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 |
|---|---|---|---|---|
3,316,970 | <p>I have been having some difficulties with this question. </p>
<p>How to find the maximum without the help of a calculator or graphing device?</p>
| Anurag A | 68,092 | <p>For each <span class="math-container">$m$</span>, Consider the line <span class="math-container">$\ell_m$</span> given by <span class="math-container">$y+mx=0$</span>. Observe that <span class="math-container">$\ell_m$</span> passes through the origin. Also consider the point <span class="math-container">$(1,1)$</sp... |
3,099,652 | <p>Can anyone help me with this!?
If <span class="math-container">$n=p_1^{k_1},p_2^{k_2},\ldots $</span>
Then I applied the given condition of divisibility of <span class="math-container">$\varphi(n)$</span> but can't reach to a conclusion.</p>
| jmerry | 619,637 | <blockquote>
<p>My question is: is there an analytical way to determine if such a function exists?</p>
</blockquote>
<p>There's a theorem for that. Specifically, the existence-uniqueness theorem for differential equations.</p>
<p><a href="https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" rel="nofo... |
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| David | 119,775 | <p>No need to do any calculations at all: since we are talking about numbers between $0$ and $1$, a cube root is larger than a square root:
$$\Bigl(\frac12\Bigr)^{1/3}>\Bigl(\frac12\Bigr)^{1/2}>\Bigl(\frac13\Bigr)^{1/2}\ .$$</p>
|
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| Alex | 38,873 | <p>What's wrong with just logging? Logarithm is a monotone increasing function, so the inequality sign stays the same. </p>
<p>First log, the multiply both sides by 2 and 3. LHS becomes $\log (\frac{1}{2})^2$, right $\log(\frac{1}{3})^3$. Now exponentiate. LHS is $\frac{1}{2} \cdot \frac{1}{2} \cdot 1$, RHS is $\frac{... |
240,669 | <p><strong>Bug introduced in version 12.0.0, and persisting through 13.2.0 on Windows. Doesn't reproduce on ARM Mac versions 13.0.0 and above.</strong></p>
<hr />
<p>Calculating the integral <span class="math-container">$$\int\limits_0^1 \frac{x^2\log(1-x^4)} {1+x^4}\,dx$$</span> symbolically</p>
<pre><code>Integrate[x... | Andreas | 69,887 | <p>The integration with help of Rubi gives after inserting the limits:</p>
<pre><code>(1/(4*Sqrt[2]))*(Pi*Log[2] - Log[(Sqrt[2] + 1)^2]*Log[2] +
PolyLog[2, -(Sqrt[2] + 1)^2] - PolyLog[2, (Sqrt[2] - 1)^4]/2 +
Re[4*PolyLog[2, I*(Sqrt[2] - 1)] - 4*PolyLog[2, I*(Sqrt[2] + 1)] +
PolyLog[2, (Sqrt[2] + 1)^2 + I/10000000000... |
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| Sugata Adhya | 36,242 | <p>Use <strong>induction</strong>. Note $k^2<k!$$\implies(k+1)!$$=k!(k+1)$$>k^2(k+1)$$>(k+1)^2.$ That $k^2>k+1$ follows from $k\geq2.$</p>
|
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| chloe_shi | 45,070 | <p>$2(n-1)>n$<br>
$3(n-2)>n$<br>
$4(n-3)>n$<br>
$\cdots$<br>
$\cdots$<br>
$(k+1)(n-k)>n$<br>
$\cdots$<br>
$\cdots$<br>
$(n-1)2>n$ </p>
<p>Multiplying all, $~$ $(n-1)^{2}\times (n-2)^{2}\times\cdots\times 2^{2}\times 1^{2}>n^{n-2}$ </p>
<p>$\therefore$ $n^{2}\times (n-1)^{2}\times \cdots 2^{2}\ti... |
52,145 | <p>I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.</p>
<h3>Question</h3>
<p>Which computer algebra system is best for what, and what is the easiest/most fun(?) way to lear... | William Stein | 8,441 | <p>Your question is "Which computer algebra system is best for what, and what is the easiest/most fun(?) way to learn how to deal with them?" </p>
<ul>
<li><p>Regarding <em>community</em>, I think Sage (<a href="http://sagemath.org" rel="noreferrer">http://sagemath.org</a>) is the best CAS, since Sage is completely ... |
2,576,307 | <p>I know that it is possible to construct extremely weird subsets of <span class="math-container">$\mathbb R$</span>, or of <span class="math-container">$\mathbb R^3$</span>, that are non-Lebesgue-measurable. For example, the ones used in the Banach-Tarski paradox.</p>
<p>Are there "less crazy" examples of n... | Robert Israel | 8,508 | <p>Solovay showed that is consistent with ZF set theory (without the full Axiom of Choice but with Dependent Choice and an inaccessible cardinal) that all sets of reals are measurable. Thus any example of a nonmeasurable set must be "crazy".</p>
|
97,914 | <p>How do we replace the first <code>Head</code> of a currying expression such as <code>a[b,c][d]</code>? When accessing the <code>0</code>th element of an expression like <code>a[b,c][d]</code>, <code>a[b,c]</code> is given as the <code>Head</code>. We would like to instead replace <code>a</code> with, say, <code>w</c... | march | 29,734 | <h3>Update 2: Fixed! (I believe)</h3>
<p>With help from <a href="https://mathematica.stackexchange.com/a/98054/29734">this answer</a>.</p>
<pre><code>Clear[replaceFirstHead]
replaceFirstHead[newHead_, expr_] := Module[{cond = True}
, expr /. _Symbol[x__] :> newHead[x] /; If[cond, cond = False; True, False]
]
re... |
97,914 | <p>How do we replace the first <code>Head</code> of a currying expression such as <code>a[b,c][d]</code>? When accessing the <code>0</code>th element of an expression like <code>a[b,c][d]</code>, <code>a[b,c]</code> is given as the <code>Head</code>. We would like to instead replace <code>a</code> with, say, <code>w</c... | Mike Honeychurch | 77 | <pre><code>y = a[b, c][d];
y[[0, 0]] = w;
y
(* w[b, c][d] *)
</code></pre>
|
686,482 | <p>In algebraic geometry, we consider the map </p>
<p>$$I:\{\text{subsets of }\mathbb{A}_k^n\}\longrightarrow\{\text{ideals of }k[x_1,\cdots,x_n]\},\qquad X\mapsto I(X)$$</p>
<p>This map is not injective, because $I(\mathbb{A_k^n})=0$. But why it is not surjective? How to find a counterexample? Thank you!</p>
| dani_s | 119,524 | <p>Remember that $f$ is a bijection if and only if $f$ is invertible. Sometimes - and this is probably the case - it is easier to find the inverse of a function rather than directly proving that it is a bijection.</p>
<p>It should be intuitively clear to you that $\alpha$ is the inverse of itself. If not, try and see ... |
320,118 | <p>Suppose $$d = un + vs$$ where $d$ is the $\gcd(n,s)$</p>
<p>Dividing $d$ both sides</p>
<p>$$1 = u(n/d) + v(s/d)$$</p>
<p>So $(n/d)$ and $(s/d)$ are integers that are relatively prime. Why does this show that for any integer dividing both of them must also divide $1$?</p>
<p>This is a part of a proof I am readin... | Math Gems | 75,092 | <blockquote>
<p>$\rm\: m A + n B = 1,\:$ so $\rm\:A\:$ and $\rm\:B\:$ are integers that are relatively prime. Why does this show that for any integer $\rm\,d\,$ dividing both $\rm\,A,B\,$ must also divide $1$?</p>
</blockquote>
<p><strong>Hint</strong> $\ $ $\rm\:d\mid A,B\:\Rightarrow\: A = ad,\ B\, =\, bd,\ $ so $... |
1,897,455 | <blockquote>
<p>If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$ then $A^2+B^2$ equals ?</p>
<p>(a) $2AB$</p>
<p>(b) $2BA$</p>
<p>(c) $A+B$</p>
<p>(d) $AB$</p>
</blockquote>
<p>I tried
$(A+B)^2=A^2+B^2+AB+BA$</p>
<p>or,$A^2+B^2=(A+B)^2-AB-BA$</p>
<p>$=(A+B)^2-A-B$</p>
<p>$
=(A+B)^... | arghbleargh | 362,140 | <p>First, here is how you can figure out the answer by process of elimination. We can see that if $A$ and $B$ are both identity, then the condition is satisfied, and $A^2 + B^2 = 2I$ in that case. This rules out option (d). Next, note that $A^2 + B^2$ is symmetric in $A$ and $B$, so if (a) were correct, then by symmetr... |
2,644,104 | <p>If $\tan^{-1} \left(\dfrac {\sqrt {1+x^2} - \sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) = \alpha$ then prove that: $x^2= \sin (2\alpha) $</p>
<p>My Attempt:
$$\tan^{-1} \left(\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) =\alpha$$
$$\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {... | Jacky Chong | 369,395 | <p>Since
\begin{align}
\tan\alpha = \frac{\sqrt {1+x^2} - \sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}} =\frac{\text{opposite}}{\text{adjacent}}
\end{align}
then it follows
\begin{align}
\sin \alpha =& \frac{\sqrt {1+x^2} - \sqrt {1-x^2}}{\sqrt{(\sqrt {1+x^2} - \sqrt {1-x^2})^2+(\sqrt {1+x^2} + \sqrt {1-x^2})^2}}\\... |
2,644,104 | <p>If $\tan^{-1} \left(\dfrac {\sqrt {1+x^2} - \sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) = \alpha$ then prove that: $x^2= \sin (2\alpha) $</p>
<p>My Attempt:
$$\tan^{-1} \left(\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {1+x^2} + \sqrt {1-x^2}}\right) =\alpha$$
$$\dfrac {\sqrt {1+x^2}-\sqrt {1-x^2}}{\sqrt {... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
2,878,699 | <p>what is the order of the splitting field of </p>
<p>$x^5 +x^4 +1 = (x^2 +x +1)( x^3 +x+1)$ over $\mathbb{Z_2}$</p>
<p>i thinks it will $6$ because $2.3 = 6$</p>
<p>Pliz help me...</p>
| dxiv | 291,201 | <p>Hint: $\;3x^2 + 4x + 12a + 9ax = x(3x+4)+3a(3x+4)= (x+3a)(3x+4)\,$.</p>
|
1,623,416 | <p>There are $n$ bins of which the $i$th contains $(i-1)$ green balls and $(n-i)$ black balls. You pick a bin at random and remove two balls at random without replacement. What is the probability that the second ball is black?</p>
| G Cab | 317,234 | <p>To me, this problem entails an interesting aspect that is worth to spell out.
