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 |
|---|---|---|---|---|
4,158,247 | <p>Give an example of sequence <span class="math-container">$\{f_n\}$</span> of continious real -value function on <span class="math-container">$[0, \infty)$</span> converging uniformly on <span class="math-container">$[0, \infty)$</span> to a function <span class="math-container">$f$</span> such that for each <sp... | Thomas Andrews | 7,933 | <p>Your sequence does not converge uniformly.</p>
<p>Try:
<span class="math-container">$$f_n(x)=\frac{1}{x+1}-\frac{1}{n+1}.$$</span> It converges uniformly to <span class="math-container">$\frac{1}{x+1}.$</span></p>
<hr />
<p><span class="math-container">$f(x)$</span> has to be either all positive or all negative. We... |
2,335,043 | <p>Let D be the ring of integers of a number field K. Let I be a non-zero ideal of D. Why D/I is finite?</p>
<p>Would you help me please? Thank you in advance.</p>
| José Carlos Santos | 446,262 | <p>This function is not defined at <span class="math-container">$0$</span>, but you can compute the limit <span class="math-container">$\lim_{x\to0}e^{x^x-1}=e^{\lim_{x\to0}x^x-1}$</span>. It happens that<span class="math-container">$$\lim_{x\to0}x^x=\lim_{x\to0}e^{x\log x}=e^{\lim_{x\to0}x\log x}$$</span>and that<span... |
2,335,043 | <p>Let D be the ring of integers of a number field K. Let I be a non-zero ideal of D. Why D/I is finite?</p>
<p>Would you help me please? Thank you in advance.</p>
| Franklin Pezzuti Dyer | 438,055 | <p>If you try to evaluate it at $x=0$, you end up getting
$$e^{0^0-1}$$
Whose value is not immediately apparent. However, by convention, $0^0$ is often said to be equal to $1$. This is because
$$\lim_{x\to 0} x^x=1$$
And if you observe the following graph of $y=x^x$, you can see that this is so:</p>
<p><a href="https:... |
9,719 | <p>I have proved this using Venn diagram but when I am trying to prove this using the rule that <strong><em>"If $ A \subset B \text{ and } B \subset A $ then $ A = B $"</em></strong>, I am having some problems with my understanding of the same,here is how I did so far:</p>
<p>Let $x \in A \cap (B-C) \Rightarrow x \in... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is a different style of proof, viz. an equational and calculational one. </p>
<p>The idea is to start at the most complex side, translate from set theory to logic by working at the element level, then expanding the definitions, and see where there leads you.</p>
<p>In other words, for all $\;x\;$ we calculat... |
335 | <p>I looked at the “accept rate” of some of the site’s top users and found that it varies widely in the range 45%–100%. I was surprised with that, as it may be taken as an indicator that either:</p>
<ol>
<li>hard questions don’t tend to get very good answers (let's assume that top users ask difficult questions), or</l... | Mr.Wizard | 121 | <p>I know that there are several of my questions for which I have upvoted multiple answers, but I haven't bothered getting around to really pinning down and accepting the best one. I hope that people do not feel that I am ungrateful for their answers. I just want to put more consideration into an eventual Accept. I ... |
335 | <p>I looked at the “accept rate” of some of the site’s top users and found that it varies widely in the range 45%–100%. I was surprised with that, as it may be taken as an indicator that either:</p>
<ol>
<li>hard questions don’t tend to get very good answers (let's assume that top users ask difficult questions), or</l... | Leonid Shifrin | 81 | <p>I would think that the top users would make this site a better place by trying to accept answers to their questions whenever possible, just because they are viewed as models by many other members of the community, particularly new ones. Seeing the high accept rate for the top users will IMO generally promote the hig... |
2,556,993 | <p>I've come across two limits which are reported to take on a range of values rather than a unique one.</p>
<p>They are:</p>
<p>$$\lim \limits_{x \to \infty} \space\frac{1+\cos x}{1-\sin x} = 0 \space to\space \infty$$</p>
<p>$$\lim \limits_{x \to \infty} \space \frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin x}} = \fra... | kingW3 | 130,953 | <p>Take first limit for example it means that you can find an increasing sequence (because $x$ goes to $\infty$, in general the sequence doesn't need to be increasing it should just converge to the what $x$ is tending to). $x_n$ for any value of $L\in[0,\infty]$ so that $$\lim_{x\to\infty}\frac{1+\cos x_n}{1-\sin x_n}=... |
2,050,078 | <p>Is there a efficient formula to calculate this sum ? $$S = 1^1+2^2+3^3+...+n^n$$ , or it simply diverges ?</p>
| Dietrich Burde | 83,966 | <p>The function $S_n=\sum_{k=1}^n k^k$ has been studied often, see <a href="https://oeis.org/search?q=1%2C5%2C32%2C288%2C3413&language=german&go=Suche" rel="nofollow noreferrer">here</a> for references. The question for a formula or an estimate is the well known <a href="http://www.jstor.org/stable/2306261?seq=... |
1,043,778 | <p>I have a list of $15$ people and I need to put them into 5 unique groups of $3$. They're going to meet once a month over the next $7$ months. I want everyone to be in different groups every month.</p>
<p>Rules:</p>
<ol>
<li><p>{1,2,3} is the same as {3,2,1}</p></li>
<li><p>By unique I mean that no two members appe... | Empy2 | 81,790 | <p>This is Kirkman's schoolgirl problem, and it has a solution which you can find here. <a href="http://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem" rel="nofollow">http://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem</a></p>
|
4,336,337 | <p>I've come across this problem:
<span class="math-container">$$
S=\sum_{i=0}^{\infty}2^i \\
S= 1+ 2\sum_{i=0}^{\infty}2^i \\
S=1+2S \\
S=-1
$$</span>
I know that you cannot rearrange this series because it is not <a href="https://en.wikipedia.org/wiki/Absolute_convergence#Rearrangements_and_unconditional_convergence"... | user21820 | 21,820 | <p>In mathematics, it is not whether something is <em>allowed</em> or not. Rather, everything is based on logic and logically <strong>precise</strong> definitions.</p>
<p><span class="math-container">$\sum_{k=0}^∞ f(k)$</span> is <strong>defined</strong> as <span class="math-container">$\lim_{n\overset{∈ℕ}→∞} \sum_{k=0... |
2,971,350 | <p>I need to show log <span class="math-container">$Fib_{n}$</span> is <span class="math-container">$\theta(n)$</span>
by the Fibonacci numbers defined as </p>
<p><span class="math-container">$$ F_n=F_{n-1}+F_{n-2}$$</span>
for
<span class="math-container">$$ n \geq 2 $$</span>
<span class="math-container">$ F_{0} =... | shiningPanther | 685,846 | <p>This is a classic case of hypothesis testing. Here, our null hypothesis is that there is no significant relationship between <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> in the simple linear regression model <span class="math-container">$Y = \beta X + \epsilon$</span>:</p>
<p>... |
585,044 | <blockquote>
<p>Let $R(n)$ be the number of ordered pairs of integers $(x,y)$ with $x^2+y^2 = n$. Using the identity $$(a+b)^2+(a-b)^2 =2(a^2 +b^2),$$
prove that for all integers $n$, $R(2n) = R(n)$.</p>
</blockquote>
<p>I'm not really sure how to start this question. Any help would be appreciated.</p>
| Adwait Kumar | 40,065 | <p>If $n$ is even and $$n = x^2 + y^2$$ for integers $x$ and $y$, then $$\frac{n}{2} = (\frac{x+y}{2})^2 + (\frac{x-y}{2})^2,$$ (using the mentioned identity) and both these terms are integers, as $x$ and $y$ must have the same parity.
