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 |
|---|---|---|---|---|
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | Soham | 242,402 | <p>Try out this book.I am sure it will help!!</p>
<p><a href="http://www.amazon.in/Challenge-Thrill-Pre-College-Mathematics-B-J-Venkatchala/dp/8122409806" rel="nofollow">http://www.amazon.in/Challenge-Thrill-Pre-College-Mathematics-B-J-Venkatchala/dp/8122409806</a></p>
|
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | Crocogator | 710,663 | <p>You can start Higher Algebra by Hall and Knight.
Elementary Number Theory by David Burton. (Great for theory, might wanna do Justin Stevens)
Circles by Dimitri would be good for beginners.
Books by Titu Andreescu are really nice for some advanced prep.</p>
<p>These books are great for Olympiad level preparation.</p>... |
4,521,661 | <p>Calls arrive according to Poisson process with parameter <span class="math-container">$$</span>. Lengths of the calls are iid with cdf <span class="math-container">$F_x(x)$</span>. What is the probability distribution of the number of calls in progress at any given time?</p>
<p>I am confused, is the answer then just... | Daniel S. | 362,911 | <p>You have an m/g/infinity queue. The distribution of the number of customers in that queue is known to be Poisson</p>
<p>See for instance Kleinrock book</p>
|
2,113,777 | <p>I have the following IVP (Initial value problem, Cauchy-Problem), and I do not know how to solve this.</p>
<p>$$y'=e^{-x}-\frac{y}{x} \qquad \qquad y(1)=2$$</p>
<p>I hope you can help me, cause I really do not know how to start.</p>
<p>Thank you! :)</p>
| projectilemotion | 323,432 | <p>Start by writing:
$$\frac{dy}{dx}+\frac{y}{x}=e^{-x} \tag{1}$$
This is a first order linear ODE. Hence, we can either use an <a href="https://en.wikipedia.org/wiki/Integrating_factor" rel="nofollow noreferrer">integrating factor</a> or use <a href="https://en.wikipedia.org/wiki/Variation_of_parameters#First_order_eq... |
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | Steven Alexis Gregory | 75,410 | <p>$6-2√3 \approx2(1.732) = 6 - 3.464 = 2.536$
$3√2-2 \approx 3(1.414) - 2 = 4.242 - 2 = 2.242$</p>
<p>This implies that $\dfrac 52$ is between the two quantities:</p>
<p>\begin{align}
\dfrac{49}{4} &> 12 \\
\dfrac 72 &> 2\sqrt 3 \\
\dfrac{12}{2} &> 2\sqrt 3 + \dfrac 52 \\
6 - 2\sqrt ... |
583,030 | <p>I have to show that the following series convergences:</p>
<p>$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$</p>
<p>I have tried the following:</p>
<ul>
<li>The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.</li>
<li>I tried splitting up the series in ... | Jean-Claude Arbaut | 43,608 | <p>If you sum two successive terms (for indices $2n-1$ and $2n$), you get</p>
<p>$$\frac{3}{2n+1} - \frac{1}{2n} = \frac{6n-2n-1}{2n(2n+1)}= \frac{4n-1}{2n(2n+1)}$$</p>
<p>And its sum is not convergent, thus your series is not either.</p>
|
41,676 | <p>Really stuck on this one....</p>
<p>$\displaystyle f(x) = \frac{x - \sin{x}}{x^{2}}$ for $x \neq 0$ and $0$ when $x = 0$</p>
<p>Using the definition of the derivative, find $f'(0)$</p>
<p>I know the definition is $$ \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$</p>
<p>The way I did it was to say $$\lim_{h\to 0}\f... | Isaac | 72 | <p>You misapplied the definition of the derivative. If you want to find $f'(0)$, you cannot apply the formula for $f$ when $x\neq 0$ in $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, then substitute $x=0$; you first have to substitute $x=0$ to get $$\lim_{h \to 0} \frac{f(0+h)-f(0)}{h},$$ then apply the definition for ... |
41,676 | <p>Really stuck on this one....</p>
<p>$\displaystyle f(x) = \frac{x - \sin{x}}{x^{2}}$ for $x \neq 0$ and $0$ when $x = 0$</p>
<p>Using the definition of the derivative, find $f'(0)$</p>
<p>I know the definition is $$ \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$</p>
<p>The way I did it was to say $$\lim_{h\to 0}\f... | Community | -1 | <p>So here you have $$f'(0)= \lim_{h \to 0} \frac{f(0+h)-f(0)}{h} = \lim_{h \to 0} \biggl[\frac{h - \sin{h}}{h^{3}}\biggr]$$</p>
<p>Keep applying the L hospitals rule or else use expansion for the $\sin$ function which is given by $$\sin{h} = h -\frac{h^{3}}{3!} + \frac{h^{5}}{5!} - \cdots $$</p>
|
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | avm | 133,055 | <p>After spending many hours trying to find the real solutions of this equation, here are the ways of proving that it has no real solutions that I found:</p>
<p>For $\sin(\cos(x))=\cos(\sin(x))$ to be true, both $\cos(x)$ and $\sin(x)$ have to be equal to $\frac{\pi}{4}$ since $\cos(x)$ and $\sin(x)$ take same value i... |
1,203,179 | <p>The problem I have is this:</p>
<p>Use suitable linear approximation to find the approximate values for given functions
at the points indicated:</p>
<p>$f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$</p>
<p>I know how to do linear approximation with just one variable (take the derivative and such), but with two variabl... | Bernard | 202,857 | <p>Yes. Denoting the partial derivatives by $f'_x$ and $f'_y$, the formula is:
$$f(x_0+h,y_0+k)=f(x_0,y_0)+f'_x(x_0,y_0)h+f'_y(x_0,y_0)k+o\bigl(\lVert(h,k)\rVert\bigr).$$</p>
|
3,038,965 | <p>Here's the question I'm puzzling over:</p>
<p><span class="math-container">$\textbf{Find the perpendicular distance of the point } (p, q, r) \textbf{ from the plane } \\ax + by + cz = d.$</span></p>
<p>I tried bringing in the idea of a dot product and attempted to get going with solving the problem, but I'm headin... | Kemono Chen | 521,015 | <p><span class="math-container">$$\frac{d}{dx}\frac1{1-x}=\frac1{(1-x)^2}.$$</span></p>
|
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | user1066 | 106 | <p>As pointed out in the <a href="https://math.stackexchange.com/a/861534">linked</a> answer at <em>Mathematica Stack Exchange</em>, this method of calculating square roots is described in <strong>Square Roots From Anywhere</strong> by Terry A. Goodman and John Bernard, <em>Mathematics Teacher</em>, Vol 73, May 1979, p... |
3,561,664 | <p>I did part of this question but am stuck and don't know how to continue</p>
<p>I let <span class="math-container">$x= 2k +1$</span></p>
<p>Also noticed that <span class="math-container">$x^3+x = x(x^2+1)$</span></p>
<p>therefore
<span class="math-container">$4m+2 = 2k+1((2k+1)^2+1)$</span></p>
<p>I simplified th... | Robert Lewis | 67,071 | <p><span class="math-container">$x = 2k + 1; \tag 1$</span></p>
<p><span class="math-container">$x ^3 = 8k^3 + 12k^2 + 6k + 1; \tag 2$</span></p>
<p><span class="math-container">$x ^3 + x = 8k^3 + 12k^2 + 6k + 1 + (2k + 1)$</span>
<span class="math-container">$ = 8k^3 + 12k^2 + 8k + 2 = 4(2k^3 + 3k^2 + 2k) + 2, \t... |
3,561,664 | <p>I did part of this question but am stuck and don't know how to continue</p>
<p>I let <span class="math-container">$x= 2k +1$</span></p>
<p>Also noticed that <span class="math-container">$x^3+x = x(x^2+1)$</span></p>
<p>therefore
<span class="math-container">$4m+2 = 2k+1((2k+1)^2+1)$</span></p>
<p>I simplified th... | J. W. Tanner | 615,567 | <p>If <span class="math-container">$x$</span> is odd, then <span class="math-container">$x\equiv1$</span> or <span class="math-container">$3\pmod4$</span>; in the former case <span class="math-container">$x^3+x\equiv2\pmod4$</span>, and in the latter case <span class="math-container">$x^3+x\equiv30\equiv2\pmod4.$</span... |
246,071 | <p>How do I solve the following equation?</p>
<p>$$x^2 + 10 = 15$$</p>
<p>Here's how I think this should be solved.
