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 |
|---|---|---|---|---|
2,414,472 | <blockquote>
<p>Let $(a_n)_{n\geq2}$ be a sequence defined as
$$
a_2=1,\qquad a_{n+1}=\frac{n^2-1}{n^2}a_n.
$$
Show that
$$
a_n=\frac{n}{2(n-1)},\quad\forall n\geq2
$$
and determine $\lim_{n\rightarrow+\infty}a_n$.</p>
</blockquote>
<p>I cannot show that $a_n$ is $\frac{1}{2}\frac{n}{n-1}$. Some helps? </p>
... | Michael Rozenberg | 190,319 | <p>Let $b_{n}=\frac{(n-1)a_n}{n}$.</p>
<p>Thus, $b_{n+1}=b_n$ and since $b_2=\frac{2-1}{2}a_2=\frac{1}{2}$, we are done!</p>
|
886,003 | <p>I have two questions:</p>
<p><strong>A)</strong> Suppose that we have $$Z=c\sum_i (X_i-a)(Y_i-b)$$ where $X_i$s and $Y_i $s are independent exponential random variables with means equal to $\mu_{X}$ and $\mu_{Y}$ (for $1\le i\le n$). That is $X_i$s are i.i.d random variables and so are $Y_i $s. Besides, $a,b$ and ... | StephanieCoding | 377,265 | <p>I don't agree with the formula you derived for <span class="math-container">$var(Z)$</span>. </p>
<p>If <span class="math-container">$Z = c \sum_i (X_i - a)(Y_i - b)$</span> where <span class="math-container">$X_i$</span> are i.i.d from distribution 1 and <span class="math-container">$Y_i$</span> are i.i.d from dis... |
2,569,267 | <p><a href="https://gowers.wordpress.com/2011/10/16/permutations/" rel="nofollow noreferrer">This</a> article claims:</p>
<blockquote>
<p>we simply replace the number 1 by 2, the number 2 by 4, and the number 4 by 1</p>
<p>....I start with the numbers arranged as follows: 1 2 3 4 5 6. After doing the permutation (124) ... | Community | -1 | <p>The third paragraph states that: </p>
<blockquote>
<p>$\ldots$ If we want to apply the permutation $(124)$, we simply replace the number $1$ by $2$, the number $2$ by $4$, and the number $4$ by $1$ $\ldots$.</p>
</blockquote>
<p>Following the rule (reading from left to right), gives us the required result.</p>
|
499,652 | <p>I saw this a lot in physics textbook but today I am curious about it and want to know if anyone can show me a formal mathematical proof of this statement? Thanks!</p>
| ftfish | 84,805 | <p>Consider $\lim_{x\to 0} \frac{\tan x}{x}$ and apply the <a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow">L'Hôpital's rule</a>.</p>
<p>You'll get a limit of $1$ as $x \to 0$, proving the statement (relative error).</p>
<p>You can even get a bound on the error term with <a href="http:/... |
281,288 | <p><img src="https://i.imgur.com/0C2Jl.jpg" alt="curved line graph"></p>
<p>In this curved line graph, I need to be able to make a formula that can tell me the interpolated value at any point on the curved path given one Data input.</p>
<p>So for example if I wanted to know what value the line was at exactly half way... | Luke Allen | 31,876 | <p>I have realized this is impossible for my scenario. The actual application of this would be for a graph with hundreds of points, all using splines, not Lagrange polynomial curves, which curve the line differently than splines somewhat, throwing off the accuracy which is all-important. Even using spline interpolation... |
79,658 | <blockquote>
<p>Let $U$ and $W$ be subspaces of an inner product space $V$. If $U$ is a subspace
of $W$, then $W^{\bot}$ is a subspace of $U^{\bot}$?.</p>
</blockquote>
<p>I don't find the above statement intuitively obvious. Could someone provide a proof?</p>
| wildildildlife | 6,490 | <p>It should be intuitive, already at the level of logic:</p>
<p>To be in $W^\perp$, you have to satisfy a certain condition $P(w)$ (namely: 'be orthogonal to $w$') for each and every element $w\in W$. </p>
<p>So given a subset $U\subseteq W$, to be in $U^\perp$ means you have to satisfy $P(w)$ <em>merely</em> for al... |
2,031,699 | <p>Let $A,B$ be open subsets of $\mathbb{R}^n$. </p>
<p>Does the following equality hold?</p>
<p>$$\partial(A\cap B)= (\bar A \cap \partial B) \cup (\partial A \cap \bar B)$$</p>
<p>Edit: Thanks for showing me in the answers that above formula fails if $A$ and $B$ are disjoint but their boundaries still intersect. I... | DanielWainfleet | 254,665 | <p>If $A$ is dense and co-dense in the non-empty space $X$ (that is, $X$ \ $A$ is also dense in $X$), suppose $B=X$ \ $A.$ Then $\emptyset=A\cap B=\partial (A\cap B)$ but $\bar A=\bar B=\partial A=\partial B=X\ne \emptyset.$</p>
<p>For example, with $X= \mathbb R^n$ let $A$ be the set of points with rational co-ordin... |
3,328,822 | <blockquote>
<p>How do I evaluate <span class="math-container">$$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$</span> </p>
</blockquote>
<p>To my knowledge the following integral should be related to the Gamma function.</p>
<p>I have tried using the substitution <span c... | Zacky | 515,527 | <p>It looks like a tricky integral, however Feynman's trick deals with it nicely.
<span class="math-container">$$I=\int^{\infty}_0 \exp\left(-\left(4x+\dfrac{9}{x}\right)\right) \sqrt{x}dx\overset{\sqrt x\to x}=2\int_0^\infty \exp\left(-\left(4x^2+\frac{9}{x^2}\right)\right)x^2 dx$$</span>
Now consider the following i... |
2,842,217 | <p>im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough ... | J.G. | 56,861 | <p>Write $\frac{\sin x}{x}=\frac{\tan x}{x}\cos x$ as a power series in $x^2$, with $\frac{\tan x}{x}=t_0+t_1 x^2+t_2 x^4+\cdots$. Equating coefficients of powers of $x^2$ one by one gives $1=t_0,\,-\frac{1}{6}=-\frac{t_0}{2}+t_1,\,\frac{1}{120}=\frac{t_0}{24}-\frac{t_1}{2}+t_2$ etc. Write down as many of those as you ... |
2,842,217 | <p>im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough ... | user5713492 | 316,404 | <p>My impression is that it's kind of backwards, in a numerical sense, to think about the coefficients of the $\tan$ series in terms of the Bernoulli numbers because it's simple and numerically stable to calculate the $\tan$ coefficients directly and in fact provides a reasonable method for computing the Bernoulli numb... |
3,581,390 | <p>The problem is as follows:</p>
<p>Mike was born on <span class="math-container">$\textrm{October 1st, 2012,}$</span> and Jack on <span class="math-container">$\textrm{December 1st, 2013}$</span>. Find the date when the triple the age of Jack is the double of Mike's age.</p>
<p>The alternatives given in my book are... | J. W. Tanner | 615,567 | <p>We have <span class="math-container">$M=14+J$</span>, where <span class="math-container">$M$</span> is Mike's age in months and <span class="math-container">$J$</span> is Jack's age in months,</p>
<p>and <span class="math-container">$2\times M=3\times J$</span>. Substitute <span class="math-container">$14+J$</span... |
2,101,756 | <p>From the power series definition of the polylogarithm and from the integral representation of the Gamma function it is easy to show that:
\begin{equation}
Li_{s}(z) := \sum\limits_{k=1}^\infty k^{-s} z^k = \frac{z}{\Gamma(s)} \int\limits_0^\infty \frac{\theta^{s-1}}{e^\theta-z} d \theta
\end{equation}
The identity ... | Ash | 407,754 | <p>As we know that $\log_33=1$ <br>$\therefore$ $$6\log_33=\log_3(y)^5-\log_3(y)$$ $$\log_3(3)^6=\log_3\left(\frac{y^5}{y}\right)$$ $$\log_3(3)^6=\log_3(y)^4$$ taking antilog both the side we can write as $$3^6=y^4$$ I don't know how did you find that equation but now from the first part of the full question $$\log_3(x... |
1,734,680 | <p>How can I find $F'(x)$ given $F(x) = \int_0^{x^3}\sin(t) dt$ ? <br>
I think that (by the fundamental theorem of calculus) since $f = \sin(x)$ is continuous in $[0, x^3]$, then $F$ is differentiable and $F'(x) = f(x) = \sin(x)$ but I'm not sure...</p>
| Community | -1 | <p>I think you might not know about antiderivatives yet, so this answer will avoid using them.</p>
<p>By the FTC,
$$ \frac{d}{dx} \int_a^x f(t) \, dt = f(x). $$</p>
<p>But you don't have $x$. You have $x^3$. So you'll need to use the chain rule:</p>
<p>$$ \frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)... |
3,991,691 | <p>I'm having some trouble proving the following:</p>
<blockquote>
<p>Let <span class="math-container">$d$</span> be the smallest positive integer such that <span class="math-container">$a^d \equiv 1 \pmod m$</span>, for <span class="math-container">$a \in \mathbb Z$</span> and <span class="math-container">$m \in \math... | poetasis | 546,655 | <p>The equation is symmetric and
it is easy to see solutions if, for example, we solve for <span class="math-container">$y$</span>.</p>
<p><span class="math-container">$$x^2 + y^2 - 5xy + 5 = 0 \implies\quad
y = \frac{5 x \pm \sqrt{21 x^2 - 20}}{2}\qquad |x|\ge 1$$</span></p>
<p>Note that the absolute value of <span c... |
3,613,950 | <blockquote>
<p>Given the set <span class="math-container">$S$</span> that is the set of all subsets of <span class="math-container">$\{1, 2, \ldots, n\}$</span>. Two different sets are chosen at random from <span class="math-container">$S$</span>. What is the probability that
the two subsets share exactly two eq... | greg | 357,854 | <p><span class="math-container">$\def\m#1{\left[\begin{array}{c}#1\end{array}\right]}\def\p#1#2{\frac{\partial #1}{\partial #2}}$</span>Let <span class="math-container">$U$</span> be an unconstrained matrix
and use a colon denote the trace function in product form, i.e.