Assume to have a bin with <em>g</em> green balls and <em>b</em> black balls, let's indicate it as [g, b].
When we extract one ball from it, we get the following scheme:
$$ \begin{gathered}
\text{prob}\text{. }\frac{g}
{{g + b}}\quad \t... |
106,309 | <p>I'm creating a big manipulate box with several sliders, and each slider must have a small explication text (units, for example) after a mathematical symbol. As a simple example, here's the code I'm currently using for a slider :</p>
<pre><code>{{phi, 0, Style[Subscript[\[CurlyPhi], 0] "(degrees)", Bold, 10]},
... | ubpdqn | 1,997 | <p>Just a variant: function of parameter and extract solution at start and the use of <code>ParametricNDSolve</code> (though not needed for this particular example but just to show an alternative):</p>
<pre><code>fun[k_] := [x] /. DSolve[{f'[x] == -2 x f[x], f[0] == k}, {f[x]}, x][[1]]
Plot[Evaluate@Table[fun[j], {j, ... |
335,383 | <p>Hello I am currently stuck at integration by parts. And I was wondering why at the second $u$ substitution I should choose $u=\cos 3x$ instead of the $\sin 2x$. Is there a specific rule which applies?</p>
<p><img src="https://i.stack.imgur.com/mH7JL.png" alt="enter image description here"></p>
<p>Best,
Sebastian</... | Brian M. Scott | 12,042 | <p>Why not try it for yourself and see: let $u=\sin 2x$ and $dv=\cos 3xdx$. Then $du=2\cos 2x dx$ and $v=\frac13\sin 3x$, so</p>
<p>$$\int\cos 3x\sin 2x dx=\frac13\sin 2x\sin 3x-\frac23\int\sin 3x\cos 2x dx\;,$$</p>
<p>and</p>
<p>$$\begin{align*}
\int\sin 3x\cos 2x dx&=\frac12\sin 3x\sin 2x-\frac32\int\cos 3x\si... |
835,111 | <blockquote>
<p>How many solutions are there for $a^{2014} +2015\cdot b! = 2014^{2015}$, with $a,b$ positive integers?</p>
</blockquote>
<p>This is another contest problem that I got from my friend.</p>
<p>Can anybody help me find the answer? Or give me a hint to solve this problem?</p>
<p>Thanks</p>
| Gabriel Romon | 66,096 | <p>Not the whole solution, but enough to get going.</p>
<p>We have that $\displaystyle a^{2014}+b!\equiv 0 \pmod{2014}$</p>
<ul>
<li>If $b\geq 53$, then $\displaystyle a^{2014}\equiv 0 \pmod{2014}$. </li>
</ul>
<p>Since $2014=2\times19\times53$, this implies $\displaystyle a\equiv 0 \pmod{2014}$</p>
<p>Hence $a=201... |
1,471,023 | <p>I'm trying to get the hang of permutations and combinations for a discrete math class. One of my questions is as follows:</p>
<p>$$\text { You have a group of 30 people, and want to divide the group into two groups of 15 people each. }$$
$$\text { How many ways can this be done? } $$</p>
<p>My intuition is tellin... | Algebraic | 278,347 | <p>This is a combination. To be precise, $_{30} C_{15}$. You can see this by realizing that having one group, say {1,2,3...15} implies that you are excluding half of the population, namely {16,17...30}. This isn't really the proper notation for counting these things, but it is appropriate nonetheless. It doesn't matter... |
4,229,288 | <p><span class="math-container">$$\int_0^1\int_0^x \sqrt{x+y^2}\,dydx = \int_0^x\int_0^1 \sqrt{x+y^2}\,dxdy$$</span></p>
<p>Are these two integrals equivalent to each other? I assumed they weren't after imagining that one should also change the order of integration (in that case, analysing the region and changing the l... | epi163sqrt | 132,007 | <p>We can write OPs left-hand side as
<span class="math-container">\begin{align*}
\color{blue}{\int_0^1\int_0^x \sqrt{x+y^2}\,dydx}&=\int_{0\leq x\leq 1}\int_{0\leq y\leq x}\sqrt{x+y^2}\,dydx\\
&=\int_{0\leq y\leq 1}\int_{y\leq x\leq 1}\sqrt{x+y^2}\,dxdy\\
&\,\,\color{blue}{=\int_0^1\int_{y}^1\sqrt{x+y^2}\,... |
1,550,541 | <p>After trying for hours I decided to ask. Please can anyone help me with this problem.</p>
<p>"Two cards are drawn at random and are thrown away from a pack of 52 cards. Find the probability of getting an Ace from the remaining 50 cards."</p>
<p>Please explain the correct method to do this.</p>
<p>I'm getting answ... | Em. | 290,196 | <p>Caution: The problem says</p>
<blockquote>
<p>Find the probability of getting an Ace from the remaining 50 cards.</p>
</blockquote>
<p>This is 100% since, in the worst case scenario, you threw away 2 Aces at the beginning. Then there are still two Aces in the remaining 50. Therefore, you always get an Ace.
Are y... |
2,436,775 | <p>Question : </p>
<p>How much work is done in pulling an object constrained to move along the portion of the curve y = $x^2$; z = $x^3$ from (0; 0; 0) to (1; 1; 1) (positions in meters), if the rope pulling it is always in the direction < 1;-3;-4 > and the tension in the rope is constant at 100 Newtons?</p>
<p>At... | Math Lover | 348,257 | <p>You do not have to compute the unit vector along $\langle 1,3,-4 \rangle$. All you need is to compute the force vector along this vector. In particular, $$\vec F = c(\mathbf{i}+3\mathbf{j}-4\mathbf{k}),$$ where $|\vec F|=100 \implies c = 100/\sqrt{1^2+3^2+4^2}$. After computing $\vec F$, obtain the work, $W$, by $$W... |
4,253,780 | <p>I have an isosceles triangle with height <code>h</code> and base <code>b</code>.</p>
<p>I need to know how long is a segment parallel to the base <code>b</code> and distant <span class="math-container">$h_i$</span> from the base <code>b</code> as depicted in the diagram below.</p>
<p><a href="https://i.stack.imgur.c... | Angus Rogers | 930,571 | <p>Consider half the angle, θ, at the top of the triangle. <span class="math-container">$$tan\big(\frac{θ}{2}\big)=\frac{1}{2}\frac{b}{h}$$</span></p>
<p>Also consider, <span class="math-container">$$tan\big(\frac{θ}{2}\big)=\frac{I_n}{h-h_n}$$</span>
Hence <span class="math-container">$$I_n=\frac{b}{2h}(h-h_n)$$</span... |
3,275,356 | <blockquote>
<p>Prove that <span class="math-container">$$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+......=12e-5$$</span></p>
</blockquote>
<p><span class="math-container">$$
e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+......
