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 |
|---|---|---|---|---|
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | Mikasa | 8,581 | <p>I tink if you work as follows, you will get the answer better(I hope so):</p>
<p>When $\alpha(x)$ is very small, then $\ln(1+\alpha(x))\sim\alpha(x)$ so $$\ln\left(\cos(2x)\right)=\ln\left(1+(-2\sin^2(x)\right)\sim -2\sin^2(x)$$</p>
<p>Now take your limit with this fact again. It is $-2$.</p>
|
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Alain Valette | 14,497 | <p>Since you have read Rudin's "Real and complex analysis", you are ready to attack Rudin's "Fourier analysis on groups", which is equally pleasant reading.</p>
<p>Still valuable for the link with Banach algebra theory, is L.H. Loomis' "An introduction to abstract harmonic analysis".</p>
<p>For connections with unita... |
2,135,918 | <p>Does L'Hopitals Rule hold for second derivative, third derivative, etc...? Assuming the function is differential up to the $k$th derivative, is the following true?</p>
<p>$$\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} = \lim_{x\to c} \frac{f''(x)}{g''(x)} = \cdots = \lim_{x\to c} \frac{f^{(k)... | Michael Hardy | 11,667 | <p>When L'Hopital's rule is stated by saying that $\displaystyle \lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)},$ one of the hypotheses is that $\lim_{x\to c}f(x) = \lim_{x\to c} g(x) = 0,$ or else both limits are among the two objects $\pm\infty.$ In order to take it one more step and say $\display... |
1,071,321 | <p>The problem is quite simple to formulate. If you have a large group of people (n > 365), and their birthdays are uniformly distributed over the year (365 days), what's the probability that every day of the year is someone's birthday?</p>
<p>I am thinking that the problem should be equivalent to finding the number o... | Valentin | 310,690 | <p>Just for sake of illustration I give some numerical results:</p>
<ul>
<li>for $1000$ people we get a propability of $1.7*10^{-10}$% to exhaust all birthdays</li>
<li>for $1500$ people we get $0.2$% </li>
<li>for $2000$ people we get $22$%</li>
<li>for $2286$ people we get $50$%</li>
<li>for $2500$ people we have $6... |
320,704 | <p>Given an irrational <span class="math-container">$a$</span>, the sequence <span class="math-container">$b_n := na$</span> is dense and equidistributed in <span class="math-container">$\mathbb S^1$</span> where we view <span class="math-container">$\mathbb S^1$</span> as <span class="math-container">$[0, 1]$</span> ... | burtonpeterj | 30,721 | <p>This is probably suboptimal, but one can use the convergents of the continued fraction expansion of <span class="math-container">$a$</span> to efficiently find <span class="math-container">$n \in \mathbb{N}$</span> such that <span class="math-container">$||na||_\mathbb{Z} \leq e$</span>, where <span class="math-cont... |
3,261,334 | <blockquote>
<p>Prove isomorphism of groups <span class="math-container">$\langle G, + , {}^{-1}\rangle$</span> and <span class="math-container">$\langle G, *,{}^{-1}\rangle$</span>, where <span class="math-container">$a*b=b+a$</span><br>
<span class="math-container">$\forall a,b \in G$</span></p>
</blockquote>
<... | Community | -1 | <p>Let <span class="math-container">$\phi (g)=g^{-1}\,,\forall g\in G$</span>. </p>
<p><span class="math-container">$\phi$</span> is a homomorphism: <span class="math-container">$\phi (g*h)=(g*h)^{-1}=h^{-1}*g^{-1}=g^{-1}+h^{-1}=\phi(g)+\phi(h)$</span>.</p>
<p>The kernel of <span class="math-container">$\phi$</spa... |
320,619 | <p>Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is
$||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. </p>
<p>Prove that $B'(0,1) = \{x \in l^\infty : ||x|| \le 1\} $ is not compact.</p>
<p>Now, we are given a hint that we can use the equivalence of sequent... | copper.hat | 27,978 | <p>Let $E = \{ e_n \}_n$ where $e_n$ is the vector with $1$ in the $n$th position and zero everywhere else. Clearly $E \subset \overline{B}(0,1)$, hence it is bounded. Since $\|e_n - e_m\| = 1$ whenever $n \neq m$, it is clear that each point in $E$ is isolated. Hence $E$ is closed.</p>
<p>Now let $U_n = B(e_n, \frac... |
1,652,747 | <p>Ok, so I think I'm getting the hang of this. Is this more or less on the right track?</p>
<p>$$e^{z-2}=-ie^2$$
$$e^ze^{-2}=-ie^2$$
$$e^z=-ie^4$$
$$\ln(e^z)=\ln(-ie^4)$$
$$z=\ln|-i|+iarg(-i)+2\pi ik+4$$
$$z=\frac{i\pi}{2}-\frac{i\pi}{2}+4+2\pi ik$$
$$z=4+2\pi ik$$</p>
| paradus_hex | 991,156 | <p>Continuing from line 4:
<span class="math-container">$$
z=ln\left(i^3\right)+4........(i)
$$</span>
We know,
<span class="math-container">$ln(z)=ln(r)+i(Arg(z)+2k\pi)$</span> where k=0,+-1,+-2....<br/></p>
<p>Applying this to <span class="math-container">$ln\left(-i\right)$</span>, we get:</p>
<p><span class="math-c... |
2,186,743 | <p>I think this is a very basic question but somehow I am unable to understand the answer.</p>
<hr>
<blockquote>
<p>Find the number of zeroes of $f(x) = x^3 + x + 1$.</p>
</blockquote>
<hr>
<p>Answer in the book : </p>
<p>$f^\prime (x) = 3x^2 + 1$ since $3x^2 + 1 \ge 0$ for all $x \in \mathbb{R}$ therefore $f^\p... | Chris Leary | 2,933 | <p>Suppose the function has two zeros, then by Rolle, $f'(x)$ must be zero for some real number $c$ strictly between the two roots. This is impossible, since $f'(x) > 0$ for all real $x.$ So, the function has at most one zero.To show it has a zero, note that $f(-1) < 0$ and $f(0)>0,$ and appeal to the intermed... |
3,743,282 | <p>One definition of complex sympletic group I have encountered is (sourced from <a href="https://en.wikipedia.org/wiki/Symplectic_group" rel="nofollow noreferrer">Wikipedia</a>):
<span class="math-container">$$Sp(2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \}$$</span></p>
<p>What is the motivation ... | Ivo Terek | 118,056 | <p>In general, if <span class="math-container">$V$</span> is a <span class="math-container">$n$</span>-dimensional <span class="math-container">$F$</span>-vector space equipped with a bilinear form <span class="math-container">$b\colon V \times V \to F$</span> and <span class="math-container">$T\colon V \to V$</span> i... |
1,690,092 | <p>$a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$</p>
<p>and I need to check whether this sequence converges to a limit without finding the limit itself. I think about using the squeeze theorem that converges to something (I suspect '$1$').</p>
<p>But I wrote $a_{n+1}$ ... | Fnacool | 318,321 | <p>On the one hand, </p>
<p>$$a_n \ge \mbox{smallest summand} \times \mbox{number of summands}= \frac{1}{\sqrt{n^2+2n-1}}\times n .$$</p>
<p>To deal with the denominator, observe that </p>
<p>$$n^2+2n-1 \le n^2+2n+1=(n+1)^2.$$</p>
<p>On the other hand, </p>
<p>$$a_n \le \mbox{largest summand}\times \mbox{number... |
3,477,795 | <p>How to use the absolute value function to translate each of the following statements into a single inequality.</p>
<p>(a) <span class="math-container">$\ x ∈ (-4,10) $</span> </p>
<p>(b) <span class="math-container">$\ x ∈ (-\infty,2] \cup[9,\infty) $</span></p>
<p>I think in the first one the absolute value of ... | Rachid Atmai | 99,885 | <p>Any bounded function with countably many discontinuities is Riemann Integrable. Your function has only 3 discontinuity points and is thus Riemann integrable. In addition, this function is bounded. One them computes the Riemann integral in a piecewise fashion.</p>
|
626,928 | <p>I took linear algebra course this semester (as you've probably noticed looking at my previously asked questions!). We had a session on preconditioning, what are they good for and how to construct them for matrices with special properties. It was a really short introduction for such an important research topic, so I'... | Yiorgos S. Smyrlis | 57,021 | <p>It is still a very hot topic of Numerical Linear Algebra - The main question is to improve the conditioning of a (usually very large and potentially ill-conditioned) matrix.</p>
<p>Take a look at the work of Dongarra: <a href="http://web.eecs.utk.edu/~dongarra/etemplates/node396.html" rel="nofollow">Preconditioning... |
891,137 | <p>Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?</p>