<span class="math-container">$$A:B = {\rm Tr}(A^T... |
400,926 | <p>Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. The reference is "Global Carleman Estimates for Waves and Applications" by Baudouin, Buhan, Ervedoza. </p>
<hr>
... | Shuhao Cao | 7,200 | <p><strong>Derivation of Euler-Lagrange equation:</strong> If $z$ minimizes $K_{s,p}(z)$, then any small perturbation on $z$ will make this functional bigger. Hence we want: $\newcommand{\e}{\epsilon}$
$$
\lim_{\e\to 0}\frac{d}{d\e} K_{s,p}(z+\e v) = 0.\tag{$\dagger$}
$$
This means the perturbation $\e v$ in the test ... |
4,004 | <p>This is related to <a href="https://math.stackexchange.com/q/133615/26306">this post</a>, please read the comments.</p>
<p>What is the usual way of dealing with that kind of problems on math.SE?
(By "that kind of problems" I mean someone posting tasks from an ongoing contest.)</p>
<p>I mean I did email the contest... | Phira | 9,325 | <p>Since the recent comments on a posted contest question links here, let me state my answer:</p>
<p>While there can be no obligation of this site to do detective work and be responsible for never answering a contest question, I strongly feel that <strong>if</strong> someone provides a link that it is a contest questi... |
1,522,216 | <p>I want to show that following:
$$\left(\frac{n^2-1}{n^2}\right)^n\sqrt{\frac{n+1}{n-1}}\leq 1; ~~n\geq 2$$ and $n$ is an integer. </p>
<p>After some simplifications, I got left hand-side as
$$LHS:\left(1-\frac{1}{n}\right)^{n-\frac{1}{2}} \left(1+\frac{1}{n}\right)^{n+\frac{1}{2}}$$
It is clear that the 1st term is... | mathlover | 281,534 | <p>There are only two possibilities, Z-X-Y or X-Z-Y.Then doing the necessary calculations we get YZ either 13 or 7 so answer is 91.</p>
|
395,685 | <p>I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't remember the exact date, but it was from after Hamilton introduced the Ricci flow but well before Perelman's work. Unfo... | Dmitri Panov | 943 | <p>There is a <a href="https://www.youtube.com/watch?v=Qzxk8VLqGcI" rel="nofollow noreferrer">video of Thurston's talk "A discussion on geometrization"</a> from May 7, 2001. In the last part of this talk he speaks about possible approaches to proving geometrization.</p>
<p>Starting from <a href="https://youtu... |
3,980,441 | <p>I need to prove this. I need your help to verify that my proof is correct (or not) please.</p>
<blockquote>
<p>Prove that this integral exists: <span class="math-container">\begin{align}
\int_{2}^{\infty}\frac{dx}{\sqrt{1+x^{3}}} \end{align}</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Fist we ne... | Mark | 470,733 | <p>Your solution is correct, though you didn't really need to use the limit comparison test. You could just stop after the first line. Since <span class="math-container">$\frac{1}{\sqrt{1+x^3}}\leq\frac{1}{\sqrt{x^3}}$</span> and the integral <span class="math-container">$\int_2^{\infty}\frac{1}{\sqrt{x^3}}dx$</span> c... |
2,129,086 | <p>I know that the total number of choosing without constraint is </p>
<p>$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$</p>
<p>Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3. </p>
<p>the textbook has the following solution </p>
<p>$\binom{3+5−1}{5}=\binom{7}{5}=21$ I can't figure out where is the 5 coming from?</... | Maczinga | 411,133 | <p>This can be solved also using the <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars method</a>.
The point is paying attention to variables that take value 0.
So you have 3 cases:</p>
<p>1) all variables $\ne 0$
This amounts to $\binom{11-1}{3-1}=45$</p>
... |
3,485,441 | <p>I don't quite understand why Burnside's lemma
<span class="math-container">$$
|X/G|=\frac1{|G|}\sum_{g\in G} |X_g|
$$</span>
should be called a "lemma". By "lemma", we should mean there is something coming after it, presumably a theorem. However, I could not find a theorem which requires Burnside as a lemma. In ever... | Math101 | 668,360 | <p>One consequence is for the necklace problem, see this post:</p>
<p><a href="https://math.stackexchange.com/questions/2016732/necklace-problem-with-burnsides-lemma">Necklace problem with Burnside's lemma</a></p>
|
1,987,507 | <p>I find this question, which comes from section 2.2 of Dummit and Foote's algebra text, to be somewhat confusing:</p>
<blockquote>
<p>Let $G = S_n$, fix $i \in \{1,...,n\}$ and let $G_i = \{\sigma \in G ~|~ \sigma(i) = i\}$ (the stabilizer of $i$ in $G$). Use group actions to prove that $G_i$ is a subgroup of $G$.... | Graham Kemp | 135,106 | <p>If $p$ is the price per ticket, then $\frac 1{20} (p−\$80)+\frac{19}{20} p$ is the expected return for selling <em>one</em> ticket.</p>
<p>You want the expected return for selling <em>twenty</em> tickets to equal $\$30$. Fortunately the Linearity of Expectation means this is:</p>
<p>$$20\times(\frac 1{20} (... |
114,895 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/21282/show-that-every-n-can-be-written-uniquely-in-the-form-n-ab-with-a-squa">Show that every $n$ can be written uniquely in the form $n = ab$, with $a$ square-free and $b$ a perfect square</a> </p>
</blockq... | André Nicolas | 6,312 | <p>The proof of existence that you gave is fine, and can be adapted to produce a proof of uniqueness by using the essential uniqueness of prime power factorization. </p>
<p>But let us prove existence and uniqueness without explicit use of the representation of natural numbers as a product of powers of primes.</p>
<p>... |
2,443,496 | <blockquote>
<p>Can someone point me in the right direction as to how to take the derivative of this function:
$$ f(x) = 2 \pi \sqrt{\frac{x^2}{c}} $$</p>
</blockquote>
<p>Thank you</p>
| Raffaele | 83,382 | <p>When $x\ge 0$ you have $f(x)=\dfrac{2\pi\,x}{\sqrt c}\to f'(x)=\dfrac{2\pi}{\sqrt c}$</p>
<p>when $x<0$ then $f(x)=.\dfrac{2\pi\,x}{\sqrt c}\to f'(x)=-\dfrac{2\pi}{\sqrt c}$</p>
<p>To put all together in one formula
$$f'(x)=\frac{2 \pi \, \text{sgn}(x)}{\sqrt{c}}$$
where $\text{sgn}(x)=\left\{
\begin{array}{rr}... |
3,251,754 | <p>Let <span class="math-container">$M$</span> be the set of all <span class="math-container">$m\times n$</span> matrices over real numbers.Which of the following statements is/are true??</p>
<ol>
<li>There exists <span class="math-container">$A\in M_{2\times 5}(\mathbb R)$</span> such that the dimension of the nulls... | Vizag | 566,333 | <p>Your argument is correct. Here is another way you could think about it: </p>
<p><span class="math-container">$$P(B^c|C) =\frac{P(B^c\cap C)}{P(C)}$$</span>
<span class="math-container">$$=\frac{P(C)-P(B\cap C)}{P(C)}$$</span>
<span class="math-container">$$=1-P(B|C)$$</span></p>
<p>Draw a Venn diagram to see that ... |
3,589,685 | <p>Can you give an example of an isomorphism mapping from <span class="math-container">$\mathbb R^3 \to \mathbb P_2(\mathbb R)$</span>(degree-2 polynomials)?</p>
<p>I understand that to show isomorphism you can show both injectivity and surjectivity, or you could also just show that an inverse matrix exists.</p>