\begin{align*}
x^2 + 10 - 10 & = 15 - 10 \\
x^2 & = 15 - 10 \\
x^2 & = 5 \\
x & = \sqrt{5}
\end{align*}
I was thinking that the square root of 5 is iregular repeating 2.23606797749979 nu... | beauby | 49,048 | <p>When you say "What we want to prove is ...", you're actually wrong, you have to "update" the $n$ to $n+1$ on every fraction, not only the one you add.</p>
|
1,386,683 | <p>I posted early but got a very tough response.</p>
<p>Point $A = 2 + 0i$ and point $B = 2 + i2\sqrt{3}$ find the point $C$ $60$ degrees ($\pm$) such that Triangle $ABC$ is equilateral. </p>
<p>Okay, so I'll begin by converting into polar form:</p>
<p>$A = 2e^{2\pi i}$ and $B = 4e^{\frac{\pi}{3}i}$</p>
<p>$\overli... | Community | -1 | <p>Let $z=re^{i\theta}$. Then this equals:</p>
<p>$$\lim_{r\to 0} \frac{(re^{-i\theta})^2}{re^{i\theta}}=\lim_{r\to 0} \frac{r^2e^{-2i\theta}}{re^{i\theta}}=\lim_{r\to 0} r e^{-3i\theta}=0$$</p>
|
4,495,044 | <p>Edit: There is an answer at the bottom by me explaining what is going on in this post.</p>
<p>Define a function <span class="math-container">$f : R \to R$</span> by <span class="math-container">$f(x) = 1$</span> if <span class="math-container">$x = 0$</span> and <span class="math-container">$f(x) = 0$</span> if <spa... | Seeker | 1,050,393 | <p>Tao defines limits using adherent points and not limit points. My previous comments regarding the use of adherent points and limit points throughout this post are completely wrong as I didn’t understand it too well myself at the time either.</p>
<p>But now that I do, here is what is going on in this post. Tao mentio... |
2,879,883 | <p>Suppose that $f$ and $g$ are differentiable functions on $(a,b)$ and suppose that $g'(x)=f'(x)$ for all $x \in (a,b)$. Prove that there is some $c \in \mathbb{R}$ such that $g(x) = f(x)+c$.</p>
<p>So far, I started with this:</p>
<p>Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in \mathbb{R}$ such that... | Lev Bahn | 523,306 | <p>Fixing your proof a little bit, I can extend your proof.</p>
<p>Suppose that your domain of function is (a,b). And let's choose any $\beta\in (a,b)$.</p>
<p>So for any $x\in (a,b)$ $\exists \alpha\in (\beta,x)$ (or $\alpha \in(x,\beta)$ if $x<\beta$) such that $h'(\alpha)=\frac{h(x)-h(\beta)}{x-\beta}=0$.</p>
... |
2,879,883 | <p>Suppose that $f$ and $g$ are differentiable functions on $(a,b)$ and suppose that $g'(x)=f'(x)$ for all $x \in (a,b)$. Prove that there is some $c \in \mathbb{R}$ such that $g(x) = f(x)+c$.</p>
<p>So far, I started with this:</p>
<p>Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in \mathbb{R}$ such that... | mfl | 148,513 | <p>You say that </p>
<blockquote>
<p>Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in
> \mathbb{R}$ such that $h'(c) = \frac{h(b)-h(a)}{b-a} =0$. Then
$h'(c)=0 \implies h(c)=c.$</p>
</blockquote>
<p>This is not correct. If $h'(c)=0$ then you have $h(b)=h(a).$ </p>
<p>But you are in the correct way... |
2,601,088 | <p>I'm new to the group theory and want to get familar with the theorems in it, so I choose a number $52$
to try making some obseveration on all group that has this rank. Below are my thoughts. I don't know if there is any better way to think of these (i.e., an experienced group theorist would think), and I still have... | Mark Bennet | 2,906 | <p>You can take a direct product of cyclic groups of order $26$ and $2$ - this is abelian, but has no element of order $52$ so can't be cyclic.</p>
<p>Look out on your travels for a structure theorem on abelian groups which tells you how to find all the different ones. If the prime factorisation of the order of $G$ i... |
2,371,108 | <p>Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very... | Soha Farhin Pine | 331,967 | <h2>One-of-a-kind cubic equations</h2>
<blockquote>
<ol>
<li><blockquote>
<p>$x^3-6x^2+12x-8=0 $</p>
</blockquote></li>
<li><blockquote>
<p>$\implies x^3-2^3-6x^2+12x=0$</p>
</blockquote></li>
<li><blockquote>
<p>$a^3-b^3=(a-b)[(a-b)^2+3ab]\\\therefore x^3-2^3=(x-2)[(x-2)^2+3\times x\times2]$</... |
22,839 | <p>Is it possible to have the text generated by <code>PlotLabel</code> (or any other function) aligned to the left side of the plot instead of in the center?</p>
| kglr | 125 | <pre><code> Labeled[Plot[Sinc[x], {x, 0, 9}], Style["plot label", "Section"], {{Top, Left}}]
</code></pre>
<p><img src="https://i.stack.imgur.com/3sCtU.png" alt="enter image description here"></p>
<p>Also</p>
<pre><code>Panel[Plot[Sinc[x], {x, 0, 9}], Style["plot label", "Section"], {{Top, Left}},
Appearance ->... |
355,740 | <p>Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = 2^x \cdot \ln 2$</p>
<p>$$\begin{array}{c | c | c | c}
x & f(x) & f'(x) & \frac{f'(x)}{f(x)}\\ \hline
-1... | Hui Yu | 19,811 | <p>Other answers have already clarified the definition of the characteristic of a ring. But I may add one definition that is completely <em>element-free</em>, so hopefully this would make things clearer that characteristics do not depend on the <em>number</em>.</p>
<p>$\mathbb{Z}$ is initial in the category $\operator... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Akhil Mathew | 344 | <p>Many books on PDE or functional analysis (e.g. Taylor's) will have a detailed coverage of distributions.</p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Tom Leinster | 586 | <p>Friedlander and Joshi's <em>Introduction to the Theory of Distributions</em> is short, elegant and efficient.</p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Anthony Pulido | 8,479 | <p>I agree with Johannes's comment, but despite this, one book that might fit your criteria is <em>Theory of distributions</em> by M.A. Al-Gwaiz. I haven't looked at it for some months, but it made the following standard texts more accessible:</p>
<ul>
<li>Friedlander and M. Joshi's <em>Introduction to the Theory of D... |
1,048,668 | <p>Let $f\colon (a,b) \to \mathbb{R}$ a non constant differentiable function. </p>
<p>Is the following statement true:</p>
<p>If $f$ has a local maximum <em>and</em> a local minimum then $f$ also does have an inflection point.</p>
<p>If so, how to prove it, if not, what would be a counterexample?</p>
<p><em>Remark<... | hmakholm left over Monica | 14,366 | <p>Consider this:</p>
<p>$$ f(x) = \begin{cases} (x+1)(x+3)-2 & \text{when }x\le -1 \\
2x & \text{when } -1 < x < 1 \\
2-(x-1)(x-3) & \text{when } x \ge 1 \end{cases} $$</p>
<p>(Graphed by <a href="http://www.wolframalpha.com/input/?i=piecewise+[%7b%7b+%28x%2B1%29%28x%2B3%29-2%2C+x%3C-1%7d%2C+%7b2x%... |
662,403 | <p>I'm working on a homework assignment concerning convex optimization and I came across a problem involving the convexity of the function and the convexity of the domain of the function.</p>
<p>Consider the function $f : [0,1]^3 \in R$ with the following form
$$
f(x,y,z) = xlnx + ylnz + zlnz + \alpha ( x + y + z - 1)... | Michael Grant | 52,878 | <p>As requested, my comments promoted to an answer, with a little extra commentary.</p>
<p>Here's a basic definition of convexity: a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is convex if its domain $\mathop{\textbf{dom}} f$ is a convex set and if
$$f(\alpha x+(1-\alpha)y)\leq \alpha f(x)+(1-\alpha) f(y) \quad \f... |