$$</span></p>
<p>I have no clue of whe... | azif00 | 680,927 | <p>Your series is apparently
<span class="math-container">$$\sum_{n=1}^{\infty} \frac{3n^2+n+5}{n!}$$</span></p>
|
3,932,156 | <p>In the previous questions I've proved that <span class="math-container">$(1+i)^{2n+1}=a_n+ib_n$</span> where <span class="math-container">$a_n$</span> and <span class="math-container">$b_n$</span> are <span class="math-container">$\pm 2^n$</span> and that <span class="math-container">$(1-i)^{2n+1}=a_n - ib_n$</span>... | clathratus | 583,016 | <p>From the binomial theorem,
<span class="math-container">$$(x+a)^{2n+1}=\sum_{m=0}^{2n+1}{2n+1\choose m}x^ma^{2n+1-m}$$</span>
so
<span class="math-container">$$\begin{align}
(x+a)^{2n+1}-(x-a)^{2n+1}&=\sum_{m=0}^{2n+1}{2n+1\choose m}x^m(a^{2n+1-m}-(-a)^{2n+1-m})\\
&=\sum_{m=0}^{2n+1}{2n+1\choose m}x^ma^{2n+1... |
186,069 | <p>If 3x + logx = 30, then what is x?</p>
<p>Solution with W-Function...
The equation is 3x+Logx=30(1) we know that accept w<em>e^w=y=>w=W(k,y)..k in Z.But if w=3x=>3x</em>e^(3x)=y=>3x=W(k,y)(2) but from (1)… Log(3x)+3x=Logy=>Log3+Logx+3x=Logy=> Logx+3x=Logy-Log3=Log(y/3). Because we have the same equation {1,2} and t... | kglr | 125 | <p>For a purely graphical approach, you can use the options <code>MeshFunctions</code> and <code>Mesh</code> and post-process the <code>Plot</code> output to add text elements:</p>
<pre><code>f[x_] = 3 x + Log[x] - 30;
Normal[Plot[f[x], {x, 0, 20},
PlotPoints -> 100,
MeshFunctions -> {#2 &},
Mesh ... |
256,278 | <p>I was thinking about the following problem :</p>
<p>Define $ f:\mathbb C\rightarrow \mathbb C$ by </p>
<p>$$f(z)=\begin{cases}0 & \text{if } Re(z)=0\text{ or }Im(z)=0\\z & \text{otherwise}.\end{cases}$$</p>
<p>Then the set of points where $f$ is analytic is:</p>
<blockquote>
<p>(a) $\{z:Re(z)\neq 0$ an... | Ash GX | 85,478 | <p>It is not even continuous elsewhere except at $z=0$.
At $z=0$, taking derivatives along the axis and along the diagonal produces different results.</p>
|
2,812,355 | <p>I have read that infinity is not an element of R, the set of all real numbers and infinity is not a number. So can we say that infinity and minus infinity does not belongs to R ? Or can we say that plus infinity and minus infinity belongs to the set of all real numbers ? Please help .</p>
| Rhys Hughes | 487,658 | <p>I'm reminded of <a href="https://www.youtube.com/watch?v=Uj3_KqkI9Zo" rel="nofollow noreferrer">this video</a> when I see this sort of question. Essentially, $\infty$ is just a <em>concept</em>. It doesn't behave in the same way that the common real number does. To see this, consider the two equations:
$$x=x+1$$
$$x... |
2,785,013 | <p>Following this <a href="https://www.derivative-calculator.net/#expr=5%5E%28xcosx%29&showsteps=1" rel="nofollow noreferrer">enter link description here</a>, where we're performing the derivative
$$\frac{d}{dx}5^{x\cos(x)},$$</p>
<p>The first step of the differentiation is: </p>
<p>$[a^u(x)]'$ = $ \ln(a) \,a^u(x... | MPW | 113,214 | <p>That rule is obtained using <em>logarithmic differentiation</em>. Taking logs essentially converts an exponentiation into a multiplication, allowing you to use the already-familiar product rule. You can use it to derive the generalized power rule for derivatives:
$$\left(f^g\right)' = gf^{g-1}\cdot f' + f^g\log f\cd... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | dtldarek | 26,306 | <p>There is already great general answer by Ittay Weiss, so I will try a different approach. In fact, I will try to explain a bit the infinite sum you stated. As for the infinity, one could write <em>a lot</em> about it (mainly because there are multiple infinities, each with different properties), and I couldn't even ... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | Emanuele Paolini | 59,304 | <p>Actually in the first development of set theory a very intereseting contradiction was introduced. It is the so called <a href="http://en.wikipedia.org/wiki/Russell%27s_paradox" rel="nofollow">Russel Paradox</a>. I say it is interesting because is one of the very few cases where our intuition on what <em>should</em> ... |
3,791,848 | <p>I'm trying to solve the following question from the <a href="https://www.math.ucla.edu/%7Echparkin/gre/GREProb.pdf" rel="nofollow noreferrer">real analysis</a> section:</p>
<blockquote>
<ol>
<li>Let <span class="math-container">$K$</span> be a nonempty subset of <span class="math-container">$\mathbb R^n$</span> wher... | Mark | 470,733 | <p>A subset of <span class="math-container">$\mathbb{R^n}$</span> is compact if and only if it is closed and bounded, this is a standard result. Now, suppose every continuous real valued function defined on <span class="math-container">$K$</span> is bounded. In particular, the function <span class="math-container">$f(x... |
97,095 | <p>I want to generate a list of <strong><em>n</em></strong> coordinate points which are on the circumference of an ellipse. I wrote this code:</p>
<pre><code>n = 150;
ellipseFunc[a_,b_,t_] := {(a*b*Cos[t]/Sqrt[b*b*Cos[t]*Cos[t] + a*a*Sin[t]*Sin[t]]), (a*b*Sin[t]/Sqrt[b*b*Cos[t]*Cos[t] + a*a*Sin[t]*Sin[t]])};
listell... | Bob Hanlon | 9,362 | <p>Assuming that <code>a</code> and <code>b</code> are integers, then <code>1 < b < 2</code> cannot be satisfied; at least one inequality must be <code><=</code>. Use <a href="http://reference.wolfram.com/language/ref/Reduce.html" rel="nofollow"><code>Reduce</code></a> or <a href="http://reference.wolfram.com/... |
3,904,381 | <p>Let M be a coadjoint orbit of dimension 6 of <span class="math-container">$SU(3)$</span>, and let T be the maximal torus in <span class="math-container">$SU(3)$</span>. If we denote <span class="math-container">$\mu : M \longrightarrow \mathbb{R}^2$</span> the moment map associated to the action of T on M, then the ... | Max | 2,633 | <p>The elements in <span class="math-container">$su(3)$</span> satisfy <span class="math-container">$X+X^T=0$</span> and are unitarily diagonalizable, with purely imaginary eigenvalues. This means that, identifying <span class="math-container">$su(3)$</span> with its dual and adjoint and coadjoint actions, the orbits a... |
2,079,953 | <p>Giving the probability space with states $i\in \Omega$, the conditional probability of starting at $i$ is $\mathbb{P}_i = \mathbb{P}(.|X_0=i)$. </p>
<p>Giving $A\in \Omega$ , let's define the hitting time : $H^{A} : \Omega \rightarrow \mathbb{N}$ :
$$H^{A}(\omega) = \{\text{inf } n, X_n(\omega) \in A\}$$</p>
<p>I'... | Landon Carter | 136,523 | <p>For any state $i\notin A$, note that $E_i[H^A]=\sum_{n=1}^\infty nP_i(H^A=n)=\sum_{n=1}^\infty n\sum_{j\notin A}P_i(H^A=n, X_1=j)+\sum_{n=1}^\infty n\sum_{j\in A}P_i(H^A=n,X_1=j)$</p>
<p>Now observe if $j\in A$, then $P_i(H^A=n,X_1=j)=0$ if $n>1$ and $=P_i(H^A=1,X_1=j)=P_i(X_1=j)=p(i,j)$ for $n=1$.</p>
<p>Hence... |
7,420 | <p>I have a stand alone <code>Manipulator</code> (it is not in a <code>Manipulate</code>) and would like to know how to alter the style of the label. There are no explicit styling options for <code>Manipulator</code> so I tried wrapping it in <code>Style</code>:</p>
<pre><code>Style[