<p>M... | Community | -1 | <p>Assuming you mean the polynomials of degree less than or equal to <span class="math-container">$2$</span>, it is a three dimensional space, with basis <span class="math-container">$\{1,x,x^2\}$</span>. So, just send basis vectors to basis vectors:</p>
<p><span class="math-container">$$e_1\to1,e_2\to x,e_3\to x^2$$... |
2,588,968 | <p>I have the double integral</p>
<p>$$\int^{10}_0 \int^0_{-\sqrt{10y-y^2}} \sqrt{x^2+y^2} \,dx\,dy$$</p>
<p>And I am asked to evaluate this by changing to polar coordinates.</p>
| Michael Hardy | 11,667 | <p>Complete the square:
$$
10y-y^2 = 25 - (5-y)^2,
$$
so the graph of $x = -\sqrt{10y-y^2} = - \sqrt{5^2 - (5-y)^2}$ is the left half of the circle $x^2 + (5-y)^2 = 5^2.$</p>
<p>Exercises with polar coordinates will have shown you that $$\tag 1 r = 5\sin\theta$$ is that circle. If you multiply both sides of $(1)$ by $... |
2,847,277 | <p>Are there primes $p=47\cdot 2^n+1$, where $n\in\mathbb Z_+$? Tested for all primes $p<100,000,000$ without equality.</p>
| Nominal Animal | 318,422 | <p>There are several options, depending on exactly what you want to do.</p>
<hr>
<p>Let $\hat{d}$ be the unit ($\lVert\hat{d}\rVert = 1$) direction vector,
$$\hat{d} = \frac{\vec{d}}{\lVert\vec{d}\rVert} \tag{1}\label{NA1}$$</p>
<p>Let $\hat{u}$ be an unit vector perpendicular to $\hat{d}$; essentially, the directio... |
3,831,073 | <p>Let <span class="math-container">$\alpha = \sqrt[3]{4+\sqrt{5}}$</span>. I would like to prove that <span class="math-container">$\left[ \mathbb{Q} \left( \alpha \right ) : \mathbb{Q} \right] = 6$</span>. We have <span class="math-container">$\alpha^3 = 4 + \sqrt{5}$</span>, and so <span class="math-container">$(\al... | Sarvesh Ravichandran Iyer | 316,409 | <p>There's of course, the wacky way as suggested by Edward above, thanks to him!</p>
<p>But there's a criterion by Osada , which fits the bill perfectly.</p>
<blockquote>
<p>Let <span class="math-container">$f(x) =x^n + a_{n-1}x^{n-1} + ... + a_1x \pm p$</span> be a monic polynomial with integer coefficients, such that... |
3,831,073 | <p>Let <span class="math-container">$\alpha = \sqrt[3]{4+\sqrt{5}}$</span>. I would like to prove that <span class="math-container">$\left[ \mathbb{Q} \left( \alpha \right ) : \mathbb{Q} \right] = 6$</span>. We have <span class="math-container">$\alpha^3 = 4 + \sqrt{5}$</span>, and so <span class="math-container">$(\al... | Mummy the turkey | 801,393 | <p>As requested by OP I am rewriting my comment as an answer. We will show that <span class="math-container">$[\mathbb{Q}(\sqrt[3]{4+\sqrt{5}}) : \mathbb{Q}(\sqrt{5})] = 3$</span> by showing that <span class="math-container">$f(x) = x^3 - (4+\sqrt{5})$</span> has no solution in <span class="math-container">$\mathbb{Q}(... |
68,563 | <p>I was wondering if there's a formula for the cardinality of the set $A_k=\{(i_1,i_2,\ldots,i_k):1\leq i_1<i_2<\cdots<i_k\leq n\}$ for some $n\in\mathbb{N}$. I calculated that $A_2$ has $n(n-1)/2$ elements, and $A_3=\sum_{j=2}^{n-2}\frac{(n-j)(n-j+1)}{2}$. As you can see, the cardinality of $A_3$ is already ... | AndJM | 16,682 | <p>The $A_k$ can also be expressed as $\{(i_1,i_2,\ldots,i_k)\;|\; 1\leq i_1\leq n-(k-1),i_1+1\leq i_2\leq n-(k-2),\ldots,i_{k-1}+1\leq i_k\leq n\}$. This way, it is clear how many choices there are for each $i_j$. Multiplying will give you the ol' $n \choose k$ formula.</p>
<p>edit: Apologies. It's not clear to me ri... |
521,500 | <p>Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof.
The theorem states in particular that for $s\geq0$ fixed, the process $(C_t:=B_{t+s}-B_{s}, t\geq0)$ is independent from $\mathcal{F}_s=\sigma(B_u, 0\leq u\leq s)$.</p>
<p>The proof starts with... | Did | 6,179 | <p>Because these sigma-algebras $\sigma(B_u;u\leqslant s)$ and $\sigma(B_{s+u}-B_s;u\geqslant0)$ are generated by the pi-systems that one suggested that you use hence if the pi-systems are independent, so are the sigma-algebras (a result often called Dynkin lambda-pi theorem).</p>
|
1,407,131 | <p>I need to prove the following integral is convergent and find an upper bound
$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$</p>
<p>I've tried integrating $\frac{1}{1+x^2+y^2} \lt \frac{1}{1+x^2+y^4}$ but it doesn't converge</p>
| David C. Ullrich | 248,223 | <p>Finding the exact value of $\int_0^\infty\frac{dx}{a^2+x^2}$ is just a calc I exercise. Let $a=\sqrt{1+y^4}$ and see what happens...</p>
|
1,190,083 | <p>A positive element x of a C*-algebra A is a self-adjoint element whose spectrum is contained in the non-negative reals. If there's a faithful finite-dimensional representation of A where the involution is conjugate transposition, I think the second condition just means that x can be thought of as a matrix with posit... | aly | 169,618 | <p>You can also have an infinite dimensional example. Take $x$ and $y$ be two distinct non-null elements in a Hilbert space $\mathcal{H}$ with dimension at least $2$ such that $\langle x,y\rangle\ge 0$. Then the rank-one operator $x\otimes y$ (defined as $z\mapsto\langle z,y\rangle x$) has the spectrum $\{0,\langle x,y... |
9,085 | <p>So as the title says I am trying to make a list where each element is determined by a users choice of an element in a PopupMenu.</p>
<p>My first attempt:</p>
<pre><code>test = Table["A", {5}];
Table[PopupMenu[Dynamic[test[[n]]], {"A", "B", "C"}], {n, 5}]
</code></pre>
<p>Returned the following error</p>
<pre><co... | kglr | 125 | <p>You can also use:</p>
<pre><code> test = Table["A", {5}];
Table[With[{n = n}, PopupMenu[Dynamic[test[[n]]], {"A", "B", "C"}]], {n, 5}]
</code></pre>
<p>or </p>
<pre><code> Table[PopupMenu[Dynamic[test[[k]]], {"A", "B", "C"}] /. k -> n, {n, 5}]
</code></pre>
|
1,598,451 | <p><em>(Sorry for the inconvenience related to the tags, please feel free to correct my post if it needs a better scope by adding some other tags).</em></p>
<p><strong>CONTEXT</strong></p>
<p>I have several (decimal) numbers shaped like this :</p>
<ul>
<li>1.081</li>
<li>289.089167</li>
<li>2.98</li>
<li>...</li>
</... | Ron Gordon | 53,268 | <p>Hint:</p>
<p>$$\begin{align}\int_1^{100} dx \frac{f(x)}{x} = \int_1^{10} dx \frac{f(x)}{x} + \int_{10}^{100} dx \frac{f(x)}{x} \end{align}$$</p>
<p>and sub $x=100/u$ in the 2nd integral.</p>
|
1,598,451 | <p><em>(Sorry for the inconvenience related to the tags, please feel free to correct my post if it needs a better scope by adding some other tags).</em></p>
<p><strong>CONTEXT</strong></p>
<p>I have several (decimal) numbers shaped like this :</p>
<ul>
<li>1.081</li>
<li>289.089167</li>
<li>2.98</li>
<li>...</li>
</... | Chappers | 221,811 | <p>Suppose more generally that $a>0$ and
$$ f(x)=f(a^2/x). $$
Then we need to look at
$$ \int_a^{a^2} \frac{f(x)}{x} \, dx = \int_a^{a^2} \frac{f(a^2/x)}{x} \, dx. $$
Use the substitution $y=a^2/x$: then $x=a \implies y=a$, $x=a^2 \implies y=1$, and $dx/x=-dy/y$, so
$$ \int_a^{a^2} \frac{f(a^2/x)}{x} \, dx = \int_1^... |
2,521,331 | <p>I need to show, that when we have $X,Y$ - any metric spaces and
<br>
$f:X \ni x \to a \in Y$ is constant , then $f$ is continuous . </p>
<p>$(X,\tau_{1}),(Y,\tau_{2}) $ - topological spaces : $f: X\to Y$.