618,339 | <p>Could somebody please check my solution?</p>
<p>I want to check, whether $\sum\limits_{n=1}^{\infty}\frac{(1+\frac{1}{n})^n}{n^2}$ converges or diverges.</p>
<p>Using the Comparison test:</p>
<p>Let $a_n = \frac{(1+\frac{1}{n})^n}{n^2},~ b_n=\frac{1}{n^2}$</p>
<p>Since $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$ c... | ncmathsadist | 4,154 | <p>Using the limit comparison test,
$$ {1\over n^2}\left(1 + {1\over n}\right)^n \sim {e\over n^2},$$
your series converges.</p>
|
214,486 | <p><a href="https://i.stack.imgur.com/rZXpG.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/rZXpG.gif" alt="enter image description here"></a></p>
<p>I made it by another software, and met some problems to change it into MMA code.</p>
<pre><code>f[x_] := Graphics[
Line[AnglePath[{90 °, -90 °}[[
1 + Nest[... | Vitaliy Kaurov | 13 | <p><a href="https://i.stack.imgur.com/w2kuu.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/w2kuu.gif" alt="enter image description here"></a></p>
<p>A simple way to make Dragon Curve is using <code>AnglePath</code>. Define a function that generates points for the Dragon curve:</p>
<pre><code>dragonPTS[k_]:... |
3,807,550 | <p>I am stuck in a true/false question. It is</p>
<p>In a finite commutative ring, every prime ideal is maximal.</p>
<p>The answer says it's false.</p>
<p>Well what I can say is (Supposing the answer is right)</p>
<p><span class="math-container">$(1)$</span> The ring can't be Integral domain since finite integral domai... | rschwieb | 29,335 | <p>There is no counterexample, because even if the ring has no identity, the quotients by primes must have identity.</p>
<p><a href="https://math.stackexchange.com/a/2882463/29335">Every nonzero finite ring without zero divisors has a multiplicative identity</a>, so the quotient would in fact be a finite domain with id... |
422,799 | <p>Maschke's theorem says that every <em>finite-dimensional</em> representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group which is not completely reducible?</p>
<p>EDIT: As mentioned in the answers, there is actually no finite-dime... | Anonymus | 258,276 | <p>Take $Z_2\times Z_2$, with generators A and B. Consider the field K(t), where K is the field of two elements. Let $\phi(A)=\left[ {\begin{array}{cc}
1 & 1 \\ 0 & 1 \ \end{array} } \right]$ and $\phi(B)=\left[ {\begin{array}{cc}
1 & t\\ 0 & 1 \ \end{array} } \right]$, where $... |
4,101,974 | <p>I'm trying to understand what does this matrix operator norm means and what it does to matrix A. <span class="math-container">$${{\left\| A \right\|}_{1,\,\infty }}:={{\max }_{{{\left\| x \right\|}_{\infty }}=1}}{{\left\| Ax \right\|}_{1}}$$</span> Can somebody help with the explanation and maybe an example?</p>
| Paul Frost | 349,785 | <p>Munkres defines a word as finite sequence <span class="math-container">$s = (x_1,\dots,x_n)$</span> of elements <span class="math-container">$x_i \in G_{\alpha_i}$</span> and says that such a sequence represents <span class="math-container">$x \in G$</span> if <span class="math-container">$x = \prod_{i=1}^n x_i = x... |
4,528,629 | <p>When doing an exercise about linear representations of finite groups I stumbled upon this Isomorphism in the comments of another <a href="https://math.stackexchange.com/questions/308680/basic-identity-of-characters?rq=1">post</a> which I was not aware of.</p>
<p>In this context <span class="math-container">$V$</span... | Nicolas | 1,015,842 | <p>This follows immediately from the definition (performing the matrix multiplication and employing linearity of the expectation). Let <span class="math-container">$\mathbf{X} \in \mathbb{R}^{m, n} $</span> and
<span class="math-container">$\mathbf{A}\in \mathbb{R}^{d, m}$</span>
<span class="math-container">$$
\begin{... |
1,392,209 | <blockquote>
<p>Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$</p>
</blockquote>
<p>My attempt </p>
<p>So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$</p>
<p>$$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$
$$=\frac{x^2}{\sin^2 x}\cdot\frac{\sin x+x\cos x}{x}\cdot\fr... | Nathan Smith | 209,892 | <p>I think this limit would be considerably easier using Taylor Series instead of LHR.</p>
<p>$ \frac{\sin^2(x)-x^2\cos^2(x)}{x^2\sin^2(x)} \approx \frac{(x-\frac{x^3}{6})^2-x^2(1-\frac{x^2}{2})^2}{x^2(x)^2}=\frac{\frac{2x^4}{3}+O(x^5)}{x^4} \rightarrow \frac{2}{3} $</p>
|
2,706,872 | <p>I'm working on my latest linear algebra assignment and one question is as follows: </p>
<p>In $\mathbb R^3$ let <em>R</em> be the reflection over the null space of the matrix </p>
<p><em>A</em> = [4 4 5]</p>
<p>Find the matrix which represents <em>R</em> using standard coordinates. </p>
<p>I am familiar with the... | trancelocation | 467,003 | <p>First of all, the formula should be $$P = B(B^TB)^{-1}B^T$$ where the columns of $B$ form of a basis of $ker(A)$. </p>
<p>Think geometrically when solving it. Points are to be reflected in a plane which is the kernel of $A$ (see third item):</p>
<ul>
<li>find a basis $v_1, v_2$ in $ker(A)$ and set up $B = (v_1 \, ... |
1,663,113 | <p>I'm having a mind wrenching question that I just cannot answer. It's been a while since I was at the school bench so I wonder if anyone can help me out? :)</p>
<p>We have 10 students with 5 cakes each to be shared amongst each other.
The students can give the cakes out, but they can’t give a piece to a person who g... | Mike Earnest | 177,399 | <p>Place the kids in a circle, numbered 1 to 10. Have the odd numbered kids pass four cakes, one to each of the four closest students on their right (so 1 passes to 2,3,4,5). Have the even numbered students pass five cakes to the five students on their left (so 2 passes to 3,4,5,6,7).</p>
<p>The odd students receive 5... |
3,143,670 | <p>I'm not entire sure how to proceed on this question. I believe I am supposed to use a triangle inequality with epsilons and <span class="math-container">$m$</span>, <span class="math-container">$n \geq N$</span> to get <span class="math-container">$N_1$</span> and <span class="math-container">$N_2$</span> before set... | String | 94,971 | <p>You must show that for <span class="math-container">$m,n\geq N$</span> we have:
<span class="math-container">$$
||x_m-y_m|-|x_n-y_n||\leq\varepsilon
$$</span>
from the triangle inequality applied twice in two different versions we know that:
<span class="math-container">$$
\begin{align}
||x_m-y_m|-|x_n-y_n||
&\l... |
947,254 | <p>The problem is part (b):</p>
<p><b>1.4.7.</b> A pair of dice is cast until either the sum of seven or eigh appears.</p>
<p> <b>(a)</b> Show that the probability of a seven before an eight is 6/11.</p>
<p> <b>(b)</b> Next, this pair of dice is cast until a seven appears twice or until each of a six and e... | Yuval Filmus | 1,277 | <p>We can think of the experiment as follows. At the start, we have a biased three-sided coin that outputs $6,7,8$ with probabilities $5/16,3/8,5/16$; we don't care about the other outcomes, so we can just ignore them. After we see $6$, we don't care about $6$, so the probabilities of $7,8$ are $6/11,5/11$.</p>
<p>Her... |
3,689,513 | <p>I've been stuck on one of my homework numbers.