| Mary Star | 80,708 | <p>We define $$f(x)=\log{(1+x)}-x$$</p>
<p>The domain of $f$ is $(-1, +\infty)$.</p>
<p>$$f'(x)=\frac{1}{1+x}-1=\frac{1-1-x}{1+x}=\frac{-x}{1+x}$$</p>
<p>$$f'(x)=0 \Rightarrow x=0$$</p>
<p>$$\text{ For } x \geq 0 : f'(x)\leq 0$$
$$\text{ For } -1<x\leq 0 : f'(x)\geq 0$$</p>
<p>That means that $f$ is decreasing ... |
35,964 | <p>This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong:</p>
<p>Question:
Show that a ball dropped from height of <em>h</em> feet and bounces in such a way that each bounce is $\frac34$ of the height of... | Bill Dubuque | 242 | <p>To get symmetry - a complete sawtooth pattern - suppose the ball is first thrown up from ground to height $h$. The total distance $x$ is that of the first tooth $= 2h$ plus the remaining sawtooth $= 3/4\ x$. Thus $x =\: 2h + 3/4\ x,\ $ so $x = 8h\:.\ $ Subtracting the intitial throw up leaves $7h.$</p>
|
40,500 | <blockquote>
<p>What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?</p>
</blockquote>
<p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a conf... | alezok | 13,388 | <p>Let $(B_t)_{t \geq 0}$ be a Brownian motion and $\mathcal B_t:=\sigma(\{B_s : s \leq t\})$ its natural filtration. Then</p>
<ul>
<li>$B_t$ is a martingale,</li>
<li>$B_t^2-t$ is a martingale,</li>
<li>$\exp(\theta B_t - \frac{1}{2}\theta^2 t)$ is a martingale, and</li>
<li>$H_n(t,B_t)$ is a martingale for every $n$... |
2,333,083 | <p>I want to know about when the infinite product form of the Riemann Zeta Function converges.</p>
<p>It is easy to show that $$\sum_{n=1}^\infty \frac 1{n^x}$$ converges for $\Bbb R(x)>1$.</p>
<p>When does $$\prod_{p=primes} \frac {p^x}{p^x-1}$$ converge? And how do you show that?</p>
| Dando18 | 274,085 | <p>The product converges for $\mathbb{R}(s)>1$ (from <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler_product_formula" rel="nofollow noreferrer">wikipedia page</a>). This product approaches $1$ from the right hand side, which follows from the Dirichlet series for $\zeta(s)$, which converges for $\m... |
1,893,609 | <p>I am trying to show that $A=\{(x,y) \in \Bbb{R} \mid -1 < x < 1, -1< y < 1 \}$ is an open set algebraically. </p>
<p>Let $a_0 = (x_o,y_o) \in A$. Suppose that $r = \min\{1-|x_o|, 1-|y_o|\}$ then choose $a = (x,y) \in D_r(a_0)$. Then</p>
<p>Edit: I am looking for the proof of the algebraic implication t... | Claude Leibovici | 82,404 | <p>Assuming that the points are not colinear, for a very simple procedure, consider <a href="https://en.wikipedia.org/wiki/Conic_section" rel="nofollow">the general equation of conics</a> the circle $$x^2+y^2+a x+b y +c=0$$ and build, for each data point, the equation $$f_i=x_i^2+y_i^2+a x_i+b y_i+c=0$$ This then reduc... |
1,916,297 | <p>Let $\{F_n\}_{n=1}^\infty $ be the Fibonacci sequence (defined by
$ F_n=F_{n-1}+F_{n-2}$ with $F_1=F_2=1$).</p>
<p>Is it true that for every integer number $N$ there exist positive integers
$n,m,k,l$ such that $N=F_n+F_m-F_k-F_l$ ?</p>
| Mr. Sigma. | 367,180 | <p>Edit: I'm wrong. </p>
<p>I'm getting yes!
Take k,l = 1. Then it follows.</p>
<p>N = F(n) + F(m) - 2
=> N + 2 = F(n) + F(m),Take N' = N+2. Therefore, we get.</p>
<p>=> N' = F(n) + F(m) ,N'>=2.
Which is possible since every natural number is a combination of two Fibonacci number. For N=0 & 1 also we have possi... |
2,975,665 | <p>I am asked to find the sum of the series <span class="math-container">$$\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}$$</span></p>
<p>For some reason (that I don't understand) I can't apply the techniques for finding the sum of the series that I usually would to this one? I think the others that I have done have been... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}= \sum_{n=2}^\infty\frac{(x+1)^{n}}{(n)!} $$</span></p>
<p><span class="math-container">$$=\sum_{n=0}^\infty\frac{(x+1)^{n}}{(n)!} -1-(x+1) $$</span></p>
<p><span class="math-container">$$=e^{x+1}-x-2$$</span></p>
|
1,179,981 | <p>As the title suggests. Let $G$ be a group, and suppose the function $\phi: G \to G$ with $\phi(g)=g^3$ for $g \in G$ is a homomorphism. Show that if $3 \nmid |G|$, $G$ must be abelian.</p>
<p>By considering $\ker(\phi)$ and Lagrange's Theorem, we have $\phi$ must be an isomorphism (right?), but I'm not really sure ... | krirkrirk | 221,594 | <p>$\phi$ is an isomoprhism, because $g^{3} = 1 \iff g=1$ for the order of $g\in G$ must divide the order of $G$.
I guess you have to consider $\phi (gh)$ and $\phi(hg)$.
If $\forall g, h\in G, \phi(gh)=\phi(hg)$, then $ghg^{-1}h^{-1}\in ker\phi$ thus $gh=hg$ and it's won. But not really sure how to do it. </p>
|
2,931,762 | <blockquote>
<p>Given a function <span class="math-container">$f$</span> is defined for integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> as given:
<span class="math-container">$$f(mn) = f(m)\,f(n) - f(m+n) + 1001$$</span>
where either <span class="math-container">$m$</s... | Batominovski | 72,152 | <p><strong>Disclaimer.</strong> This is an attempt to see what happens if I drop the conditions that <span class="math-container">$f(1)=2$</span> and that at least one of <span class="math-container">$m$</span> and <span class="math-container">$n$</span> in the functional equation must be <span class="math-container... |
230,126 | <p>A family of functions is known as <span class="math-container">$\left(\varphi_{0}, \varphi_{1}, \cdots, \varphi_{n}\right)$</span>.</p>
<p>I'd like to know how to express their inner product conveniently as follows:</p>
<p><span class="math-container">$$\left(\begin{array}{cccc}
\left(\varphi_{0}, \varphi_{0}\right)... | A little mouse on the pampas | 42,417 | <p>The results of the two methods are the same <span class="math-container">$\color{Gray} {\text{(2001 武汉 岩石 数值分析 4)}} $</span>:</p>
<pre><code>ClearAll["Global`*"]
f[x_] := x^3
L[a_, b_, c_] := Integrate[(f[x] - (a*x^2 + b*x + c))^2, {x, 0, 1}]
StagnationPoints =
Solve[{D[L[a, b, c], a] == 0, D[L[a, b, c]... |
2,304,332 | <p>This is related to <a href="https://math.stackexchange.com/questions/2162294/prove-that-there-exists-f-g-mathbbr-to-mathbbr-such-that-fgx-i/2294324#2294324">Prove that there exists $f,g : \mathbb{R}$ to $\mathbb{R}$ such that $f(g(x))$ is strictly increasing and $g(f(x))$ is strictly decreasing.</a></p>
<p>But acco... | robjohn | 13,854 | <p>Consider
$$
g(x)=g(f(g(x)))=-g(x)
$$</p>
|
2,547,888 | <p>A $\subset$ B, A and B closed then B-A is open</p>
<p>How can I prove this statement? Is that correct? I need it because is used to prove that a closed subset of a compact space is compact. </p>
<p>Imagine B=[0,4] and A=[1,2] contained in A, then B-A is neither open or closed, what am I doing wrong??</p>
<p>PS: ... | Mr. X | 508,297 | <p>If you are considering $B$ as your ambient space, then $B-A$ would definitely be open but if your ambient space is something else then $B-A$ may or may not be open.</p>
|
2,547,888 | <p>A $\subset$ B, A and B closed then B-A is open</p>
<p>How can I prove this statement? Is that correct? I need it because is used to prove that a closed subset of a compact space is compact. </p>
<p>Imagine B=[0,4] and A=[1,2] contained in A, then B-A is neither open or closed, what am I doing wrong??</p>
<p>PS: ... | Believer | 409,319 | <p>Remember this:
If A is open and B is closed,then,</p>
<p>$1.$A-B is $\color{red}{open}$</p>
<p>$2.$B-A is $\color{red}{closed}$</p>
|
176,987 | <p>Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ whose entries are also chosen independently and uniformly from $\{0,1\}$. Let $N = Mv$ where multiplication is performed... | Samuel Lelièvre | 38,711 | <p>One approach is computer exploration: write a program to plot a few points, and plot of the functions you want to compare to on the same graphics.</p>
<p>I wrote a quick Sage program to compute and plot the entropy H(Z(n,m)) for all 0 <= m <= n <= 10, and I added the plots of (x=t, y=t, z=t) and (x=t, y=sq... |
176,987 | <p>Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ whose entries are also chosen independently and uniformly from $\{0,1\}$. Let $N = Mv$ where multiplication is performed... | domotorp | 955 | <p>This is not an answer, just a long comment.</p>
<p>As pointed out by John Mangual, this game is similar to Mastermind.