I know a definition : $f: X\to Y $ is continuous if $ \forall_{W \in \tau_{2}}\ f^{-1}[W] \in \tau_{1} $ .... | Hayfisher | 503,715 | <p>This holds for every topological space, not just for metric spaces. Let $a \in Y$ be fixed. Since</p>
<p>$$f:X \to Y, x \mapsto a$$ holds, the preimage of any $V \subseteq Y$ is</p>
<ul>
<li>$\emptyset$, iff $a \notin V$,</li>
<li>whole $X$, iff $a \in V$, like you already denoted.</li>
</ul>
<p>Thus the preimage... |
3,969,943 | <p>It's been a few years since doing any type of trigonometry questions and I've seemed to forgotten everything about it. Below is a question with the solution. You're not supposed to use a calculator.</p>
<p><span class="math-container">$$\begin{align}
&\cos\frac{2\pi}{3}+\tan\frac{7\pi}{4}-\sin\frac{7\pi}{6} \\[4... | Kavi Rama Murthy | 142,385 | <p>For (a) what you have done is correct.</p>
<p>For (b) your argument is not valid. Note that <span class="math-container">$\sum \ln (1+a_n) <\infty$</span>. This implies that <span class="math-container">$\ln (1+a_n) \to 0$</span> so <span class="math-container">$a_n \to 0$</span>. Now, there exists <span class="m... |
52,657 | <p>I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). Given this information, is there a way to rapidly derive the formula for this parabolic curve? My issue is that I ... | Shaun Ault | 13,074 | <p>To answer question found in the title: "... an equation for a parabola from its $x$ and $y$ intercepts", the correct equation is:</p>
<p>$$y = \frac{c}{ab}(x-a)(x-b),$$</p>
<p>where $a, b$ are the $x$-intercepts and $c$ is the $y$-intercept. We can prove this is correct by noting that $y = 0$ when $x=a$ or $x=b$ ... |
52,657 | <p>I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). Given this information, is there a way to rapidly derive the formula for this parabolic curve? My issue is that I ... | Zar | 14,450 | <p>the equation would look like this</p>
<p>$$ y = k(x-a)(x-b)$$</p>
<p>now we have to figure out what k is. We know what the maximum value is, call it c, and that it's x value is 0. Therefor we can plug this into the equation so that we get the following</p>
<p>$$c = k(-a)(-b)$$
$$c = kab$$
therefor
$k = c... |
1,797,712 | <p>Let $G = \Bbb{Z}_{360} \oplus \Bbb{Z}_{150} \oplus \Bbb{Z}_{75} \oplus \Bbb{Z}_{3}$</p>
<p>a. How many elments of order 5 in $G$</p>
<p>b. How many elments of order 25 in $G$</p>
<p>c. How many elments of order 35 in $G$</p>
<p>d. How many subgroups of order 25 in $G$</p>
<p>I think I have done a,b,c correctly ... | Josh Hunt | 282,747 | <p>What method did you use?</p>
<p>In general the cyclic group of order $n$ has $\phi(d)$ elements of order $d$ whenever $d | n$. (You have $\phi(n)$ elements of order $n$, any element generates a cyclic subgroup, and summing these up gives you the order of the group.)</p>
<p>Also, if you have two elements of order $... |
2,366,610 | <p>Let $U$ be an $n \times n$ unitary matrix and $X$ an $n \times n$ real symmetric matrix. Suppose that $$U^\dagger X U = X$$ for all real symmetric $X$, then what are the allowed unitaries $U$? It seems that the only possible $U$ is some phase multiple of the identity $U=aI$ where $|a|=1$ but I'm not able to show th... | Bernard | 202,857 | <p>You can use <em>equivalents</em> and expansion in power series.</p>
<p>Let's begin with the denominator:
$$\sinh x-\sin x=x+\frac{x^3}{3!}+o(x^3)-\Bigl(x-\frac{x^3}{3!}+o(x^3)\Bigr)=\frac{x^3}3+o(x^3)\sim_0\frac{x^3}3.$$
Now for the numerator:</p>
<p>First, by definition, $\;\operatorname{arsinh}(\sinh(x))=x$.</p... |
4,444,669 | <p>I'm unsure about the problem below</p>
<hr>
Under which conditions is the following linear equation system solvable ?
<span class="math-container">$$x_1 + 2x_2 - 3x_3 = a$$</span>
<span class="math-container">$$3x_1 - x_2 + 2x_3 = b$$</span>
<span class="math-container">$$x_1 - 5x_2 + 8x_3 = c$$</span>
<hr>
<p>We se... | John Bentin | 875 | <p>The proof will depend on which model you take for the real numbers (e.g. Dedekind cuts, equivalence classes of Cauchy sequences, etc.). Perhaps the easiest model for this question is (despite its arbitrariness and its awkwardness in other respects) the traditional one of decimal expansions. Thus, any positive real n... |
3,736,706 | <p>Let <span class="math-container">$M$</span> be an <span class="math-container">$A$</span>-module and let <span class="math-container">$\mathfrak{a}$</span> and <span class="math-container">$\mathfrak{b}$</span> be coprime ideals of A.</p>
<p>I must show that <span class="math-container">$M/ \mathfrak{a}M \oplus M/ \... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Consider the short exact sequence:
<span class="math-container">$$0\longrightarrow A/\mathfrak a\cap\mathfrak b\longrightarrow A/\mathfrak a\times A/\mathfrak b\longrightarrow A/\mathfrak a+\mathfrak b\longrightarrow 0 $$</span>
and tensor by <span class="math-container">$M$</span>.</p>... |
3,766,585 | <p>Let <span class="math-container">$X_1,X_2,...,X_n$</span> be random sample from a DF <span class="math-container">$F$</span>, and let <span class="math-container">$F_n^* (x)$</span> be the sample distribution function.</p>
<p>We have to find <span class="math-container">$\operatorname{Cov}(F_n^* (x), F_n^* (y))$</sp... | VIVID | 752,069 | <p>It comes down to two known limits as follows:
<span class="math-container">$$\lim_{x\to 0} \frac{\ln |1+x^3|}{\sin^3 x}=\lim_{x\to 0} \frac{\ln |1+x^3|}{x^3}\frac{x^3}{\sin^3 x}=1\cdot 1=1$$</span></p>
|
3,766,585 | <p>Let <span class="math-container">$X_1,X_2,...,X_n$</span> be random sample from a DF <span class="math-container">$F$</span>, and let <span class="math-container">$F_n^* (x)$</span> be the sample distribution function.</p>
<p>We have to find <span class="math-container">$\operatorname{Cov}(F_n^* (x), F_n^* (y))$</sp... | Fred | 380,717 | <ol>
<li>if <span class="math-container">$|x|$</span> is "small", then <span class="math-container">$1+x^3 >0.$</span> Hence we have to compute</li>
</ol>
<p><span class="math-container">$$\lim_{x\to 0} \frac{\ln (1+x^3)}{\sin^3 x}.$$</span></p>
<ol start="2">
<li><span class="math-container">$\frac{\ln (1... |
3,766,585 | <p>Let <span class="math-container">$X_1,X_2,...,X_n$</span> be random sample from a DF <span class="math-container">$F$</span>, and let <span class="math-container">$F_n^* (x)$</span> be the sample distribution function.</p>
<p>We have to find <span class="math-container">$\operatorname{Cov}(F_n^* (x), F_n^* (y))$</sp... | Z Ahmed | 671,540 | <p><span class="math-container">$$\ln(1+z)=z-z^2/2+\cdots, \sin z=z-z^3/6+\cdots$$</span>
<span class="math-container">$$\lim_{x\to 0} \frac{x^3-x^6/2+\cdots}{(x-x^3/6+\cdots)^3} =\lim_{x \to 0} \frac{1-x^3/2+\cdots}{1-3x^2/6+\cdots}=1.$$</span>
Lastly, we have used the binomial series: <span class="math-container">$(1... |
69,902 | <p>I'm VERY new to Mathematica programming (and by new I mean two days), and was solving Project Euler question 12, which states:</p>
<blockquote>
<p>Which starting number, under one million, produces the longest [Collatz] chain?</p>
