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 |
|---|---|---|---|---|
719,055 | <p>I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends on the choice of homeomorphism up to, I think, isotopy ( Please forgive the rambling; I'm trying to put together a l... | Abhishek Verma | 94,307 | <p>There is a $C$ which <strong>looks like</strong> $A/B$ (suppose) in some german notation. And now in some rule you denote some value by $C$ and others by $A$ and $B$ and $C = A/B$ is the rule. You won't say you broke the notation $C$ into $A/B$ because it was looking so.</p>
<p>You will deal more such cases in inte... |
2,912,376 | <p>I understood why he chose the positive square root in the sin but why the tan is also positive ? Isn't the tan positive and negative in this interval ?
<a href="https://i.stack.imgur.com/2rBFs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rBFs.png" alt="enter image description here"></a></p>
| egreg | 62,967 | <p>Note that, for $\theta\in[0,\pi]$ and $\theta\ne\pi/2$,
$$
\tan\theta=\frac{\sin\theta}{\cos\theta}
$$
Since you have already established that $\sin\arccos x=\sqrt{1-x^2}$, you can directly conclude that
$$
\tan\arccos x=\frac{\sqrt{1-x^2}}{x}
$$
because $\cos\arccos x=x$ by definition.</p>
<p>Using $\pm$ is mislea... |
697 | <p><a href="https://mathoverflow.net/questions/36307/why-cant-i-post-a-question-on-math-stackexchange-com">This question</a> was posted on MO about not being able to post on math.SE. While MO wasn't the right place for the question, I have to wonder what is. New users who are experiencing difficulty using math.SE can... | Robert Cartaino | 69 | <p>The footer of every single page on the site has a <code>contact us</code> link listed in bold.</p>
|
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | msteve | 67,412 | <p>To compare two complex numbers, we usually look at their modulus: if $z = x+iy$, then the modulus of $z$ is $|z| := \sqrt{x^2 + y^2}$. Regarding $z$ as a point in the complex plane, the modulus of $z$ is the distance to the origin. We can now compare two complex numbers such as $5+2i$ and $3$: notice that $|5+2i| = ... |
1,113,415 | <p>Is there a website or a book with a calculus theorems list?
Or what are the ways remembering calculus theorems list?</p>
| Michael Albanese | 39,599 | <p>An expression is said to be undefined if its meaning or value is not defined. In some cases, an expression is undefined because it is impossible to define it in a consistent or meaningful way; this is the case for the expression $\frac{1}{0}$. No matter how you choose to define the expression $\frac{1}{0}$, it leads... |
1,860,267 | <blockquote>
<p>Prove the convergence of</p>
<p><span class="math-container">$$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$</span></p>
</blockquote>
<p>First I thought the integral does not converge because</p>
<p><span class="math-container">$$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \le \int\lim... | Olivier Oloa | 118,798 | <p>You might want to use integration by parts, obtaining for <span class="math-container">$M\ge1$</span>,
<span class="math-container">$$
\int_{1}^M \frac{\cos x}{x}\: dx=\left[\frac{\sin x}{ x}\right]_1^M+ \int_1^M \frac{\sin x}{x^2}\: dx
$$</span> letting <span class="math-container">$M \to \infty$</span> gives
<span... |
4,469,583 | <p>How can I construct/define arbitrary semi-computable (but not computable) sets?</p>
<p>Recall that a set A is semi-computable if it is domain of a computable function f. Recall also that a set A is computable if and only if both A and the complement set (A<sup>c</sup>) are semi-computable.</p>
<p>In particular, I am... | SHC MostWanted | 905,738 | <p>The answer is provided by @Mitchell Spector in comments.</p>
<p>Pick any semi-computable, but not computable set <span class="math-container">$K$</span>. Then, a new set can be defined as <span class="math-container">$\{ 2x \mid x \in K\}$</span>.</p>
<p>Such a set is a proper subset of <span class="math-container">... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | alephalpha | 6,652 | <pre><code>nar[m_] :=
ToExpression[
"Compile[{$},Do[With[{n=0" <>
StringJoin[
Table["+1" <> Array["0" &, m - 1 - i, 1, StringJoin] <> "a" <>
ToString[m - 1 - i], {i, 0, m - 1}]] <> ",n2=0" <>
Table["+a" <> ToString[m - 1 - i] <> "^" <&g... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | ubpdqn | 1,997 | <p>This may not be efficient but it is terse:</p>
<pre><code>narc[n_] := Module[{r, l, t},
r = Range[n];
t = Total@(#^Length[#]) & /@ (IntegerDigits /@ r);
Pick[r, r - t, 0]
]
</code></pre>
<p><code>narc[10000000]</code> yields:</p>
<pre><code>{1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9... |
70,728 | <p>I've started taking an <a href="http://www.ml-class.org/" rel="noreferrer">online machine learning class</a>, and the first learning algorithm that we are going to be using is a form of linear regression using gradient descent. I don't have much of a background in high level math, but here is what I understand so fa... | Roy Alilin | 535,881 | <p>The idea behind partial derivatives is finding the slope of the function with regards to a variable while other variables value remains constant (does not change). Or what's the slope of the function in the coordinate of a variable of the function while other variable values remains constant. As what I understood f... |