In fact, it is even more similar to the game defined by Erdos and Renyi here:
<a href="http://www.renyi.hu/~p_erdos/1963-12.pdf" rel="nofollow">http://www.renyi.hu/~p_erdos/1963-12.pdf</a>.
In your... |
853,878 | <p>I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and revenue. Why would I want to find the marginal revenue (aka rate of change for revenue or cost)? What does this rate of ch... | Kamster | 159,813 | <p>Think of it in the case of maximizing profit. Let profit be represented by $\pi$. we see $\pi$ is generally a function of our cost(C) and revenue (R) since our profit is just revenue minus cost. Letting $x$ be the goods we are selling we see revenue and cost are a function of $x$ (i.e. $R(x)$ and $C(x)$).Thus
$$\pi... |
1,303,183 | <p>If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as $l=\sum_i\alpha_iv_i$. Here $\alpha_1,\alpha_2...\alpha_n$ are the coefficients that define the vector $l$ according to th... | Matthias Kaul | 244,244 | <p>They are unique. Think about what happens if they aren't and you subtract two different representations from each other.</p>
|
3,259,193 | <p>I have a simple question about notation regarding limits, specifically, <span class="math-container">$$\lim_{\|x\| \rightarrow \infty}f(x).$$</span> </p>
<p><strong>Question:</strong> </p>
<p><span class="math-container">$\lim_{\|x\| \rightarrow \infty}f(x)$</span>:</p>
<p>In words what we are doing is taking the... | copper.hat | 27,978 | <p>It is just notation. We say <span class="math-container">$\lim_{\|x\| \to \infty} f(x) = L$</span> <strong>iff</strong>
for any <span class="math-container">$\epsilon>0$</span> there is some <span class="math-container">$B$</span> such that if <span class="math-container">$\|x\| > B$</span> then <span class="m... |
588,488 | <p>I know that for the harmonic series
$\lim_{n \to \infty} \frac1n = 0$ and
$\sum_{n=1}^{\infty} \frac1n = \infty$.</p>
<p>I was just wondering, is there a sequence ($a_n =\dots$) that converges "faster" (I am not entirely sure what's the exact definition here, but I think you know what I mean...) than $\frac1n$ to ... | Andrés E. Caicedo | 462 | <p>There is no slowest divergent series. Let me take this to mean that given any sequence <span class="math-container">$a_n$</span> of positive numbers converging to zero whose series diverges, there is a sequence <span class="math-container">$b_n$</span> that converges to zero faster and the series also diverges, wher... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Charles E. Leiserson | 629,584 | <p>Here's a rigorous solution that doesn't require calculus or reasoning about distributions.</p>
<p>Let's pick the first point <span class="math-container">$A_0$</span> at random, then pick two <strong>diameters</strong> <span class="math-container">$B$</span> and <span class="math-container">$C$</span> (instead of t... |
250,484 | <p>For a fixed set $X$ and a finite collection $E_1,E_2,\ldots,E_k\subseteq X$, define the binary relation <em>adjacency</em> as follows: $E_i,E_j$ are adjacent
if their intersection is nonempty.
We term the transitive closure of this relation by
<em>transitive adjacency</em>
and define the <em>adjacent union</em> by
$... | Bjørn Kjos-Hanssen | 4,600 | <p>In philosophy, this would be called <a href="https://en.wikipedia.org/wiki/Family_resemblance" rel="noreferrer">family resemblance</a> -- if $E_i\cap E_j\ne\emptyset$ and $E_j\cap E_k\ne\emptyset$ then $E_i$ and $E_k$ have a family resemblance.</p>
<p>That is, perhaps I have no common feature with my second cousin,... |
2,701,658 | <p>If <span class="math-container">$1^2+2^2+3^2 + ... + 10^2=385$</span>, then what is the value of <span class="math-container">$2^2+4^2+6^2 + ... + 20^2$</span>?</p>
<p>Options are <span class="math-container">$770$</span>, <span class="math-container">$1155$</span>, <span class="math-container">$1540$</span>, <span... | Mr Pie | 477,343 | <p>You are given the equation, $$1^2 + 2^2 + 3^2 +\cdots + 10^2 = 385.$$ From this equation, you want to find the value of $x$ in the following equation: $$2^2 + 4^2 + 6^2 +\cdots + 20^2 = x.$$ I know the problem you have been given does not include a variable $x$, but we will assign the unknown expression to such a va... |
1,058,515 | <p>determine whether $v_1=(1,1,2)$ , $v_2=(1,0,1)$ and $v_3=(2,2,3)$ span $\mathbb{R}^3$</p>
<p>I think that it must prove that it linear Independence </p>
<p>let $α_1v_1+α_2v_2+α_3v_3=(0,0,0)$</p>
<p>so what next , is this a correct way or I'm completely wrong </p>
<p>please guide me to understand this</p>
<p>t... | JohnD | 52,893 | <p>Put their columns in a matrix $A$. You want to know if for all vectors $\mathbf{b}=(a,b,c)$ if $A\mathbf{x}=\mathbf{b}$ has a solution. Do you know how to answer that question?</p>
|
1,058,515 | <p>determine whether $v_1=(1,1,2)$ , $v_2=(1,0,1)$ and $v_3=(2,2,3)$ span $\mathbb{R}^3$</p>
<p>I think that it must prove that it linear Independence </p>
<p>let $α_1v_1+α_2v_2+α_3v_3=(0,0,0)$</p>
<p>so what next , is this a correct way or I'm completely wrong </p>
<p>please guide me to understand this</p>
<p>t... | Empiricist | 189,188 | <p>You are on the right way. That yields</p>
<p>$$\alpha_1 (1,1,2) + \alpha_2 (1,0,1) + \alpha_3 (2,2,3) = (0,0,0)$$</p>
<p>$$
\left( \begin{array}{ccc}
1 & 1 & 2 \\
1 & 0 & 2 \\
2 & 1 & 3 \end{array} \right)
\left( \begin{array}{c}
\alpha_1 \\
\alpha_2 \\
\alpha_3 \end{array} \right)
=
\left(... |
3,080,664 | <p>Hi I've almost completed a maths question I am stuck on I just can't seem to get to the final result. The question is:</p>
<p>Find the Maclaurin series of <span class="math-container">$g(x) = \cos(\ln(x+1))$</span> up to order 3.</p>
<p>I have used the formulas which I won't type out as I'm not great with Mathjax ... | heropup | 118,193 | <p>Since you also don't need very high orders, the straightforward calculation of derivatives is tractable, though not preferable computationally. If <span class="math-container">$f(x) = \cos \log(1+x)$</span>, then <span class="math-container">$$f'(x) = -\frac{\sin \log (1+x)}{1+x}, \quad f'(0) = 0.$$</span> Then <s... |
3,080,664 | <p>Hi I've almost completed a maths question I am stuck on I just can't seem to get to the final result. The question is:</p>
<p>Find the Maclaurin series of <span class="math-container">$g(x) = \cos(\ln(x+1))$</span> up to order 3.</p>
<p>I have used the formulas which I won't type out as I'm not great with Mathjax ... | K B Dave | 534,616 | <p>A heuristic way using complex numbers that's related to one of the other answers: define
<span class="math-container">$$\begin{align}
f(x)&=\cos\ln(1+x) \\
g(x)&=\sin \ln(1+x)\text{.}
\end{align}$$</span>