</blockquote>
<p>Now don't take this question wrong. <strong>I am not asking for a... | KennyColnago | 3,246 | <p>Your <code>collatzLength</code> function is fast on an individual integer, but when you map it to all integers from 1 to a million, the function recalculates values repeatedly. For example, the Collatz series for $n=10$ is $\{10,5,16,8,4,2,1\}$. But the length for $n=5$ would have been already calculated to be 6. He... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | walkingtonowhere | 5,158 | <p>Sometimes learning in general is not about the actual usefulness of the subject matter in question, but how it changes and expands your thinking. Inspiration can also stem from many different places. </p>
<p>The main issue is that in many High school classes, students are taught to look at a problem and work it l... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Benjamin Dickman | 262 | <p><strong>Edit (Feb 2016):</strong> Since the OP mentioned Hacker's <em>Algebra</em> opinion piece in the NYTimes, perhaps this is a good place to point out his most recent follow-up in a similar direction (I exclude here my own assessment of either): <a href="http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Jeff | 6,016 | <p>I am not a formal teacher or a mathematician, but a mechanical engineer who loves to learn and derives great satisfaction from mentoring other up-and-coming “S.T.E.M.” professionals/students. Due to my lack of educator credentials or particular knowledge on the subject, my response reflects personal views and experi... |
2,412,959 | <p>In <a href="https://math.stackexchange.com/questions/170362/pointwise-convergence-implies-lp-convergence">this</a> question a user asks if pointwise convergence implies convergence in $L^p$. I would have thought that the answer is yes. I am not experienced with measure theory, which is how that question is framed. T... | Fred | 380,717 | <p>In general $ \lim_{n\to \infty} \int_\Omega |f_n(x)-f(x)|^p dx = \int_\Omega |\lim_{n\to \infty} f_n(x)-f(x)|^p dx$ is false !</p>
<p>Example: Let $p=1$ , $ \Omega =[0,1]$ and let $f_n$ be defined as follows $( n \ge 3)$:</p>
<p>$f_n(x)=n^2x$ , if $0 \le x \le 1/n$, $f_n(x)=-n^2x+2n$, if $1/n \le x \le 2/n$ and $f... |
4,478,486 | <p>I have just started to read Stein's Singular Integrals and Differentiability properties of functions.</p>
<p>The Hardy-Littlewood maximal function has just been introduced i.e. <span class="math-container">$$M(f)(x):= \sup_{r > 0} \frac{1}{m(B(x,r))}\int_{B(x,r)}|f(y)|dy$$</span></p>
<p>where <span class="math-co... | The_Sympathizer | 11,172 | <p><span class="math-container">$\otimes$</span>, also called <strong>tensing</strong>, is something you get bundled with the tensor product that you don't have in an ordinary vector space. How the tensor product vector space and tensing work together are what the real "meat" behind the tensor product is. Con... |
1,379,188 | <p>The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all <strong>piecewise</strong> smooth paths between $p$ and $q$.</p>
<p><strong>Does it change if we take the infimum only over smooth paths?</strong>
(Note that if a smooth manifold is connected, <a href="https://math.stacke... | ASCII Advocate | 260,903 | <p>An additional remark to the answer.</p>
<p>On a Riemannian manifold (without "missing" points, e.g., complete) the minimum length in any homotopy class of path exists and is attained by a geodesic path, which is necessarily smooth. If the manifold is of some reasonably finite topological type (compact is much more... |
2,326,564 | <p>Is it true that iff CardA = Card A then A is a set of distinct terms? </p>
<p>[This questions is actually from a confusion on what a set versus multiset is]</p>
| jgsmath | 455,126 | <p>I will use $\bar A$ for ~A.</p>
<p>$A + \bar A B = \overline{\overline{A + \bar{A} B}} = \overline{\bar A \cdot \overline{\bar A B}} =\overline{\bar A \cdot(\bar {\bar A} + \bar B)} = \overline{\bar A \cdot (A + \bar B)} = \overline{\bar A \cdot A + \bar A \cdot \bar B} = \overline{0+\bar A \cdot \bar B} = \overlin... |
2,326,564 | <p>Is it true that iff CardA = Card A then A is a set of distinct terms? </p>
<p>[This questions is actually from a confusion on what a set versus multiset is]</p>
| Axel Kemper | 58,610 | <p>From </p>
<p>$A \lor \bar{A} = T$</p>
<p>$T \lor B = T$</p>
<p>$B = B \land T = B \land (A \lor \bar{A}) = BA \lor B\bar{A}$</p>
<p>we can rewrite</p>
<p>$A \lor B = A \lor BA \lor B\bar{A} = A(T \lor B) \lor B\bar{A} = A \lor B\bar{A}$</p>
<p>In plain English:<br>
Regardless of $B$, $A \lor \bar{A}B$ is tr... |
1,687,714 | <p><a href="https://i.stack.imgur.com/nZEAy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nZEAy.jpg" alt=""></a></p>
<p>I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem th... | reuns | 276,986 | <p>it seems correct, but you didn't talk about <strong>the region of convergence</strong>. I personally consider the distribution $h(t) = \sum_{n=0}^\infty \delta(t-2an)$ (one peak at every $t = 2an$) whose Laplace transform is $$\sum_{n=0}^\infty e^{-2asn} = \frac{1}{1-e^{-2as}}$$ (only for $Re(s) > 0$ !! for $Re(s... |
1,014,987 | <p>I need to solve the bound for $n$ from this inequality: </p>
<p>$$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$</p>
<p>where $c$ is some known constant value. How can I solve this? At first I was going to take the logarithm, but the difference of the two exponentials trouble me...</p>
<p>Any hints? :) Thnx for any help !... | Empy2 | 81,790 | <p>To solve $$c=\phi^n-(-\phi)^{-n}$$
If $n$ is even, then $$c=\phi^n-\phi^{-n}\\(\phi^n)^2-c(\phi^n)-1=0$$
and you can solve a quadratic for $\phi^n$ as a function of $c$. Similar if $n$ is odd.</p>
|
1,014,987 | <p>I need to solve the bound for $n$ from this inequality: </p>
<p>$$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$</p>
<p>where $c$ is some known constant value. How can I solve this? At first I was going to take the logarithm, but the difference of the two exponentials trouble me...</p>
<p>Any hints? :) Thnx for any help !... | jjepsuomi | 53,500 | <p>Here is the answer I got by using the hints given to me: </p>
<p>First I select $c = \frac{\sqrt{5}}{0.05}$, so my equation becomes: </p>
<p>$$\frac{\sqrt{5}}{0.05} =1.618^{n+1} - (-0.618)^{n+1}$$</p>
<p>I set $\phi = 1.618$ and $\displaystyle -\frac{1}{\phi} = -0.618$ and I get </p>
<p>$$\frac{\sqrt{5}}{0.05} =... |
3,712,094 | <p><a href="https://i.stack.imgur.com/S3n1g.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S3n1g.jpg" alt="enter image description here"></a></p>
<p>For part (a) these are clearly two parallel lines so no points of intersection.<br>
For part (b) this has one point of intersection because these two ... | Z Ahmed | 671,540 | <p><span class="math-container">$(a)$</span>: two parallel non-intersecting lines, (b) intesecting lines.</p>
<p>In the case of three planes (equations) when non two are parallel and no two are idendtical: Take <span class="math-container">$z=k$</span> and solve for %x, y$ punt these in the third equation.