On the other hand, if $n$ is an integer with $$n = a^2 +b^2$$ for integers $a$ and $... |
201,103 | <p>When comparing datasets using <code>DistributionChart</code>, is it possible to add a horizontal line to the chart that lies over all of the individual distributions within the chart?</p>
| OkkesDulgerci | 23,291 | <p>You can use <code>GridLines</code></p>
<pre><code>SeedRandom@2; DistributionChart[
RandomReal[BetaDistribution[2, 1/4], {2, 100}],
GridLines -> {None, {1}},
GridLinesStyle -> Directive[Red, AbsoluteThickness@2],
Method -> {"GridLinesInFront" -> True}]
</code></pre>
<p><a href="https://i.stack.im... |
2,257,601 | <p>Let $$x^4-4x^3+mx^2+nx+1=0$$ have four roots positive real number. Then : what is $n,m \in \Bbb R$</p>
<p>My try </p>
<p>Let four roots be $x=a,b,c,d \in \Bbb R^+$ so, $$x^4-4x^3+mx^2+nx+1=0\\(x-a)(x-b)(x-c)(x-d)=0$$
Next?</p>
| achille hui | 59,379 | <p>The only solution is $(m,n) = (6,-4)$.</p>
<p>Let $a_1,\ldots,a_4$ be the four roots of the polynomial
$$x^4 - 4x^3 + mx^2 + nx + 1 = (x-a_1)(x-a_2)(x-a_3)(x-a_4)$$
Let $\sigma_1, \ldots,\sigma_4$ be the elementary symmetric polynomials associated to $a_1, \ldots, a_4$.
$$
\sigma_1 = \sum_{i=1}^4 a_i,\quad
\sigma_2... |
679,343 | <p>I have been researching this for the past hour or so, but I have not really found a sufficient answer. What I have found is that vaguely, math education PhDs set policies for math education. However, universities almost uniformly have math professors teach mathematics courses. I believe that usually the math prof... | GEdgar | 442 | <p>Example. </p>
<p>Someone (probably the salesman) says: Student clickers! the greatest thing for low-level large lecture math classes since the invention of the overhead projector! So, the math education PhD (Professor Z) sets up a study. Some classes done with clickers, others without. Pre- and post-testing, t... |
1,740,221 | <p>If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1?</p>
<p>I'm concerned with this expression in the context of a geometric series. Normally if we have $\sum_{i=0}^{\infty} r^{q}$ with |r| < ... | Yakov Shklarov | 222,048 | <p>So I'm a bit late to the party but it seems to me that the other answers, while correct, are incomplete. Reading between the lines it seems like you're trying to evaluate the series</p>
<p>$$\sum_{k = 0}^{\infty} x^{\lfloor \frac{k}{a-b} \rfloor},\quad\text{integers }a,b \text{ and } 0 \leqslant x < 1.$$</p>
<p... |
3,685,965 | <p>How do you solve a problem such as <span class="math-container">$$3^n\mod 10 = 9$$</span> for every integer solution for n? There is obviously a trivial solution for n=2 and I know from experimentation that the general solution would be <span class="math-container">$$n=2+4k$$</span> where k is an integer but how do ... | Quanto | 686,284 | <p>Let <span class="math-container">$x\to -x$</span> over <span class="math-container">$(-1,0)$</span></p>
<p><span class="math-container">\begin{align}
\int_{-1}^1\frac{\ln(1-x^2)}{(1-\beta x)^2}dx
= & \ 2\int_0^1\ln(1-x^2) \frac{1+\beta^2x^2}{(1-\beta^2x^2)^2} dx\\
= & \ 2\int_0^1 \ln(1-x^2)\ d\left( \frac{x... |
1,869,843 | <p>We know that:
$$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ </p>
<p>My question is there other examples that satisfies:
$$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$</p>
| Michael Hardy | 11,667 | <p>Let $a=1+2+3+\cdots+n = \dfrac{n(n+1)} 2$.</p>
<p>Then whenever $p$ is an <b>odd</b> number, the sum
$$
1^p+2^p+3^p+\cdots + n^p
$$
can be written as a polynomial function of $a$.</p>
<p><a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials" rel="nofollow">https://en.wikipedia.org/wiki... |
3,837,815 | <blockquote>
<p>Prove <span class="math-container">$\log_{4}6$</span> is irrational.</p>
</blockquote>
<p>I've seen proofs of this which boil down to:
<span class="math-container">$$4^{m} = 6^{n}$$</span></p>
<p>But how does this prove that it is irrational?
For example it is possible to have <span class="math-containe... | Joshua Wang | 773,061 | <p>We will prove by contradiction. Assume <span class="math-container">$\log_{4}(6)$</span> is rational, and equal to <span class="math-container">$\frac{m}{n}$</span> for integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span>. Then, we have:</p>
<p><span class="math-container">$6 ... |
12,239 | <p>I know this is a trivial problem but am stuck with this thing.</p>
<p>$A \in \mathbb{R}^{m \times m}$ is a symmetric positive definite matrix and $C \in \mathbb{R}^{m \times p}$
</p>
<p>How do I show that the matrix $$B = \begin{bmatrix} A & C \\ C^T & 0 \end{bmatrix}$$ is indefinite?</p>
<p>I can show th... | Community | -1 | <p>Ok. So I figured it out. But I think my argument is really convoluted. I would appreciate if someone could come up with a simpler argument.</p>
<p>Without loss of generality, we can assume that the matrix $C$ is of full-rank $p$.</p>
<p>First let $B = \begin{bmatrix} A & C \\ C^T & 0 \end{bmatrix}$, where ... |
1,101,520 | <blockquote>
<p>Evaluate the integral
$$\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Let $f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now differentiate both side with respect to $x$, and we get</p>
<p>$$
\begin{align}
f'(x) &= \frac{(x^3+3x+1)^2\cdot a-2\cdot (x^3+... | Tolaso | 203,035 | <p>Here is another solution... <em>I also happen to think that it cannot be done with substitution.</em></p>
<p>$$\begin{aligned}
\int \frac{5x^3+3x-1}{\left ( x^3+3x+1 \right )^3}\,dx &=\int \frac{-x^3-3x-1+6x^3+6x}{\left ( x^3+3x+1 \right )^3}\,dx \\
&= \int \frac{-x^3-3x-1+2x\left ( 3x^2+3 \right )}{\le... |
2,176,815 | <p>Find all collision points between the parametric curves $x_1=\cos t$, $y_1=2 \sin t + \sin 2t$ and $x_2=2t$, $y_2= t - \frac12$.</p>
<p>I have no idea of how to proceed from the starting step of setting the $x$ values equal to each other, and the $y$ values equal to each other.</p>
<p>I also don't know what the an... | Dando18 | 274,085 | <p>$x_1 = x_2 \,\Rightarrow \cos t=2t \,\Rightarrow t=0.450184$</p>
<p>$y_1=y_2 \, \Rightarrow 2\sin t + \sin(2t) = t-\frac{1}{2} \,\Rightarrow
t=-1.547, -0.169352, 1.863$</p>
<p>They are only ever at the same x-coordinate when $t=0.4501284$ and only at the same y-coordinate when $t=-1.547, -0.169352, 1.863$, so sinc... |
343,549 | <p>For <span class="math-container">$\Omega$</span> a bounded open set of <span class="math-container">$\mathbf{R}^d$</span> and <span class="math-container">$f\in L^p(\Omega)$</span> the infimum
<span class="math-container">\begin{align*}
\inf_{C\in\mathbf{R}} \|f-C\|_p
\end{align*}</span>
is reached (by compactness)... | Robert Israel | 13,650 | <p>Consider the case <span class="math-container">$p=6$</span> where <span class="math-container">$\Omega$</span> consists of three sets of equal measure on which <span class="math-container">$f$</span> takes values <span class="math-container">$0, 1, 3$</span> respectively. Then <span class="math-container">$C_6(f)$<... |
2,595,650 | <p>Lets say we want to try to show $f(n) \in O(g(n))$, which may or may not be true.</p>
<p>Then by definition we find a large enough $n$ and a constant $k > 0$ such that $|f(n)| \le k|g(n)|$, but lets say $$\lim_{n \to \infty}\frac{|f(n)|}{|g(n)|} = \infty$$ what can I conclude from this? That $f(n) \not\in O(g(n)... | zwim | 399,263 | <p>Provided $g(n)\neq 0$ you can divide the big O.</p>
<p>$f(n)=O(g(n))$ means also $\dfrac{f(n)}{g(n)}=O(1)$</p>
<p>Being $O(1)$ means $\le k$ for some constant $k$, in other words, it is <strong>bounded</strong>.</p>
<p>Obviously an infinite limit is not bounded, thus $f(n)\notin O(g(n))$</p>
|
1,135,416 | <p>I have two arrays with 16 unsigned integers. I compute the sum of the first array = x and the sum of the second array = y. What is the chance they will be the same?