Then
<span class="math-container">$$\begin{align}f(0)+\mathrm{i}g(0)&=1\\
(f+\mathrm{i}g)'&=\fra... |
4,184,248 | <p>I'm trying to prove an inequality from an old book named "Algebra and elementary functions" by Kochetkov (ru - "Алгебра и элементарные функции" Е. С. Кочетков). This is an old book from 1970 (URSS era). The book is not copyrighted anymore, and it is freely available over the internet.</p>
<p>The ... | Huy Nguyen | 922,437 | <p>This polynomial has degree 2, therefore it can not have 3 distinct roots, hence all the coefficients must be zero</p>
|
155,373 | <p>Consider $X,Y \subseteq \mathbb{N}$. </p>
<p>We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$.</p>
<p>We say that $X \equiv_c Y$ iff there exist a bijective computable function between $X$ and $Y$. </p>
<p>Can you show me some examples in which the two concepts disagree?</p>
| William | 13,579 | <p>Let $X = \omega$, the natural number. Let $\bar{K}$ be the complement of the $K = \{e : \phi_e(e)\downarrow\}$, the halting problem set. That is, $K$ is set of $e$ such that the $e^\text{th}$ Turing Machine on input $e$ converges. </p>
<p>Suppose there exists a computable bijection from $\omega \rightarrow \bar{K}$... |
44,306 | <p>This morning I realized I have never understood a technical issue about Cauchy's theorem (homotopy form) of complex analysis. To illustrate, let me first give a definition.</p>
<p>(In what follows $\Omega$ will always denote an open subset of the complex plane.)</p>
<p><em>Definition</em> Let $\gamma, \eta\colon [... | Sam | 3,208 | <p>As has already been pointed out by Akhil in the comments, any two smooth curves $\gamma_0, \; \gamma_1$ which are continuously homotopic are also smoothly homotopic. The point here is to approximate a continuous homotopy, rather than the curves themselves.</p>
<p>More concretely, given homotopic smooth curves $\gam... |
164,321 | <p>I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:</p>
<p>$$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$</p>
<p>Where $n$, $\nu$, and $\mu$ are all integers, and $a$, $b$, and $c$ are all real and positive.</p>
<p>If not generally, a specific ca... | rubenvb | 7,606 | <p>Although I'm sure @Zurab's answer will indeed give me a solution (and I've heard the suggestion before in other situations), I'm not very familiar with the technique.</p>
<p>Following the suggestion by @Johannes in the comments, there is a much more straightforward way that, although not fully general, fits my prob... |
4,330,852 | <blockquote>
<p>Let <span class="math-container">$A_1 ,\ldots ,A_m$</span> be subsets of an <span class="math-container">$n$</span>-element set such that <span class="math-container">$|A_i \cap A_j |\leq t$</span> for all <span class="math-container">$i\neq j$</span>. Prove that <span class="math-container">$$\sum_{i=... | Youem | 468,504 | <p><em>Hint</em> Use inclusion-exclusion formula to compute the cardinality of the union of <span class="math-container">$A_i$</span> and use the fact that this cardinality is at most <span class="math-container">$n$</span></p>
|
1,897,771 | <p>Let $(a, b)$ be the open interval $\left\{z\in\mathbb{R} : a < z < b\right\}$. </p>
<p>Write the theorem "If $x,y\in(a,b)$ then $|x − y| < b − a$" in logic form, and then prove the theorem.</p>
| MarnixKlooster ReinstateMonica | 11,994 | <p>Here is an alternative form of the first answer, with some suggestions on how this proof could be designed.$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} }
\newcommand{\hint}[... |
1,856,409 | <p>In business with 80 workers, 7 of them are angry. If the business leader visits and picks 12 randomly, what is the probability of picking 12 where exactly 1 is angry? </p>
<p>(7/80)<em>(73/79)</em>(72/78)<em>(71/77)</em>(70/76)<em>(69/75)</em>(68/74)<em>(67/73)</em>(66/72)<em>(65/71)</em>(64/70)*(63/69)*12=0.413458... | Joffan | 206,402 | <p>The denominators of the fractions stay constant. The total multiplication across the denominators is all the ways to pick 12 people from 80, where the order is retained:
$$ 80\cdot 79\cdot 78\cdot 77\cdot 76\cdot 75\cdot 74\cdot 73\cdot 72\cdot 71\cdot 70\cdot 69 = \frac{80!}{68!}
$$</p>
<p>We say that order is uni... |
2,372 | <p>How can I find the number of <span class="math-container">$k$</span>-permutations of <span class="math-container">$n$</span> objects, where there are <span class="math-container">$x$</span> types of objects, and <span class="math-container">$r_1, r_2, r_3, \cdots , r_x$</span> give the number of each type of object?... | robjohn | 13,854 | <p>Consider the sum defined by
<span class="math-container">$$
\begin{align}
&\sum_{k=0}^\infty s_k\frac{x^k}{k!}\\
&=\textstyle\underbrace{\left(1{+}x{+}\dots{+}\frac{x^8}{8!}\right)}_{\substack{\text{the exponent of $x$}\\\text{is the number of c's}}}\underbrace{\left(1{+}x{+}\dots{+}\frac{x^5}{5!}\right)}_{\... |
2,015,809 | <blockquote>
<p>If $$f(x)=3 f(1-x)+1$$
for all $x$, what is the value of $f(2016)$?</p>
</blockquote>
<p>I am not sure how to do this, because I see two "$f$"s.</p>
<p>All I could try is substituting,</p>
<p>$$f(x)=3(1-2016)+1\\
=-6044$$</p>
<p>Which I am pretty sure wrong.</p>
<p>How do I deal with this quest... | hamam_Abdallah | 369,188 | <p>We have</p>
<p>$$f(x)=3(3f(1-(1-x))+1)+1$$</p>
<p>$$=9f(x)+4$$</p>
<p>$\implies$</p>
<p>$$f(x)=-\frac12=f(2016)$$</p>
|
275,527 | <pre><code>Sqrt[Matrix[( {
{0, 1},
{-1, 0}
} )]] /. f_[Matrix[x__]] :> Matrix[MatrixFunction[f, x]]
</code></pre>
<p><code>Matrix</code> is an undefined symbol but I want to define some substitutions with it.</p>
| lericr | 84,894 | <p>I assume this is related to that other question you recently posted--maybe merge them? Anyway, I'm going to change MatrixFunction for now to hopefully get a clearer answer. Your pattern matched an expression whose body only had one argument. You need to add something to the pattern to match more arguments. Try somet... |
1,384,203 | <p>The additive character of a finite field $\mathbb F_q$ is obtained by using the trace function to base field $F_p$.</p>
<p>Can we write some of them into a middle field. Using trace function from $\mathbb F_{q^n}$ to $\mathbb F_q$? If yes, how?</p>
| Jyrki Lahtonen | 11,619 | <p>Morgan gave you already the negative answer. For the sake of completeness...</p>
<p>There are plenty of trace maps around here. There is the relative trace
$$
tr^{q^n}_q:\Bbb{F}_{q^n}\to\Bbb{F}_q, x\mapsto x+x^q+x^{q^2}+\cdots+x^{q^{n-1}}.
$$
Then we have the two (absolute) trace maps of both these fields all the w... |
1,804,360 | <p>The following is a classically valid deduction for any propositions <span class="math-container">$A,B,C$</span>.