The number precise that the following equation is a non-linear equation of order 1 with x>0.</p>
<p><span class="math-container">$$y' + {{y\ln(y)}\over x}= xy$$</span></p>
<p>So far, I tried 2 different methods to solve them. As suggested by internet (link below): Be... | Jan Eerland | 226,665 | <p>Well, we have the following first-order nonlinear ordinary differential equation:</p>
<p><span class="math-container">$$\text{y}'\left(x\right)+\frac{\text{y}\left(x\right)\ln\left(\text{y}\left(x\right)\right)}{x}=x\text{y}\left(x\right)\tag1$$</span></p>
<p>This can be rewritten in the following form:</p>
<p><s... |
4,086,485 | <blockquote>
<p>We can regard <span class="math-container">$\pi_1(X,x_0)$</span> as the set of basepoint-preserving homotopy classes of maps <span class="math-container">$(S^1,s_0)\rightarrow(X,x_0$</span>). Let <span class="math-container">$[S^1,X]$</span> be the set of homotopy classes of maps <span class="math-conta... | Paul Frost | 349,785 | <p>Loops based at <span class="math-container">$x_0$</span> <em>can be (freely) homotopic</em> to the pink loop. As an example consider the loop starting at <span class="math-container">$x_0$</span> which travels along the upper half of the left circle until it reaches the intersection of both circles, then travels onc... |
3,417,001 | <blockquote>
<p>I am wondering if the ring of polynomials <span class="math-container">$F[x]$</span> with coefficients in the field <span class="math-container">$F$</span> is ever isomorphic to <span class="math-container">$\mathbb{Z}$</span> for some field <span class="math-container">$F$</span>. </p>
</blockquote>
... | lhf | 589 | <p>No such fields exist.</p>
<p>If <span class="math-container">$\phi: F[x] \to \mathbb Z$</span> is a ring homomorphism, then so is its restriction to <span class="math-container">$F$</span>. This restriction must be an injection. Therefore, <span class="math-container">$\phi(F)$</span> is a subfield of <span class="... |
2,544,261 | <p>The question:</p>
<blockquote>
<p>Find values of $x$ such that $2^x+3^x-4^x+6^x-9^x=1$, $\forall x \in \mathbb R$.</p>
</blockquote>
<p>Notice the numbers $4$, $6$ and $9$ can be expressed as powers of $2$ and/or $3$. Hence let $a = 2^x$ and $b=3^x$.</p>
<p>\begin{align}
1 & = 2^x+3^x-4^x+6^x-9^x \\
& =... | Servaes | 30,382 | <p>A sum of squares equals zero if and only if each of the squares equals zero. So you get $a=b=1$.</p>
|
3,191,233 | <p>Why is <span class="math-container">$ce^λ=1$</span> equal to <span class="math-container">$c=e^{-λ}$</span>?</p>
| Michael Stachowsky | 337,044 | <p>You are looking for a solution to your differential equation between those two times. In general you are going to have to recompute things every time step. From the theory of ODEs, your state equation is solved thus:</p>
<p><span class="math-container">$$\vec{x}((k+1)T) = \vec{x}[k+1] = e^{AT}\vec{x}(kT) + \int_{... |
2,113,413 | <p>A man on the top of the tower, standing in the seashore finds that a boat coming towards him makes 10 minutes to change the angle of depression from $30$ to $60$. How soon will the boat reach the seashore.
<img src="https://i.stack.imgur.com/iHWdl.jpg" alt="enter image description here"></p>
<p>My Attempt,
We kno... | Vishnu V.S | 397,349 | <p>Let's consider $AB$ to be $h$. Then,
$$
BD=\frac{h}{\tan30^\circ}=h\sqrt{3} \\
BC=\frac{h}{\tan60^\circ}=\frac{h}{\sqrt{3}}\\
CD=BD-BC=\frac{2h}{\sqrt{3}}
$$
If the speed of the boat is $x$ metres per second,
$$
CD=\frac{2h}{\sqrt{3}}=600x
\implies \frac{h}{x\sqrt{3}}=300
$$
Time in which the boat travels distanc... |
2,797,717 | <p>Prove that exist function $\varphi :\left( {0,\varepsilon } \right) \to \mathbb{R}$ such that
$$\mathop {\lim }\limits_{x \to {0^ + }} \varphi \left( x \right) = 0,\mathop {\lim }\limits_{x \to {0^ + }} \varphi \left( x \right)\ln x = - \infty .$$
I think $\varphi \left( x \right) = \frac{1}{{\ln \left( {\ln \left(... | Kavi Rama Murthy | 142,385 | <p>Let $f$ be such that $\lim_{x\to0+} f(x)=-\infty$ Let $\phi (x)=\frac 1 {\sqrt |f(x)|}$. Then $\phi (x) \to 0$ and $\phi (x) f(x) \to -\infty$ (because $|\phi (x) f(x)|=\sqrt {|f(x)|}$ and $\phi (x) f(x)$ has the same sign as $f(x))$. This answers the second part and the first part is a special case when $f(x)=\ln (... |
3,552,555 | <p>Let <span class="math-container">$S$</span> be the set of all column matrices
<span class="math-container">$
\begin{bmatrix}
b_1 \\
b_2 \\
b_3
\end{bmatrix}
$</span>
such that <span class="math-container">$b_1,b_2,b_3 \in \mathbb{R}$</span> and the system of equations (in real variables)
<span class... | nonuser | 463,553 | <p>So <span class="math-container">$$\frac a{b+\sqrt c}+\frac d{\sqrt c} = q,\;\;\;\;\; q\in \mathbb Q$$</span></p>
<p>Let <span class="math-container">$x=\sqrt{c}\notin \mathbb Q$</span>, then <span class="math-container">$$\frac a{b+x}+\frac d{x} = q\;\;\;\;/\cdot x(b+x)$$</span></p>
<p><span class="math-containe... |
3,800,521 | <p>Let <span class="math-container">$x=\tan y$</span>, then
<span class="math-container">$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$</span></p>
<p>Given answer is <span class="math-container">$0$</span></p>
<p>What’s wrong here?</p>
| lab bhattacharjee | 33,337 | <p>From <a href="https://math.stackexchange.com/questions/523625/showing-arctan-frac23-frac12-arctan-frac125/523626#523626">showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$</a>
<span class="math-container">$$2\arctan x=\begin{cases}\arctan\dfrac{2x}{1-x^2}\text{ if } x^2\le1\\\pi+\arctan\dfrac{2x}{1-x... |
345,310 | <p>This is computed based on the following recursive formula <span class="math-container">$$w_n=\frac{\lambda_nw_{n+1}+\mu_nw_{n-1}+1}{\lambda_n+\mu_n}$$</span> where: <span class="math-container">$n$</span> is the inital state, State <span class="math-container">$0$</span> is absorbing, <span class="math-container">$\... | Iosif Pinelis | 36,721 | <p>Clearly, the expected time <span class="math-container">$w_n$</span> till extinction from the initial state <span class="math-container">$n$</span> is nondecreasing in <span class="math-container">$n$</span>. So, if <span class="math-container">$w_1=\infty$</span>, then <span class="math-container">$w_n=\infty$</spa... |
2,138,009 | <p>Let $f(z)=(1+i)z+1$. Then $f(z)=\sqrt 2 e^{i\pi/4}z+1$ and thus $f=t\circ h\circ r$ where $t$ is the translation of vector $1$, $r$ the rotation of center $0$ and angle $\pi/4$ and $h$ the homothetic of parameter $\sqrt 2$. I found a fix point $z=\frac{-1}{2}+\frac{i}{2}$.</p>
<p>1) What if $f\circ f\circ f\circ f\... | lhf | 589 | <p><em>Hint:</em> If $f(z)=az+b$, with $a\ne1$, then $f^n(z)=a^n z + b\dfrac{a^n-1}{a-1}$.</p>
|
329,600 | <p>Let $U,V,W$ be vector spaces over $F$ and $S: U \to V$, and $T: V \to W$ linear maps.</p>
<p>(a) Show that if $S$ and $T$ are isomorphisms, then $T\circ S$ is an isomorphism, too.</p>
<p>(b) Show that if $U$ is isomorphic to $V$ and $V$ is isomorphic to $W$, then $U$ is isomorphic to $W$.</p>
| xavierm02 | 10,385 | <p>$a)$</p>
<p>$S\in GL(U,V), T\in GL(V,W)$</p>
<p>Since $GL(U,V) \subset L(U,V)$ and $GL(V,W) \subset L(V,W)$, $S\in L(U,V) $ and $t\in L(V,W)$ from which you can deduce $T\circ S \in L(U,W)$.</p>
<p>Then you must know that given two vector spaces $A$ and $B$, $\forall f \in L(A,B), [f\in GL(A,B) \Leftrightarrow f ... |
1,178,361 | <p>The surface with equation $z = x^{3} + xy^{2} $ intersects the plane with equation $2x-2y = 1$ in a curve. What is the slope of that curve at $x=1$ and $ y = \frac{1}{2} $</p>
<p>So I put $ x^{3} + xy^{2} = 2x - 2y - 1 $</p>
<p>We have $ x^{3} + xy^{2} - 2x + 2y + 1 $</p>
<p>Do I then differentiate wrt x and y si... | kobe | 190,421 | <p>The statement is true, because under the given assumptions, $f(f^{-1}(A)) = A$ for all $A \subseteq Y$. Indeed, let $A \subseteq Y$. If $y \in A$, then $y = f(x)$ for some $x \in X$ (since $f$ is onto), which implies $x \in f^{-1}(A)$ and hence $y \in f(f^{-1}(A))$. Conversely, if $y \in f(f^{-1}(A))$, then $y = f(z... |
1,282,489 | <p>I have a simple problem that I need to solve. Given a height (in blue), and an angle (eg: 60-degrees), I need to determine the length of the line in red, based on where the green line ends. The green line comes from the top of the blue line and is always 90-degrees.</p>
<p>The height of the blue line is variable.... | Rob | 241,939 | <p>Since you're dealing with right triangle, you can just use cosine function:</p>
<p><span class="math-container">$$\cos(\theta)=\frac{blue}{red}$$</span></p>
<p>Or, substituting <span class="math-container">$60^o$</span> for <span class="math-container">$\theta$</span> and 10 for <em>blue</em>:</p>
<p><span class="ma... |
427,835 | <p>Which website/journal/magazine would you recommend to keep up with advances in applied mathematics?