2,702,060 | <p>A little confusion on my part. Study of multi variable calculus and we are using the formula for length of a parameterized curve. The equation makes intuitive sense and I can work it OK. But I also recall using the same integral with out the parameterizing to find the length of a curve where the first term of the sq... | Sri-Amirthan Theivendran | 302,692 | <p>A combinatorial proof. Consider the two element subsets of $\Omega=\{0,1,\dotsc,n\}$. There are $\binom{n+1}{2}$ of them (corresponding to the right hand side of the equality). But we can count in another way. Classify the two element subsets based on their maximum element. For $1\leq k \leq n$, there are $\binom{k}... |
2,702,060 | <p>A little confusion on my part. Study of multi variable calculus and we are using the formula for length of a parameterized curve. The equation makes intuitive sense and I can work it OK. But I also recall using the same integral with out the parameterizing to find the length of a curve where the first term of the sq... | fleablood | 280,126 | <p>Method 1: (requires you to consider whether $n$ is odd or even.)</p>
<p>$S = 1 + 2 + ...... + n$.</p>
<p>Join up the first to term to the last term and second to second to last and so on.</p>
<p>$S = \underbrace{1 + \underbrace{2 + \underbrace{3 +....+(n-2)} + (n-1)} + n}$.</p>
<p>$= (n+1) + (n+1) + .....$.</p>
... |
3,680,864 | <p>I'm trying to understand the relation between the following conditions. I will assume that <span class="math-container">$X$</span> is a Hausdorff topological space and <span class="math-container">$A \subset X$</span>.</p>
<ol>
<li><span class="math-container">$\overline{A}$</span> is compact;</li>
<li>Every net <s... | Henno Brandsma | 4,280 | <p>IIRC the proof is along these lines: </p>
<p>If we have a net <span class="math-container">$x_i, i \in I$</span> that is defined on <span class="math-container">$\overline{A}$</span>, we need to proof it has a convergent subnet (or cluster point) in <span class="math-container">$\overline{A}$</span>. For each <span... |
9,111 | <p>What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$? </p>
<p>I want to evaluate It and I've tried to use the most obvious way: simply typing and evaluating $(x+y)^2$, But it gives me only $(x+y)^2$ as output. I've been searching for it in the last minutes but I still got no clue, can you help me?</p>... | Vitaliy Kaurov | 13 | <p>Short answer is</p>
<pre><code> Expand[(x + y)^2]
</code></pre>
<blockquote>
<p>x^2 + 2 x y+ y^2</p>
</blockquote>
<p>But I recommend you to look at the following tutorials.</p>
<ul>
<li><p><a href="http://reference.wolfram.com/mathematica/tutorial/TransformingAlgebraicExpressions.html" rel="noreferrer">Transf... |
2,148,861 | <p>In one of my junior classes, my Mathematics teacher, while teaching Mensuration, told us that <strong>metres square</strong> and <strong>square metres</strong> have a difference between them and <strong>metres cube</strong> and <strong>cubic metres</strong> too have a difference between them and that we should not m... | Khalid parvaz | 1,047,841 | <p>"Square metre" means area, while "metre square" means a square having all sides 1 metre.</p>
<p>So, "8 square metres" is an area, while an "8-metre square" means a square of side 8 metres (and area 64 square metres).</p>
|
179,223 | <p>I have posted the same question on the community (<a href="http://community.wolfram.com/groups/-/m/t/1394441?p_p_auth=YV2a4wzw" rel="nofollow noreferrer">http://community.wolfram.com/groups/-/m/t/1394441?p_p_auth=YV2a4wzw</a>).</p>
<p>I tried to register the movie posted below (compressed version here) using Mathem... | b3m2a1 | 38,205 | <p>I know nothing about <code>ImageAlign</code> but thought it'd be fun to imitate what StackReg did in the video in your link.</p>
<p>Here's strategy based on the fact that the central blob will remain approximately circular throughout. Note that I use a bizzarro way to get the initial blob form--this isn't really ne... |
2,336,988 | <blockquote>
<p>Let $a,b,c>0 ,2b+2c-a\ge 0,2c+2a-b\ge 0,2a+2b-c\ge 0$ show that
$$\sqrt{\dfrac{2b+2c}{a}-1}+\sqrt{\dfrac{2c+2a}{b}-1}+\sqrt{\dfrac{2a+2b}{c}-1}\ge 3\sqrt{3}$$</p>
</blockquote>
<p>I try use AM-GM and Cauchy-Schwarz inequality and from here I don't see what to do</p>
| Michael Rozenberg | 190,319 | <p>Since our inequality is homogeneous, we can assume that $a+b+c=3$.</p>
<p>Hence, $2b+2c-a=2(3-a)-a=3(2-a)\geq0$, which gives $\{a,b,c\}\subset(0,2]$.</p>
<p>Thus, we need to prove that
$$\sum_{cyc}\sqrt{\frac{2(b+c)}{a}-1}\geq3\sqrt3$$ or
$$\sum_{cyc}\sqrt{\frac{2(3-a)}{a}-1}\geq3\sqrt3$$ or
$$\sum_{cyc}\sqrt{\fra... |
2,379,405 | <blockquote>
<p>Determine convergence or divergence of $$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx$$</p>
</blockquote>