Manipulator[0.5, AppearanceElemen... | Vitaliy Kaurov | 13 | <p>To add to the spectrum of solutions here I suggest these approaches.</p>
<h3>1) ----------- <code>Magnify</code> -----------</h3>
<p><code>Magnify</code> will uniformly change the size of <code>Manipulator</code>, its numeric label and everything that is related to that. It will work inside an interface too:</p>
... |
1,337 | <p>Is there a way to post answers without gaining reputation points but retaining the higher editing barrier (or "authorship" if that concept applies here) of an ordinary, non-community-wiki posting? I don't mean comments but full length answers, editable more than 5 minutes later.</p>
<p>Often it is desirable to an... | Community | -1 | <p>If you do not do it too often, it wouldn't be too much of a bother to others if you set bounties. You can set a bounty on one of your old questions, quickly accept some answer and the bounty will be awarded soon(if I understand correctly); that way the bounty notification won't stay in the front page for too long.<... |
2,709,696 | <p>stuck on a question and can't seem to make any progress:</p>
<p>We have an insurance company who expects the number of accidents their policy holders will have is Poisson distributed. The Poisson mean $\Delta$ follows a Gamma distribution with the $\Gamma$(2,1) density function being $f_{\Delta}(\lambda) = \lambda ... | an4s | 533,556 | <p>Using the following code in MATLAB shows that besides $n = 0$ and $n = 4$, there is no other solution, at least up till $n = 512$ which is when MATLAB declares the LHS of the equation to be infinite. The use of functions <code>digits</code> and <code>vpa</code> allow for greater precision of numbers. These are helpf... |
2,036,864 | <p>Here is the question. </p>
<p>From past records, a clothing store finds that 55% of the people who enter the store will make a purchase. During a one hour period, 20 people enter the store. The random variable x represents the number of people who make the purchase. </p>
<p>Find the probability that :</p>
<ol>
<... | Momo | 384,029 | <p>${20 \choose 7} (.55)^7 (.45)^{20-7}$</p>
|
2,863,179 | <p>I'll state the Cantor's theorem proof as is it is in my study texts:</p>
<p>Theorem (Cantor): Let $X$ be any set. Then $|X|<|\mathcal{P}(X)|$</p>
<p>Proof: Define map $\varphi:X\rightarrow\mathcal{P}(X)$ by $\varphi:x\mapsto\{x\}$. $\varphi$ is injective, thus $|X|\leq|\mathcal{P}(X)|$. Now suppose there is a b... | Especially Lime | 341,019 | <p>Mees de Vries' answer gives an explanation of where your misunderstanding of the usual proof was. </p>
<p>To address your alternative proof attempt: you show that there is a particular injection from $X$ to $\mathcal{P}(X)$, but that it's not surjective. This isn't enough to prove that $|X|<|\mathcal{P}(X)|$. Fo... |
28,361 | <p>My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?</p>
<p>I more or less understand that if $G=SL_1(D)$ where $D$ is a ... | Victor Protsak | 5,740 | <p>As Charles said, for a semisimple group over a local field compact $\iff$ anisotropic.</p>
<p>I confirm that Chapter 6 of Platonov-Rapinchuk contains the proof of (a) the vanishing of the Galois cohomology $H^1(K, G)$ for a connected simply connected semisimple group $G$ over a local field $K$ and (b) Classificatio... |
2,981,450 | <blockquote>
<p>Prove that there do not exist natural <span class="math-container">$n$</span> such that <span class="math-container">$(1+i)^n=1$</span>.</p>
</blockquote>
<p>I try to prove with the binomial and proving by induction but it isn't working</p>
<p><img src="https://i.stack.imgur.com/erGAb.png" alt="try ... | Kavi Rama Murthy | 142,385 | <p>Take modulus. <span class="math-container">$1=|(1+i)^{n}|=(\sqrt 2)^{n}$</span> which is impossible except when <span class="math-container">$n=0$</span>. </p>
|
2,981,450 | <blockquote>
<p>Prove that there do not exist natural <span class="math-container">$n$</span> such that <span class="math-container">$(1+i)^n=1$</span>.</p>
</blockquote>
<p>I try to prove with the binomial and proving by induction but it isn't working</p>
<p><img src="https://i.stack.imgur.com/erGAb.png" alt="try ... | Mark Bennet | 2,906 | <p>Let's take <span class="math-container">$n$</span> a positive integer.</p>
<p>Compute <span class="math-container">$(1+i)^1=1+i; (1+i)^2=2i; (1+i)^3=2(i-1); (1+i)^4=-4$</span></p>
<p>You can then easily express <span class="math-container">$(1+i)^{4n+r}: 0\le r\le3$</span> as one of four simple expressions which d... |
302,519 | <p>My teacher gave us an homework. I solved it, but I don't think I have the right answer.</p>
<p><strong>PROBLEM</strong></p>
<p>We have three coins identical in appearance.</p>
<ul>
<li>Coin A falls on tails and heads with equal probability</li>
<li>Coin B falls twice as much on tails as heads</li>
<li>Coin C alwa... | CogitoErgoCogitoSum | 52,938 | <p>Well, there's your problem. Its in French.</p>
<p>$P(T|A) = 1/2,
P(T|B) = 2/3,
P(T|C) = 1/1$</p>
<p>$P(A) = P(B) = P(C) = 1/3$</p>
<p>$P(TA) = P(T|A) P(A) = 1/6$</p>
<p>$P(TB) = P(T|B) P(B) = 2/9$</p>
<p>$P(TC) = P(T|C) P(C) = 1/3$</p>
<p>$P(TA \lor TB \lor TC) = P(TA) + P(TB) + P(TC) = 1/6 + 2/9 + 1/3 = 13/1... |
654,398 | <p>For each $a \in \mathbb{Z}^+$ let the following ODE</p>
<p>$$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$</p>
<ul>
<li>Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a polinomial with degree $a+1$</li>
<li>Using an appropiate reduction order method, show that the general ... | Claude Leibovici | 82,404 | <p>I can be totally wrong here but let me give you my feeling : you have a second order differential equation <strong>without</strong> any boundary conditions. So, you can express all the coefficients as functions of $a_0$,$a_1$ and $a$. You have established the recurrence relaion.</p>
|
654,398 | <p>For each $a \in \mathbb{Z}^+$ let the following ODE</p>
<p>$$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$</p>
<ul>
<li>Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a polinomial with degree $a+1$</li>
<li>Using an appropiate reduction order method, show that the general ... | Dmoreno | 121,008 | <p>I think you are not using Fröbenius method correctly, shouldn't the solution be expressed as it follows:</p>
<p>$$x(t) = \sum^\infty_{l=0} b_l t^{l+s}?$$</p>
<p>where $s$ is to be solved from the indicial equation, which is $s (s-1) = 0$ and $b_l$ can be obtained by the recurrence relation. If you give me some min... |
83,882 | <p>Suppose I have a quadratic polynomial in two variables <code>x</code> and <code>y</code> in which the squares with respect to <code>x</code> and <code>y</code> have already been completed:</p>
<pre><code> q = -72 + 9 (-2 + x)^2 + 4 (3 + y)^2 ;
</code></pre>
<p>How might I extract the "constant" part <code>-72</cod... | J. M.'s persistent exhaustion | 50 | <p>Apparently, nobody brought up this possibility:</p>
<pre><code>SeriesCoefficient[-72 + 9 (-2 + x)^2 + 4 (3 + y)^2, {x, 2, 0}, {y, -3, 0}]
-72
</code></pre>
<p>where it is assumed that you know the terms subtracted from the corresponding variables.</p>
|
3,777,750 | <p>I am analyzing the following system, where <span class="math-container">$I_{in}$</span> is a scalar parameter:
<span class="math-container">$$