More specifically my interest are:</p>
<ul>
<li>multivariate/spatial interpolation</li>
<li>numerical methods</li>
<li>computational geometry</li>
<li>geostatistics</li>
<li>etc</li>
</ul>
<p>I am looking for a fair... | user35959 | 35,959 | <p>For multivariate/spatial interpolation (I'm interested in RBFs and meshfree methods), I see things published in SIAM Journal of Numerical Analysis, Mathematics of Computation (Math. Comp), Foundations of Computational Mathematics (FoCM), Constructive Approximation, and Journal of Approximation Theory.</p>
|
172,131 | <p>Given <span class="math-container">$P$</span>, a polynomial of degree <span class="math-container">$n$</span>, such that <span class="math-container">$P(x) = r^x$</span> for <span class="math-container">$x = 0,1, \ldots, n$</span> and some real number <span class="math-container">$r$</span>, I need to calculate <spa... | Michael Hardy | 11,667 | <p>Via <a href="http://en.wikipedia.org/wiki/Lagrange_polynomial" rel="nofollow">Lagrange polynomials</a> you can fit a finite number of points exactly, and then plug $n+1$ into the polynomial you get.</p>
<p>I'm not sure that fully answers the question, even though it gives you a way to find the right number in every... |
2,286,540 | <p>$(x,y)=(x,\sqrt{x})$
$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$
$=\sqrt{x^4-5x^2+9}$</p>
<p>$g(x)=x^4-5x^2+9$ , $g'(x)=4x^3-10x=0$ $: x=0, x=+-\sqrt{10}/2$</p>
<p>The question is: Should I put those x-values in g(x) or the orginal graph,$
y=\sqrt{x}$. To me, it's logical to put it into g(x), but in an example it was re... | imranfat | 64,546 | <p>An alternative way is the use of derivative. At any point $(a,\sqrt{a})$, the value of the derivative is $\frac{1}{2\sqrt{a}}$ and so its perpendicular value at that point would be $-2\sqrt{a}$ The perpendicular line follows $y-\sqrt{a}=-2\sqrt{a}(x-a)$ and we want this point to pass through $(3,0)$ so substitute th... |
4,098,682 | <p>I am trying to prove this following theorem about multiplying left cosets.</p>
<blockquote>
<p>Let <span class="math-container">$H \subset G$</span> a subgroup and <span class="math-container">$G/H$</span> the set of left cosets of <span class="math-container">$H$</span> in <span class="math-container">$G$</span>. W... | Mark Saving | 798,694 | <p>This isn't what they're looking for. Recall that <span class="math-container">$S = \{aH | a \in G\}$</span> is the set of left cosets, where <span class="math-container">$aH := \{ab | b \in H\}$</span>. We are trying to show that the multiplication operator <span class="math-container">$\cdot : S^2 \to S$</span> def... |
783,502 | <p>Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, it's quite imprecise.
$$f(x)={|x+1|\over x}$$</p>
<p>Thank you!!!</p>
| Caleb Stanford | 68,107 | <p>Here is another way.</p>
<ul>
<li><p>First, draw the graph of $\frac{x + 1}{x}$.
This should be pretty simple since $\frac{x + 1}{x} = 1 + \frac{1}{x}$, and you know what $\frac{1}{x}$ looks like. You also know that $\frac{x+1}{x}$ has a zero at $x = -1$.</p></li>
<li><p>Second, since $|x + 1|$ switches the sign f... |
1,162,147 | <p>The definition of open set is different in metric space and topological space, though metric space is a special case of topological space. The definition in metric space seems to convey the idea that all the points isolated from outside from outside, while the definition in topological space is intended to separate ... | Ittay Weiss | 30,953 | <p>A metric space has a metric function which can be used to define the notion of open set. Thus, in a metric space the notion of open space is derived from the metric. In a topological space the open sets are not derived from anything, they are given axiomatically. The axioms for a topology are chosen to capture some ... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | cgiovanardi | 230,653 | <p>If this is a riddle, I would do : 13,1 + 7,9 + 9</p>
|
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | Darshan Jain | 620,036 | <p>It's
<span class="math-container">$$3_3+3_3+3_3=30.$$</span>
Where <span class="math-container">$3_3$</span> is read as 3 base 3.</p>
|
1,555,697 | <p>So, I'm working out one of my assignments and I'm a little bit stuck on this problem:</p>
<blockquote>
<p>A fish store is having a sale on guppies, tiger barbs, neons,
swordtails, angelfish, and siamese fighting fish (6 kinds). How many
ways are there to choose 24 fish with at least 1 guppy, at least 2
tige... | mathochist | 215,292 | <p>It seems like you could just sum up all the cases where you have $0$, $1$, $2$, and $3$ siamese fighting fish separately. If there are no siamese fighting fish, then we can choose $15$ out of $4$ types, if there is exactly one siamese fighting fish, then we can choose $14$ fish out of $4$ types, if there are exactly... |
106,131 | <blockquote>
<p>Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by<br>
$\
f(x) =
\begin{cases}
1/q
& \text{if } x =p/q \space(\mathrm{lowest}\space \mathrm{terms},\space\mathrm{nonzero})\\
0 & \text{if } x = 0\space\mathrm{or}\space x\not\in\mathbb{Q}
\end{cases}
$<br>
Show that f is continuo... | DonAntonio | 31,254 | <p>I think the following can help you a lot:</p>
<p><strong>Proposition:</strong> Let $\,\displaystyle{\left\{\frac{p_n}{q_n}\right\}}\subset\mathbb Q\,$ be a rational sequence, with $\,(p_n,q_n)=1\,\,\text{and}\,\,q_n>0\,\,,\,\forall n\,$ , and s.t.