Also, how many combinations out of the total combinations are the same?</p>
<p>An example:
Lets say an array1 contains these numbers (low values just f... | FireGarden | 87,896 | <p>A game of Tetris (on a 10×20 grid) consisting of alternating S and Z tetrominoes will necessarily end before 70000 tetrominoes are played.</p>
<p><a href="http://euclid.trentu.ca/aejm/V4N1/Tsuruda.V4N1.pdf" rel="nofollow noreferrer">http://euclid.trentu.ca/aejm/V4N1/Tsuruda.V4N1.pdf</a> — Lemma 2.</p>
|
4,363,566 | <p>I am trying to prove Euler's equation for the case of <span class="math-container">$4x$</span>, that is:
<span class="math-container">$$e^{i4x} = \cos 4x+i \sin 4x.$$</span>
I can get the imaginary part in a form that is equivalent to <span class="math-container">$\sin 4x$</span>, but I am struggling with the real p... | Darshan P. | 992,496 | <p><span class="math-container">$$\left(\frac {e^{ix} + e^{-x}}{2}\right) =\left(\frac {u+ \frac 1u}{2}\right)= \cos x$$</span>
<span class="math-container">$4$</span>rth power:</p>
<p><span class="math-container">$$\begin{align*}
\left(\frac {u + \frac 1u}{2}\right)^4
&= \cos^4 x = \frac {u^4 + 4u^3\frac 1u + 6u^2... |
3,276,351 | <blockquote>
<p>A particle is allowed to move in the <span class="math-container">$\mathbb{Z}\times \mathbb{Z}$</span> grid by choosing any of the two jumps:</p>
<p>1) Move two units to right and one unit up</p>
<p>2) Move two units up and one unit to right.</p>
<p>Let <span class="math-container">$P=(... | Acccumulation | 476,070 | <p>If <span class="math-container">$U$</span> is the number of move up twice moves, and <span class="math-container">$R$</span> is the number of move right twice moves, we have that the final position is <span class="math-container">$(2R+U,R+2U)$</span></p>
<p>For <span class="math-container">$P$</span>, we have:</p>
... |
1,037,061 | <blockquote>
<p>Find the Consumer Surplus, given the demand and supply equations $$ D(x)=\frac{405}{\sqrt{x}} $$
$$ S(x)=5\sqrt{x} $$
The equilibrium point is $(81,45)$.</p>
</blockquote>
<p>I know the formula for <a href="http://en.wikipedia.org/wiki/Economic_surplus#Calculation_from_supply_and_demand" rel="nof... | mfl | 148,513 | <p><strong>Hint</strong></p>
<p>$$\int xe^{-x^2}dx =-\frac12 e^{-x^2}$$</p>
|
619,519 | <p>The equation $\log_3(\sqrt{x+1}+1)=(3^{x+1}-1)^2\,$
has two solutions, but I can't solve the equation.</p>
| Szabolcs Sipos | 115,826 | <p>As Constructor wrote the equation is:
$f(t)=f^{-1}(t)$ where $t=x+1$ and $f(t)=log_3(\sqrt{t}+1)$</p>
<p>If there is such $t_0$ that $f(t_0)=f^{-1}(t_0)$ <=> $f(t_0)=t_0 => log_3(\sqrt{t_0}+1)=t_0 => 3^{t_0}=\sqrt{t_0}+1$</p>
<p>$3^t$ is convex, $\sqrt{t}+1$ is concave so cannot be more than 2 solutions:... |
1,045,920 | <p>Let T: $M_{2x2} \rightarrow M_{2x2}$ be defined by</p>
<p>T(A) = $\begin{bmatrix}
1 & 3 \\[0.3em]
-1 & 1 \\[0.3em]
\end{bmatrix} A $</p>
<p>Is T a linear transformation?</p>
<p>I want to show that the operations for linear transformations hold. However, I am not sure if I am... | ml0105 | 135,298 | <p>We have:</p>
<p>$$T(A + B) = \begin{bmatrix} 1 & 3 \\ -1 & 1 \end{bmatrix}(A + B)$$</p>
<p>Note that matrix multiplication distributes, so we get:</p>
<p>$$T(A + B) = \begin{bmatrix} 1 & 3 \\ -1 & 1 \end{bmatrix}A + \begin{bmatrix} 1 & 3 \\ -1 & 1 \end{bmatrix}B = T(A) + T(B)$$ </p>
|
211,625 | <p>The problem here is that the matrix
$$\begin{pmatrix}
2 & 3 \\
1 & -1 \\
\end{pmatrix}$$
is invertible but not unimodular and hence the elements $2x+3y$ and $x-y$ generate a free abelian group of rank 2 but still a proper subgroup of $\langle x,y\rangle$.
But the question was to prove that $2x+3y$ and $x-y... | Aaron | 9,863 | <p>If they form a basis for $G$, then you can find integers $A, B, C, D$ such that $x=A(2x+3y)+B(x-y)$ and $y=C(2x+3y)+D(x-y)$. There are unique rational numbers $A, B, C, D$ satisfying those equations. Are they integers?</p>
|
65,260 | <p>Trying to learn how <code>Opacity</code> works, I found a possible issue.
Generating points in a square spiral with:</p>
<pre><code>sqrSpiral[rev_, inc_] := Module[{x, y, z, dx, dy, list = {}},
x = y = z = 0;
dx = {0, 1, 0, -1}; dy = {1, 0, -1, 0};
Do[
Do[
list = Append[list, {x += dx[[i]], y += dy[[i... | Simon Woods | 862 | <p>This appears to be a problem with the hardware rendering engine. A workaround is to use the "BSPTree" renderer:</p>
<pre><code>Style[
Graphics3D[{Thickness[0.01], Opacity[0.5], Red, line}, Boxed -> False],
"RenderingOptions" -> {"Graphics3DRenderingEngine" -> "BSPTree"}]
</code></pre>
<p><img src="http... |
2,899,316 | <p>Let $(a_n)$ be a sequence of real numbers. Consider the two points,</p>
<ul>
<li>1.$\quad$ $(a_n)$ diverges,</li>
<li>2.$\quad$ $(a_n)$ <a href="https://proofwiki.org/wiki/Definition:Divergent_Sequence_to_Infinity" rel="nofollow noreferrer">tends to <strong>plus</strong> infinity</a>.</li>
</ul>
<p>Using the f... | Community | -1 | <p>"Eventually non-negative" is necessary but not sufficient.</p>
<p>"Eventually increasing" is not necessary and not sufficient.</p>
<p>"Unbounded" is necessary but not sufficient.</p>
<p>"Eventually increasing and unbounded" is not necessary but sufficient.</p>
<p>As said in comments, "larger than any number $u$ ... |
3,187,856 | <p>Find all complex-valued <span class="math-container">$n \times n$</span> matrices <span class="math-container">$A$</span> such that <span class="math-container">$\det A = 0$</span> and <span class="math-container">$\text{rank}(AB) = \text{rank}(BA)$</span> for any <span class="math-container">$n \times n$</span> com... | Eclipse Sun | 119,490 | <p>Here is a simple way. Suppose <span class="math-container">$A\ne 0$</span> and <span class="math-container">$\det A=0$</span>. Then its null space <span class="math-container">$N(A)$</span> and its range <span class="math-container">$R(A)$</span> are both nontrivial. So we can pick any nonzero <span class="math-cont... |
3,930,878 | <p>Can someone show me how to solve <span class="math-container">$$\arg z^3 = 0$$</span>
I don't know even how to start because of the zero.</p>
| the_candyman | 51,370 | <p>A complex number <span class="math-container">$z \neq 0$</span> can be always rewritten as:</p>
<p><span class="math-container">$$z = M e^{i\theta},$$</span></p>
<p>with <span class="math-container">$M > 0$</span> and <span class="math-container">$\theta \in [0, 2\pi]$</span>.</p>
<p>Recall that <span class="math... |
116,196 | <p>Simple Question from a Stumped Rookie: </p>
<p>Using ListPlot, I can't get PlotStyle to look below the first level of a nested list to assign individual saturation values for each point; it assigns the first three saturation values to the first three groups of points. </p>
<pre><code>pts = {{{0.10, 485}, {0.22, 4... | David G. Stork | 9,735 | <p>Delete all vertices of degree = 2:</p>
<pre><code>g = RandomGraph[{20,30}, VertexLabels-> "Name"];
myVertexDegrees = VertexDegree[g, #] & /@ VertexList[g];
vertexestoremove = Flatten@Position[myVertexDegrees, 2];
mygraph = VertexDelete[g, vertexestoremove];
Graph[mygraph, VertexLabels -> "Name"]
</code></... |
116,196 | <p>Simple Question from a Stumped Rookie: </p>
<p>Using ListPlot, I can't get PlotStyle to look below the first level of a nested list to assign individual saturation values for each point; it assigns the first three saturation values to the first three groups of points. </p>