<p>As the graph of the function suggests convergence, Let's find an upper bound that converges.</p>
<p>$$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx \leq
\int_0^{\infty} \frac{2... | Simply Beautiful Art | 272,831 | <p>Showing that there is an upper bound that diverges to $+\infty$ doesn't actually prove anything. What we want is a lower bound:</p>
<p>$$x>1\implies\frac{1+\cos^2x}{\sqrt{1+x^2}}\ge\frac1{\sqrt{x^2+x^2}}=\frac{2^{-1/2}}x$$</p>
<p>And it seems you already know that $\int_0^\infty\frac1x~\mathrm dx$ diverges.</p... |
2,379,405 | <blockquote>
<p>Determine convergence or divergence of $$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx$$</p>
</blockquote>
<p>As the graph of the function suggests convergence, Let's find an upper bound that converges.</p>
<p>$$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx \leq
\int_0^{\infty} \frac{2... | hamam_Abdallah | 369,188 | <p><strong>hint</strong>
$$1+\cos^2 (x)\ge 1$$</p>
<p>$$\sqrt {1+x^2}\sim x \;(x\to +\infty) $$</p>
|
2,146,911 | <blockquote>
<p>Given natural numbers <span class="math-container">$m,n,$</span> and a real number <span class="math-container">$a>1$</span>, prove the inequality :</p>
<p><span class="math-container">$$\displaystyle a^{\frac{2n}{m}} - 1 \geq n\big(a^{\frac{n+1}m} - a^{\frac{n-1}{m}}\big)$$</span></p>
<p><strong>SOU... | Martin R | 42,969 | <p>Let $x = a^{\frac 1m} > 1$. Using
$$
x^{2n} - 1 = (x-1)(1+ x+x^2 + \ldots + x^{2n-1}) \\
x^{n+1} - x^{n-1} = (x-1) (x^{n-1}+x^n)
$$
we get
$$
x^{2n} - 1 - n(x^{n+1} - x^{n-1}) =
(x-1)\left( 1+ x+x^2 + \ldots + x^{2n-1} - n(x^{n-1}+x^n) \right) \\
= (x-1)\sum_{k=1}^n \left( x^{k-1} + x^{2n-k} - x^{n-1}-x^n\right)... |
2,307,021 | <p>I am struggling with a confusing differentials' problem. It seems like there is a key piece of information missing:</p>
<p><strong>The problem:</strong></p>
<blockquote>
<p>The electrical resistance $ R $ of a copper wire is given by $ R = \frac{k}{r^2} $ where $ k $ is a constant and $ r $ is the radius of the... | Archis Welankar | 275,884 | <p>Taking logs we have $ln (R)=ln (k)-ln (r^2) $ thus differentiating we have $\frac {dR}{R}=-2\frac {dr}{r} $ now multiplying by $100$ we have $\text {percent error in R}=-2\text {percent error in r}$ thus $\text {percent error in R}=- (\pm 10)$(as radius increases resistance decreases and vice-versa.)</p>
|
1,903,717 | <p>This is actually from an Analysis text but i feel its a set theory question.</p>
<p>Proposition for ever rational number $\epsilon > 0$ there exists a non-negative number x s.t $x^2 < 2 < (x+ \epsilon )^2 $</p>
<p>It provides a proof that im having trouble understanding.</p>
<p>Proof: let $ \epsilon >... | Noah Schweber | 28,111 | <p>Before the bolded passage, you've concluded that if the statement you're trying to prove <em>fails</em>, then it must be the case that $x^2<2$ implies $(x+\epsilon)^2<2$. </p>
<p>Now, just take $x=0$. $0^2=0<2$, so we must have $(0+\epsilon)^2=\epsilon^2<2$. </p>
<p>So $\epsilon^2<2$. Now take $x=\e... |
2,025,934 | <p>May $V$ be an $n$ dimensional Vektorspace such that $\dim (V) =: n \ge 2$.</p>
<p>We shall prove, that there are infinitely many $k$-dimensional subspaces of $V$, $\forall k \in \{1, 2, ..., n-1\}$.</p>
<p>So first, I thought about using induction, the base step is not that hard, for $n=2$ we take two vectors, say... | Alex M. | 164,025 | <p>In fact, induction does work, even though there are other, more direct approaches. I am going to assume that $V$ is a vector space over some infinite field, otherwise your result is false.</p>
<p>If $n=2$, then $k=1$: consider then all the straight lines passing through the origin; it is obvious and it doesn't requ... |
1,073,459 | <p>I'd like to better understand states on C*-algebras.</p>
<p>What properties should I investigate and in which order?</p>
<p><em>(Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what else?)</em></p>
<p>I suppose basic facts about functional analysis.</p>
<p><em>(C*-algebras, spe... | Simon S | 21,495 | <p>If you were integrating over the volume</p>
<p>$$\left(\frac{x}{a}\right)^{2} + \left(\frac{y}{b}\right)^{2} + \left(\frac{z}{c}\right)^{2} \leq 1$$</p>
<p>you would use spherical polars with $x = ar\sin\theta\cos\phi$, $y = br...$, $z = cr...$.</p>
<p>Now try and modify those so they fit your shape by taking the... |
1,073,459 | <p>I'd like to better understand states on C*-algebras.</p>
<p>What properties should I investigate and in which order?</p>
<p><em>(Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what else?)</em></p>
<p>I suppose basic facts about functional analysis.</p>
<p><em>(C*-algebras, spe... | Mark McClure | 21,361 | <p>Let's write
\begin{align}
x &= a(\rho\sin(\phi)\sin(\theta))^3 \\
y &= b(\rho\sin(\phi)\cos(\theta))^3 \\
z &= c(\rho\cos(\phi))^3,
\end{align}
for then, $(x/a)^{2/3} + (y/b)^{2/3} + (z/c)^{2/3} = \rho^2$. Furthermore, the Jacobian of change of variables is
$$648 \rho ^8 \sin ^2(\theta ) \cos ^2(\t... |
1,619,292 | <p>Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. </p>