$$\frac{p_n}{q_n}\xrightarrow [n\to\infty]{}x\in\mathbb R-\mathb... |
122,274 | <p>I have a question, I think it concerns with field theory.</p>
<blockquote>
<p>Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?</p>
</blockquote>
<p>Thanks in advance. It bothers me for several days. </p>
| Thomas Andrews | 7,933 | <p>We will use fairly liberally the result that if $q(x)\in\mathbb F_p[x]$ is irreducible, then, for any $k$, $q(x)\mid x^{p^k}-x$ if and only if $\deg q\mid k$.</p>
<p>If $q_n(x)=x^{p^n}-x+1$ is irreducible, then there is a automorphism, $\phi$ of the field $\mathbb F_p[x]/\left<q_n(x)\right>$ which sends $\bar... |
3,396,882 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a non-negative random variable, and suppose that <span class="math-container">$P(X \geq
n) \geq 1/n$</span> for each <span class="math-container">$n \in \mathbb{N}$</span>. Prove that <span class="math-container">$E(X) = \infty$</span>.</p>
</blockquote>... | Henry | 6,460 | <p>Suppose we have a discrete random variable <span class="math-container">$Y$</span> with <span class="math-container">$\mathbb P(Y=n) = \frac{1}{n(n+1)}$</span> for all positive integers <span class="math-container">$n$</span> </p>
<p>then <span class="math-container">$\mathbb P(Y\ge n) = \frac1n \le \mathbb P(X \g... |
1,168,446 | <p>I have the following nonlinear differential equation (I am using $y$ as shorthand $f(x)$):</p>
<p>$$\sin(y - y') = y''$$</p>
<p>I have tried the following</p>
<p>$$\cos(y - y')(y'-y'') = y'''$$
$$-\sin(y - y')(y'-y'')^2 + \cos(y - y')(y''-y''') = y''''$$
$$-y''(y'-y'')^2 + \dfrac{y'''}{y'-y''}(y''-y''') = y''''$$... | Robert Israel | 8,508 | <p>There's not much hope of closed-form solutions. You could use numerical methods or series.</p>
|
251,705 | <p>I would like to find the residue of $$f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent expansion of the function.</p>
<p>I managed to do it using the first way, and the answer is $-3/(4e)$. However, I'm ou... | robjohn | 13,854 | <p>Substitute $z=w+i$ to get
$$
\begin{align}
\frac{e^{iz}}{z\,(z^2+1)^2}
&=\frac{e^{iw-1}}{(w+i)\,(w^2+2iw)^2}\\
&=\frac{i}{4ew^2}\frac{e^{iw}}{(1-iw)(1-\frac{i}{2}w)^2}\\
&=\frac{i}{4ew^2}\frac{1+iw+\dots}{(1-iw)(1-iw+\dots)}\\
&=\frac{i}{4ew^2}(1+3iw+\dots)
\end{align}
$$
Thus, the residue is the coe... |
3,679,806 | <p>this is Probability Density Function(pdf)
if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent, how to prove <span class="math-container">$X^2$</span> and <span class="math-container">$Y^2$</span> are also independent</p>
| Toni | 763,699 | <p>Here is another way, if you are not familiar with measure theory. I assume we are talking about real-valued random variables.
Let <span class="math-container">$a,b\geq0$</span>
<span class="math-container">$$P(X^2 \leq a,\,Y^2 \leq b)=P(-\sqrt{a}\leq X \leq \sqrt{a},\,-\sqrt{b}\leq Y \leq \sqrt{b}) \\= P(-\sqrt{a}\... |
1,118,259 | <p>Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the remaining solid?</p>
<p>Can someone help me at least setting up the integral? I know that there is a similar problem... | user84413 | 84,413 | <p>This answer combines the ideas of the two previous answers: </p>
<p>It implements Christian Blatter's suggestion, and makes extensive use of Achille Hui's techniques.</p>
<hr>
<p>Let $V$ be the volume of the sphere with 2 holes, let $V_{H}$ be the volume of each of the cylindrical holes, and let $V_{I}$ be the v... |
368,461 | <p>Let <span class="math-container">$G=(V,E)$</span> be a finite simple graph. We say a map <span class="math-container">$p:V\to [n]:=\{1,\ldots,n\}$</span> is a <em>pseudo-coloring</em> if for all <span class="math-container">$a\neq b\in[n]$</span> there is <span class="math-container">$v\in\psi^{-1}(\{a\})$</span> an... | Markiian Khylynskyi | 144,883 | <p>It is not true. Let <span class="math-container">$G$</span> and <span class="math-container">$H$</span> be graphs, and let <span class="math-container">$p_{max}$</span> be maximal pseudo-coloring of the graph <span class="math-container">$H$</span>. Show that the map <span class="math-container">$p((x,y))=p_{max}(y)... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | lone student | 460,967 | <p>You can easily observe that, the expression <span class="math-container">$(x-1)^2$</span> is almost included in the polynomial <span class="math-container">$P(x):=x^4+x^3-2x+1$</span>.</p>
<p>Let's rewrite your polynomial as follows:</p>
<p><span class="math-container">$$
\begin{align}P(x)=x^4+x^3-\color{red}{x^2}+\... |
1,890,040 | <p>This is related to <a href="https://math.stackexchange.com/questions/1888881/expanding-a-potential-function-via-the-generating-function-for-legendre-polynomi">this previous question of mine</a> where (with lots of help) I show that $$\sum_{l=0}^\infty \frac{R^l}{a^{l+1}}P_l(\cos\theta)=\frac{1}{\sqrt{a^2-2aR\cos\the... | Ivan Neretin | 269,518 | <p>OK, Henning Makholm did the explanation, and did it good, but I'll do my part anyway.</p>
<p>First we extend the faces of our dodecahedron just a bit beyond the edges, until it grows nice little spikes on all 12 faces and thus becomes a <a href="https://en.wikipedia.org/wiki/Small_stellated_dodecahedron" rel="nofol... |
1,811,528 | <p>The definition of the order of an element in a group is:</p>
<blockquote>
<p>The order of an element $x$ of a group $G$ is the smallest positive integer $n$ such that $x^{n}=e$.</p>
</blockquote>
<p>Doesn't this definition assume that the integers are somehow relevant to every group? </p>
<p>All of the other de... | Stefan Perko | 166,694 | <p>Here is an alternative viewpoint:</p>
<p>Actually, the common definition of a group with an operation $\cdot : G\times G\to G$ is not the only possible definition of a group. In fact it is <em>biased</em>, because for any group there a alot of $n$-ary operations besides the group multiplication.