<pre><code>pts = {{{0.10, 485}, {0.22, 4... | yode | 21,532 | <hr />
<h1>Method 1</h1>
<pre><code>input = {13 -> 7, 7 -> 0, 0 -> 16, 16 -> 2, 2 -> 15, 10 -> 5, 5 -> 12,
12 -> 18, 18 -> 15, 17 -> 18, 15 -> 6, 6 -> 8, 8 -> 4, 9 -> 8,
4 -> 19, 19 -> 11, 11 -> 1, 1 -> 20, 20 -> 3, 3 -> 4, 14 -> 19};
g = Graph[inpu... |
2,050,484 | <p>Let $f$ be $\in L[a,b]$ Assume for any subinterval $I \subset [a,b]$ we have $\int_I f \geq |I|$ show that $f \geq 1$ a.e on $[a,b]$.</p>
<p>I started with a proof by contradiction. Assume there exists a set $E$ that has positive measure where $f < 1$ on E. Since f $\in L[a,b]$ we can find $|f| < N$ a.e. I... | daw | 136,544 | <p>$T-\lambda I$ is invertible if and only if $(T-\lambda I)^*=T^*-\lambda I$ is invertible: Since for every linear operator $A$ invertibility of $A$ and of $A^*$ are equivalent, which follows by taking the adjoints of, e.g., $AA^{-1}=I$ and $A^{-1}A=I$.</p>
|
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | CiaPan | 152,299 | <p>See Trapezoidal rule, e.g. at <a href="https://en.wikipedia.org/wiki/Trapezoidal_rule" rel="nofollow noreferrer">Wikipedia</a>.</p>
<p>You can show the function to be integrated is concave, so any approximation of its graph with a polygonal chain gives you a lower bound of the integral. Try to split the integration... |
3,141,004 | <p>I am just a high school student, so please excuse me if this question sound silly to you.</p>
<p>How do I graph this : <span class="math-container">$\dfrac{9}{x} - \dfrac{4}{y} = 8$</span></p>
<p>I could do trial and error, but is there a more systematic way that I can use to graph this equation.</p>
| SNEHIL SANYAL | 636,469 | <p>Notice the coefficients of <span class="math-container">$\frac{1}{x}$</span> and <span class="math-container">$\frac{1}{y}$</span>, we can alter <span class="math-container">$x$</span> in multiples of <span class="math-container">$3$</span> or <span class="math-container">$9$</span> to get corresponding values of <s... |
1,588,827 | <p>I'm trying to understand an algorithm, to solve $Ax = b$ linear equations.</p>
<p>But there is an equation, which I can not understand: $e^{-iAt}$
What does it mean, to calculate the $-iAt$ power of $e$ ?</p>
| MaxW | 23,782 | <p>It is a little unclear, but for the $-iAt$ power of $e$, I'm going to assume that $i$ is the $\sqrt{-1}$ so that the expression is an combination of a real and an <a href="https://en.wikipedia.org/wiki/Imaginary_number" rel="nofollow">imaginary number</a> as you can determine from the series expansion. Think of the ... |
1,945,144 | <p>I'm trying to figure out which kind of matrices can be written as $A^{k_{1}}B^{k_{2}}...A^{k_{n}}B^{k_{n}}$ where </p>
<p>A=$\begin{pmatrix} 1&1\\0&1\end{pmatrix}$ and B=$\begin{pmatrix} 2&1\\1&1\end{pmatrix}$.</p>
<p>I did it for $C=\begin{pmatrix} 1&2\\0&1\end{pmatrix}$ and $D=\begin{pmat... | reuns | 276,986 | <p>$A = \begin{pmatrix}1&1 \\ 0 & 1 \end{pmatrix}$ and $A^{-2}BA^{-1} = \begin{pmatrix}0&-1 \\ 1 & 0 \end{pmatrix}$ so the subgroup generated by $A,B$ is $SL_2(\mathbb{Z})$</p>
<p>Now for the monoid generated by $A,B$, I don't know.</p>
|
3,106,223 | <blockquote>
<p><span class="math-container">$$A = \left( \begin{array} { c c c c c } { 1 } & { 4 } & { 15 } & { 1
} & { - 9 } \\ { 0 } & { 1 } & { 4 } & { 0 } & { - 3 } \\ { 2 } & { -
4 } & { - 18 } & { - 1 } & { 15 } \\ { 1 } & { 0 } & { - 1 } & { 0 } &a... | YiFan | 496,634 | <p>The procedure is exactly the same as for square matrices. I'll do a simpler example for the nullspace, see if you can apply the idea to your case. Let's try to find the nullspace for
<span class="math-container">$$ A=\begin{bmatrix}1&2&3&4\\1&1&1&4\\0&1&2&3\end{bmatrix}. $$</span>... |
3,306,110 | <p>the expression <span class="math-container">$x(x-9)$</span> is resolved in the manner
<span class="math-container">$$x(x-9)=x^2-9x$$</span>
but why not like this?
<span class="math-container">$$x(x-9)=x^2-9$$</span>
where does the <span class="math-container">$9x$</span> come from?</p>
| John Joy | 140,156 | <p>What's the area of the dark green rectangle? The area of the light green rectangle is <span class="math-container">$A = x(x-9)$</span>, but it's also the difference between the square <span class="math-container">$x^2$</span> and the dark green rectangle. What is this difference? Can you see where the distributive p... |
1,904,786 | <p>How can I get a good approximation of the sum</p>
<p>$$\sum_{s=1}^\infty x^s\ln(s)$$</p>
<p>by hand ?</p>
<p>If I only consider the part of the function, where it is strictly decreasing, I can bound the sum by integrals. But how can I approximate the sum upto this point ?</p>
| Jack D'Aurizio | 44,121 | <p>Assuming $|x|<1$, by <a href="http://mathworld.wolfram.com/FrullanisIntegral.html" rel="nofollow">Frullani's theorem</a> we have
$$ \log(s) = \int_{0}^{+\infty}\frac{e^{-z}-e^{-sz}}{z}\,dz \tag{1}$$
hence
$$ \sum_{s\geq 1}x^s \log(s) = \int_{0}^{+\infty}\sum_{s\geq 1}\frac{(e^{-z}-e^{-sz})x^s}{z}\,dz = \frac{x^2}... |
1,922,325 | <p>How do you prove that if $m\phi(m)=n\phi(n)$ then $m=n$?</p>
<p>Here, $\phi$ is the Euler Phi function.</p>
| Robert Z | 299,698 | <p>Hint. Assume that $n\phi(n)=m\phi(m)$. Let $p$ be the largest prime factor of $n$ and let $p^k$ be the highest power of $p$ dividing $n$. Since $\phi(n)=\prod_{p^k \| n} p^{k-1} (p-1)$ it follows that $p^{2k-1}$ is the highest power of $p$ dividing $n\phi(n)$. Now, $p$ is the largest prime factor of $n\phi(n)$ which... |
2,316,076 | <p>I was given the following permutation
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
6 & 5 & 2 & 4 & 3 & 1 \\
\end{pmatrix}</p>
<p>and I was asked to write it as the product of disjoint cycles.</p>
<p>The disjoint cycles I found were $(1, 6)$ and $(2, 5, 3)$, which my ... | Marcus Andrews | 97,648 | <p>You can have $(4)$ if you want. It may be "fixed," but it's also a "$1$-cycle" disjoint from the rest, so it still fits within the definition.</p>
<p>It's really just a matter of notation and clarification based on what's convenient. If you are omitting $1$-cycles, however, it's usually a good idea to state the per... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | Jessica B | 4,746 | <p>A purely personal opinion:</p>
<p>Your title asks about the importance of exact pictures in geometry. I would say <em>exact</em> pictures are really not needed, because the point of geometry is to learn to reason from geometrical principles without needing to rely on precise measurements. Indeed, I would say that o... |
2,230,507 | <p>I want to compute the smallest nonzero singular value of the matrix A, which is defined as follows. </p>
<p>Let $B = rand(500, 250)$, $A = B*B^t$, where $t$ denotes the transpose of the matrix. </p>
<p>I found the following matlab code to compute singular values of the matrix A which is based on the Singular val... | dantopa | 206,581 | <p><strong>Summary</strong></p>
<p>The answer is...use <code>svds</code>. </p>
<p><strong>What are the singular values?</strong></p>
<p>There may be some confusion over how you get the singular values. The command <code>svd</code> computes the singular values and the components that you don't want. The command <code... |
3,990,110 | <p>I was seeking the other day for intuition about the Hölder inequality and more precisely the geometric intuition behind the relation <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>.</p>