<p>Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ and put $Y_i:=p_i(Y)$ where $p_i:\bigoplus_{i\in I}X_i\longrightarrow X$
is the obvious projection. </p>
<p>I... | Martin Peters | 185,067 | <p>My recommendations are:</p>
<blockquote>
<p>Ivar Ekeland: <em>The broken dice, and other mathematical tales of chance</em></p>
<p>Vladimir Arnold: <em>Catastrophe Theory</em>.</p>
</blockquote>
|
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | Stefan4024 | 67,746 | <p>Divide by $\sqrt{x}$ to get </p>
<p>$$\lim_{x \to \infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}} = \lim_{x \to \infty} \frac{1}{\sqrt{1 + \sqrt{\frac 1x + \sqrt{\frac{1}{x^3}}}}} = 1$$</p>
|
3,842,653 | <p>I'm working on a problem for a class and I'm a bit confused on what exactly the question is asking, the question is as follows,</p>
<p>Suppose <span class="math-container">$(x_n)$</span> is a sequence in <span class="math-container">$\Bbb R$</span>. Prove that <span class="math-container">$\bigl\{a \in \Bbb R : \te... | NicholasLP | 829,598 | <p>I think that you need to prove that the set of the <span class="math-container">$a$</span>'s, defined as the limits of subsequences of <span class="math-container">$x_n$</span>, is equal to the infinite intersections of the closures of the sets defined by the subsequences <span class="math-container">$\{ x_n, x_{n+1... |
2,968,655 | <p>My numerical calculations suggest that the equation
<span class="math-container">$$x = \frac{1}{1+e^{-a+bx}}$$</span>
has a unique solution for any <span class="math-container">$a,b \in \mathbb R$</span>. How would one go about showing this?</p>
| B. Goddard | 362,009 | <p>(Someday, I'll learn how to include pictures on here.)</p>
<p>Let <span class="math-container">$r=e^{-a}$</span> and note that <span class="math-container">$r$</span> is positive.</p>
<p>We solve the equation for <span class="math-container">$e^{bx$}$</span> to get</p>
<p><span class="math-container">$$e^{bx} = \... |
2,179,289 | <p>Every valuation ring is an integrally closed local domain, and the integral closure of a local ring is the intersection of all valuation rings containing it. It would be useful for me to know when integrally closed local domains are valuation rings.</p>
<p>To be more specific,</p>
<blockquote>
<p>is there a prop... | mzafrullah | 60,902 | <p>Note the following statements.</p>
<p>I. A quasi local domain $(D,M)$ is a valuation domain if and only if $D$ is
a Bezout domain (i.e. for every pair $a,b$ in $D,$ the ideal $(a,b)$ is
principal or, equivalently, every finitely generated ideal of $D$ is
principal).</p>
<p>If $D$ is a valuation domain then as for ... |
2,828,205 | <p><a href="https://i.stack.imgur.com/JJRaZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JJRaZ.png" alt="enter image description here"></a></p>
<p>First we take identity element from set which is Identity matrix so S=I
for which b(σ(x),σ(y))=b(x,y) which is identity transformation in O(V,b)
so k... | Federico Fallucca | 531,470 | <p>$O(V,b)$ is the set of linear maps $\sigma$ that verify $b(\sigma(x),\sigma(y))=b(x,y)$ for every $x,y\in V$ and $B$ is the matrix associated to $b$ in the canonical base (for example)
Now you can prove that when you fix a base on $V$ </p>
<p>$b(x,y)=x^tBy$</p>
<p>And so if you consider $A_\sigma$ the matrix asso... |
1,204,745 | <p>Let $(\Omega, A, \mathbb{P} )$ be a probability space. Let $f: \Omega \rightarrow [-\infty, \infty]$ an $A$-measurable function. </p>
<p>If $f$ is bounded on the positive side and unbounded on the negative side. Is it possible that $\mathbb{E}[f]$ (the expectation with probability measure $\mathbb{P}$ ) is finite?... | Zach466920 | 219,489 | <p>For your first question, possibly. As for your second question. Yes, it can be finite, most symetric distributions are examples of this. The measure is bounded, but the space can be infinite.</p>
|
3,457,876 | <p>How should I even begin to attempt to show that:
<span class="math-container">$$\frac{\|\bf{x} - \tilde{x} \|}{\|\bf{x}\|} \leq \frac{cond(\bf{A})}{1 - \|\bf{A}^{-1} (\bf{A} - \bf{\tilde{A}}) \|} \left( \frac{\|\bf{b} - \bf{\tilde{b}} \|}{\|\bf{b}\|} + \frac{\|\bf{A} - \bf{\tilde{A}} \|}{\|\bf{A}\|} \right)$$</span>... | Algebraic Pavel | 90,996 | <p><strong>HINT:</strong> From
<span class="math-container">$$\begin{split}
x-\tilde{x}
&=
A^{-1}b-\tilde{A}^{-1}\tilde{b}
\\&=A^{-1}b-\tilde{A}^{-1}b+\tilde{A}^{-1}b-\tilde{A}^{-1}\tilde{b}
\\&=\tilde{A}^{-1}(\tilde{A}-A)A^{-1}b+\tilde{A}^{-1}(b-\tilde{b})
\\&=\tilde{A}^{-1}[(\tilde{A}-A)x+(b-\tilde{b... |
3,457,876 | <p>How should I even begin to attempt to show that:
<span class="math-container">$$\frac{\|\bf{x} - \tilde{x} \|}{\|\bf{x}\|} \leq \frac{cond(\bf{A})}{1 - \|\bf{A}^{-1} (\bf{A} - \bf{\tilde{A}}) \|} \left( \frac{\|\bf{b} - \bf{\tilde{b}} \|}{\|\bf{b}\|} + \frac{\|\bf{A} - \bf{\tilde{A}} \|}{\|\bf{A}\|} \right)$$</span>... | Dat Minh Ha | 690,489 | <p>I might edit this answer into a proper one once I'm a bit more free, but for now, please accept these screenshots since I don't have enough time to convert my custom LaTeX commands into normal ones.