<span class="math-container">$\def\imp{\rightarrow}$</span></p>
<blockquote>
<p><span class="math-container">$A \imp B \lor C \vdash ( A \imp B ) \lor ( A \imp C )$</span>.</p>
</blockquote>
<p>But I'... | Hanno | 81,567 | <p>Are you familiar with Kripke semantic of intuitionistic logic (see <em>Semantics of intuitionistic logic</em> <a href="https://en.wikipedia.org/wiki/Kripke_semantics" rel="noreferrer">here</a>)?<span class="math-container">$\def\imp{\rightarrow}$</span></p>
<p><em>Intuitively</em>, a proposition is not interpreted ... |
2,247,445 | <p>Find isomorphism between <span class="math-container">$\mathbb F_2[x]/(x^3+x+1)$</span> and <span class="math-container">$\mathbb F_2[x]/(x^3+x^2+1)$</span>.</p>
<hr />
<p>It is easy to construct an injection <span class="math-container">$f$</span> satisfying <span class="math-container">$f(a+b)=f(a)+f(b)$</span> an... | Kanwaljit Singh | 401,635 | <p>Hint -</p>
<p>Make a right angle triangle. We know that $\tan \theta = \frac{\text{perpendicular}} {\text{base}}$.</p>
<p>Comparing it with $\tan \alpha$ you get perpendicular = 1 and base = 3.</p>
<p>Now find 3rd side of triangle using phythagorous theorem. Then you can find value of $\cos \alpha$ and $\sin \alp... |
1,297,023 | <p>I have a question mainly in functional analysis. </p>
<p>Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows:</p>
<p>Does $a_n \in H^1(\mathbb R^n)$, $|a_n| <1$, $a_nb_n \in L^1(\mathbb R^n)$, and $b_n \rightharpoonup 0$ weakly in $H^1(\mathbb R^n)$ imply $\... | daw | 136,544 | <p>No.</p>
<p>Let $b$ be a smooth function with support of $b$ included in the unit ball. Assume that $\|b\|_{H^1}=1$. Then define $b_n$ to be translates of $b$:
$$
b_n(x) = b(x + 2n e_1).
$$
Then $b_n\rightharpoonup 0$ in $H^1$, but $b_n$ does not converge stronlgy to zero in $H^1$ or $L^2$. Set $a_n:=b_n$. Then $\|a... |
1,297,023 | <p>I have a question mainly in functional analysis. </p>
<p>Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows:</p>
<p>Does $a_n \in H^1(\mathbb R^n)$, $|a_n| <1$, $a_nb_n \in L^1(\mathbb R^n)$, and $b_n \rightharpoonup 0$ weakly in $H^1(\mathbb R^n)$ imply $\... | Peter | 111,826 | <p>I think it is not true. The following should work.
Take $g_n:=a_n = b_n \in C_c^\infty$ with $\text{supp} g_n \in [n,n+1]$ and $0\leq g_n \leq 1$ and $g_n = 1$ on $[n+\frac 14, n+\frac 34]$. </p>
|
3,817,760 | <p>In geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance point and center of mass of the triangle. Why is that the case?</p>
<p>This is the best explanation I could think of. I ho... | CiaPan | 152,299 | <p>Split a triangle into narrow stripes, parallel to one side.</p>
<p><a href="https://i.stack.imgur.com/OKf4c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OKf4c.png" alt="enter image description here" /></a></p>
<p>A center of mass of each stripe is close to the middle of its length, which in tur... |
4,046,265 | <p>I'm evaluating a line integral of the function <span class="math-container">$T= x^2 + 4xy + 2yz^3$</span> from <span class="math-container">$a = (0,0,0)$</span> to <span class="math-container">$b=(1,1,1)$</span> on the path <span class="math-container">$z = x^2$</span>, and <span class="math-container">$y = x$</span... | Andreas Cap | 202,204 | <p>It seems to me that you got the computation of the derivative wrong. As you have written (with a small simplification as observed in the comment by @guidoar and using the chain rule), you get
<span class="math-container">$$
\Phi\circ\Psi^{-1}(x,v)=((\phi\circ\psi^{-1})(x), D(\phi\circ\psi^{-1})(x)(v)).
$$</span>
(I... |
158,469 | <p>This is another exercise from Golan's book.</p>
<p><strong>Problem:</strong> Let $V$ be an inner product space over $\mathbb{C}$ and let $\alpha$ be an endomorphism of $V$ satisfying $\alpha^*=-\alpha$, where $\alpha^*$ denotes the adjoint. Show that every eigenvalue of $\alpha$ is purely imaginary.</p>
<p>My prop... | Martin Argerami | 22,857 | <p>Let me show another argument which applies to a more general setting: if $\alpha$ is a linear operator on a Hilbert space satisfying $\alpha^*=-\alpha$, then the spectrum of $\alpha$ is purely imaginary (i.e. real part equal zero). </p>
<p>Indeed, one simply needs to notice that $\alpha-\lambda\,\text{id}$ is inver... |
3,586,001 | <p>Let <span class="math-container">$k$</span> be a positive integer. Is it true that there are infinitely many primes of the form <span class="math-container">$p=6r-1$</span> such that <span class="math-container">$r$</span> and <span class="math-container">$k$</span> are coprime? Feel free to assume any well known (e... | Sungjin Kim | 67,070 | <p>Denote <span class="math-container">$(a,b)$</span> by the greatest common divisor of <span class="math-container">$a$</span> and <span class="math-container">$b$</span>. Denote also <span class="math-container">$q^a||k$</span> when <span class="math-container">$q^a$</span> is the largest power of a prime <span clas... |
4,642,772 | <p>I am trying to understand this proof:
<a href="https://math.stackexchange.com/questions/3201004/prove-that-varphipk-pk-pk-1-for-prime-p">Prove that $\varphi(p^k)=p^k-p^{k-1}$ for prime $p$</a></p>
<p>At some point it is stated that the number of multiples of p in range <span class="math-container">$[1,p^k)$</span> i... | cigar | 1,070,376 | <p>Anything less than or equal to <span class="math-container">$p^{k-1}$</span> can be multiplied by <span class="math-container">$p$</span> and not go over <span class="math-container">$p^k.$</span></p>
<p>So there's exactly <span class="math-container">$p^{k-1}$</span> non-totatives.</p>
<p>Subtract those out to get ... |
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | kglr | 125 | <p>Also related to Mike's solution:</p>
<pre><code>ClearAll[irregPartition];
irregPartition[list_List, sizes_List, offsets_List] :=
Module[{offsetlist, sizelist},
offsetlist = (NestWhile[Join[offsets, #] &,offsets, (Tr@# < Length@list) &] //
NestWhile[Most, #, (Tr@# >= Length@list) &] &... |
855,463 | <p>I have the following question regarding support vector machines: So we are given a set of
training points $\{x_i\}$ and a set of binary labels $\{y_i\}$.</p>
<p>Now usually the hyperplane classifying the points is defined as:</p>
<p>$ w \cdot x + b = 0 $</p>
<p><strong>First question</strong>: Here $x$ does not d... | Joel Bosveld | 108,482 | <p>For the first equation, $w\cdot x+b=0$, $w$ is the direction normal (orthogonal/perpendicular) to the hyperplane. You are correct that the $x$ (satisfying this equation) are the points on the hyperplane.</p>
<p>In the second equation, the $x$ are the training (or test) data. These should not lie on the hyperplane, ... |
297,251 | <p>In his well-known <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101110.pdf" rel="nofollow noreferrer">paper</a> Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:</p>
<p>We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \omega_1\}$ of m... | Vít Tuček | 6,818 | <p>The <a href="http://inspirehep.net/record/1591349" rel="nofollow noreferrer">article</a> that was linked by David G. Stork actually contains information about how one can get these pictures. Namely, the <a href="http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/lattice_polytope.html#sage.geom... |
3,361,224 | <blockquote>
<p>Show that a dense subspace <span class="math-container">$Y$</span> of a first countable separable topological space <span class="math-container">$X$</span> is separable.</p>
</blockquote>
<p><em>Proof:</em></p>
<p><span class="math-container">$X$</span> is separable.