<p>I found <a href="http://www.tricki.org/article/Geometric_view_of_H%C3%B6lders_inequality" rel="nofollow nore... | Daniele Tampieri | 317,063 | <p>In my opinion, the correct answer to the question has been already given by John Dawkins: thus the following one is simply an illustration of his answer based on the proof of Hölder's inequality given in reference [1], §4.1, pp. 100-101, the best I've seen in print.</p>
<p>The proof given by Vol'pert and Hudjiaev st... |
77,958 | <p>I would like to ask the following: if for a group <span class="math-container">$G$</span> the homology <span class="math-container">$H_n(G,\mathbb{Z})$</span> is <span class="math-container">$\mathbb{Z}$</span>-torsion for every <span class="math-container">$n\geq n_0$</span>, then what can be said concerning <span ... | Chris Gerig | 12,310 | <p>This Kunneth formula still holds, I proved it here:</p>
<p><a href="https://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient/75485#75485">Kuenneth-formula for group cohomology with nontrivial action on the coefficient</a></p>
<p>which holds in our sit... |
8,263 | <p>1.What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this.</p>
<p>2.For how many two-digit numbers, the sum of the digits is greater than the product of the digits?</p>
| yunone | 1,583 | <p>A good approach to these is to reduce your base number according to the modulus. So since $15\equiv 1\pmod{7}$, you would have $15^{89}\equiv 1^{89}\equiv 1\pmod{7}$. In other words, $15^{89}$ is 1 greater than a multiple of $7$, or 6 less than a multiple of $7$. So to get a complete solution set, you could add any ... |
270,683 | <p>How can I make a function, say, <code>R[expression]</code> such that it works on arguments like this <code>Integrate[f[x],{x,0,Infinity}]</code>, applying some operator <code>r[]</code> to the function under integral, and leaving the epression unevaluated if the first level is not an integral from zero to infinity?<... | Sjoerd Smit | 43,522 | <p>To elaborate on my comment:</p>
<pre><code>ClearAll[R]
SetAttributes[R, HoldFirst]
R[Integrate[f_[x_], {x_, 0, Infinity}]] := Integrate[r[f[x]], {x, 0, Infinity}]
R[Integrate[Sin[x], {x, 0, Infinity}]]
R[Integrate[Sin[x], {x, 0, 1}]]
R[Integrate[Sin[x], x]]
R[Sin[x]]
</code></pre>
<p>You specified that the expressi... |
2,031,779 | <p>Let $0<\beta <1$ and $a_0, a_1 \in \mathbb R$. I want to show that this sequence converges: </p>
<p>$$a_{n+1} = \beta a_n + (1-\beta) a_{n-1}$$</p>
<p>Furthermore, it is given that there you can explicitly formulate the sequence as $$c_1\lambda_1^n+c_2\lambda_2^n$$</p>
<p>I played it through with $a_5$ and... | Dr. Wolfgang Hintze | 198,592 | <p>Obviously, any constant $B$ is a solution. A second independent solution is found by letting</p>
<p>$$a(n)=A q^n$$</p>
<p>Inserting this into the recursion gives </p>
<p>$$q=\beta -1$$</p>
<p>Hence the general solution is</p>
<p>$$a(n)=A (\beta -1)^n+B$$</p>
<p>As $0<\beta < 1$ the term with $A$ goes to ... |
3,759,708 | <p>I already know that <span class="math-container">$b_{n+1} = a_n +3b_n$</span> and <span class="math-container">$a_{n+1} = 3a_n - b_n$</span>. So</p>
<p><span class="math-container">$a_{n+2} = 3(3a_n-b_n)-(3b_n+a_n) = 9a_n-3b_n-3b_n-a_n = 8a_n-6b_n$</span> and</p>
<p><span class="math-container">$b_{n+2} = 8b_n+6a_n$... | TYeung | 579,878 | <p>Consider <span class="math-container">$b_{n+1}=a_n+3b_n$</span>, rearranging gives
<span class="math-container">$$a_n=b_{n+1}-3b_n \tag{1}$$</span>
Also consider <span class="math-container">$a_{n+1}=3a_n-b_n $</span> rearranging gives
<span class="math-container">$$b_n=3a_n-a_{n+1} \tag{2}$$</span>
Subsitute <span ... |
3,759,708 | <p>I already know that <span class="math-container">$b_{n+1} = a_n +3b_n$</span> and <span class="math-container">$a_{n+1} = 3a_n - b_n$</span>. So</p>
<p><span class="math-container">$a_{n+2} = 3(3a_n-b_n)-(3b_n+a_n) = 9a_n-3b_n-3b_n-a_n = 8a_n-6b_n$</span> and</p>
<p><span class="math-container">$b_{n+2} = 8b_n+6a_n$... | doobdood | 800,483 | <p>It looks like you made a substitution error when writing
<span class="math-container">$$8a_n−6b_n+r(3a_n−b_n)+sa_n=8a_n−6b_n+r(3b_n+a_n)+sb_n$$</span>
Here, you substituted <span class="math-container">$8a_n-6b_n$</span> for <span class="math-container">$b_{n+2}$</span> instead of <span class="math-container">$8a_n+... |
2,330,368 | <p>I have calculate the sum of the following power series: $$\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n$$</p>
<p>I've determined the convergence radius $R = 1$convergence interval $I_c = (-1,1)$ and the convergence set $A = (-1,1)$ for this power series. Furthermore, I've tired to bring it to a from which can be easily comp... | user254433 | 254,433 | <p>Recall that $F(x)=\frac{1}{1+x}=\sum_{n=0}^\infty (-1)^n x^n$. Differentiating, we get
$$
F'(x)=\sum_{n=0}^\infty (-1)^n n x^{n-1}=-\sum_{n=0}^\infty(-1)^{n}(n+1)x^n.
$$
Differentiating again:
$$
F''(x)=\sum_{n=0}^\infty (-1)^{n+1}(n+1)n x^{n-1}=\sum_{n=0}^\infty(-1)^{n}(n+2)(n+1)x^n.
$$
Thus,
$$
F''(x)+F'(x)=\sum_... |
1,770,180 | <p>I was solving this limit from the Demidovich's book of exercises:
$$\lim_{n\to\infty} \frac{\sqrt[n]{\vphantom{\Large a}\, n!\,}}{n}$$</p>
<p>and I managed to get it to this state but then I got stuck:</p>
<p>$$e^{\frac{1}{n}(\log(n) + \log(n - 1) + ... + \log(1)) - \log(n)}$$</p>
<p>where $\log x$ is a natural l... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
1,104,450 | <p>The truth table for a logical disjunction shows that there is only one situation where the result can be false, being when both statements are false. As long as one statement is true, the result is also true. </p>
<p>This makes perfect sense to me, but I am wondering why this is the case. Could it not have been equ... | Lord_Farin | 43,351 | <p>You are right that there is no reason to have this precedence in propositional logic.</p>
<p>Therefore, we have both the definitions of conjunction and disjunction:</p>
<p>$$p \lor q \leadsto \text{At least one of the following is }\color{red}{\text{true}}: p, q.\\
p \land q \leadsto \text{At least one of the foll... |
1,938,341 | <p>Prove, that $\forall n \in \mathbb{N}$ the following identity holds:
$$a^n - b^n = (a-b)\sum_{k=0}^{n-1} a^k b^{n-k-1}.$$</p>
<hr>
<p>For $n = 1$ we get $a^1 - b^1 = a - b$ on LHS, and on RHS we get $(a-b)\sum_{k=0}^{0}a^k b^{n-k-1} = a-b.$ Both sides are equal, so identity hold for $n = 1$.</p>
<hr>
<p>Assume t... | Aysha A. | 202,076 | <p>To prove that
$$b^n-a^n =(b-a)\sum_{k=0}^{n-1}b^ka^{n-1-k}$$
It suffices to show that $$b^n-1 = (b-1)\sum_{k=0}^{n-1}b^k,$$
as follows:
$$\begin{align}
(b-1)\sum_{k=0}^{n-1}b^k &=b\sum_{k=0}^{n-1}b^k-\sum_{k=0}^{n-1}b^k\\
&=\sum_{k=0}^{n-1}b^{k+1}-\sum_{k=0}^{n-1}b^k\\
&=\sum_{k=1}^{n}b^{k}-\sum_{k=0}^{... |
2,014,156 | <p><a href="http://www.wolframalpha.com/input/?i=lim%20(sum%201%2F(a%2Bn)%5E2,%20n%20%3D%201%20to%20infinity),%20a%20to%20infinity" rel="nofollow noreferrer">WolframAlpha</a> says that the sequence
$$
S_n = \sum\limits_{i=1}^{\infty}\frac{1}{(n+i)^2}
$$
tends to $0$. The proof given by the site uses notations unfamilia... | Rob Arthan | 23,171 | <p>Your thinking is along the right lines, but you are not reversing the order of two limiting processes here. There is just one limiting process and you are just reindexing. Here is how I would express it: If $\sum_{m=1}^{\infty}x_m$ is any convergent series with sum $x$ say, then the "tail end" sequence $t_n= \sum_{m... |