<a href="https://i.stack.imgur.com/dBzi1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dBzi1.png"... |
2,613,410 | <blockquote>
<p>What is the value of <span class="math-container">$2x+3y$</span> if</p>
<p><span class="math-container">$x+y=6$</span> & <span class="math-container">$x^2+3xy+2y=60$</span> ?</p>
</blockquote>
<p>My trial:
from given conditions: substitute <span class="math-container">$y=6-x$</span> in <span class="... | Community | -1 | <p>Assuming a typo ($y$ instead of $y^2$), we restore</p>
<p>$$\begin{cases}x+y=6,\\x^2+3xy+2y^2=(x+y)(x+2y)=60,\end{cases}$$ then</p>
<p>$$2x+3y=\frac{60}6+6.$$</p>
|
4,403,081 | <p><span class="math-container">$$\int \dfrac{dx}{x\sqrt{x^4-1}}$$</span></p>
<p>I need to solve this integration.
I solved and got <span class="math-container">$\dfrac12\tan^{-1}(\sqrt{x^4-1}) + C$</span>, however the answer given in my textbook is <span class="math-container">$\dfrac12\sec^{-1}(x^2) + C$</span></p>
<... | B. Goddard | 362,009 | <p>I tell my students that inverse trig functions are <em>angles.</em> So if you write</p>
<p><span class="math-container">$$\tan^{-1}\sqrt{x^4-1} = \theta,$$</span></p>
<p>then</p>
<p><span class="math-container">$$\tan\theta = \sqrt{x^4-1}.$$</span></p>
<p>A right triangle that tells this story has <span class="math... |
4,474,806 | <p>I use the following method to calculate <span class="math-container">$b$</span>, which is <span class="math-container">$a$</span> <strong>increased</strong> by <span class="math-container">$x$</span> percent:</p>
<p><span class="math-container">$\begin{align}
a = 200
\end{align}$</span></p>
<p><span class="math-cont... | GEdgar | 442 | <p>It is true that: increase by <span class="math-container">$x$</span> percent then decrease the result by <span class="math-container">$x$</span> percent does not get you back where you started.</p>
<p>Let's do a case where it is clearer, say <span class="math-container">$x=100$</span>. Start with <span class="math-... |
1,296,420 | <p>I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?</p>
| Seirios | 36,434 | <p>I think it is an open problem whether or not there exists a finitely presented group $G$ satisfying $G \simeq G \times G$. However, several such finitely generated groups are known. Probably the first example was given by Jones in <a href="http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ17_02%2FS14467887... |
2,658,195 | <p>I have the following problem with which I cannot solve. I have a very large population of birds e.g. 10 000. There are only 8 species of birds in this population. The size of each species is the same.</p>
<p>I would like to calculate how many birds I have to catch, to be sure in 80% that I caught one bird of each s... | Community | -1 | <p>First of all, you procedure is exactly like flipping a coin and getting heads or tails, so you should phrase your question that way. If you were only to throw the coin three times, everyone would have an equal chance of winning, but because you throw continuesly until someone gets their letters, certain letters have... |
2,399,842 | <p>I ran $\frac{d^n}{dx^n}[(x!)!]$ through <em><a href="https://www.wolframalpha.com/input/?i=nth%20derivative%20of%20(x!)!" rel="nofollow noreferrer">Wolfram|Alpha</a></em>, which returned</p>
<blockquote>
<p>$$\frac{\partial^n(x!)!}{\partial x^n} = \Gamma(1+x!)\,R(n,1+x!)$$
for</p>
<ul>
<li><p>$R(n,x)=\ps... | Simply Beautiful Art | 272,831 | <p>Define $f_n(x)$ as follows and show that$$f_n(x)=\Gamma(x)R(n,x)=\frac d{dx}\Gamma(x)R(n-1,x)=f'_{n-1}(x)\\f_0(x)=\Gamma(x)$$</p>
<p>From this, you can see that we actually have</p>
<p>$$f_n(x)=\Gamma^{(n)}(x)$$</p>
<p>And particularly,</p>
<blockquote>
<p>$$R(n,x)=\frac{f_n(x)}{\Gamma(x)}=\frac{\Gamma^{(n)}(x... |
1,335,950 | <p>I have the following sum ($n\in \Bbb N)$:
$$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$
It can be proved that the sum is equal to
$$ \frac{n}{3n + 1} \tag{2}$$
My question is, how do I get the equality? I mean, if I hadn't knew the formula $(2... | Satish Ramanathan | 99,745 | <p>Hint:</p>
<p>Use partial fraction of the ratio</p>
<p>$\frac{1}{(3n-2)(3n+1)}=\frac{1}{3}(\frac{1}{3n-2}-\frac{1}{3n+1})$</p>
<p>You will see mass cancellation such as below:
$\frac{1}{3}\left[1-\frac{1}{4}\right]$</p>
<p>$\frac{1}{3}\left[\frac{1}{4}-\frac{1}{7}\right]$</p>
<p>..</p>
<p>$\frac{1}{(3n-2)(3n+1)... |
1,335,950 | <p>I have the following sum ($n\in \Bbb N)$:
$$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$
It can be proved that the sum is equal to
$$ \frac{n}{3n + 1} \tag{2}$$
My question is, how do I get the equality? I mean, if I hadn't knew the formula $(2... | Arthur | 250,056 | <p>This is as nice task for induction, isn't it?</p>
<p>for $n=1$ we clearly have</p>
<p>$$\sum_{i=1}^1 \frac{1}{(3k-2)(3k+1)} = \frac{1}{4} = \frac{1}{3\cdot 1+1}$$</p>
<p>The induction step is not so difficult as well - it was 2 lines on my paper.</p>
|
1,335,950 | <p>I have the following sum ($n\in \Bbb N)$:
$$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$
It can be proved that the sum is equal to
$$ \frac{n}{3n + 1} \tag{2}$$
My question is, how do I get the equality? I mean, if I hadn't knew the formula $(2... | Vikram | 11,309 | <p>To find the sum of $n$ terms of a series each term of which is composed of the reciprocal of the product of $r$ factors in arithmetical progression, the first factors of the several terms being in the same arithmetical progression, use following</p>
<p>Write down the $n^{th}$ term, strike off a factor from the begi... |
