrix}
a_{12} & a_{13} \\
a_{32} & a_{... | Luis Vera | 178,730 | <p>The only place I remember seeing special notation is in Steven Roman's "Advanced Linear Algebra".</p>
<blockquote>
<p>Let $A \in \mathcal{M}_{n \times m}(\mathbb{F}),\, A=[a_{ij}].$ If $B \subseteq\{1,\ldots,n\}$ and $C\subseteq\{1,\ldots,m\},$ then
$\,A[B,C]\,$ denotes the submatrix of $A$ that results in rem... |
2,423,055 | <p>I am sort of baffled by this thing, already real number has every thing in it why is this concept of $\Bbb R^2$ ? What does it mean? What is its advantage?</p>
| md2perpe | 168,433 | <p>$\mathbb R $ doesn't have everything. For example, it doesn't have an $x $ such that $x^2 = -1$. The complex numbers contain such objects.</p>
<p>$\mathbb R^2$ can be used for a lot of things. The most obvious is to describe points in a plane.</p>
|
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | aryaman | 362,622 | <p>The area of a triangle $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$ is given by
$$\triangle= \left|\frac{1}{2}(x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2}))\right| $$
If the area of triangle is zero, it means the points are collinear.</p>
<p>If we code this in <code>Python3</code>, it ... |
101,929 | <p>I am trying to do the following :-</p>
<pre><code>f[x_] := {x, 2};
Do[f[x_] :=Append[f[x], {x, 4}],{3}] % runs into recursion limit
</code></pre>
<p>because I want to grow my function (which is a list of functions) in a loop ? What is the right way to do this ?</p>
| cleanplay | 8,107 | <p>I can do something like :-</p>
<pre><code> f[x_] := {x, 2};
Do[g[x_] = Join[f[x], {{x, 4}}]; f[x_] = g[x]; Clear[g], {3}]
</code></pre>
<p>I will accept better answers if there are. Thanks</p>
|
310,651 | <p>$\mathbb{Z}_{30}$* $= \{1,7,11,13,17,19,23,29\}$</p>
<p>The number of elements is $8$ and $8$ is not prime, therefore $\mathbb{Z}_{30}$* is not cyclic.</p>
<p>and the generators are $7,11,13,17,19,23,29$.</p>
<p>can anyone correct me please?</p>
| Ittay Weiss | 30,953 | <p>A group of prime order must indeed be cyclic. But a group of non-prime order, may or may not be cyclic. For instance $\mathbb Z_2 \times \mathbb Z_2$ is a non-cyclic group of order $4$, while $\mathbb Z_4$ is a cyclic group of order $4$. If the order of the group is not a prime number then you simply don't automatic... |
4,446,352 | <p>Consider two different ways to formally define a predicate, say the predicate <span class="math-container">$E(x)$</span> which is to be interpreted as "<span class="math-container">$x$</span> is even."</p>
<p><strong>CASE <span class="math-container">$~1$</span></strong></p>
<p><span class="math-container"... | Tankut Beygu | 754,923 | <p>It may be helpful to look into the statements in terms of <em>restricted quantifiers</em>. Restricted quantification is used when the range of a quantified variable is desired to be restricted to a part of the domain of discourse. Usual notations for the general case are (simpler notations are also used):</p>
<p><sp... |
3,654,315 | <blockquote>
<p>Let <span class="math-container">$m$</span> be an odd positive integer. Prove that</p>
<p><span class="math-container">$$ \dfrac{ \sin (mx) }{\sin x } = (-4)^{\frac{m-1}{2}} \prod_{1 \leq j
\leq \frac{(m-1)}{2} } \left( \sin^2 x - \sin^2 \left( \dfrac{ 2 \pi