\begin{aligned}
&\dot{V} = 10 \left( V - \frac{V^3}{3} - R + I_{in} \right) \\
&\dot{R} = 0.8 \left( -R +1.25V + 1.5 \right)
\end{aligned}
$$</span></p>
<p>It is a si... | Cesareo | 397,348 | <p>Not an answer. I leave this MATHEMATICA script as a calculation process to the critical data.</p>
<pre><code>f[i0_, v_, r_] := {10 (v - v^3/3 - r + i0), 0.8 (-r + 1.25 v + 1.5)};
sols = Quiet@Solve[f[i0, x, y] == 0, {x, y}];
J0 = Grad[f[i0, x, y], {x, y}] /. sols[[1]];
eig = Eigenvalues[J0];
Plot[Re[eig], {i0, 0, 3}... |
2,910,860 | <p>Question: [For which positive real numbers $a$ and $b$ is</p>
<p>$u(x,y) = \cosh(ax)\sin(by)$</p>
<p>harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$]</p>
<hr>
<p>I got $a=b$ for the condition so that</p>
<p>$u(x,y) = \cosh(ax)\sin(ay)$ and $v(x,y) = -\s... | saulspatz | 235,128 | <p>We know that the volume of a tetrahedron is one-third the area of the base times the height, so if we can find the are of $\triangle ABC$ and the volume of the tetrahedron, we are home free. We can find the area of a triangle, given the lengths of the sides, by <a href="https://en.wikipedia.org/wiki/Heron%27s_formu... |
3,744,077 | <p>When a group of people need to decide a winner or leader between them, one approach would be that a random hidden integer is chosen with uniform distribution on <span class="math-container">$\{0, 1, ..., n\}$</span> and all <span class="math-container">$p$</span> participants publicly choose a number.</p>
<p>Then, t... | Alex Ravsky | 71,850 | <p>I tried to analyze the game, but, according to antkam’s <a href="https://math.stackexchange.com/questions/3744077/what-is-the-optimal-strategy-of-guessing-a-number-where-closest-without-going-ov#comment7699993_3744077">guesses</a>, the analysis became more and more technical, so I decided to stop it. My findings ar... |
562,763 | <p>Let's say we have chess table $n^2$ and we want to put 8 rooks on the table, so that none of them are under eachother's fire.</p>
<p>I've come up with this:
$$n^2 + n^2(n-1)+n^2(n-1)(n-2) ...$$
I think that I'm not taking into account that the pieces are indifferentiable. That means I should substract number of per... | Marc van Leeuwen | 18,880 | <p>You can choose $8$ rows and $8$ columns in $\binom n8^2$ ways, and then there is a standard rook placement (permutation) problem left, giving a factor $8!$. So you get
$$
\binom n8^28!=\frac{n^2(n-1)^2\ldots(n-7)^2}{8!}\quad\text{solutions.}
$$
This is also what you get from your method if you divide out the $8!$ ... |
2,035,987 | <p>$$e^{x^2}=e^{18x} \cdot 1/e^{80}$$</p>
<p>I dropped the e since it was the same base and solved:</p>
<p>$$x^2=18x-80$$</p>
<p>I then moved $ 18x$ and $80$ to the other side and got:
$$x^2-18x+80=0$$</p>
<p>The roots are $ 8 $ and $ 10 $ but its not possible with the signs I got. Where did I go wrong with the sig... | wpkzz | 245,835 | <p>Okey, there are a couple of mistakes on your procedure:
$$e^{18x}/e^{80}=e^{(18x-80)}$$,
so, when you "drop the $e$", that is to say, when you take the logarhythm of both sides, you have:
$$x^2=18x-80$$.
Then you only have to be carefull with your rearrangement of the equation. </p>
|
3,883,504 | <p>I am trying to express the following: I have a set <span class="math-container">$A$</span> and the powerset (set of all subsets of <span class="math-container">$A$</span>) <span class="math-container">$P(A)$</span>. I have another set <span class="math-container">$S \in P(A)$</span>, and I want to get the sets in <s... | Oğuzhan Kılıç | 481,167 | <p>So <span class="math-container">$P(A\setminus S)$</span> could do the job ...</p>
|
1,561,316 | <p>A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m is less than n. These polynomials are multiplied and all like terms are combined. The resulting polynomial has a maximum of how many nonzero terms? How would you prove that the answer is mn?</p>
| Michael Hardy | 11,667 | <p>\begin{align}
& (A+B+C+D)(X+Y+Z) \\[12pt]
= {} & \phantom{{}+{}} A(X+Y+Z) \\
& {} + B(X+Y+Z) \\
& {} + C(X+Y+Z) \\
& {} + D(X+Y+Z) \\[12pt]
= {} & \phantom{{}+{}} AX+AY+AZ \\
& {} + BX+BY+BZ \\
& {} + CX+CY+CZ \\
& {} + DX+DY+DZ \\[12pt]
= {} & \text{a sum of }4\times3\text{ t... |
1,683,977 | <p>How should I compute the derivative of $e^{x\sin x}$ ?
I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles )
I know that derivative of $e^x$ is $e^x$, but I cannot understand what to do with that $\sin x$?</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Logarithmic differentiation is very useful $$y=e^{x\sin( x)}$$ Take logarithms $$\log(y)=x \sin(x)$$ Differentiate $$\frac{y'}y=x \cos(x)+\sin(x)$$ So $$y'=\big(x \cos(x)+\sin(x)\big)e^{x\sin x}$$</p>
|
1,907,275 | <p>Me and a friend could need some assistance:</p>
<p><a href="https://i.stack.imgur.com/UBkDm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UBkDm.png" alt="Given function + Plot"></a></p>
<blockquote>
<p>$$x(t)=t\cos t$$
$$y(t)=t\sin t$$</p>
</blockquote>
<p>We've been given the assignment ... | mvw | 86,776 | <p>Intersection with the $x$-axis means $y = 0$:
$$
x = t \cos t \\
0 = t \sin t
$$
The easiest solution is at $t = 0$. If $t \ne 0$ you can use
$$
0 = \sin(t) \Rightarrow t \in \{ \pm \pi, \pm 2 \pi, \pm 3 \pi \}
$$
(because you can divide the second equation by $t$ on both sides) to have $t \in [-10, 10]$.</p>
<p>I... |
1,907,275 | <p>Me and a friend could need some assistance:</p>
<p><a href="https://i.stack.imgur.com/UBkDm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UBkDm.png" alt="Given function + Plot"></a></p>
<blockquote>
<p>$$x(t)=t\cos t$$
$$y(t)=t\sin t$$</p>
</blockquote>
<p>We've been given the assignment ... | Dr. Sonnhard Graubner | 175,066 | <p>the equation $$t\sin(t)=0$$ gives us $$t=0$$ or $$t=k\pi$$ with $$k \in \mathbb{Z}$$</p>
|
107,972 | <p>The <code>Sign</code> function works in a very straightforward manner:</p>
<pre><code>Sign[8]
Sign[-8]
Sign[0]
</code></pre>
<blockquote>
<pre><code>1
-1
0
</code></pre>
</blockquote>
<p>Unfortunately, in my code I need <code>Sign[0]</code> to equal 1 or negative 1. I tried this but get a funny error:</p>
<pre><... | m_goldberg | 3,066 | <p>The simplest way I can think of to define your <code>ssign</code> function is:</p>
<pre><code>ssign[0 | 0.]= 1;
ssign[x_] := Sign[x]
Plot[ssign[x], {x, -5, 5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/jrM0H.png"><img src="https://i.stack.imgur.com/jrM0H.png" alt="plot"></a></p>
|
147,932 | <p>I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that $ y = 0$ mean for the solutions, the intercept at the $x$ axis? Why aren't the solutions at $f(x) = 403045$ or some o... | David Allen | 431,233 | <p>I believe this is another way of looking at it.</p>
<blockquote>
<p>In most cases, "<strong>roots</strong>" are a subset of all possible solutions (all the $x$, $y$ coordinates that satisfy the equation). If an equation does not have a root solution, then there is no direct, algebraic way of solving for $x$, give... |
1,368,310 | <p>Suppose that a large lot with 10000 manufactured items has 30 percent defective items and 70 percent nondefective. You choose a subset of 10 items to test.
(a) What is the probability that at most 1 of the 10 test items is defective?