1,121,492 | <p>We define e to be a number which satisfies the following condition</p>
<p>$$\lim _{a \to 0} \frac{e^a-1}{a}=1. $$</p>
<p>How did we arrive to the following from above equation</p>
<p>$$e=\lim _{n \to \infty} \bigg(1+\frac{1}{n}\bigg)^n ? $$</p>
<p>so that we get the value of n</p>
| Paramanand Singh | 72,031 | <p>The approach you mention is difficult, but possible. I have presented it in my <a href="http://paramanands.blogspot.com/2014/05/theories-of-exponential-and-logarithmic-functions-part-3.html" rel="nofollow">blog post</a>. The main steps are as follows:</p>
<p>1) Define $a^{b}$ (without using any logs or $e$) rigorou... |
434,543 | <p>I have an $m \ge5$ dimensional vector $x$. How i can construct a rotation matrix $A$ s.t: $x \cdot A$ would be a vector rotated by $\theta$ angle. This is trivial for 2 dimensions, but what for this case? For instance will it work if A is this:
$A = \begin{bmatrix}
cos \theta & -\sin\theta&0&0&0 \\
... | Muphrid | 45,296 | <p>It's relatively easy to construct such a rotation map using clifford algebra; the hardest part is probably learning the clifford algebra itself. However, if you can wrap your head around it, it's a very powerful tool to use for geometric problems in higher dimensions.</p>
<hr>
<p>Clifford, or <em>geometric</em> a... |
3,988,390 | <p>I am trying to solve the problem: <span class="math-container">$x^{2}+xy+y^{3}=0$</span> using implicit differentiation.</p>
<p>My workings:</p>
<p><span class="math-container">$(1)\,\,\,{d\over dx}{[x^2]}\,+{d\over dx}{[xy]}\,+\,{d\over dx}{[y^3]}={d\over dx}{[0]}$</span></p>
<p><span class="math-container">$(2)\,\... | GossipM | 874,994 | <p>You can't conclude that <span class="math-container">$\frac{d}{dx}(xy)=y$</span>, since this is effectively the product of the functions <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. Hence, using the product rule gives
<span class="math-container">$$\frac{d}{dx}(xy)=y\frac{d}{d... |
4,236,301 | <p>Taking derivatives both sides with respect to <span class="math-container">$a$</span> in the definition of homogeneous distribution equation <span class="math-container">$$a^{-d}\int f(x)\phi(\frac{x}{a})=a^k\int f(x)\phi(x)$$</span> for all <span class="math-container">$a>0$</span> and the test function <span cl... | Petra Axolotl | 291,842 | <p>Only the "only if" part was treated in the original question and the previous answer. Here is the "if" part.</p>
<p>Let <span class="math-container">$\Phi(a) = F_a(\varphi) = F(\varphi^a)$</span> for <span class="math-container">$a>0$</span>, <span class="math-container">$\varphi \in \mathcal{... |
1,627,830 | <p>I have to calcuate this limit: $$\lim _{x\to 0}\left(\frac{1-\frac{\left(1+x\right)^{\frac{1}{x}}}{e}}{x}\right)$$</p>
<p>I have no idea how to start, maybe taylor series?</p>
<p>Thanks.</p>
| Clement C. | 75,808 | <p>Detailed approach, using Taylor series. You will need the following two expansions (to order 2 and 1):
$$\begin{align}
\ln(1+x) &= x-\frac{x^2}{2} + o(x^2) \\
e^x &= 1+x + o(x)
\end{align}
$$
around $0$.</p>
<p>Now, starting from inside the parenthesis and working our way to the full expression:
$$
\frac{1}... |
894,969 | <p>Broadly speaking, when <em>numerically</em> minimizing a <em>d</em>-dimensional objective function:</p>
<ul>
<li><p>Gradient descent generally requires more iterations, but each iteration is fast<br />
(we only need to compute 1st derivatives)</p>
</li>
<li><p>Newton's method generally requires fewer iterations, but... | Mikhail Katz | 72,694 | <p>In general a function one is trying to find a zero of is given by an explicit formula so the calculation of the derivative is a single-occurrence event and need not to be incorporated into the iteration. For the advantages of Newton's method see <a href="http://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow"... |
856,814 | <p>For which x does the function $f(x) = x^3-6x^2-5x+5$ assume its maximum value on the interval $[-5,5]$?</p>
<p>The critical points for this function are $\frac{12 + \sqrt{204}}{6}$ and $\frac{12 - \sqrt{204}}{6}$.
The end points are -5 and 5.</p>
<p>The maximum value is solved to be for $\frac{12 - \sqrt{204}}{6}\... | André Nicolas | 6,312 | <p>We have to make some assumptions. The technically easiest assumption is that the events "hit the bottle" on the various trials are <strong>independent</strong>. This is not necessarily realistic. Maybe one gets rattled if one misses a few times. Or maybe one learns to correct one's aim. </p>
<p>Let us assume indepe... |
1,731,227 | <p>Find he local maximum and minimum value and saddle points of the function: $$f(x,y)=x^2-xy+y^2-9x+6y+10$$</p>
<p>The answer is a min of $(-4,1), f(-4,1)=73$</p>
<p>I got a min of $(12/5,-21/5)$
my
$$
f_x=2x-y-9\\
f_y=-x+2y+6
$$
set $f_x = 0 = f_y$ and we get $x=(9+y)/2$ so $y=-21/5$ and $x=12/5$</p>
<p>I found ou... | gt6989b | 16,192 | <p>You are solving the system, and your solution does not satisfy the 2 equations.
The second is easier to use, you get $x=2y+6$ so $4y+12-y-9=0$, or equivalently, $y=-1$ and then $x = 6-2=4$...</p>
|
51,810 | <p>How do you individually rotate labels on a pie chart? For example, I have this chart detailing money composition.</p>
<pre><code>Chart2 = PieChart[{10, 5, 22, 63}, ImageSize -> Large,
ChartLabels ->
Placed[{"Gold (10%)", "Silver (5%)", "Currency (22%)",
"Demand Deposits (63%)"}, "RadialCenter", ... | Nasser | 70 | <p>Use Rotate on the specific Labels instead of on all them as group? And use <code>Style</code> on each to change font color and size</p>
<pre><code> Chart2 = PieChart[{10, 5, 22, 63},ImageSize -> Large,
ChartLabels -> Placed[{Rotate[Style["Gold (10%)", Red, 20], 45 Degree],
Rotate["Silver (5%)", -30 Degree... |
51,810 | <p>How do you individually rotate labels on a pie chart? For example, I have this chart detailing money composition.</p>
<pre><code>Chart2 = PieChart[{10, 5, 22, 63}, ImageSize -> Large,
ChartLabels ->
Placed[{"Gold (10%)", "Silver (5%)", "Currency (22%)",
"Demand Deposits (63%)"}, "RadialCenter", ... | kglr | 125 | <p>With a custom <code>ChartElementFunction</code> you can automate the angles of the labels (using tricks similar to the ones used in <a href="https://mathematica.stackexchange.com/a/10939/125">this Q/A</a>)</p>
<pre><code>ClearAll[cedFunc];
cEDF = ChartElementDataFunction;
cedFunc[datafunc_: cEDF["Sector"]][{{t0_, t... |
4,295,884 | <p>The challenge is to solve this equation <span class="math-container">$2^{x}+7^{y}=9^{z}$</span> in positive integers. The obvious solution is <span class="math-container">$x=y=z=1$</span>. Using brute force, I found <span class="math-container">$3$</span> possible solutions:
<span class="math-container">\begin{eqnar... | Derek Luna | 567,882 | <p>Refer to Henri Cohen's <em>Number Theory Volume I: Tools for Diophantine Equations</em>
p. <span class="math-container">$410-411$</span> and also <span class="math-container">$461$</span> for exercise <span class="math-container">$35$</span>.</p>
<p>For <span class="math-container">$x = 1$</span>, the equation becom... |
2,029,326 | <p>In the latest episode of Survivor there were 6 people and they had to draw rocks from a bag. In the bag were 6 rocks, 5 white rocks and 1 black rock.
First person drew a rock and kept it in his hand.
Then the second person drew a rock.