For example you hav... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Henry Towsner | 62 | <p>I think your student pointed out the key issue: "Well, obviously I didn't know that zero was an answer when I was doing the problem". That's exactly right: at the time the student was dividing, they didn't know whether or not x is 0, which means they didn't know whether or not it made sense to divide by x.</p>
<p>... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Jared | 2,204 | <p>Frankly, I think your reasoning is the best. You initially have a quadratic, thus you should <em>expect</em> to find two solutions (although you may not). By dividing by $x$, you get only a single solution: so what is the other? You clearly cannot find it by dividing by $x$, but you <em>can</em> find it by either... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Steven Gubkin | 117 | <p>This is more of a long comment than an answer.</p>
<p>Some of the other answers here are advocating for factoring rather than case analysis. I just want to point out that proving the theorem that for any two real numbers <span class="math-container">$ab = 0$</span> if and only if <span class="math-container">$a=0$<... |
1,818,764 | <p>I think everything I have done is kosher, but unless I am missing an identity it is a different answer than the online quiz and wolfram alpha give.</p>
<p>I tried to use the trig substitution
$$ x=2\sin(\theta)\Rightarrow dx=2\cos(\theta)$$</p>
<p>Which yields
$$\int\frac{x^2}{\sqrt{4-x^2}}dx=\int\frac{4\sin^2(\t... | Micheal Brain Hurts | 324,389 | <p>for $\displaystyle\int\dfrac{x^2}{\sqrt{4-x^2}}dx$ ;</p>
<p>Integral by parts;</p>
<p>Let be $\quad du=\dfrac{-x}{\sqrt{4-x^2}}dx$</p>
<p>$u=\sqrt{4-x^2}$</p>
<p>And $\quad(-x)=v\longrightarrow -dx=dv$</p>
<p>$\displaystyle\int\dfrac{x^2}{\sqrt{4-x^2}}dx=-x.\sqrt{4-x^2}+\displaystyle\int\sqrt{4-x^2}dx$</p>
<p... |
121,450 | <p>I am trying to prove that the series <span class="math-container">$\sum \dfrac {1} {\left( m_{1}^{2}+m_{2}^{2}+\cdots +m_{r }^{2}\right)^{\mu} } $</span> in which the summation extends over all positive and negative integral values and zero values of <span class="math-container">$m_1, m_2,\dots, m_r$</span>, except ... | Did | 6,179 | <p>Comparing the series with an integral, one sees that the series converges if and only if the $r$-dimensional integral
$$
I_r=\int_{\mathbb R^r}[\|x\|\geqslant1]\,\frac{\mathrm dx}{\|x\|^{2\mu}}
$$
converges. Consider the spherical coordinates $(s,\alpha)$ with $s\geqslant0$ and $\alpha$ in the sphere $S^{r-1}$. Th... |
3,103,372 | <p>The following is a system of quadratic congruences:
<span class="math-container">$$\left\{\begin{array}{cl}x^{2}\equiv a&\pmod{3}\\x^{2}\equiv b&\pmod{7}\end{array}\right.$$</span>
If <span class="math-container">$\left(\frac{a}{3}\right)=1=\left(\frac{b}{7}\right)$</span>, where <span class="math-container"... | Wolfgang Kais | 640,973 | <p>Let's assume that <span class="math-container">$\gcd(n,m)=1$</span> and that we have integers <span class="math-container">$\alpha,\beta \in \mathbb Z$</span> each solving one of the quadratic congruences such that</p>
<p><span class="math-container">$$\alpha^2 \equiv a \pmod n \quad \land \quad \beta^2 \equiv b \p... |
3,103,372 | <p>The following is a system of quadratic congruences:
<span class="math-container">$$\left\{\begin{array}{cl}x^{2}\equiv a&\pmod{3}\\x^{2}\equiv b&\pmod{7}\end{array}\right.$$</span>
If <span class="math-container">$\left(\frac{a}{3}\right)=1=\left(\frac{b}{7}\right)$</span>, where <span class="math-container"... | Servaes | 30,382 | <p>Let <span class="math-container">$x$</span> be any solution to <span class="math-container">$x^2\equiv a\pmod{3}$</span> and <span class="math-container">$y$</span> any solution to <span class="math-container">$y^2\equiv b\pmod{7}$</span>. Then <span class="math-container">$x+3m$</span> is also a solution for any in... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Fakemistake | 173,351 | <p>This answer uses the formula for the sum of odd numbers, which is
<span class="math-container">$$i^2=\sum_{k=1}^i 2k-1$$</span>
First insert this into the considered formula
<span class="math-container">$$s_n\overset{\mathrm{def}}{=}\sum_{i=1}^ni^2=\sum_{i=1}^{n}\sum_{k=1}^i (2k-1)$$</span></p>
<p>Definitely in the... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Patrick Sheehan | 599,007 | <p>This has a few parts, but should bridge the gap between what you were looking for (hopefully?) and the answers from Tyler and Pedro.</p>
<p>If I remember right, that method of Gauss' has a geometric interpretation that Gauss provided along with it. Namely that the sum 1+2+...+n can be visualized as a "staircase... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | Community | -1 | <p>The general formula derives from the simple one.</p>
<p>The general sequence is $$a=a-b+b,a+b=a-b+2b,a+2b,\cdots a-b+nb,$$ i.e. $n$ terms from $a$ to $a+(n-1)b=a+c$.</p>
<p>Then</p>
<p>$$\sum_{k=1}^n(a-b+kb)=n(a-b)+\frac{n(n+1)}2b=n\frac{a+a+(n-1)b}2=n\frac{a+c}2.$$</p>
<hr>
<p>With $11$ to $11n$ in steps $11$,... |
290,132 | <p>Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function.</p>
<p>How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ? </p>
<p>In particular I wonder most about the case $a=1$ and $b=e$. (I already know the trivial cases $a=-1$ and $a=0$)</p>
<p>I know how to express $f(x+1)$ int... | Haskell Curry | 39,362 | <p>Pick $ f(x) = \lambda e^{kx} $. Then
$$
f(x) + f(x + 1) = \lambda \left[ e^{kx} + e^{k(x + 1)} \right] = \lambda (1 + e^{k}) e^{kx}.
$$
By choosing $ k = 1 $ and $ \lambda = \dfrac{1}{1 + e} $, you get a solution.</p>
|
104,375 | <p>How I am supposed to transform the following function in order to apply the laplace transform.</p>
<p>$f(t) = t[u(t)-u(t-1)]+2t[u(t-1) - u(t-2)]$</p>
<p>I know that it has to be like this</p>
<p>$L\{f(t-t_0)u(t-t_0)\} = e^{-st_0}F(s), F(s) = L\{f(t)\}$</p>
| 000 | 22,144 | <p>This question is old, but I'd like to share my view of it. I hope this offends no one.</p>
<p>What you are dealing with are equivalence classes modulo an integer. That is, you are dealing with equivalence classes defined by $[a]=\{b \in \mathbb{Z}:b \cong a \pmod 3\}$ for an element $a$.</p>
<p>In the case of the ... |
2,965,717 | <p>How would you prove that <span class="math-container">$$\displaystyle \prod_{k=1}^\infty \left(1+\dfrac{1}{2^k}\right) \lt e ?$$</span></p>
<p>Wolfram|Alpha shows that the product evaluates to <span class="math-container">$2.384231 \dots$</span> but is there a nice way to write this number? </p>
<p>A hint about so... | Jack D'Aurizio | 44,121 | <p>It can be easily shown that for any <span class="math-container">$x\in(0,1)$</span> we have</p>
<p><span class="math-container">$$ \frac{1+2x+\frac{4x^2}{3}+\frac{8x^3}{21}+\frac{16 x^4}{315}+\frac{32 x^5}{9765}}{1+x+\frac{x^2}{3}+\frac{x^3}{21}+\frac{x^4}{315}+\frac{x^5}{9765}} <1+x < \frac{1+2x+\frac{4x^2}{... |
1,522,929 | <p>For every fixed $t\ge 0$ I need to prove that the sequence $\big\{n\big(t^{\frac{1}{n}}-1\big) \big\}_{n\in \Bbb N}$ is non-increasing, i.e.