(b) Approximate the previous answer using the binomial distribution.</p>
<p>I am ... | Michael Hardy | 11,667 | <p>The $10$ items are not chosen independently since they are chosen without replacement. If it were with replacement, with a tiny chance that the same item might be chosen more than once, then the binomial distribution would be exact rather than a very close approximation. As it is, part (a) must use a hypergeometri... |
931,951 | <p>How can I find all solutions of $Ax = 0$ in parametric vector form where A is row equivalent to the matrix </p>
<p>$\begin{pmatrix}
-1&-4&0&-4\\2&-8&0&8
\end{pmatrix}$</p>
| ChrisD | 176,117 | <p>What you want to do first is put your matrix A into RREF (reduced row echelon form). The reduced form of your matrix A is:
$\begin{pmatrix}
1&0&0&4\\0&1&0&0
\end{pmatrix}$
$\\$ So what can we do from here? We can write the general equation, which provides us with a parametric description o... |
3,108,432 | <p>I was practicing limit and I came across this question: </p>
<blockquote>
<p><span class="math-container">$$
\lim_{x \to 0} \frac{|x|^{1/2} \cos(\pi^{1/x^2})}{2 + (x^2 + 3)^{1/2}}
$$</span></p>
</blockquote>
<p>I reasoned the absolute value of <span class="math-container">$x$</span> goes to zero, and the denomi... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$|\cos \theta | \leq 1$</span> for all <span class="math-container">$\theta$</span> so the limit is <span class="math-container">$0$</span>. </p>
|
2,326,050 | <p>This is a question from RMO 2015.</p>
<p>Show that there are infinitely many triples (x,y,z) of integers such that $x^3+y^4=z^{31}.$</p>
<p>This is how I did my proof:
Suppose $z=0,$ which is possible because $0$ is an integer. Then $x^3+y^4=0 \Rightarrow y^4=-x^3.$ Now, suppose $y$ is a perfect cube such that it ... | JohnnyC | 453,248 | <p>A small variation on the 'official' solution: instead of proving that the equation $12r+1=31k$ has infinitely many positive integer solutions we find a particular one, let's say $k=7,r=18$; it follows that $x=2^{4\cdot 18},y=2^{3\cdot 18},z=2^7$ is a particular solution for our equation and from here it is easy to n... |
2,581,622 | <p><strong>Question:</strong> </p>
<blockquote>
<p>Solve $y'(t)=\operatorname{sin}(t)+\int_0^t y(x)\operatorname{cos}(t-x)dx$ such that $y(0)=0$ </p>
</blockquote>
<p><strong>My try:</strong><br>
I applied Laplace transform on both sides of the equation. </p>
<p>$
sL\{y(t)\} = \frac{1}{s^2+1}+L\{cos(t)*y(t)\} \... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. You are on the right track. But please check your results, since from your identity
$$
sL\{y(t)\}(s)=\frac{1}{s^2+1}+L\{cos(t)\}(s)\times L\{y(t)\} (s)
$$ using
$$
L\{cos(t)\}(s)=\frac{s}{s^2+1}
$$ I rather get
$$
L\{y(t)\}(s)=\frac1{s^3}
$$ which is now standard to solve.</p>
|
716,364 | <p>I am just beginning to study fields and for whatever reason am finding their presentation to be completely baffling - moreso than I think anything I have ever studied. I am reading out of chapter 21 of this free book: <a href="http://abstract.ups.edu/download.html" rel="nofollow">http://abstract.ups.edu/download.htm... | user46372819 | 86,425 | <p>We see that as $g$ changes from $-3$ to $-7$, $x$ changes from $0$ to roughly $1.25$. So we have $\frac{\partial g}{\partial x}= \frac{-7-(-3)}{1.25}=-\frac{4}{1.25}=3.2$, and equivalently for the change in $y$: $\frac{\partial g}{\partial y}= \frac{1-(-3)}{1.25}=\frac4{1.25}=3.2$</p>
<p>Using the equation of a pla... |
2,947,806 | <p>Clearly it isn't, a quick sketch would show it, but I need an analytical proof. The obvious suggestion would be to view the limit on both sides and try to find a inconsistency but how does that work exactly in this case.</p>
<p>Evaluating from both sides the definition of differentiability yields already an intuiti... | Mason | 552,184 | <p>We can argue that <span class="math-container">$f(x)=\cos(\sqrt{|x|} )$</span> is differentiable for <span class="math-container">$x<0$</span> and for <span class="math-container">$x>0$</span> we find that in both these cases </p>
<p><span class="math-container">$f'(x)= -\operatorname{sgn}\left(x\right)\frac{... |
766,779 | <p>Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$.</p>
<p>I started by the standard lagrangian method:$$L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(1-x^2-y^2-4z^2)$$
Which subsequently gives: $${∂L\over∂x}=2x-... | Amzoti | 38,839 | <p>Write $\left(\dfrac{L}{2}\right)^3 = \dfrac{L^3}{8}$. </p>
<p>Now, we divide that by $6$ and get $\dfrac{\dfrac{L^3}{8}}{\dfrac{6}{1}}$.</p>
<p>We invert and multiply and have:</p>
<p>$$ \dfrac{L^3}{48}$$</p>
|
2,115,640 | <p>Let $a, k, p, r$ and $n$ be all members of $\mathbb{Z}$. $p$ is an arbitrary number such that $a+pr=n$</p>
<p>Why does the statement below hold true?</p>
<p>\begin{align}
\sum_{k=0}^{p}(a+kr)
&=a+(a+r)+(a+2r)+\cdots+\left(a+(p-2)r\right)+\left(a+(p-1)r\right)+(a+pr)\\
&=a+(a+r)+(a+2r)+\cdots+(n-2r)+(n-r)+n... | Simply Beautiful Art | 272,831 | <p>One may note that</p>
<p>$$\sum_{k=0}^pa=\underbrace{a+a+a+\dots+a}_{p+1}=a(p+1)$$</p>
<p>and likewise,</p>
<p>$$\sum_{k=0}^pkr=r\left(1+2+3+\dots+p\right)\\=r\frac{p(p+1)}2$$</p>
|
2,115,640 | <p>Let $a, k, p, r$ and $n$ be all members of $\mathbb{Z}$. $p$ is an arbitrary number such that $a+pr=n$</p>
<p>Why does the statement below hold true?</p>
<p>\begin{align}
\sum_{k=0}^{p}(a+kr)
&=a+(a+r)+(a+2r)+\cdots+\left(a+(p-2)r\right)+\left(a+(p-1)r\right)+(a+pr)\\
&=a+(a+r)+(a+2r)+\cdots+(n-2r)+(n-r)+n... | Kiran | 82,744 | <p>Use formulas of arithmetic progrssion. The given sequence is an arithmetic progression; first term $=a$, last term $=n$, common difference $=r$</p>
<p>$\text{number of terms }\\=\dfrac{\text{last term}-\text{first term}}{\text{common difference}}+1\\=\dfrac{n-a}{r}+1$</p>
<p>$\text{Sum }\\=\dfrac{\text{number of t... |
2,487,589 | <p>I came across some engineering paper and equation in the paper doesn't seem to make sense unless cot(A)=cot(-B) in the image below.</p>
<p>But I never heard of such identity before.</p>
<p>Can cot(A)=cot(-B) be true or it can't be true in general?</p>
<p><a href="https://i.stack.imgur.com/xRDw7.png" rel="nofollow... | JJacquelin | 108,514 | <p>$$\frac{dP}{dt} = cP\ln(\frac{K}{P}) =cP\left(\ln(K)-\ln(P) \right) $$</p>
<p>THE ARDUOUS WAY :</p>
<p>Change of function :
$$P(t)=e^{y(t)} \quad\to\quad \frac{dP}{dt} = e^{y(t)}\frac{dy}{dt} $$
and $\quad\frac{dP}{dt}=cP\left(\ln(K)-\ln(P) \right) =c e^{y(t)} \left(\ln(K)-y(t) \right) $
$$e^{y(t)}\frac{dy}{dt}=c ... |
116,727 | <p>I was thinking a bit about isometric embeddings into Hilbert spaces and got the following idea.</p>
<p>First, as we recall, many vector spaces over the reals are isomorphic to $\mathbb{R}^{\alpha}$ for some cardinal number $\alpha$. [EDIT: As Carl Mummert remarked below, not every vector space is of this form, as I... | Arturo Magidin | 742 | <p>In the <em>usual topology</em> of $\mathbb{R}$, $\mathbb{Q}$ is neither open nor closed.</p>
<p>The interior of $\mathbb{Q}$ is empty (any nonempty interval contains irrationals, so no nonempty open set can be contained in $\mathbb{Q}$). Since $\mathbb{Q}$ does not equal its interior, $\mathbb{Q}$ is not open.</p>
... |
2,161,420 | <p>I'm thinking about how to compute $\int_{\frac{3}{2n}}^1 nx^{n-1}(x-\frac{3}{2n})^n dx$ or give limsup of it as n tends to infinity? It seems integrate by part may work, but it is complicated. I try to use dominated convergence theorem, since the function converges to $nx^{2n-1}$, but it seems there is not a uniform... | jJjjJ | 404,893 | <p>I think you can upper bound the integral using Holder inequality, in fact you have to notice that $nx^{n-1}$ is integrable over $[\frac{3}{2n}, 1]$ and that $\Vert (x-\frac{3}{2n})^n \Vert_{\infty} = (1-\frac{2}{3n})$ (on $[\frac{3}{2n}, 1]$). You get</p>
<p>$\int_{\frac{3}{2n}}^1 nx^{n-1}(x-\frac{3}{2n}))^n dx \l... |