One of the... |
3,576,026 | <p>I was solving a problem and got down to this:
<span class="math-container">$$\lim_{n \to \infty} \arctan\left(\frac{\sum_{k=0}^n-\frac{1}{1+k^2}}{\sum_{k=0}^n \frac{k}{1+k^2}}\right)$$</span>
After this, I said that, since the bottom series diverges and the upper one converges, the result is <span class="math-contai... | Z Ahmed | 671,540 | <p>As <span class="math-container">$$\sum_{k=0}^{\infty}\frac{1}{k^2+1}=\frac{1}{2}(1+\pi \coth \pi)$$</span> is finite
<span class="math-container">$\sum_{k=0}^{\infty} \frac{k}{k^2+1}$</span> is divergent so the required limit needs to be
<span class="math-container">$L=\tan^{-1}(0)=0.$</span></p>
|
2,216,778 | <p>My question is if there exists a way to evaluate the sum</p>
<p>$$
{{s}\choose{s}}^{\!2} + {{s + 1}\choose{s}}^{\!2} + \ldots {{s+r}\choose{s}}^{\!2}.
$$</p>
<p>In other words, it's the sum of the squares of the first r binomial coefficients on the s-th right-to-left diagonal of Pascal's triangle. Moreover, is it... | User | 329,924 | <p>Both of your two ideas work and are indeed based on the fact that the polynomial set of that form is not closed under addition(obvious, for $t^2+t^2=2t^2$). Hence it's not a subspace.</p>
|
2,216,778 | <p>My question is if there exists a way to evaluate the sum</p>
<p>$$
{{s}\choose{s}}^{\!2} + {{s + 1}\choose{s}}^{\!2} + \ldots {{s+r}\choose{s}}^{\!2}.
$$</p>
<p>In other words, it's the sum of the squares of the first r binomial coefficients on the s-th right-to-left diagonal of Pascal's triangle. Moreover, is it... | Kernel_Dirichlet | 368,019 | <p>Let <span class="math-container">$V$</span> be a vector space and <span class="math-container">$W\subset V$</span>. We want <span class="math-container">$W$</span> to satisfy three key axioms for it to fit the definition of subspace.</p>
<p><span class="math-container">$1$</span>. <span class="math-container">$\{0\}... |
221,428 | <p>Is there any pair of random variables (X,Y) such that Expected value of X goes to infinity, Expected value of Y goes to minus infinity but expected value of X+Y goes again to infinity?</p>
| Community | -1 | <p>It would probably be easier to start as follows: Notice that the $G_i$ being dense and open guarantees that $G_1 \cap G_2 \neq \emptyset$. Now choose an $x$ in so that there is a ball $E_1$ in completely contained in the intersection. Shrinking the ball if necessary, you can assume that $\overline{E_1}$ is completel... |
2,838,037 | <p>For the set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$, I understand that $\{\frac 1n \mid n \in \mathbb N\}$ is open and closed in $A$ because it is a union of all the connected components $\{\frac 1n\}$ in $A$ for all $n \in \mathbb N$. Even though $\{0\}$ is also a connected component of $A$, why is $\{0\}$... | William Elliot | 426,203 | <p>Viewing A as a subspace of R, since {0} is closed,
within A, B = A - {0} is open. B is not closed within A
because 0 is an adherance point of B that is not in B. </p>
<p>Using the clumbsy definition of closed, B is not closed<br>
within A because 0 is a limit point of B that is not in B. </p>
|
2,838,037 | <p>For the set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$, I understand that $\{\frac 1n \mid n \in \mathbb N\}$ is open and closed in $A$ because it is a union of all the connected components $\{\frac 1n\}$ in $A$ for all $n \in \mathbb N$. Even though $\{0\}$ is also a connected component of $A$, why is $\{0\}$... | Mostafa Ayaz | 518,023 | <p>If $\{0\}$ is open then there must exist some $\epsilon>0$ such that $$\{x\in A:|x|<\epsilon\}\subseteq\{0\}$$if such an $\epsilon$ exists we must have $$\{\dfrac{1}{n}:n>\dfrac{1}{\epsilon}\}\subseteq\{0\}$$which doesn't hold at all then the set is not open</p>
|
1,725,337 | <p>How does the following definition of Taylor polynomials:</p>
<p>$f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ </p>
<p>where $R_k(x_0,h)=\int^{x_0+h}_{x_0} \frac{(x_0+h-\tau)^k}{k!}f^{k+1}(\tau) d\tau$</p>
<p>where I guess $\lim_{h\to 0} \frac{R_k(x... | André Nicolas | 6,312 | <p>Hint: To find a series expression for $\frac{2}{(8+x)^2}$, differentiate the power series of (more or less) $\frac{1}{8+x}$. Note that $\frac{1}{8+x}$ has derivative $-\frac{1}{(8+x)^2}$.</p>
<p>To find the series for $\frac{1}{8+x}$, rewrite as $\frac{1}{8}\cdot \frac{1}{1+x/8}$, and use the familiar series for $\... |
2,636,931 | <p>Consider the ellipse given by:</p>
<p>$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0.
$$</p>
<p>What is the equation of an ellipse which has major and minor axis equal to $p$ times the major and minor axis length of the above ellipse.</p>
<p>My attempt is as follows:
We can remove rotation, increase axis length and then r... | Community | -1 | <p>You obtain this effect by rescaling the coordinate axis by the factor $p$, and the equation becomes</p>
<p>$$
A\frac{x^2}{p^2} + B\frac{xy}{p^2} + C\frac{y^2}{p^2} + D\frac{x}{p} + E\frac{y}{p} + F =0.
$$</p>
<p>If the center must remain unchanged, translate the center to the origin (the center is found by solving... |
3,787,167 | <p>Let <span class="math-container">$\{a_{jk}\}$</span> be an infinite matrix such that corresponding mapping <span class="math-container">$$A:(x_i) \mapsto (\sum_{j=1}^\infty a_{ij}x_j)$$</span> is well defined linear operator <span class="math-container">$A:l^2\to l^2$</span>.
I need help with showing that this ope... | crush3dice | 765,780 | <p>The basic idea is that <span class="math-container">$A(l^2)$</span> will not lie in <span class="math-container">$l^2$</span> if <span class="math-container">$A$</span> was not bounded. The proof:</p>
<p>If <span class="math-container">$A$</span> was not bounded then for every <span class="math-container">$C>0$</... |
61,798 | <p>Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ?</p>
<p>For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than $m=3 , n=2$ ?</p>
<p>If not, is there a deeper reason for this identity to be true only for the case $m=3 , ... | J. M. ain't a mathematician | 498 | <p>The <a href="http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials">Faulhaber polynomials</a> are expressions of sums of <em>odd</em> powers as a polynomial of triangular numbers $T_n=\frac{n(n+1)}{2}$. Nicomachus's theorem, $\sum\limits_{k\leq n} k^3=T_n^2$, is a particular special case.</p>
<p>... |
61,798 | <p>Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ?</p>
<p>For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than $m=3 , n=2$ ?</p>
<p>If not, is there a deeper reason for this identity to be true only for the case $m=3 , ... | user02138 | 2,720 | <p>Here is a curious (and related) identity which might be of interest to you. Let $D_{k} = ${ $d$ } be the set of <a href="http://en.wikipedia.org/wiki/Unitary_divisor">unitary divisors</a> of a positive integer $k$, and let $\sigma_{0}^{*} \colon \mathbb{N} \to \mathbb{N}$ denote the number-of-unitary-divisors (arith... |
888,101 | <p>Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?</p>
| Tomasz Kania | 17,929 | <p>Since you've used the tag <em>Functional analysis</em>, you might be interested in non-metrisability of certain topologies ubiquitous in analysis:</p>
<ul>
<li><p><a href="https://math.stackexchange.com/questions/424876/weak-topology-on-an-infinite-dimensional-normed-vector-space-is-not-metrizable">Weak topology on... |