1,612,220 | <p>This is an exercise page 7 from Sutherland's book Introduction to Metric and Topological Spaces.</p>
<blockquote>
<p>Suppose that <span class="math-container">$V,X,Y$</span> are sets with <span class="math-container">$V\subseteq X\subseteq Y$</span> and suppose that <span class="math-container">$U$</span> is a subse... | gebruiker | 145,141 | <p>You start to reason in circles beyond the point "So I am wondering if I can say:..." </p>
<p>You need to keep in mind what you are trying to do. Namely you want to show that $x\in X\cap (Y\backslash U)\implies x\in V$. So once you've showed that $x\in V$ you can just stop. If I were your teacher I would ask you to ... |
3,755,355 | <p>I wanted to prove that every group or order <span class="math-container">$4$</span> is isomorphic to <span class="math-container">$\mathbb{Z}_{4}$</span> or to the Klein group. I also wanted to prove that every group of order <span class="math-container">$6$</span> is isomorphic to <span class="math-container">$\ma... | Nicky Hekster | 9,605 | <p>As @rain1 pointed out, we have a group <span class="math-container">$G=\{1,a,b,ab\}$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are different, commute and are not equal to <span class="math-container">$1$</span>. Let us call <span class="math-container">$ab=ba=c... |
1,047,544 | <p>I'm doing some research and I'm trying to compute a closed form for $ \mathbb{E}[ X \mid X > Y] $ where $X$, $Y$ are independent normal (but not identical) random variables. Is this known?</p>
| heropup | 118,193 | <p>Explicitly, we have for $X \sim \operatorname{Normal}(\mu_x, \sigma_x^2)$ and $Y \sim \operatorname{Normal}(\mu_y, \sigma_y^2)$, $$\operatorname{E}[X \mid X > Y] = \int_{y=-\infty}^\infty \int_{x=y}^\infty x f_{X,Y}(x,y) \, dx \, dy$$ where $$f_{X,Y}(x,y) = \frac{1}{2\pi \sigma_x \sigma_y} \exp \biggl(-\frac{(x-\... |
476,147 | <p>I am working on a problem and I need help getting started. Any pointers would be greatly appreciate it</p>
<p>My problem: Given a $50,000 purse and 20/20 hindsight, and a particular stock, what are the best buying and selling points if the the only requirement is to maximize net profit. The stock is a daily chart g... | Ross Millikan | 1,827 | <p>You want to own the stock any day it goes up and not on any day it goes down, assuming you are not worrying about commissions. So take the difference between today and tomorrow and use the sign of it to decide whether to buy or sell.</p>
|
1,224,202 | <p>Does the following equation makes any sense at all?</p>
<p>$$
\frac{1}{|X|\cdot|Y|}\sum\limits_{x \in X}\sum\limits_{y \in Y}\begin{cases}
1 & \mathrm{if~} x > y\\
0.5 & \mathrm{if~} x = y\\
0 & \mathrm{if~} x < y
\end{cases}
$$</p>
<p>For every comparison of $x$ a... | Kitegi | 120,267 | <p>It looks correct to me. I don't know what you mean by something more beautiful, but you could use the sign function.
$$\frac{1}{|X|\cdot |Y|} \sum_{\substack{x\in X \\ y \in Y}} \frac{1+\operatorname{sgn}(x-y)}{2} $$</p>
|
11,353 | <p>Thinking about the counterintuitive <em>Monty Hall Problem</em> (stick or switch?),
revisited in <a href="https://matheducators.stackexchange.com/a/11346/511">this ME question</a>,
I thought I would issue a challenge:</p>
<blockquote>
<p>Give in one (perhaps long) sentence a convincing explanation of why <em>swit... | JTP - Apologise to Monica | 64 | <p>My preferred explanation - </p>
<p>The key thing to understand is that MH knows the correct door. Say there were a thousand doors. Your chance of choosing the correct door is 1/1000. Now, MH has 999 doors, and after opening 998, there's one left. In effect, he has reduced all the chance, the .999 to that one door. ... |
562,802 | <p>I have been recently investigating the sequence 1,11,111,... I found, contrary to my initial preconception, that the elements of the sequence can have a very interesting multiplicative structure. There are for example elements of the sequence that are divisible by primes like 7 or 2003.</p>
<p>What I am interested ... | N. S. | 9,176 | <p>As you can see in the answers to this <a href="https://math.stackexchange.com/questions/83932/proof-that-a-natural-number-multiplied-by-some-integer-results-in-a-number-with/83968#83968">question</a>, a number has a multiple of the form $111...1$ if the number is not divisible by $2$ and $5$ (i.e. relatively prime ... |
562,802 | <p>I have been recently investigating the sequence 1,11,111,... I found, contrary to my initial preconception, that the elements of the sequence can have a very interesting multiplicative structure. There are for example elements of the sequence that are divisible by primes like 7 or 2003.</p>
<p>What I am interested ... | Ahaan S. Rungta | 85,039 | <p>Indeed, there are such nice properties. To start, consider the following exercise from <em>The Art and Craft of Problem Solving</em> by Paul Zeitz. </p>
<p><strong>Example 1.2</strong>: There is an element in the sequence $ 7, 77, 777, \cdots $ that is divisible by $2003$. </p>
<p><strong>Proof</strong>: We prove ... |
1,245,651 | <p>In algebra, I learned that if <span class="math-container">$\lambda$</span> is an eigenvalue of a linear operator <span class="math-container">$T$</span>, I can have
<span class="math-container">\begin{equation}
Tx = \lambda x
\tag{1}
\end{equation}</span>
for some <span class="math-container">$x\neq 0$</span>, whic... | cfh | 164,698 | <p>Spectral theory in infinite-dimensional spaces is quite a bit more complicated than in the finite-dimensional case.