Let <span class="math-container"... | bof | 111,012 | <p>Looks good to me. Here's a slightly different way to look at it. Consider the following properties of a topological space <span class="math-container">$X$</span>:</p>
<p>(1) <span class="math-container">$X$</span> is separable and first countable;</p>
<p>(2) <span class="math-container">$X$</span> has a countable ... |
989,118 | <p>The sum is $$\sum_{n>0} \mathrm{i}^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ and $$\sum_{n>0} (\mathrm{-i})^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$</p>
<p>I suspect they are zero because I am working on a project which from symmetry consideration they should be zero. </p>
<p><st... | Saketh Malyala | 250,220 | <p>So we know that $i^2$ = -1, and that squared is $i^4$ which is 1. As aforementioned, $i^2 = -i^4$. Because, $i^4=1$, $i^2=i^6=i^{10}...i^{38}$, and $i^4=i^8..i^{36}$. Therefore, canceling out terms leaves us with $(i^{38})^2$, which evaluates to $(-1)^2$, which we know to be one. </p>
<p>==> $(((i^2-i^4)+(i^6-i^8).... |
51,903 | <p>Does anyone know of a tool which</p>
<ol>
<li>Can display formulas neatly, preferably like this website without hassle. (Unlike wikipedia with <code>:<math></code>) </li>
<li>Has a wiki like structure: i.e categories of pages, individual articles with hyperlinked sections, subsections etc. </li>
<li>Preferrab... | Ilmari Karonen | 9,602 | <p>It <a href="http://michaelnielsen.org/polymath1/index.php?title=Special%3aVersion" rel="nofollow">looks like</a> the polymath wiki you mention is just a plain old no-frills MediaWiki installation. Getting MediaWiki installed on a webhost isn't actually very hard, although admittedly getting the <a href="http://www.... |
180,121 | <p>Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\Psi_p:\mathcal{F}_p \rightarrow f_{\ast}f^{\ast}(\mathcal{F})_p=f_{p,\ast}f_p^{\ast}(\mathcal{F}_p)$$</p>
<p>I'm inter... | Sasha | 4,428 | <p>If $f_*O_X=O_Y$ then this is true for any locally free (or flat) sheaf (and if $f$ is flat then for any quasicoherent sheaf) by projection formula. </p>
|
2,228,635 | <p>I have tried to start with $\epsilon> 0$, and with $f$ uniformly continuous at $(a,b)$, so then there exists a $\delta>0$ such that for $x,u \in (a,b)$ with $|x - u|<δ$ then $|f(x)-f(u)|<\epsilon$. Since $x,u < b$ then $b-x >0$ so we have $0 < b - x <\delta$ then $|f(x)-f(b)|<\epsilon$. I... | skyking | 265,767 | <p>As said, you can always prove things like this with Venn diagrams, truth tables or Karnaugh diagram. In this case I'd say that would be the easiest way. </p>
<p>Otherwise you just use the distributive law together with absorbtion:</p>
<p>$$(A\cup B)\cap(B\cup C)\cap(C\cup A)
\\= ((A\cap B)\cup (A\cap C)\cup (B\cap... |
2,228,635 | <p>I have tried to start with $\epsilon> 0$, and with $f$ uniformly continuous at $(a,b)$, so then there exists a $\delta>0$ such that for $x,u \in (a,b)$ with $|x - u|<δ$ then $|f(x)-f(u)|<\epsilon$. Since $x,u < b$ then $b-x >0$ so we have $0 < b - x <\delta$ then $|f(x)-f(b)|<\epsilon$. I... | Henno Brandsma | 4,280 | <p>I managed to convince myself of this equality as follows: $x$ is in the left hand side, iff $x$ is in at least two of the sets $A,B,C$ (by definition of union, and then intersection ). In that case it will be in all unions $A \cup B, B \cup C, A \cup C$, because to avoid that you have to miss at least two distinct s... |
412,944 | <p>Is it mathematically acceptable to use <a href="https://math.stackexchange.com/questions/403346/prove-if-n2-is-even-then-n-is-even">Prove if $n^2$ is even, then $n$ is even.</a> to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?<... | Key Ideas | 78,535 | <p>Yes, essentially analogous arguments work to extend the notion of <strong>parity</strong> from the ring of integers to many rings of algebraic integers such as $\,\Bbb Z[\sqrt[n]{k}].\ $ </p>
<p>The key ideas are: one can apply parity arguments in any ring that has $\ \mathbb Z/2\ $ as an image, e.g. the ring of al... |
1,413,212 | <blockquote>
<p>Given $x$ the number of circles of radius $r$,
what is a good approximate size of the radius of a bigger circle which they fit in?</p>
</blockquote>
<p>To explain in actual problem terms. I want to move units in a video games which take up a certain amount of area around, I don't want those units ... | Community | -1 | <p>By illegal calculations, we get $\lim_{x \to 1}f(x) = 4$. To prove that now, use the $\epsilon - \delta$ definition of limits, with $\delta = \epsilon/10$. </p>
|
1,413,212 | <blockquote>
<p>Given $x$ the number of circles of radius $r$,
what is a good approximate size of the radius of a bigger circle which they fit in?</p>
</blockquote>
<p>To explain in actual problem terms. I want to move units in a video games which take up a certain amount of area around, I don't want those units ... | Paramanand Singh | 72,031 | <p>Use the algebra of limits to get $$\lim_{x \to 1}f(x) = \lim_{x \to 1}f(x) - 4 + 4 = \lim_{x \to 1}\frac{f(x) - 4}{x - 1}\cdot (x - 1) + 4 = 10\cdot 0 + 4 = 4$$</p>
|
2,086,598 | <p>Let's start with a well known limit:
$$\lim_{n \to \infty} \left(1 + {1\over n}\right)^n=e$$</p>
<p>As $n$ approaches infinity, the expression evaluates Euler's number $e\approx 2.7182$. Why that number? What properties does the limit have that leads it to 2.71828$\dots$ and why is this number important? Where can ... | Travis | 404,488 | <p>$e$ is used in recursive interest.</p>
<p>It's related to the fact that 5% interest ever 6 months is better than 10% every 12 months</p>
<p>Euler's number is the limit of intrest, ie: if you made an infinitesimal amount of interest constantly (at an infinitesimal interval, in the end, you would have $x*e$ money, x... |
2,086,598 | <p>Let's start with a well known limit:
$$\lim_{n \to \infty} \left(1 + {1\over n}\right)^n=e$$</p>
<p>As $n$ approaches infinity, the expression evaluates Euler's number $e\approx 2.7182$. Why that number? What properties does the limit have that leads it to 2.71828$\dots$ and why is this number important? Where can ... | Doug M | 317,162 | <p>It is not so important that it equals $2.718\cdots$ but that it converges to some number. </p>
<p>When Napier,Jacob Bernoulli, et. al. were looking at it, they were thinking about the implications of the compound interest calculation. And the effect of increasing the compounding frequency has an upper bound.</p>
... |
3,476,181 | <p>I'm trying to solve this group theory problem, and I'm really not sure how to approach this. The question is:</p>
<p>Show that <span class="math-container">$\mathbb{F}_7[\sqrt{-1}] := \{a+b\sqrt{-1}$</span> | <span class="math-container">$a,b \in \mathbb{F}_7\}$</span> is a ring.</p>
<p>I have been stuck on this p... | bof | 111,012 | <p>Your intuition is correct. Quoting Wikipedia on <a href="https://en.wikipedia.org/wiki/Higman%27s_lemma" rel="noreferrer">Higman's lemma</a>:</p>
<blockquote>
<p>Higman's lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is <a href="https://en.... |
3,684,158 | <p>You are given two circles:</p>
<p>Circle G: <span class="math-container">$(x-3)^2 + y^2 = 9$</span></p>
<p>Circle H: <span class="math-container">$(x+3)^2 + y^2 = 9$</span></p>
<p>Two lines that are tangents to the circles at point <span class="math-container">$A$</span> and <span class="math-container">$B$</span... | Ng Chung Tak | 299,599 | <p>Not answering the question but giving further observation,</p>
<p><span class="math-container">\begin{align}
\sqrt{(x-r)^2+y^2-r^2} \pm \sqrt{(x+r)^2+y^2-r^2} &= 2s \\
\sqrt{x^2-2rx+y^2} \pm \sqrt{x^2+2rx+y^2} &= 2s \\
2(x^2+y^2) \pm 2\sqrt{(x^2+y^2)^2-4r^2x^2} &= 4s^2 \\
(x^2+y^2)^2-4r^2x^2 &am... |
2,796,854 | <p>what happens if both the second and first derivatives at a certain point are $0$? Is it an infliction point, an extremum point, or neither? Can we say anything at all about a point in such a case? </p>
| tien lee | 557,074 | <p>• if $\frac{df}{dx}(p) = 0$, then $x = p$ is called a critical point of $f(x)$, and we do not know anything new about the behavior of $f( x) $ at $x = p$.</p>
<p>• if $\frac{d^2f}{dx^2} (p) = 0$ at $x = p$, then we do not know anything new about the behavior of $f(x)$ at $x = p$.</p>
<p>As @arugula mentioned $f(x... |
3,407,174 | <p>I'm dealing with a specific polynomial function. The first derivative of it is displayed below. As you can see, it has the shape of an asymetric function. But what does this tells us about the initial function in general and in terms of finding a maximum or a minimum for it? I know it is a general question, but I fi... | ling | 670,949 | <p>Cauchy inequality
<span class="math-container">$$(a_1+a_2+\cdots +a_m)^2\leq m(a_1^2+a_2^2+\cdots +a_m^2).$$</span>
Hence
<span class="math-container">$$(1+\frac{1}{\gamma})(x_1^2+\cdots +x_m^2)-\frac{(x_1+\cdots +x_m)^2}{m}\geq \frac{1}{\gamma}|x|^2>0.$$</span>
That is positive definite.</p>
|
1,638,490 | <p>Yes, I know the title is bizarre. I was urinating and forgot to lift the seat up. That made me wonder: assuming I maintain my current position, is it possible for the toilet seat (assume it is a closed, but otherwise freely deformable curve) to be moved/deformed such that stream does not pass through the hole anymor... | Ross Millikan | 1,827 | <p>If it is deformable, pass it under your feet, up your back, and over your head. It takes a bunch of deformation to make that work.</p>
|
55,071 | <p><strong>Bug introduced in 10.0.0 and persisting through 10.3.0 or later</strong></p>
<hr>
<p>I've upgraded my home installation of <em>Mathematica</em> from version 9 to 10 today on a Windows 8.1 machine, and I'm getting a weird font issue - the fonts are not anti-aliased, and look unbalanced and weird. Just look:... | Karsten 7. | 18,476 | <p>I analyzed which fonts are loaded when starting Mma v10 compared to v9.