48,626 | <p>In <code>ListPointPlot3D</code>, it seems the only point style available is the default, because there is no <code>PlotMarkers</code> option for this function. Is there a way to change the point style? For example, what if I want to draw the points as small cubes?</p>
| kglr | 125 | <pre><code>lpdata = Table[(4 π - t) {Cos[t + π/2], Sin[t + π/2], 0} + {0, 0, t}, {t, 0, 4 π, .1}];
lpp1 = ListPointPlot3D[lpdata,
Filling -> Bottom, ColorFunction -> "Rainbow", BoxRatios -> 1,
FillingStyle -> Directive[LightGreen, Thick, Opacity[.5]], ImageSize -> 400];
</code></pre>... |
2,981,063 | <p>I have seen this statement in a quiz:</p>
<blockquote>
<p>Let <span class="math-container">$X_i$</span> denote state <span class="math-container">$i$</span> in a Markov chain. It is necessarily true
that <span class="math-container">$X_{i+1}$</span> and <span class="math-container">$X_{i-1}$</span> are uncorrel... | E-A | 499,337 | <p>Conditioned on <span class="math-container">$X_i$</span>, <span class="math-container">$X_{i+1}$</span> and <span class="math-container">$X_{i-1}$</span> are indeed uncorrelated (and actually are much stronger: they are independent; you can check this).</p>
<p>However, think of the following chain: <span class="mat... |
1,419,209 | <p>How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus.</p>
<p>$$\sum_{i=0}^\infty \frac{i}{4^i}$$</p>
| Mark Viola | 218,419 | <p>Another approach is to write</p>
<p>$$\begin{align}
\sum_{i=0}^{\infty}\frac{i}{4^i}&=\sum_{i=1}^{\infty}\frac{1}{4^i}\left(\sum_{j=1}^{i}1\right)\\\\
&=\sum_{j=1}^{\infty}\sum_{i=j}^{\infty}\frac{1}{4^i}\\\\
&=\sum_{j=1}^{\infty}\frac{1}{4^j}\frac{1}{1-\frac14}\\\\
&=\frac{1/4}{(1-\frac14)^2}\\\\
&... |
3,536,061 | <p>Find the number of ways you can invite <span class="math-container">$3$</span> of your friends on <span class="math-container">$5$</span> consecutive days, exactly one friend a day, such that no friend is invited on more than two days. </p>
<p>My approach: Let <span class="math-container">$d_A,d_B$</span> and <span... | Giovanny Soto | 721,759 | <p>You can solve the problem as follows: </p>
<p>Let's call the three friends as <span class="math-container">$A,B,C$</span>, we need to invite them in such way that none of them go to your house more than 2 days. Obviously there are two friends (lest say <span class="math-container">$A$</span> and <span class="math-c... |
939,237 | <p>Prove $n^2 < n!$.</p>
<p>This is what I have gotten so far</p>
<p>basis step: $p(4)$ is true
Inductive Hypothesis assume $p(k)$ true for $k \ge 4$</p>
<p>Inductive Step $p(k+1)$ : $(k+1)^2 < (k+1)!$</p>
<p>$$(k+1)^2 =k^2 + 2k + 1 < k! + 2k +1$$</p>
<p>Can someone please explain the last step this is fr... | IAmNoOne | 117,818 | <p>Inductive Step:</p>
<p>Assume the case for $n$ is true, then for $n \geq 4$ $$(n + 1)^2 = n^2 + 2n + 1 < n! + 2n + 1 < n! + n^2 \leq n! + n!n = n!(n+1) = (n+1)!.$$</p>
|
1,006,562 | <p>So I am trying to figure out the limit</p>
<p>$$\lim_{x\to 0} \tan x \csc (2x)$$</p>
<p>I am not sure what action needs to be done to solve this and would appreciate any help to solving this. </p>
| Mark Fischler | 150,362 | <p>$$
\csc{2x} = \frac{1}{\sin 2x} = \frac{1}{2 \sin x \cos x}
$$
Then
$$
\lim_{x\rightarrow 0} \frac{\sin x}{\cos x} \frac{ 1}{2 \sin x \cos x}=
\lim_{x\rightarrow 0} \frac{1}{2 \cos^2 x} = \frac{1}{2}
$$</p>
|
489,562 | <p>I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first order logic. After that we prove simple math statements via dir... | Asaf Karagila | 622 | <p>Be excited about sets and logic, and generally what you are talking about.</p>
<p>When I was a freshman I had a TA in calculus 2 that was totally awesome. Not because he was particularly good, and I was particularly uninterested in the topic. But to hear him talk about the theorems was inspiring.</p>
<p>I took fro... |
489,562 | <p>I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first order logic. After that we prove simple math statements via dir... | Peter Smith | 35,151 | <p>Can I echo @dfeur's suggestion that a bit of history and conceptual commentary could be intriguing/fun/motivational (at least for more intellectually curious students)? </p>
<p><em>Sets</em> How did sets get into the story in the nineteenth century (the arithmetization of analysis)? Frege's disaster and Russell's p... |
2,373,073 | <p>Let $a, b, c$ be distinct integers, and let $P$ be a polynomial with integer coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$ at the same time. </p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>.Hint : By the remainder theorem, <span class="math-container">$P(x) - P(y)$</span> is divisible by <span class="math-container">$x-y$</span>, for all <span class="math-container">$x,y$</span>. </p>
<p>Assume that <span class="math-container">$a < b < c$</span>, since they are distinct, and see that putting <... |
422,233 | <p>I was asked to find a minimal polynomial of $$\alpha = \frac{3\sqrt{5} - 2\sqrt{7} + \sqrt{35}}{1 - \sqrt{5} + \sqrt{7}}$$ over <strong>Q</strong>.</p>
<p>I'm not able to find it without the help of WolframAlpha, which says that the minimal polynomial of $\alpha$ is $$19x^4 - 156x^3 - 280x^2 + 2312x + 3596.$$ (True... | Zhen Lin | 5,191 | <p>To begin, clear denominators:
<span class="math-container">$$(1 - \sqrt{5} + \sqrt{7}) \alpha = 3 \sqrt{5} - 2 \sqrt{7} + \sqrt{35}$$</span>
We need to make the coefficient of <span class="math-container">$\alpha$</span> rational, so use a difference-of-squares trick to get rid of the <span class="math-container">$\... |
422,233 | <p>I was asked to find a minimal polynomial of $$\alpha = \frac{3\sqrt{5} - 2\sqrt{7} + \sqrt{35}}{1 - \sqrt{5} + \sqrt{7}}$$ over <strong>Q</strong>.</p>
<p>I'm not able to find it without the help of WolframAlpha, which says that the minimal polynomial of $\alpha$ is $$19x^4 - 156x^3 - 280x^2 + 2312x + 3596.$$ (True... | Community | -1 | <p>A general purpose method is that the equation</p>
<p>$$ \sum_{k=0}^n c_k \alpha^k = 0 $$</p>
<p>is a <em>linear equation</em> in the unknowns $c_k$, and thus this can be solved with linear algebra.</p>
<p>Since the number itself is a rational linear combination of the four <em>linearly independent</em> numbers $1... |
3,743,673 | <p>Using calculus to find the minima:</p>
<p><span class="math-container">$$y(x) = x^x$$</span></p>
<p><span class="math-container">$$ln(y) = x*ln(x)$$</span></p>
<p><span class="math-container">$$(1/y)*\frac{dy}{dx} = ln(x) + x*\left(\frac{1}{x}\right) = ln(x) + 1$$</span></p>
<p><span class="math-container">$$\frac{d... | Community | -1 | <p>From the definition,</p>
<p><span class="math-container">$$x^x=e^{x\log x}>0.$$</span></p>
<p>An exponential is always positive.</p>
<hr />
<p>The case of <span class="math-container">$x=0$</span> is debatable and in fact <span class="math-container">$x^x$</span> is not really defined at zero. But for this discus... |
1,085,511 | <p>What would be the irrational number $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers given by this expression:
$$
\left(
\begin{array}{@{}c@{}}2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\dotsb}}}\end{array}
\right)^{1/8}
$$</p>
| RE60K | 67,609 | <blockquote>
<p>$$z = 2207-\dfrac{1}{2207-\dfrac{1}{2207-\dfrac{1}{2207...}}}$$</p>
</blockquote>
<p>So:
$$z=2207-\frac1z$$
Note that $z$ is less than $2207$, so we used the (minus) sign.