j }{m } \right) \right) $$</span></p>
</bl... | Conrad | 298,272 | <p>Note that <span class="math-container">$\sin (x-\frac{2\pi j }{m})=-\sin(x+\frac{(m-2j)\pi}{m})$</span> and <span class="math-container">$m-2j$</span> goes through the odd numbers <span class="math-container">$1,...m-2$</span> when <span class="math-container">$ 1\le j \le \frac{m-1}{2}$</span></p>
<p>By the parale... |
4,582,378 | <p><em>Defintition:</em> A real sequence <span class="math-container">$\ (x_n)_n\ $</span> is <em>convex</em> if <span class="math-container">$\ x_n - x_{n+1} \geq x_{n+1} - x_{n+2}\quad \forall\ n\in\mathbb{N}. $</span></p>
<p>Continuing on from this question <a href="https://math.stackexchange.com/questions/4581894/i... | Adam Rubinson | 29,156 | <p>Given any strictly positive, convex, decreasing, summable <span class="math-container">$\ (a_n).$</span></p>
<p>Let: <span class="math-container">$\ k_1 = 1,\quad k_{j+1} = 2 \left\lceil \frac{1}{a_{k_j}} \right \rceil + k_j,\qquad \forall\ j\in\mathbb{N}$</span></p>
<p>Then for each <span class="math-container">$\ ... |
11,435 | <p>I know what they are and how to define them. They have to serve some purpose - else why include them? But I never used them and all examples on this site and the docs never show a practical use of <code>UpValues</code>.</p>
<p>Can someone give me an example to see when they are actually needed?</p>
| Verbeia | 8 | <p>I am embarrassed to admit that I completely forgot about my own <a href="http://www.verbeia.com/mathematica/mma/upvalues.nb" rel="noreferrer">tutorial explaining how to use <code>UpValues</code> to create a lag operator</a>, which I wrote sometime in the late 1990s and have had on my web site since then.</p>
<p>The... |
2,121,611 | <p>Compute a quadrature of</p>
<p>$\int_c^d\int_a^b f(x,y)dxdy$</p>
<p>using the Simpson rule and estimate the error.</p>
<p>So the Simpson rule says </p>
<p>$S(f) = (b-a)/6(f(a)+4f((a+b)/2) +f(b))$</p>
<p>So i get </p>
<p>$\int_c^d(b-a)/6(f(a)+4f((a+b)/2) +f(b))dy$</p>
<p>Is that even correct? How do I go on?<... | slitvinov | 78,215 | <p>Something to get you started. Define an operator:
$$
I_x(f) := \int_a^b f(x) \, dx
$$
We can see that $I_x(f + g) = I_x(f) + I_x(g)$ and if $\alpha$ does not depend on $x$, $I_x(\alpha f) = \alpha I_x(f)$. In other words the operator is linear.</p>
<p>If it acts on $f(x, y)$ it returns a function of $y$ only.</p>
... |
626,920 | <p>$a_n=\sum_{k=1}^{n} \frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$</p>
<p>How to find $\lim a_n$?</p>
| Lucian | 93,448 | <p>Brutally speaking, $$\sum_{k=1}^{n}\frac1{n+k}=\left(\sum_{k=1}^{2n}\frac1k\right)-\left(\sum_{k=1}^{n}\frac1k\right)\simeq\ln2n-\ln n=\ln\frac{2n}n=\ln2.$$ Rigourously, we have $$\sum_{k=1}^{n}\frac1{n+k}=\frac1n\cdot\sum_{k=1}^{n}\frac1{1+\frac kn}=\int_0^1\frac{dx}{1+x}=\ln(1+x)|_0^1=\ln2.$$</p>
|
293,110 | <p>How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way. </p>
<p>I would love to get de... | Flash Sheridan | 13,530 | <p>Church’s “Set Theory with a Universal Set” and its variants have complements, with Replacement restricted to sets equinumerous to a well-founded set:</p>
<ul>
<li>Alonzo Church 1974a. “Set Theory with a Universal Set,” <em>Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV,</em> ed... |