1,334,923 | <p>So since my high school algebra tells me that $361°$ is basically rotating the whole axis around and adding $1$ more degree, I would assume that at least in trigs they are the same.</p>
<p>But then $1°$ is acute, while $361°$ is not classified as either acute, obtuse or right. I understand this difference.</p>
<p>... | Plutoro | 108,709 | <p>The ratio test states that the series $$\sum_{n=1}^\infty a_n$$ converges as long as $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1.$$
So we can use the ratio test here:
$$\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{n+1}}{(n+1)!}}{\frac{(-1)^nx^n}{n!}}\right|=\lim_{n\to\infty}\left|-\frac{x^{n+1}n!}{x^... |
1,334,923 | <p>So since my high school algebra tells me that $361°$ is basically rotating the whole axis around and adding $1$ more degree, I would assume that at least in trigs they are the same.</p>
<p>But then $1°$ is acute, while $361°$ is not classified as either acute, obtuse or right. I understand this difference.</p>
<p>... | Aleksandar | 240,930 | <p>This series is equivalent to:</p>
<p>$\sum_{k=1}^\infty \frac{(-x)^n}{n!}$.</p>
<p>This series is equivalent to $e^{-x}-1$. This clearly has no singularities. Therefore the radius of the disc of convergence is infinite.</p>
|
103,536 | <p>I'm reading the book <em>Graphs and Their Uses</em> which contains the following theorem and proof:</p>
<blockquote>
<p>THEOREM 2.3. A connected graph with 2k odd vertices contains a family
of k distinct trails which, together, traverse all edges of the graph
exactly once. </p>
<p>PROOF. Let the odd vert... | Community | -1 | <p>Here's an alternative proof. Take an odd vertex, and construct a trail from that to another odd vertex. Remove the edges of that trail; the resulting graph has $2k-2$ odd vertices. Repeat.</p>
|
4,377,170 | <p>I have the following system:</p>
<p><span class="math-container">$$
(a + tb)\cos \theta = x \\
(b + ta) \sin \theta = y \\
$$</span></p>
<p>with the constraints</p>
<p><span class="math-container">$$a > b$$</span> <span class="math-container">$$\theta \in (-\pi, \pi]$$</span> <span class="math-container">$$t >... | Claude Leibovici | 82,404 | <p><span class="math-container">$$(a + b\,t)\cos (\theta) = x \tag 1$$</span>
<span class="math-container">$$(b + a\,t) \sin(\theta) = y \tag 2$$</span></p>
<p>Let <span class="math-container">$X=\frac x b$</span>, <span class="math-container">$Y=\frac y a$</span>, <span class="math-container">$\alpha=\frac ab$</span>,... |
3,431,903 | <p>If there is some continuous function in <span class="math-container">$\mathbb{R}$</span> that satisfies <span class="math-container">$f(x)\notin \mathbb{Q}$</span> for every x. Is f then a constant function? How would I show this?</p>
| MSIS | 678,294 | <p>Yes. Equivalently to franklin's answer, the Reals are connected and a continuous function preserves connectedness, while the Irrationals ( the intended range/codomain of your function) are (totally) disconnected in that maximal connected components are singletons. For your image to be connected, the image must be a ... |
1,868,044 | <blockquote>
<p>For given positive integers $r,v,n$ let $S(r,v,n)$ denote the number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$ and such that $x_i \leq v$ for $i = 1,\ldots,n$. Prove that $$S(r,v,n) = \sum_{k=0}^m (-1)^k \binom{n}{k} \binom{r-(v+1)k+n-1}{n-1}... | Nate | 91,364 | <p>This is inclusion exclusion. The $k = 0$ term corresponds to all non-negative solutions to the equation, the $k = 1$ term corresponds to all non-negative solutions where at least 1 of the $x_i$ is larger than $v$, the $k = 2$ term corresponds to all non-negative solutions where at least 2 of the $x_i$ are larger tha... |
1,889,002 | <p>I have been going through and doing some (non-assessed) homework questions, but am getting really stuck on conditional probability. The following problem is one that I simply cannot get my head around.</p>
<p>Question: Die A has four red and two blue faces, and die B has two red and four blue faces. One of the dice... | kviiri | 187,461 | <p>For part two, start by considering the question: "What is the probability that two throws result in two reds?"</p>
<p>Since Die A has a $2/3$ probability of providing red, and Die B has a $1/3$ probability of providing red. So:</p>
<p>$P(\text{2 reds}\ |\ A) = (2/3) \times (2/3) = 4/9$</p>
<p>$P(\text{2 reds}\ |\... |
2,617,244 | <p>I'm trying to get the <em>least x</em> from a system of congruences by applying the Chinese Remainder Theorem. Keep running into issues.</p>
<p>System of congruences:
$$
x \equiv 0 (_{mod} 7) \\
x \equiv 5 (_{mod} 6) \\
x \equiv 4 (_{mod} 5) \\
x \equiv 3 (_{mod} 4) \\
x \equiv 2 (_{mod} 3) \\
x \equiv 1 (_{mod} 2... | jnyan | 365,230 | <p>I know this is bad method, but the number you want must be multiple of 7, it should be odd number, last digit must end in 9(why?!!) quickly you will find that smallest such number is 119.</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Pete L. Clark | 1,149 | <p>I am very fond of Narkiewicz's <em>Elementary and Analytic Theory of Algebraic Numbers</em>. The bibliography is more than 170 pages long. The end of the chapter notes are wonderful. I find Chapter 1 to be such a marvel of scholarly exposition that I keep reading it over and over again -- it's hard for me to go o... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Akhil Mathew | 344 | <p>Pretty much anything by Serge Lang, as far as I can tell. For instance, his <em>Differential and Riemannian Manifolds</em> contains asides (especially in the preface) with references to all sorts of papers and other books in differential geometry and topology---which I find all the more remarkable since Lang was a ... |
1,961,176 | <p>Proof. Let $A$ be nonempty and bounded from above. Let $s_{1}$, $s_{2}$ be two supremums of A. Since $s_{1}$ is an upper bound and $s_{2}$ is less than equal to any upper bound. We have $s_{2}\leq s_{1}$.
Similarly, $s_{1}\leq s_{2}$. Therefore, $s_{1}=s_{2}$.</p>
<p><strong>My question is:</strong> in proof, ''... | Jack D'Aurizio | 44,121 | <p>We have
$$ \sum_{n\geq 1}\left(\frac{1}{n}-\frac{1}{n+z}\right)=\gamma+\psi(1+z)\tag{1}$$
where $\psi(z)=\frac{d}{dz}\log\Gamma(z)$, by the <a href="https://en.wikipedia.org/wiki/Gamma_function#Weierstrass.27s_definition" rel="nofollow">Weierstrass product for the $\Gamma$ function</a>. It follows that</p>
<p>$$ \s... |
3,618,029 | <p>In <em>Cormen's Introduction to Algorithm's book</em>, I'm attempting to work the following problem:</p>
<blockquote>
<p>Show that the solution to the recurrence relation <span class="math-container">$T(n) = T(n-1) + n$</span> is <span class="math-container">$O(n_2)$</span> using substitution
<span class="math-... | vonbrand | 43,946 | <p>The idea is to assume that <span class="math-container">$T(n) = O(n^2)$</span>, i.e. there is some <span class="math-container">$c$</span> such that for some <span class="math-container">$n_0$</span> for all <span class="math-container">$n \ge n_0$</span> you have <span class="math-container">$T(n) \le c n^2$</span>... |
720,368 | <p>I want to express
$$\prod_{k=1}^n \left( k - \frac{1}{2} \right)$$
using the gamma function. I think this is equivalent to $\left(k-\frac{1}{2}\right)!$ so I set $a=k-1$
and then used the identity $$\Gamma \left(n+\frac{1}{2}\right) = {(2n)! \over 4^n n!} \sqrt{\pi}$$
to get
$$\prod_{k=1}^n \left( k - \frac{1}{2} \r... | Davide Giraudo | 9,849 | <p>Define $a_n:=\prod_{k=1}^n\left(k-\frac 12\right)$. From the relationship $\Gamma(x+1)=x\Gamma(x)$ for $x$ positive, we derive
$$k-\frac 12=\frac{\Gamma(k+1-1/2)}{\Gamma(k-1/2)},$$
hence the product which defines $a_n$ is telescopic. We obtain
$$a_n=\frac{\Gamma(n+1/2)}{\Gamma(1/2)}.$$
Since $\Gamma(1/2)=\sqrt\pi$... |
242,177 | <p>Show that exactly one of:
\begin{cases} B^Tv = 0\\ d^Tv = 1 \end{cases}
or
$$Bu=d$$
has a solution.