In particular, we have to distinguish between the spectrum $\sigma(A)$ of an operator and its eigenvalues. Let $A$ be a linear operator on a Banach space $X$ over the scalar field $C$.
We have
$$
\si... |
3,583,330 | <p>I've approached the problem the following way : </p>
<p>Out of the 7 dice, I select any 6 which will have distinct numbers : 7C6.</p>
<p>In the 6 dice, there can be 6! ways in which distinct numbers appear.</p>
<p>And lastly, the last dice will have 6 possible ways in which it can show a number.</p>
<p>So the re... | Vincent | 101,420 | <p>You probably noticed that your answer differs from the correct answer by a factor 2, so apparently you count everything twice.</p>
<p>Suppose your dice are labeled A, B, C, D, E, F, G and you throw:</p>
<p>A:1</p>
<p>B:2</p>
<p>C: 3</p>
<p>D:4</p>
<p>E: 5</p>
<p>F: 6</p>
<p>G: 1</p>
<p>Then you count this t... |
4,279,076 | <p>I have seen in wikipedia that irrational numbers have infinite continued fraction but I also found <span class="math-container">$$1=\frac{2}{3-\frac{2}{3-\ddots}}$$</span> so my question is that does that mean <span class="math-container">$1$</span> is irrational because it can be written as an infinite continued fr... | jjagmath | 571,433 | <p>The theorem about irrationals and and infinite continued fractions is for <strong>simple</strong> continued fractions. See <a href="https://en.m.wikipedia.org/wiki/Continued_fraction" rel="nofollow noreferrer">here</a></p>
|
33,387 | <p>I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies</p>
<p>$r \leq \nu (A^{2} -4B) +\nu(B) -1$</p>
<p>where $\nu(n)$ is the number of distinct positive prime divisors of $n$.</p>
<p>I can not find a n... | GeoffDS | 8,671 | <p>It's in Alvaro Lozano-Robledo's book. In fact, you can find it online.</p>
<p><a href="http://www.math.uic.edu/~wgarci4/pcmi/PCMI_Lectures.pdf" rel="nofollow">http://www.math.uic.edu/~wgarci4/pcmi/PCMI_Lectures.pdf</a></p>
<p>It's Theorem 2.6.4 on page 42.</p>
|
33,387 | <p>I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies</p>
<p>$r \leq \nu (A^{2} -4B) +\nu(B) -1$</p>
<p>where $\nu(n)$ is the number of distinct positive prime divisors of $n$.</p>
<p>I can not find a n... | Álvaro Lozano-Robledo | 14,699 | <p>Two co-authors and I included a proof of this fact in <a href="http://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf" rel="nofollow">our paper</a>, in order to make our article self-contained (but we do not claim to be the first ones to point this out). As Pete Clark explains, it follows... |
1,533,646 | <p>Dying someone appointed in the will the following: If his pregnant wife giving birth to a son , then she will inherit 1/3 of the estate and his son 2/3 . If giving birth to daughter , then she would inherit 2/3 of the property and the daughter 1/3 . The woman gave birth to twins after the death of her husband , a b... | hmakholm left over Monica | 14,366 | <p>Classical and intuitionistic propositional logic do not prove the same formulas, even in the purely implicational fragment.</p>
<p>Most famously, <em>Peirce's Law</em> $((P\to Q)\to P)\to P$ is a classical tautology, but is not intuitionistically valid. (That is, classical logic proves it, but intuitionistic logic ... |
349,177 | <p>I have the following functional equation:
<span class="math-container">$$f(a+b)=f^{(n-1)}(a)f(b)+f^{(n-2)}(a)f^{(1)}(b)+\dots+f(a)f^{(n-1)}(b)=\sum_{k=0}^{n-1}f^{(n-1-k)}(a)f^{(k)}(b)$$</span>
where <span class="math-container">$a,b$</span> can be any complex numbers, <span class="math-container">$f$</span> is an en... | Community | -1 | <p>If $n=1$, we have $f(a+b) = f(a)f(b)$. If $f$ is assume to be continuous, then we have $f(x) = e^{kx}$.</p>
<p>If $n=2$, we have $f(a+b) = f'(a)f(b) + f(a) f'(b)$. Taking a cue from the above, if we let $f(x) = ce^{kx}$, we get
$$ce^{k(a+b)} = 2c^2ke^{k(a+b)}$$
This gives us $2ck = 1 \implies c = \dfrac1{2k}$.</p>
... |
349,177 | <p>I have the following functional equation:
<span class="math-container">$$f(a+b)=f^{(n-1)}(a)f(b)+f^{(n-2)}(a)f^{(1)}(b)+\dots+f(a)f^{(n-1)}(b)=\sum_{k=0}^{n-1}f^{(n-1-k)}(a)f^{(k)}(b)$$</span>
where <span class="math-container">$a,b$</span> can be any complex numbers, <span class="math-container">$f$</span> is an en... | achille hui | 59,379 | <p>Let $\partial_a$ and $\partial_b$ stand for the shorthand of $\frac{\partial}{\partial a}$ and $\frac{\partial}{\partial b}$. The functional equation can be rewritten as:</p>