$$n\big(t^{\frac{1}{n}}-1\big)\ge (n+1)\big(t^{\frac{1}{n+1}}-1\big)\;\ \forall n\in \Bbb N$$
I'm trying by induction over $n$, but got stuck in the proof for $n+1$:
<br/>
For ... | user279043 | 224,391 | <p>All right, let's do some complex analysis! Let's integrate <span class="math-container">$e^{iz^2}$</span> over the closed contour defined in three pieces (the arrows indicating the direction of contour integration)
<span class="math-container">$$
\begin{cases}
\Gamma_1: & |z|:0\rightarrow R, & \theta=0 \\
\... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | 4pie0 | 29,621 | <p>The sum and multiplication of two vectors $u,v$ from a linear space $V$ has to belong to $V$ because we want to consider elements that share some common properties, in particular they all can be expressed as a linear combination of vectors that form any base of $V$.</p>
<p>It is the same as in case of groups. In ex... |
4,001,031 | <p>(For all those that it may concern, this is not a duplicate of my previous post, But starts in a similar way.)</p>
<p>A triangle with side lengths a, b, c with a height(h) that intercepts the hypotenuse(c) at (x , y) such that it is split into two side lengths, c = m + n, we can find Pythagoras theorem using the ar... | Community | -1 | <p>It would have been much simpler to use your initial equations as follows.</p>
<p>Since <span class="math-container">$hc=ab$</span>, you have <span class="math-container">$m=\frac{bh}{a}=\frac{b^2}{c}$</span></p>
<p>Similarly, <span class="math-container">$n=\frac{ba^2}{c}$</span>.</p>
<p>Now express <span class="mat... |
1,737,674 | <p>I am trying to understand how to find all congruence classes in $\mathbb{F}_2[x]$ modulo $x^2$. How can I compute them ? Can someone get me started with this? I am having trouble understanding $\mathbb{F}_2[x] $ is it the set $\{ f(x) = a_nx^n + ...+ a_1 x + a_0 : a_i = 0,1 \} $?</p>
| ashi | 299,991 | <p>I think the correct is answer equal to $75°$</p>
<p>I did it by construction</p>
<p>First of all I made triangle $BDC$ with ur provided information and doing some calculations ie ($\measuredangle DBC=15°$, $\measuredangle BDC=120^{\circ}$ ,$\measuredangle BCD=45°$) than I continued line $CD$ till point $A$ with ur... |
1,737,674 | <p>I am trying to understand how to find all congruence classes in $\mathbb{F}_2[x]$ modulo $x^2$. How can I compute them ? Can someone get me started with this? I am having trouble understanding $\mathbb{F}_2[x] $ is it the set $\{ f(x) = a_nx^n + ...+ a_1 x + a_0 : a_i = 0,1 \} $?</p>
| Piquito | 219,998 | <p>In the figure below we have $$\frac ac=\frac{\sin x}{\sin 45^\circ}\qquad(1)$$ Furthermore
$$\frac{z}{\sin 15^\circ}=\frac{a}{\sin 120^\circ}\qquad(2)$$
$$\frac{2z}{\sin (120^\circ-x)}=\frac{c}{\sin 60^\circ}\qquad(3)$$</p>
<p>From $(1),(2),(3)$ we get $$1.46407\sin x=1.7320\cos x+\sin x\iff \tan x=3.73219$$</p>
<... |
2,245,631 | <blockquote>
<p>$x+x\sqrt{(2x+2)}=3$</p>
</blockquote>
<p>I must solve this, but I always get to a point where I don't know what to do. The answer is 1.</p>
<p>Here is what I did: </p>
<p>$$\begin{align}
3&=x(1+\sqrt{2(x+1)}) \\
\frac{3}{x}&=1+\sqrt{2(x+1)} \\
\frac{3}{x}-1&=\sqrt{2(x+1)} \\
\frac{(3-x... | Javi | 434,862 | <p>I usually check if the integer divisors of the independent term of the polynomial is a root in order to decompose it in factors. </p>
|
3,840,692 | <p>The equation is <span class="math-container">$2z^2w''+3zw'-w=0$</span></p>
<p><span class="math-container">$z_0=0$</span> is a regular singular point, so <span class="math-container">$w(z)=\sum_{n=0}^{\infty} a_nz^{n+r}$</span></p>
<p>then <span class="math-container">$w'(z)=\sum_{n=0}^{\infty} (n+r)a_nz^{n+r-1}$</s... | Claude Leibovici | 82,404 | <p>Notice that, for this problem, Frobelius method is not required. Let <span class="math-container">$w(z)=z u(z)$</span> to end with
<span class="math-container">$$6u''+13u'=0$$</span> Just reduction of order and a quite simple problem.</p>
|
155,024 | <p>I am trying to plot a polynomial inside <code>Manipulate</code> with indexed coefficients.</p>
<p>Surprisingly, only an empty plot is generated.</p>
<p>Any help welcome.</p>
<pre><code>Block[{n = 3},
Manipulate @@ {
Column[{
Sum[a[i] x^i, {i, 0, n}],
Plot[Evaluate@Sum[a[i] x^i, {i, 0, n}], {x, -5, 5}]
}],
... | jkuczm | 14,303 | <p>On my computer simple <code>LibraryFunction</code>, that compares subsequent elements in a loop, is fastest.</p>
<pre><code>pairwiseBooleJkuczm = Last@Compile[{{data, _Real, 1}},
Table[
Boole[Compile`GetElement[data, i] === Compile`GetElement[data, i + 1]],
{i, Length@data - 1}
],
CompilationTarget -&... |
155,024 | <p>I am trying to plot a polynomial inside <code>Manipulate</code> with indexed coefficients.</p>
<p>Surprisingly, only an empty plot is generated.</p>
<p>Any help welcome.</p>
<pre><code>Block[{n = 3},
Manipulate @@ {
Column[{
Sum[a[i] x^i, {i, 0, n}],
Plot[Evaluate@Sum[a[i] x^i, {i, 0, n}], {x, -5, 5}]
}],
... | Carl Woll | 45,431 | <p>Here is an approach assuming that the <a href="http://reference.wolfram.com/language/ref/SameQ" rel="noreferrer"><code>SameQ</code></a> tolerance is not changed from its default.</p>
<p><strong>\$MachineEpsilon</strong></p>
<p><code>$MachineEpsilon</code> is the smallest number that when added to <code>1.</code> p... |
254,695 | <p>The concept of dimension seems to be:</p>
<blockquote>
<p>In physics and mathematics, the dimension of a space or object is
informally defined as the minimum number of coordinates needed to
specify any point within it.</p>
</blockquote>
<p>According to <a href="http://en.wikipedia.org/wiki/Dimension_%28mathe... | Nigel Galloway | 63,512 | <p>A big difference between 4d space and spacetime is its size. Consider the size (Cardinality) of all the sets which have a one to one mapping between the digits on my hand and their members to be 5. Define the Cardinality of the set of all integers as $\aleph_0$, said as aleph-null. Note that it is possible to produc... |
4,339,772 | <p>The problem is stated as:</p>
<blockquote>
<p>Show that <span class="math-container">$\int_{0}^{n} \left (1-\frac{x}{n} \right ) ^n \ln(x) dx = \frac{n}{n+1} \left (\ln(n) - 1 - 1/2 -...- 1/{(n+1)} \right )$</span></p>
</blockquote>
<p><strong>My attempt</strong></p>
<p>First of all, we make the substitution <span ... | Claude Leibovici | 82,404 | <p><em>Just for your curiosity</em></p>
<p>Assuming that you enjoy the gaussian hypergeometric function, there is an antiderivative
<span class="math-container">$$I_n(x)=\int \left (1-\frac{x}{n} \right ) ^n \log(x)\, dx=\frac 1{n^n}\int(n-x)^n \log(x)\,dx $$</span>
<span class="math-container">$$I_n(x)=-\frac{(n-x)^{n... |
3,096,115 | <p>By using a sieve created by Prime Number Tables set up by the formula PN+(PNx6) for numbers generated by 6n+or-1, takes 182 calculations to identify 170 composite numbers. Using the Sieve of Eratosthenes would take around 1600 calculations. The Prime Number Tables identify all the composite numbers on the the list o... | Keith Backman | 29,783 | <p>Here is the problem with your method and its objective: You use the Sieve of Eratosthenes as a standard, and apply it to a list of <span class="math-container">$1000$</span> numbers to determine a minimum number of calculations to identify all prime numbers in that range. But you start with a sublist (Prime Number T... |
2,688,608 | <p>Assume matrix </p>
<p>$$A=
\begin{bmatrix}
-1&0&0&0&0\\
-1&1&-2&0&1\\
-1&0&-1&0&1\\
0&1&-1&1&0\\
0&0&0&0&-1
\end{bmatrix}
$$</p>
<p>Its Jordan Canonical Form is
$$J=
\begin{bmatrix}
-1&1&0&0&0\\
0&-1&0&0&... | Will Jagy | 10,400 | <p>I got the Jordan blocks in slightly different order.</p>
<p>What you seem to be missing is the consistency part: in my</p>
<p>$$ P =
\left(
\begin{array}{rrrrr}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
1 &am... |
838,690 | <p>True or false question</p>
<p>If B is a subset of A then {B} is an element of power set A. </p>
<p>I think this is true.</p>
<p>Because B is {1,2} say A {1,2,3} then power set of includes </p>
<p>$\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{3,2\},\{1,2,3\},\emptyset\}$</p>
<p>Unless {B} means $\{\{1,2\}\}$</p>
| amWhy | 9,003 | <p>It is true that $B$ is in the power set of a set $A$ (we'll call the powerset $P(A)$) is the set of all <strong><em>subsets</em></strong> of $A$, so the elements of $P(A)$ include all subsets of $A$.</p>
<p>Since $B$ is given to be a subset of $A$, then it is an element in the powerset of $A$.</p>
<p>However, $\{B... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.