I tried with Farkas lemma, but I run into trouble. </p>
| Sebastien B | 38,383 | <p>A solution which does not involve Farkas lemma.</p>
<ol>
<li><p>First observe that it is not possible to have solutions $u$ and $v$ to both equations at the same time.
If there were such a solutions, then
$$0=u^T 0 = u^T B^T v=(Bu)^T v=d^T v=1,$$
hence a contradiction.</p></li>
<li><p>Then if the equation $Bu=d$ ha... |
4,467,187 | <p>In <a href="https://math.stackexchange.com/questions/4564289/lemniscate-numbers-and-others-what-would-be-the-properties">this post</a> user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".</p>
<p>... | Sourav Ghosh | 977,780 | <p>Alternative:</p>
<p>Let <span class="math-container">$\overline{z_0}\in \overline{A}$</span> .</p>
<p>Then <span class="math-container">$f$</span> is analytic at <span class="math-container">$z_0\in A$</span> implies <span class="math-container">$\exists B_{R}(z_0)(R>0)$</span> such that <span class="math-contain... |
2,461,551 | <p>How can I prove by induction that $3^n ≥ 1 + 2^n$ for every $n\in\mathbb{N}$?</p>
| Bernard | 202,857 | <p>You don't really need induction for this: $\;3^n\ge 1+2^n\iff 3^n-2^n\ge 1$. Now
$$3^n-2^n=\underbrace{(3-2)}_{=\,1}(\underbrace{3^{n-1}+3^{n-2}\cdot 2+\dots +2\cdot3^{n-2}+2^{n-1}}_{\text{each term in this sum is }\ge 1}).$$</p>
|
2,599,982 | <p>Can anybody help me finding a good way to (approximately) figure out the first, lets say $200$, <strong>positive</strong> roots of $$\tan(x) + 2 \ell x - \ell ^2 x^2 \tan(x) = 0,$$
where $\ell$ is just a constant?</p>
<p>I believe there will be no analytic expression, so is there a better idea than just running <e... | Andrew | 313,017 | <p>You can simplify the equation somewhat by treating the equation as a quadratic equation and "solving" for x in terms of trig functions. Then if you can generate a list of good initial guesses, for example for large values of x the zeros should approximately satisfy lx*tanx=2, you can possibly use this form to iter... |
152,949 | <p>I noticed that a factorization over algebraic fields is useless in Mathematica. Here is the example over the field containing I*Sqrt[3]:</p>
<pre><code>Pol=4 (3 I Sqrt[3] (-12 + 6 x - 4 x^2 + x^3) y^3 z +
9 (12 - 12 x + 12 x^2 - 6 x^3 + x^4) y^4 z^2 +
I Sqrt[3] (8 + x^3) y z^3 + (4 - 2 x + x^2)^2 z^6 -
3 y^2 (4 ... | Daniel Lichtblau | 51 | <p>This is far from a complete answer but it does contain parts of a general method. The idea is to do it numerically, using a univariate factorization with specific values plugged in for two variables, then "lifting" one variable at a time to a full factorization. The list step is not very efficient (it's done with Gr... |
2,363,624 | <p>A friend told me his colleague estimated 0.95^32 using nothing, just approximating it in her head to be about 0.2.
My calculator gives the answer 0.1937114844585.</p>
<p>How would one go about doing something like that?
I've been burried in contemplation all morning.</p>
<p>Thanks in advance for any insight!</p>
| Shuri2060 | 243,059 | <p>$$\left(0.95^2\right)^{16}\approx\left(0.9^2\right)^{8}\approx\left(0.8^2\right)^{4}\approx\left(0.65^2\right)^{2}\approx0.4^2\approx0.16\approx0.2$$</p>
<p>Note that in most of the steps, the number was rounded down, so an estimate of $0.2$ is reasonable here. I'd argue this is still a 'reasonable' approximation, ... |
1,774,181 | <p>I searched leap years online and found that 1900 is not, contrary to what <em>I</em> thought, a leap year. But, why is it not if 1900 is divisible by 4:<br><br>
$\frac{1900}{4} = 475$<br>
<br> My brother was working on his math (and he obviously got it wrong and asked me for help, so.. here I am), and the question w... | Hagen von Eitzen | 39,174 | <p>Leap years under the Julian calendar were those that are divisible by 4. </p>
<p>Leap years under the Gregorian calendar are those that are divisible by 4, except that those divisible by 100 are not, except that those divisible by 400 are.</p>
|
1,774,181 | <p>I searched leap years online and found that 1900 is not, contrary to what <em>I</em> thought, a leap year. But, why is it not if 1900 is divisible by 4:<br><br>
$\frac{1900}{4} = 475$<br>
<br> My brother was working on his math (and he obviously got it wrong and asked me for help, so.. here I am), and the question w... | robjohn | 13,854 | <p>The length of a <a href="https://en.wikipedia.org/wiki/Tropical_year">Tropical Year</a> is approximately $365.2422$ mean solar days.</p>
<hr>
<p><strong>Julian Calendar</strong></p>
<p>The <a href="https://en.wikipedia.org/wiki/Julian_calendar">Julian Calendar</a> approximates the length of a tropical year as
$$
... |
118,878 | <p><code>FrameTicks</code> are plotted all round the frame, but I only want them on the left and bottom.</p>
<pre><code>ListLinePlot[Prime[Range[400]*800],
Frame -> True,
FrameTicks ->
{{{49, 4}, {97, 8}, {145, 12}, {193, 16}, {241, 20},
{289, 24}, {337, 28}, {385, 32}, {433, 36}, {481, 40}},
... | e.doroskevic | 18,696 | <p><strong>Example</strong></p>
<pre><code>ListPlot[data, Frame -> True, FrameTicks -> {{True, False}, {True, False}}]
</code></pre>
<p><strong>Note:</strong> <code>data</code> is some arbitrary data you want to plot</p>
|
118,878 | <p><code>FrameTicks</code> are plotted all round the frame, but I only want them on the left and bottom.</p>
<pre><code>ListLinePlot[Prime[Range[400]*800],
Frame -> True,
FrameTicks ->
{{{49, 4}, {97, 8}, {145, 12}, {193, 16}, {241, 20},
{289, 24}, {337, 28}, {385, 32}, {433, 36}, {481, 40}},
... | Ymareth | 880 | <p>Try this form...</p>
<pre><code>ListLinePlot[Prime[Range[400]*800], Frame -> True,
FrameTicks -> {{{{-1000000, "-1m"}, {0, "0"}, {1000000,
"1m"}, {2000000, "2m"}, {3000000, "3m"}},
None}, {{{49, 4}, {97, 8}, {145, 12}, {193, 16}, {241, 20}, {289,
24}, {337, 28}, {385, 32}, {433, 36}, {481... |
4,617,225 | <p>Is there a generalized method to constructing primes through sums using the set <span class="math-container">$[2, 2, 2, ..., 3]$</span> given its elements are <span class="math-container">$n$</span>- many 2s and a 3. This question obviously requires knowledge on differences between the primes you are constructing, a... | jp boucheron | 1,138,754 | <p>Write <span class="math-container">$z=re^{i\theta}=r(\cos\theta + i \sin\theta)$</span> with <span class="math-container">$r,\theta\in\Bbb{R}$</span>; your equation becomes <span class="math-container">$r^2+r\times r(\cos\theta+i\sin\theta)+2=2ir\sin\theta$</span>.
Identifying real and imaginary parts of the left &a... |
1,393,694 | <p>I am having a rather tough time wrapping my head around any possible logical fallacy in my solution to the following question as my answer is wrong:</p>
<p>1 percent of children have autism. A test for autism is developed such that 90% of autistic children are correctly identified as having autism but 3% of non-aut... | haqnatural | 247,767 | <p>I try to write another explanation.</p>
<p>Let's look at it as the function,which is the domain is time and range is place:when you write at follows "$$f\left( x \right) =\pm \sqrt { x } $$ it means for instance at $4$ o'clock you can be in two different places$\left\{ -2,2 \right\} $ which is imposible,and it is ... |
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