The problem could be tracked down to the loading of the fonts in the Folder </p>
<p><code>$InstallationDirectory\SystemFiles\Fonts\TrueType</code></p>
<p>If you open <code>Mathematica-Bold.ttf</code> or <code>MathematicaMono-Bold.ttf</code>, y... |
3,885,025 | <p>Let be <span class="math-container">$(A,M)$</span> a local ring (if it needs noetherian ring), <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> two prime ideals with <span class="math-container">$P_1\subseteq P_2$</span>, <span class="math-container">$\ a_1\in M\setminus P_2$</... | Eric Wofsey | 86,856 | <p>Here is an example to show that this need not be true if <span class="math-container">$A$</span> is not Noetherian. Let <span class="math-container">$A$</span> be a valuation ring with value group <span class="math-container">$\mathbb{Z}^2$</span> with the lexicographic order. Let <span class="math-container">$P_1... |
3,097,640 | <p>So I have the following problem: $a + b = c + c.
I want to prove that the equation has infinitely many relatively prime integer solutions.</p>
<p>What I did first was factor the right side to get:
(</p>
| Mark Bennet | 2,906 | <p>Suppose <span class="math-container">$c=u^2+v^2$</span> then you have <span class="math-container">$$(uc^2+v)^2+(vc^2-u)^2=c(c^4+1)$$</span> and you want <span class="math-container">$uc^2+v$</span> and <span class="math-container">$vc^2-u$</span> to be coprime.</p>
<p>Now just choose <span class="math-container">$... |
2,219,266 | <blockquote>
<p>Compute integral $\displaystyle \int_{-\infty}^{\infty}e^{-k^2t+ikx}\, dk$.</p>
</blockquote>
<p>Hint: Complete the square in the Exponent.</p>
<p>Okay, for the Exponent, we have
$$
-k^2t+ikx=-t\cdot\left(k-\frac{ix}{2t}\right)^2-\frac{x^2}{4t}.
$$</p>
<p>Now, is it easier to compute
$$
\int_{-\inf... | Bram28 | 256,001 | <p>The answer depends on <em>how</em> "you know that at least one of them is male".</p>
<p>Here are two different types of scenarios:</p>
<ol>
<li><p>Someone one day just told you that this woman had a triplet and that 'at least one of them is a boy'</p></li>
<li><p>You met this woman some day before on the street (o... |
903,117 | <p>I am trying to evaluate
$$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$
Would a contour work? I have tried using a contour but had no success.
Thanks.</p>
<p>Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anywa... | SuperAbound | 140,590 | <p>This is an approach using contour integration.
\begin{align}
\int_{\mathbb{R}}\frac{\sin^2{x}}{x^2}{\rm d}x
&=-\frac{1}{4}\lim_{\epsilon \to 0}\int_{\mathbb{R}}\frac{e^{2ix}-2+e^{-2ix}}{(x-i\epsilon)^2}{\rm d}x\tag1\\
&=-\frac{1}{4}\lim_{\epsilon \to 0}2\pi i\lim_{z \to i\epsilon}\frac{{\rm d}}{{\rm d}z}(e^{... |
119,532 | <p>I have a <code>ListAnimate</code> where the frames consist of graphics made of lines and points. I want to make sure that, when the animation is paused and then resumed, all frames appear normal again, so if you stop it to rotate the plot (it's 3D), after a loop it looks normal again.</p>
<p>My code is the followin... | MarcoB | 27,951 | <p><code>PlusMinus</code> formats nicely, but it does not have a built-in meaning. You may work around that:</p>
<pre><code>h = 0.08; t = 0.13;
Plot[
Evaluate[
Sqrt[(1/4) + (1/64 h^2) - x^2] - (1/8 h) + {-1, 1} ((3/8) t (1 - 2 x) Sqrt[1 - (2 x)^2])
], {x, -.5, .5}
]
</code></pre>
<p><img src="https://i.stack.img... |
3,777,049 | <p>Suppose that <span class="math-container">$(H_i)_{i \in I}$</span> is a collection of closed <strong>orthogonal</strong> subspaces of the Hilbert space <span class="math-container">$H$</span>. Suppose that <span class="math-container">$\sum_{i \in I} \Vert x_i \Vert^2 < \infty$</span>. Prove that <span class="mat... | Paul Frost | 349,785 | <p>I guess you forgot to mention that <span class="math-container">$x_i \in H_i$</span>. It would also be sufficient to assume that the <span class="math-container">$x_i$</span> are pairwise orthogonal. This implies that for finite <span class="math-container">$A \subset I$</span> and <span class="math-container">$\sig... |
4,320,209 | <p><em>I am stuck on the following two questions on matrix Algebra:</em></p>
<p>Let <span class="math-container">$x$</span> be a vector. Find the following:</p>
<ul>
<li><span class="math-container">$\frac{\partial}{\partial x}||x\otimes x||^2$</span>, here ||.|| denotes Euclidean norm of a vector and <span class="math... | greg | 357,854 | <p><span class="math-container">$
\def\a{\alpha}\def\b{\beta}\def\l{\lambda}\def\o{{\tt1}}
\def\B{\Big}\def\L{\left}\def\R{\right}
\def\LR#1{\L(#1\R)}
\def\BR#1{\B(#1\B)}
\def\trace#1{\operatorname{Tr}\LR{#1}}
\def\v#1{\operatorname{vec}\LR{#1}}
\def\qiq{\quad\implies\quad}
\def\t{\otimes}
\def\p{\partial}
\def\grad#1#... |
3,745,097 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<p>If <span class="math-container">$F$</span> is a non-empty countable subset of <span class="math-container">$\mathbb R$</span>, prove that <span class="math-container">$F$</span> is not an open set, but that <span class="math-contain... | atul ganju | 788,520 | <p>A countably large set consisting of the points in this sequence <span class="math-container">$a_n = \frac{1}{2^n}$</span> is not closed. It does not contain the limit point <span class="math-container">$0$</span>.</p>
|
3,962,058 | <p>Let <span class="math-container">$A$</span> be a commutative ring with identity, and <span class="math-container">$P$</span> a prime ideal of <span class="math-container">$A$</span>. I want to show that <span class="math-container">$\lim_{f \notin P} A_{f} = A_{P}$</span>. <span class="math-container">$A_f$</span> i... | Karl Kroningfeld | 67,848 | <p>Take an element <span class="math-container">$\xi$</span> of the kernel of <span class="math-container">$\Phi$</span>, represented by <span class="math-container">$\frac a{f^n}$</span> for some <span class="math-container">$f\not\in P$</span>. We know that <span class="math-container">$$\frac a{f^n}=0\quad\text{in t... |
2,102,356 | <blockquote>
<p>Under the assumption that a pair of dice is fair, the probability is
0.88 that the number of 9's appearing in 288 throws of the dice will lie within 32 $\pm$ K. What is K?</p>
</blockquote>
<p>I have found the standard deviation using </p>
<p>$\sqrt{mean\:\cdot \:trials\:\cdot \:possibility\:of\... | M.Diggerson | 401,465 | <p><strong>Hint</strong>
$\Pr$(pair of fair dice show 9) = $4/36 = 1/9$</p>
<p>$(1/9 + 8/9)^{288}$ models the outcomes of 288 trials</p>
|
2,102,356 | <blockquote>
<p>Under the assumption that a pair of dice is fair, the probability is