Last person took the only rock that left. </p>
<p>Then they all showed what the... | Nicolas FRANCOIS | 288,125 | <p>The answer is yes.</p>
<p>First point of view is they can all draw the rocks at the same time without changing the outcome.</p>
<p>Another way of seeing this is to compute the probability that the $k$th person picks the black stone. The $k-1$ persons before must have picked a white stone, and the $k$th picks the b... |
2,148,196 | <p>The below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right? </p>
<p><a href="https://i.stack.imgur.com/yvWQY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yvWQY.png" alt="enter image description here"></a></p>
| GEdgar | 442 | <p>Reese is right.</p>
<p>Plus: you cannot do IC or IL or VC or XM, etc. Only certain very specific cases are allowed for subtraction. </p>
<p>These restrictions are how we do it nowadays. But the classical Romans themselves may not have been so strict. I posted <a href="https://hsm.stackexchange.com/a/776/229">HE... |
2,410,118 | <p>Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?</p>
<p>The question is as follows :</p>
<blockquote>
<p>n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities if</p>
<p>b) O... | epi163sqrt | 132,007 | <p>A variation:
\begin{align*}
&a_0=1,\ a_2=k+1,\ a_4=2k+1,\ \ \ldots,\qquad\qquad &a_{2n}&=nk+1\qquad\qquad &n\geq 0\\
&a_1=k,\ a_3=2k,\ \ \ \ \ \ a_5=3k,\ \ \ \ \ \ \ \ \ \ldots,\qquad\qquad &a_{2n-1}&=nk\qquad\qquad &n\geq 1\\
\end{align*}</p>
<blockquote>
<p>It follows for $n... |
2,322,938 | <p>The powerset of the empty set is {$\emptyset$}. Attempt:</p>
<p>The power set of the powerset of the empty set is { { {$\emptyset$} } }? </p>
<p>Next I want to find the powerset of the power set of the powerset of the empty set P(P(P($\emptyset$))). </p>
| fleablood | 280,126 | <p>If it makes things easier you can use labels. The "whooo, trippy, power set of a power set of an empty set" nature shouldn't distract us from what is actually straightforward.</p>
<p>$A = \emptyset = \{\}$ is a set with zero elements.</p>
<p>$B= P(A) = \{A\} = \{\emptyset\} = \{\{\}\}$ is a set with $2^0 = 1$ el... |
2,046,729 | <p>In my book there is a solved question . <a href="https://i.stack.imgur.com/O4LpL.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/O4LpL.gif" alt="enter image description here" /></a></p>
<p>In that question I could not understand from where does <span class="math-container">$(3\lambda + 2)$</span> ... | user160738 | 160,738 | <p>If $3n-2=7m+2$, then adding $5$ to both sides</p>
<p>$$
3(n+1)=7(m+1)
$$</p>
<p>Hence $3|(m+1)$ so that $m=3\lambda +2$ for some integer $\lambda$.</p>
|
219,793 | <p>I have the following function </p>
<p><code>f[z_] := Piecewise[{{Cos[z] - 17 Sin[z]/z, z > 0}, {Cosh[z] - 17 Sinh[z]/z, z <= 0}}]</code></p>
<p>I am interested in the regions where <span class="math-container">$\vert f(z)\vert \leq 1$</span>, I tried to shade that region using <code>Filling</code>, but my re... | chyanog | 2,090 | <pre><code>Cases[Permutations[Range[0,9],{3}],
v:{a_,b_,c_}/;
And[
(a==6||b==8||c==2)&&Length[v⋂{6,8,2}]==1,
(a!=6&&b!=1&&c!=4)&&Length[v⋂{6,1,4}]==1,
(a!=2&&b!=0&&c!=6)&&Length[v⋂{2,0,6}]==2,
(a!=7&&b!=3&&c!=8),
(a!=7&&... |
2,980,201 | <p>I'm trying to prove that:</p>
<p><span class="math-container">$$
\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty
$$</span></p>
<p>where <span class="math-container">$|r_1|, |r_2| < 1$</span> and <span class="math-container">$c_1, c_2$</span> are some arbitraty real numbers (<span class="math-container">$r_... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that by <a href="https://en.wikipedia.org/wiki/Triangle_inequality" rel="nofollow noreferrer">triangle inequality</a></p>
<p><span class="math-container">$$|c_1r_1^j + c_2r_2^j| \le |c_1r_1^j| + |c_2r_2^j|$$</span></p>
|
204,037 | <p>Let $H$ and $K$ be two subgroups of $G$ of finite order. Can $HK$ be infinite?</p>
| Yoni Rozenshein | 36,650 | <p>Let $G$ be the group of words made up of the characters 0 and 1, such that no character appears twice in a row. We define the group operation to be concatenation and also $0^2 = 1^2 = \epsilon$ where $\epsilon$ is the empty word.</p>
<p>Let $H$ be the subgroup $\{ \epsilon, 0 \}$ and $K$ be the subgroup $\{ \epsilo... |
204,037 | <p>Let $H$ and $K$ be two subgroups of $G$ of finite order. Can $HK$ be infinite?</p>
| M Turgeon | 19,379 | <p>Ok, since it seems that the general consensus is to look at the subgroup generated by $HK$, here is another example.</p>
<p>The modular group $SL_2(\mathbb Z)$ is generated by two matrices of finite order:
$$\begin{pmatrix} 0&1\\-1&0\end{pmatrix},\text{ and }\begin{pmatrix} 0&1\\-1&0\end{pmatrix}\be... |
952,922 | <p>Evaluate <span class="math-container">$\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$</span></p>
<p>May I verify if my solution is correct? Thank you. </p>
<p>Consider <span class="math-container">$\gamma_1:=\{x:-\dfrac{1}{r}\leq x \leq-r\}, \ \gamma_2:=\{re^{it}:... | Arian | 172,588 | <p>I am providing an alternative route towards the evaluation of this integral.
Consider a change of variable $x^2=u$ then the integral can be expressed as
$$\int^{\infty}_{0}\frac{x^{\alpha}}{(1+x^2)^2}\,dx=\int^{\infty}_{0}\frac{u^{(\alpha-1)/2}}{2(1+u)^2}\,du=\frac{1}{2}\int^{\infty}_{0}\frac{u^{(\alpha-1)/2}}{(1+u)... |
1,552,292 | <p>I would like to share a point on which I was really stuck with you, in order for you to help me to see my mistake:</p>
<p>Here is the theorem:</p>
<p><strong>A set X is infinite if and only if it is equipotent (equivalent) to a proper subset of itself</strong></p>
<p>And here is my question.