503,691 | <p>I'm a bit confused about a statement that I see often in the $\lambda$-calculus literature. Namely, what exactly does the following statement mean: "By induction on the length of $M\in\Lambda$." ?</p>
<p>In such literature, $\Lambda$ is defined as the smallest subset of $\mathcal{V}\cup\{(,),\lambda\}$ such that:</... | Trismegistos | 23,730 | <p>For induction on length inductive hypothesis is that statement holds for lambda term of length n. Induction step is to infer that statement holds for lambda term of length n + 1 when it holds for lambda term of length n.</p>
<p>Structural induction is described in this topic <a href="https://math.stackexchange.com/... |
403,344 | <p>I'm trying to determine whether this statement is true or false:</p>
<p>Assume that $dim V = n$ and let $T \in L(V)$. Let $f(z) \in P_n(F)$ be a
monic polynomial of degree
$n$ such that $f(T) = 0$. Then $f(z)$ is the characteristic polynomial of $T$.</p>
<p>I know that because $f(T) = 0$, the minimal polynomial of... | N. S. | 9,176 | <p><strong>Hint</strong> Take $T$ to be the $0$ operator. Then $f(T)=0$ for all $f(z)$ divisible by $z$.</p>
|
2,421,421 | <blockquote>
<p>Evaluate
$$ \lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$$ </p>
</blockquote>
<p>I tried to solve this by L'Hospital's rule..but that doesn't give a solution..appreciate if you can give a clue.</p>
| neonpokharkar | 477,567 | <p>Dividing numerator and denominator by $\sqrt{2}$
$$\lim_{x\to \frac{π}{2}} \frac {\sqrt{\frac{1+\cos 2x}{2}}}{\sqrt {\frac{π}{2}} -\sqrt { x}}$$
$$$$
Multiplying and dividing by $(\sqrt {\frac{π}{2} } + \sqrt {x})$
$$=\lim_{x\to \frac {π}{2}}(\sqrt {\frac{π}{2}} + \sqrt x)(\frac {\sqrt {\frac {1+ \cos 2x}{2}}}{\frac... |
488,498 | <p>Find $dy/dx$ in terms of $x$ and $y$ when
$$x^2y^3=7x−3y$$</p>
<p>Not sure how to start here, would be nice with some pointers </p>
| JP McCarthy | 19,352 | <p>I am going to go a little bit further than the previous answer:</p>
<p>$$x^2y^3=7x-3y\,\,(*)$$</p>
<p>Now what does $\displaystyle \frac{dy}{dx}$ have to do with this equation? The derivative is concerned with slopes of tangents to curves. Where is the curve here? </p>
<p>Define a curve $\mathcal{C}$ on the plane... |
2,262,846 | <p>This question is somewhat naive. Please see the following proof before reading the question itself. </p>
<p>Let $L$ be a vector space over a field $\Bbb{F}.$ Assume there is an associative multiplication on $L$ so that $L$ is an associative ring, and denote this multiplication simply as $xy$ for $x,y\in L$. Further... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Using the AM-GM inequality, we have $x^6+y^6\ge 2\sqrt{x^6y^6}=2|x|^3|y|^3$, which provides an upper bound of $|y|$.</p>
|
2,300,386 | <blockquote>
<p>Consider the initial value problem:</p>
<p>\begin{cases}
u_{tt} &= c^2 u_{xx} \ \ & \text{for} \ -\infty < x < \infty, \ 0 \leq t < \infty\\
u(x,0) &= \phi(x) \ \ & \text{for} \ -\infty < x < \infty\\
u_t(x,0) &= \psi(x) \ \ & \text{for} \ -\infty < x &... | Wolfy | 217,910 | <p>We have
\begin{equation}
\begin{cases}
u_{tt} = c^2 u_{xx} \ \ & \text{for} \ -\infty < x < \infty, \ 0 \leq t < \infty\\
u(x,0) = \phi(x) \ \ & \text{for} \ -\infty < x < \infty\\
u_t(x,0) = \psi(x) \ \ & \text{for} \ -\infty < x < \infty
\end{cases}
\end{equation}
Therefore
\be... |