0.88 that the number of 9's appearing in 288 throws of the dice will lie within 32 $\pm$ K. What is K?</p>
</blockquote>
<p>I have found the standard deviation using </p>
<p>$\sqrt{mean\:\cdot \:trials\:\cdot \:possibility\:of\... | lulu | 252,071 | <p>We note that this is a standard Binomial distribution with probability $p=\frac 19$. </p>
<p>If we like, we can approximate this with a normal distribution..with mean $\mu=32$ and $\sigma =\sqrt {288\times \frac 19\times 89}=\frac {16}3$ </p>
<p>We then seek $k$ such that $$\Phi(32+K,\mu,\sigma)-\Phi(32-K,\mu,\s... |
2,010,329 | <p>The function under consideration is:</p>
<p>$$y = \int_{x^2}^{\sin x}\cos(t^2)\mathrm d t$$</p>
<p>Question asks to find the derivative of the following function. I let $u=\sin(x)$ and then $\tfrac{\mathrm d u}{\mathrm d x}=\cos(x)$. Solved accordingly but only to get the answer as
$$ \cos(x)\cos(\sin^2(x))-\cos(... | Hugo | 41,494 | <p>Well, when i to solve this kind of problem i usually do the following.</p>
<p>1) I define a two variable function like:</p>
<p>$$F(u,v) = \int_{v}^{u} \cos(t^2)dt.$$</p>
<p>2) What we want to calculate is $$\frac{dy}{dx}(x)$$ where $y(x) = F(\sin(x),x^2)$. Then we have:</p>
<p>$$\frac{dy}{dx}(x) = \frac{d}{dx}\... |
215,061 | <p>Is it possible to do a 2D plot taking the point coordinates from separate lists of x and y values? These lists were produced by calculations within the same notebook. For a small number of points it is easy to enter x & y values by hand but it gets tedious as the number of points grows (I've done it.). If this i... | kglr | 125 | <p>Make each element of <code>replace</code> a function by wrapping it with <code>Function</code>. </p>
<pre><code>functions = Function /@ replace
</code></pre>
<blockquote>
<p>{(αu #1)/(ku + #1) &, Du #1 &, (αv #1)/(kv + #1) &}</p>
</blockquote>
<p>Use <code>functions</code> with <code>Through</code>... |
1,350,062 | <p>In almost all of the physics textbooks I have ever read, the author will write the oscillating function as</p>
<blockquote>
<p>$$x(t)=\cos\left(\omega t+\phi\right)$$</p>
</blockquote>
<p>My question is that, is there any practical or historical reason why we should prefer $\cos$ to $\sin$ here? One possible e... | Sergei M. | 264,641 | <p>Direct current (I = const, U = const, etc.) can be assumed as "alternating current with ω = 0 and ϕ = 0" only when using cosine, but not sine. This reason went from the electrical engineering.</p>
|
189,308 | <p>I have this problem: Find integration limits and compute the following integral.</p>
<p>$$\iiint_D(1-z)\,dx\,dy\,dz \\ D = \{\;(x, y, z) \in R^3\;\ |\;\; x^2 + y^2 + z^2 \le a^2, z\gt0\;\}$$</p>
<p>I can compute this as an indefinite integral but finding integration limits beats me. As indefinite integral the resu... | DonAntonio | 31,254 | <p>Another approach, with cylindrical coordinates:
$$x=r\cos\theta\,\,,\,\,y=r\sin\theta\,\,,\,\,z=z\Longrightarrow |J|=r\,\, (\,\text{Jacobian})$$
Since we want to integrate on the uper half sphere $\,x^2+y^2+z^2\leq a^2\,\,,\,\,z\geq 0$ , we get:
$$0\leq r\leq a\,\,,\,\,0\leq\theta\leq 2\pi\,\,,\,\,0\leq z\leq \sqrt ... |
832,206 | <p>I want to approximate the derivative of f(x)</p>
<h2>Finite difference</h2>
<p>$f'(x) \approx \frac{f(x+h)-f(x)}{h}$</p>
<p>I was taught that the error from the subtraction is blown up for small h. This I can verify with MATLAB.</p>
<h2>Smartass method</h2>
<p>So I though maybe the following would fix the probl... | Community | -1 | <p>This is proven in these 3 textbooks. 1. Saber Elaydi, <em>An Introduction to Difference Equations</em> (2005 3 ed). Pages 66, 126.</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/JWuLj.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JWuLj.jpg" alt="enter image description here" /></a></p>
<... |
827,072 | <p>How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.</p>
<p>I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is right.</p>
| AMANE otti | 155,953 | <p>if the problem has a solution (the three points are on a circle) then you only need to calculate the equations of the two mediators and the intersection should be the center and the distance between one of the points and this center gives you r.</p>
|
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| robjohn | 13,854 | <p>$$
\begin{align}
\lim_{k\to\infty}\frac{\sum_{n=1}^k2^{2\cdot3^n}}{2^{2\cdot3^k}}
&=\lim_{k\to\infty}\sum_{n=1}^k2^{-2\left(3^k-3^n\right)}\\
&=1+\lim_{k\to\infty}\sum_{n=1}^{k-1}2^{-2\left(3^k-3^n\right)}\\
&\le1+\lim_{k\to\infty}(k-1)2^{-2\left(3^k-3^{k-1}\right)}\\[6pt]
&=1+\lim_{k\to\infty}(k-1)2... |
3,407,474 | <p>I can understand that The set <span class="math-container">$M_2(2\mathbb{Z})$</span> of <span class="math-container">$2 \times 2$</span>
matrices with even integer entries is an infinite non commutative ring. But why It doesn't have any unity? I think <span class="math-container">$I(2\times2)$</span> is a unity for ... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$I(2\times2)=\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$</span>, then <span class="math-container">$I(2\times2)\notin M_2(2\mathbb Z)$</span> since <span class="math-container">$1$</span> is odd.</p>
|
1,975,199 | <blockquote>
<p>In this problem, construct a bijection to show the identity. Use the definitions of these quantities, not formulas for them.</p>
<p>For <span class="math-container">$0≤j≤n$</span>
<span class="math-container">$$\binom{n}{j}=\binom{n}{n-j}$$</span></p>
</blockquote>
<p>I know how to prove this by inducti... | Graham Kemp | 135,106 | <p>A bijection is a one-to-one correspondence between sets. If there is a bijection between two sets they have the same cardinal size (which, for finite sets that is basically: "count of elements").</p>
<p>In this case, $\binom n j$ is the count of ways to select $j$ elements from a set of size $n$.</p>
<p>Mea... |
2,649,756 | <p>I was given the equation</p>
<p>$x=y^4$</p>
<p>and asked to find the points where the curvature was the biggest and the smallest. I know the curvature equation:</p>
<p>$κ(x)= \frac{|y″|}{\left(1+\left(y′\right)^2\right)^{\frac{3}{2}}}$</p>
<p>and I know $f(x)=y=\sqrt[4]{x}$ </p>
<p>but I am not sure how to mini... | Lukas Heger | 348,926 | <p>First, some obvervations: For two subgroups $A, B$ of an abelian group, the set $A+B:= \{a+b \mid a \in A, b \in B\}$ is again a subgroup</p>
<p>We will prove by induction on $n$, that for any $n$ rational numbers $r_1, \dots, r_n$, the set $\{\sum_{i=1}^nn_ir_i|n_i\in\mathbb{Z}\}$ is a cyclic group.</p>
<p>The ca... |
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