$$z^2-2207z+1=0\implies z=\frac{2207-\sqrt{2207^2-4}}{2}=\frac{2207-\sqrt{4870845}}{2}$$
Now we need to find:
$$\left(\frac{2207-... |
666,217 | <p>If $a^2+b^2 \le 2$ then show that $a+b \le2$</p>
<p>I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without result</p>
| Empy2 | 81,790 | <p>$(a+b)^2+(a-b)^2=2(a^2+b^2)\leq 4$, so $|a+b|\leq 2$</p>
|
1,827,080 | <p>Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$. Then there exists $C$ in $\mathbb R $ such that </p>
<ol>
<li>$|f(x)|\leq C \sqrt |x|$ for all $ x$ with $|x|\geq 1$</li>
<li>$|f(x)|\leq C |x|^2$ for all $ x$ with $|x|\geq 1$</li>
<li>$f(x)=x+... | Nizar | 227,505 | <ol>
<li><p>Does not hold, infact take $f(x)=\frac{x}{2} $. Suppose there exists $C$ $\in \mathbb{R}$ such that: $$ |f(x)|\leq C \sqrt |x| \text{ for all } x \text{ with } |x|\geq 1$$
Clearly from the inequality $C$ should be non negative. Then take $x=(2C+2)^2$, then $x \geq 1$, and so we get $$ \frac{(2C+... |
1,637,879 | <p>can you help me identify the mistake I'm making while integrating?</p>
<p>Question:</p>
<p>$$\int{\frac{2dx}{x\sqrt{4x^2-1}}}, x>\frac{1}{2}$$</p>
<p>my solution</p>
<p>$$\int{\frac{2dx}{x\sqrt{4x^2-1}}}=2\int{\frac{dx}{x\sqrt{(2x)^2-1}}}$$</p>
<p>let $$u=2x, x=1/2u, du=2dx, 1/2du=dx$$</p>
<p>$$=\frac{2}{2}... | Ben | 27,458 | <p>It's useful to consider finite-state <a href="https://en.wikipedia.org/wiki/Markov_chain" rel="noreferrer">Markov chains</a> with states $\{ 1, \ldots, N \}$. Such a Markov chain is defined by its transitions matrix $P = (P_{ij})_{i,j=1}^N$. We require that $0 \leq P_{ij} \leq 1$ for each $i, j = 1, \ldots, N$ and t... |
4,310,003 | <p>Suppose you have a non empty set <span class="math-container">$X$</span>, and suppose that for every function <span class="math-container">$f : X \rightarrow X$</span>, if <span class="math-container">$f$</span> is surjective, then it is also injective. Does it necessarily follow that <span class="math-container">$... | Laxmi Narayan Bhandari | 931,957 | <p>As @2 is even prime proceeds, that method is a bit complicated, imo. Here is a similar alternative.</p>
<p>Without separating the integral, we substitute <span class="math-container">$e^x=t$</span>.</p>
<p><span class="math-container">$$I = \int\limits_0^\infty \frac{\mathrm dt}{1+t^4}$$</span></p>
<p>Now we substit... |
4,310,003 | <p>Suppose you have a non empty set <span class="math-container">$X$</span>, and suppose that for every function <span class="math-container">$f : X \rightarrow X$</span>, if <span class="math-container">$f$</span> is surjective, then it is also injective. Does it necessarily follow that <span class="math-container">$... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
I &=\int_{-\infty}^{\infty} \frac{d y}{1+y^{4}}, \quad \text{where } y=e^{x} \\
&=\int_{-\infty}^{\infty} \frac{\frac{1}{y^{2}}}{y^{2}+\frac{1}{y^{2}}} d y \\
&=\frac{1}{2} \int_{-\infty}^{\infty} \frac{\left(1+\frac{1}{y^{2}}\right)-\left(1+\frac{1}{y^{2}}... |
151,430 | <p>Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore ho... | andrewBee | 40,349 | <p>Here is an extrinsic definition of the class of $Y$. The kernel of the natural map
$$
H^*(X) \to H^*(X \setminus Y)
$$
is a graded ideal in $H^*(X)$. The lowest degree that this ideal is non-zero in is precisely $codim(Y)$ and in this degree the image is a free $\mathbb{Z}$-module on one generator, which is $\pm[Y]$... |
1,787,806 | <p>I've recently had this problem in an exam and couldn't solve it.</p>
<p>Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd:</p>
<p>$$\sum_{i=0}^{99} 2^{i^2}$$</p>
<p>I know the basic modular arithmetic properties but this escapes my capabilities. In our algebra... | Jack D'Aurizio | 44,121 | <p>By Fermat's little theorem
$$ 2^{i^2}\!\!\!\pmod{7}=\left\{\begin{array}{ll}\color{green}{1}&\text{if } i\equiv 0\pmod{6}\\\color{blue}{2}&\text{if } i\equiv 1\pmod{6}\\\color{blue}{2}&\text{if } i\equiv 2\pmod{6}\\\color{green}{1}&\text{if } i\equiv 3\pmod{6}\\\color{blue}{2}&\text{if } i\equiv ... |
325,186 | <p>If <span class="math-container">$p$</span> is a prime then the zeta function for an algebraic curve <span class="math-container">$V$</span> over <span class="math-container">$\mathbb{F}_p$</span> is defined to be
<span class="math-container">$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right)... | Richard Stanley | 2,807 | <p>Exercise 4.8 of <em>Enumerative Combinatorics</em>, vol. 1, second
ed., and Exercise 5.2(b) in volume 2 give an explanation of
sorts for general varieties over finite fields. According
to Exercise 4.8, a generating function <span class="math-container">$\exp \sum_{n\geq 1}
a_n\frac{x^n}{n}$</span> is rational i... |
3,578,740 | <p>Good day everybody,</p>
<p>I would like to ask a question about undecidability.
May I ask You, if we have some problem that is undecidable but true, for example if RH would be found out to be undecidable it would mean that it is true, does that mean that such undecidable problem is true for no reason at all, or is ... | johnnyb | 298,360 | <p>It depends on exactly what you are asking. If you take a Turing machine, if you have an intelligently coded tape with an infinite number of possibilities encoded on it, then the halting status of all finite programs will be decidable. For a description of how this works, see Eric Holloway's "The Logical Possibilit... |
1,517,456 | <blockquote>
<p>Rudin Chp. 5 q. 13:</p>
<p>Suppose <span class="math-container">$a$</span> and <span class="math-container">$c$</span> are real numbers, <span class="math-container">$c > 0$</span>, and <span class="math-container">$f$</span> is defined on <span class="math-container">$[-1, 1]$</span> by</p>
<p><span... | Hayden | 27,496 | <p>In cases like these, you may want to use the fact that if $M/L/K$ is a tower of field extensions, then $[M:L][L:K]=[M:K]$, where $[L:K]$ is the dimension of the $K$-vector space $L$ (with the scalar product given just be multiplication of elements in $L$ by elements in $K$).</p>
<p>At this point, one can show that ... |
3,312,780 | <p>Compute <span class="math-container">$f(x) = \sum_{k = 1}^{\infty} \Bigg(\frac{1}{(k-1)!} + k\Bigg)x^{k-1}$</span></p>
<p><strong>Approach</strong></p>
<p>I'm not exactly sure how to do this, but just shooting around ideas due to this question appearing in a chapter on power series and uniform convergence, by idea... | José Carlos Santos | 446,262 | <p>The series <span class="math-container">$\sum_{n=1}^\infty\frac8n$</span> diverges, but <span class="math-container">$\frac8{3^n+2}<\frac8{3^n}$</span> and <span class="math-container">$\sum_{n=1}^\infty\frac8{3^n}$</span> converges (apply the ratio test). And <span class="math-container">$\frac1{2^n+3^n}<\fra... |
3,312,780 | <p>Compute <span class="math-container">$f(x) = \sum_{k = 1}^{\infty} \Bigg(\frac{1}{(k-1)!} + k\Bigg)x^{k-1}$</span></p>
<p><strong>Approach</strong></p>
<p>I'm not exactly sure how to do this, but just shooting around ideas due to this question appearing in a chapter on power series and uniform convergence, by idea... | azif00 | 680,927 | <p><span class="math-container">$3^n +2 > 3^n$</span> implies that <span class="math-container">$$\frac{8}{3^n+2} < \frac{8}{3^n}$$</span>
and the series <span class="math-container">$$\sum_{n=0}^\infty \frac{1}{3^n}$$</span> clearly converges.</p>
|
3,312,780 | <p>Compute <span class="math-container">$f(x) = \sum_{k = 1}^{\infty} \Bigg(\frac{1}{(k-1)!} + k\Bigg)x^{k-1}$</span></p>
<p><strong>Approach</strong></p>
<p>I'm not exactly sure how to do this, but just shooting around ideas due to this question appearing in a chapter on power series and uniform convergence, by idea... | Community | -1 | <p>Note the <em>harmonic series</em> <span class="math-container">$\sum_n\dfrac 1n$</span> diverges. But <span class="math-container">$\sum_n\dfrac 8n=8\sum_n\dfrac1n$</span>, thus it also diverges. </p>
<p>The geometric series <span class="math-container">$\sum_n x^n$</span> converges (to <span class="math-container... |
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