2,037,652 | <p>I'm taking a course on smooth manifolds and the following exercise was given to me:</p>
<blockquote>
<p>If $P\in\mathbb{R}[X_1, ..., X_{n+1}]$ is a homogeneous polynomial of degree $d$ such that $\left(\frac{\partial P}{\partial X_1}, ..., \frac{\partial P}{\partial X_{n+1}}\right)$ is never zero, prove that $Z(P... | ziggurism | 16,490 | <p>Dimension is another word for the number of degrees of freedom in a space. If you have <span class="math-container">$n$</span> dimensions and you impose one equation, you typically reduce the dimension by 1 (so we say the resulting space has codimension 1). For only the count of degrees of freedom, it does not matte... |
206,636 | <p>What's the fastest way to find the local maxima of a 2D list? <em>E.g.</em></p>
<pre><code>nx = ny = 100;
dat = Table[Sin[2. \[Pi] x/nx] (0.1 + Cos[2. \[Pi] y/ny]), {y, 0, ny}, {x, 0, nx}];
ListPlot3D[dat]
</code></pre>
<p><img src="https://i.stack.imgur.com/MyGBl.png" alt="Mathematica graphics"></p>
<p>This (u... | Ulrich Neumann | 53,677 | <p>A little bit faster than @kglr 's tricky answer is</p>
<pre><code>Position[dat, Max[dat]]
</code></pre>
|
2,245,010 | <p>Does there exist a topological group which can be covered by (nontrivial and proper) open subgroups of itself? If so, what are groups of these types called and is this a nice property for a topological group to have? Or is this just impossible?</p>
| Najib Idrissi | 10,014 | <p>It's certainly possible, consider $G = \mathbb{Z}^2$ with the discrete topology. If $x \in G$ is any element, consider $H$ to be the subgroup generated by $x$ (unless $x = 0$ in which case consider $H = \mathbb{Z} \times \{0\}$ for example). It's proper because $G$ has rank $2 > 1$, nontrivial, and open because $... |
38,480 | <p>(Before reading, I apologize for my poor English ability.)</p>
<p>I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral below
$$ \int_0^{\frac{\pi}{2}} \arctan (1 - \sin^2 x \; \cos^2 x... | J. M. ain't a mathematician | 498 | <p>I think it would be appropriate at this point to quote <a href="http://books.google.com/books?id=YHXU4W3Ez2MC&pg=PA247">Forman Acton</a>:</p>
<blockquote>
<p>...at a more difficult but less pernicious level we have the
inefficiencies engendered by exact analytic integrations where a sensible
approximation... |
144,709 | <p>Recently, I was informed that we can verify the famous formula about <strong>$\mathrm{lcm}(a,b)$</strong> and <strong>$\gcd(a,b)$</strong> which is $$\mathrm{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)} $$ via group theory. </p>
<p>The least common multiple of two integers $a$ and $b$, usually denoted by <strong>$\mathrm{lcm}(... | Arturo Magidin | 742 | <p><strong>Lemma.</strong> Let $G$ be a group, written multiplicatively, and let $H$ and $K$ be two subgroups. If $HK = \{hk\mid h\in H, k\in K\}$, then
$$|HK||H\cap K| = |H||K|$$in the sense of cardinalities.</p>
<p><em>Proof.</em> Consider the map $H\times K\to HK$ given by $(h,k)\mapsto hk$. I claim that the map is... |
2,232,677 | <p>A parallellogram $ABCD$ is given, where $E$ is the midpoint of $BC$ and $F$ is a point on $AD$ so that $|FD| = 3|AF|$. $G$ is the point where $BF$ and $AE$ intersects.</p>
<p>Express the vector $AG$ in terms of vectors $AB$ and $AD$.</p>