The set of positive i... | Mark Bennet | 2,906 | <p>Every set other than the empty set has a subset to which it is not equipotent. So having a subset "smaller" than the original set is a property equivalent to being non-empty (in standard versions of set theory, anyhow).</p>
<p>The property of being infinite is characterised by having at least one proper subset whic... |
1,469,737 | <p>I think I have a solution so this sequence in:</p>
<p>$a_n = \lfloor \frac{n-1}{2}\rfloor + n$.</p>
<p>I have been plugging numbers in and so far it works. However, there must be a better way to check if this indeed a solution.</p>
<p>My questions are:</p>
<p>1) Is this a solution, and</p>
<p>2) What is the bes... | Sam Weatherhog | 258,916 | <p>There are at least two ways to do this. Firstly, because you are looking at divisibility, you can generally split the proof into cases. The second option you have is to use induction since it seems you are trying to prove this for $n\ge 1$ (or some other integer).</p>
<p>The first way is more general in that you ca... |
2,617,388 | <p>I've an interesting question on my hand which I've approach several others but all of them gives me different insights to this probability question. </p>
<p>Here it is,</p>
<blockquote>
<p>The incidence of a suspicious transaction in a bank is <code>1 in 149</code>. They are able to correctly identify a legitima... | 5xum | 112,884 | <p>Let $S$ denote the event "the transaction is suspicious", and $I$ denote the event "the transaction is identified as suspicious"</p>
<p>What the question is asking for is $$P(\neg S| I)$$ (since $\neg S$ is the event "the transaction is not suspicious", i.e. "the transaction is legitimate").</p>
<p>Now, we use goo... |
2,617,388 | <p>I've an interesting question on my hand which I've approach several others but all of them gives me different insights to this probability question. </p>
<p>Here it is,</p>
<blockquote>
<p>The incidence of a suspicious transaction in a bank is <code>1 in 149</code>. They are able to correctly identify a legitima... | paw88789 | 147,810 | <p>On average for $14900$ transactions:</p>
<p>$100$ are suspicious, and of these $92$ are identified as suspicious;</p>
<p>and</p>
<p>$14800$ are legitimate, and of these $1184$ are (mis)identified as suspicious.</p>
<p>So out of $1276$ transactions identified as suspicious, $1184$ are in fact legitimate (on avera... |
4,321,423 | <p>Let <span class="math-container">$(X, \mathcal{A}, \mu)$</span> be a finite measure space. Fix <span class="math-container">$1 < p, q < \infty$</span> with <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>. Let <span class="math-container">$\{f_n\}$</span> be a sequence in <span class="math-c... | Umberto P. | 67,536 | <p><strong>If</strong> <span class="math-container">$\phi$</span> is a bounded linear functional on <span class="math-container">$L^q(\mu)$</span> then there exists <span class="math-container">$f \in L^p(\mu)$</span> with <span class="math-container">$\phi(g) = \displaystyle \int fg \, d\mu$</span>. You don't have to ... |
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | Avitus | 80,800 | <p>Do like this:</p>
<p>$\int \frac{y'}{y}dy=\int\frac{1}{4x-x^2}dx$, </p>
<p>i.e.</p>
<p>$ln(y)=\int\frac{1}{4x-x^2}dx+C$,</p>
<p>where $C$ denotes an arbitrary constant.</p>
<p>As</p>
<p>$\frac{1}{4x-x^2}=\frac{1}{4}(\frac{1}{x}+\frac{1}{4-x})$ (check it!)</p>
<p>then</p>
<p>$ln(y)=\int(\frac{1}{4}(\frac{1}{x... |
3,051,452 | <p>How can I prove that <span class="math-container">$\mathbb Q \times \mathbb Q$</span> is connected? I was trying to prove that it isn't because <span class="math-container">$\mathbb Q$</span> is not connected but I can't find two open <span class="math-container">$A,B$</span> to prove it.</p>
| zhw. | 228,045 | <p>The map <span class="math-container">$f(x,y)=x$</span> is continuous on <span class="math-container">$\mathbb R^2.$</span> If <span class="math-container">$\mathbb Q\times \mathbb Q $</span> were connected, then <span class="math-container">$f(\mathbb Q\times \mathbb Q)$</span> would be connected. But <span class="m... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| bill s | 1,783 | <p>There is a function meant to build structures like this -- it's called <code>Upsample</code>. First you build the nonzero stuff and then upsample. So for instance:</p>
<pre><code>xs = Range[10] x;
Upsample[xs, 3, 3]
</code></pre>
<p>{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, 0, 4 x, 0, 0, 5 x,
0, 0, 6 x, 0, 0, 7... |
1,046,828 | <p>Suppose we have 2 lines $$l_1 : x = 5 , \frac{y}{3-\alpha}=\frac{z}{-2}$$ and $$ l_2: x= \alpha , \frac{y}{-1}= \frac{z}{2-\alpha}$$ so what will be value of $\alpha$ for lines to be coplaner ? </p>
<p>I believe that this is unsymmetrical form of lines equation and we have lines from intersecting planes in each of ... | Frank Mayer | 196,771 | <p>Both lines have constant $x$ values. If their $x$ values differ, then they can only be coplanar if the slopes in yz are equal. This is the equation you have set up, which gives $\alpha = 1 \mbox{ or } 4$</p>
<p>Another solution is given by forcing the two lines to both be in the plane $x = 5$, which comes by settin... |
1,780,751 | <p>If $$I=\int\sin 2x\, \mathrm{d}x$$
since $$(\sin^2x)'=2\cos x\sin x=\sin 2x$$ then $$I=\sin^2x+C$$</p>
<p>However using substitution
$$I=\int \frac{\sin2x}{-2\sin 2x}\,\mathrm{d}u,~~~~\textrm{s.t}~~~ u=\cos2x$$
giving $$I=-\frac{1}{2}\int\,\mathrm{d}u=-\frac{1}{2}\cos2x+C=\sin^2x-\frac{1}{2}+C$$</p>
<p>I have chec... | DonAntonio | 31,254 | <p>The difference between both results is <em>a constant</em>: they both are correct since indefinite integral is determined <strong>only up to a constant</strong>, so for example</p>
<p>$$\int x\;dx=\frac12x^2+C=\frac12x^2-567+K=\frac12x^2+1000+C_1\;,\;\;\text{etc.}$$</p>
|
340,485 | <p>If $f^{'}(3) = 2$ then find</p>
<p>$$\lim_{h \rightarrow 0}\left(\frac{f(3+h^2)-f(3-h^2)}{2h^2} \right)$$</p>
| Ron Gordon | 53,268 | <p>Write </p>
<p>$$f(3 \pm h^2) = f(3) \pm h^2 f'(3) + O(h^3)$$</p>
<p>Then</p>
<p>$$\lim_{h \rightarrow 0} \frac{f(3+h^2)-f(3-h^2)}{2h^2} = f'(3)$$</p>
|
1,763,377 | <p>Let $T:V \to V$ be a linear operator such that $T^5 \neq 0,$ but $T^6 = 0.$ Suppose $V$ is isomorphic to $\mathbb{R}^6.$ Prove that there does not exist an $S:V \to V$ such that $S^2 = T.$ Does the answer change if $V$ is isomorphic to $\mathbb{R}^{12}?$</p>
<p>My thoughts for the first part: Suppose such an $S$ ... | Alex Wertheim | 73,817 | <p>Your argument is mainly correct, but you ought to make rigorous the argument that $S^{12} = 0$ implies $S^{6} = 0$. The way that I know of to do this is using the minimal polynomial: $S$ is a root of $f(X) = X^{12}$, so the minimal polynomial of $S$ divides $f(X)$. Thus, the minimal polynomial is $X^{n}$ for some $n... |
1,068,183 | <blockquote>
<p>If $x,y \in \mathbb Z$ , then find all the solutions of<br>
$$y^2=3x^4+3x^2+1$$ </p>
</blockquote>
<p>I was asked this question by my friend who said that he encountered this while solving another problem. I have tried several things but am unable to solve this question. Moreover, this has to be... | Kieren MacMillan | 93,271 | <p>The only solutions are $(x,y)=(0,-1)$ and $(x,y)=(0,+1)$. I gave an elementary proof <a href="https://math.stackexchange.com/a/1012777/93271">here</a>.</p>
|
2,748,162 | <blockquote>
<p>An elevator with capacity for 10 people is designed to support a maximum load of 750kg.If the weights of people that use the elevator are distribuited with mean 70kg and standard deviation 8kg. Which is the probability that a group of 10 people exceed the maximum load ? </p>
</blockquote>
<p>Attempt... | Community | -1 | <p>The answers here love to indulge in normal distributions but that is a red herring. Obviously weight is not normally distributed since the probability of one person weighing more than double of another is non-zero while the probabily of one person weighing negative (again, a deviation of more than the other person'... |
1,771,845 | <p>The problem is to find the number of real solutions of a cubic equation. This exercise is in a book, in the chapter about functions, limits and continuity. This chapter is before the chapter about derivatives therefore I assume that they cannot be used to solve the exercise.</p>
<p>$$f(x) = x^3-9x^2+24x-17 = 0$$</p... | un umile appassionato | 281,249 | <p>$f(2) > 0$ and $f(4)<0$, so $f$ has a zero in $[2, 4]$.</p>
|
257,279 | <p>The <strong>Proper Forcing Axiom</strong> kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is known, for example, that it implies the <em>Singular Cardinal Hypothesis</em>, SCH.</p>
<p>In <a href="... | Joel David Hamkins | 1,946 | <p>If you replace the reals $\mathbb{R}$ with Cantor space $2^{\mathbb{N}}$ or with Baire space $\mathbb{N}^{\mathbb{N}}$ (homeomorphic to the space of irrationals), then the answer is yes. Indeed, one can have the function defined on the whole space of sequences, not just the injective ones. </p>
<p>To see this, defi... |
1,358,140 | <p>I haven't been able to solve this limit, so help would be appreciated.
L'Hopital is not allowed.
Thanks in advance.</p>
| zhw. | 228,045 | <p>Let $f(x) = xe^{x^2}.$ The limit we want is</p>
<p>$$\lim_{x\to 1}\frac{f(x)- f(1)}{x-1}.$$</p>
<p>By definition, this is $f'(1).$ So just calculate $f'(1)$ (and no, we're not using L'Hopital here).</p>
|
3,134,179 | <p>Let <span class="math-container">$f \colon R \to R $</span> a continuous and periodic function, with period <span class="math-container">$T$</span>.
Show that <span class="math-container">$\lim_{n \to \infty} (\int_{a}^{b} f(nx)dx)=\frac{b-a}{T}\int_0^T f(x)dx$</span>, con <span class="math-container">$a < b$</s... | Kavi Rama Murthy | 142,385 | <p>This is very easy if you know that <span class="math-container">$f$</span> can be approximated uniformly by linear combinations of the functions <span class="math-container">$e^{i\frac {2\pi} T kx}$</span>. For this last function both sides are <span class="math-container">$0$</span> of <span class="math-container">... |
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