<p>My solution to the problem is the following: Impose an <em>affine transfor... | Soroush khoubyarian | 339,053 | <p>Firstly, the triangles AFG and BEG are similar. since AF is a quarter of AD, and BE is one half of BC(which is equal to AD) we can say that $\dfrac{AG}{EG} = \dfrac{1}{2}$ so it is obvious that $\dfrac{AG}{AE} = \dfrac{1}{3}$</p>
<p>Know we know that AE is just the sum of the vectors: $AB + BE = AB + \dfrac{AD}{2}$... |
77,980 | <p>How do I check whether a give function is 'algebraic' or not? I have a function $m(z) = 2\pi i z^n$ where $z \in \mathbb{C} \backslash \mathbb{R}$ and $n \in \mathbb{Z}$. I can write this as $w - 2\pi i z^n = 0$ where $w = m(z)$ and it is a polynomial. I think this is an algebraic function? Is my understanding corre... | Gerry Myerson | 8,269 | <p>If $n$ is a non-negative integer then $m$ is a polynomial, a fortiori, algebraic. As you say, it is a zero of the equation $w-2\pi iz^n=0$, an equation with polynomial coefficients. If $n$ is a negative integer then it is a zero of $z^{-n}w-2\pi i=0$ which again is an equation with polynomial coefficients, so in thi... |
138,173 | <p>$f(x, y) = 0$ and $g(x, y) = 0$,
both $f$ and $g$ are cubic polynomial equation (at most 10 coefficients for each).</p>
<p>Is there any fixed method to solve this degenerate equation system?
thanks.</p>
| Igor Rivin | 11,142 | <p>The magic word is "resultant" (which see, on, e.g., Wikipedia). The resultant of your equations will be a single one variable equation of degree nine. There is not much hope of it being solvable in radicals, but it will be easily solved numerically. On the other hand, if you have software which will solve it numeric... |
2,581,361 | <p>For even $n \in \mathbb{N}$, prove $\binom{n}{i}< \binom{n}{j} $ if $0\leq i<j\leq \frac{n}{2}$.</p>
<p>So far all I have been able to come up with are a bunch of seemingly useless inequalities. </p>
<p>Any hints would be greatly appreciated.</p>
| Robert Israel | 8,508 | <p>HINT:
$${n \choose j+1} = \frac{n-j}{j+1} {n \choose j} $$</p>
|
1,250,322 | <p>This is the recurrence relation I am trying to solve:
\begin{align}
T(n) & = 2 \cdot T \left( \frac{n}{4} \right) + 16, \\
T(1) & = c.
\end{align}
I broke this down (i.e., solved this recurrence relation) to $ \sqrt{2} * c * n + 32 * \sqrt{2} * n - 32 $, which runs in tight bounds $ \Theta(n) $. Can you guys... | Ian | 83,396 | <p>Following Wikipedia's notation for the master theorem, you have $a=2,b=4,f(n)=16$. So $\log_b(a)=\log_4(2)=1/2$, so $f(n)=O(n^{\log_b(a)})$. So we are in case 1, and $T(n)=\Theta(n^{1/2})$. So somewhere you made a mistake.</p>
|
4,017,790 | <p>We consider the equation <span class="math-container">$y^2-4y+4+x^2y+\arctan(x)=0$</span>, the exercice of my book say that the normal vector <span class="math-container">$n$</span> of the cartesian curve implicitally defined by the previous equation is <span class="math-container">$n=l(1,0)$</span> for <span class=... | Leoncino | 571,776 | <p>The only possible way for a person to discover an empty box and a full one, is if eight slices have been taken (not more). The second condition is, that every slice was taken from the same box. This adds up to your result <span class="math-container">$(\frac{1}{2})^8$</span> since every person must take a slice fro... |
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