<p>$$f(a+b) = \left(\partial_a^{n-1} + \partial_a^{n-2}\partial_b + \cdots + \partial_a\partial_b^{n-2} + \partial_b^{n-1}\right)(f(a)f(b))$$... |
3,498,809 | <p>Consider Ito's lemma in the following standard version
<span class="math-container">$$h(W_t) = h(W_0) + \int_0^t \nabla h(W_s) dW_s + \frac{1}{2} \int_0^t \Delta h(W_s) ds.$$</span></p>
<p>I am asking myself under which conditions, the deterministic time <span class="math-container">$t$</span> can be replaced by <s... | zhoraster | 262,269 | <p>Under the same conditions under which the Itô formula is valid. Indeed, the process <span class="math-container">$X_s = W_{\tau \wedge s}$</span> is an Itô process with stochastic differential <span class="math-container">$dX_s = \mathbf{1}_{[0,\tau]}(s) dW_s$</span> (see e.g. our <a href="https://www.wi... |
3,498,809 | <p>Consider Ito's lemma in the following standard version
<span class="math-container">$$h(W_t) = h(W_0) + \int_0^t \nabla h(W_s) dW_s + \frac{1}{2} \int_0^t \Delta h(W_s) ds.$$</span></p>
<p>I am asking myself under which conditions, the deterministic time <span class="math-container">$t$</span> can be replaced by <s... | John Dawkins | 189,130 | <p>If (as is customary) the stochastic integral is understood to be continuous in <span class="math-container">$t$</span> (a.s.) then the equality holds for all <span class="math-container">$t$</span> simultaneously, with probability 1. As such, <span class="math-container">$t$</span> can be replaced throughout by <em... |
2,291,540 | <p>Is it possible to have a sequence of continuous functions $\{f_n\}_{n=1}^\infty$ on $[a,b]$ that converges uniformly to a function $f$ but $f$ is not bounded on $[a,b]$?</p>
| Affineline | 448,123 | <p>Uniform limit of continuous functions is necessarily continuous. Also, continuous functions on an interval are necessarily bounded. Combining the two we get that there doesn't exist such a function as required. </p>
|
9,696 | <p>I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example: </p>
<p>$$\;\;14\\
+27\\
=41$$ </p>
<p>I asked her to write this out horizontally, and this is what she produced. </p>
<p>$$12+47=41$$ </p>
<p>She evidently is failing to see the n... | Amy B | 5,321 | <p>Does she have difficulty with reading or other taking in other visual information? Her problem might have nothing to do with understanding numbers and everything to do with a learning difference in how she perceives visual information. I've had students who could answer questions but couldn't handle a worksheet (... |
2,625,763 | <p>I am having trouble with factoring $2x^3 + 21x^2 +27x$. The answer is $x(x+9)(2x+3)$ but not sure how that was done. Obviously I factored out the $x$ to get $x(2x^2+21x+27)$ then from there I am lost. I tried the AC method and grouping. Can someone show the steps? Thanks! </p>
| Mostafa Ayaz | 518,023 | <p>$$2x^3+21x^2+27x\\=x(2x^2+21x+27)\\=\dfrac{x}{2}(4x^2+42x+54)\\=\dfrac{x}{2}(2x+18)(2x+3)\\=x(x+9)(2x+3)$$for factorizing $4x^2+42x+54$ we know that it must in form of $(2x+a)(2x+b)$. So:$$4x^2+42x+54=(2x+a)(2x+b)=4x^2+(a+b)2x+ab$$which implies$$ab=54\\a+b=21$$clearly the only numbers satisfying those equalities are... |
2,625,763 | <p>I am having trouble with factoring $2x^3 + 21x^2 +27x$. The answer is $x(x+9)(2x+3)$ but not sure how that was done. Obviously I factored out the $x$ to get $x(2x^2+21x+27)$ then from there I am lost. I tried the AC method and grouping. Can someone show the steps? Thanks! </p>
| Arnav Borborah | 520,392 | <p>Quite simple; The AC Method <em>can</em> be used here:</p>
<p>$$2x^3 + 21x^2 +27x =$$
$$x(2x^2 + 21x +27) =$$
Now we have to find two factors that multiply to $54$ ($2 \times 27$), and add up to $21$. Two such numbers are $3$ and $18$, which are now used to split apart the polynomial.
$$x((2x^2 + 3x) + (18x + 27)) ... |
3,473,944 | <p>So i have an object that moves in a straight line with initial velocity <span class="math-container">$v_0$</span> and starting position <span class="math-container">$x_0$</span>. I can give it constant acceleration <span class="math-container">$a$</span> over a fixed time interval <span class="math-container">$t$</s... | Manuel Pena | 341,519 | <p>You are impossing to much things. Think about it this way. If you begin at speed <span class="math-container">$v_0$</span> and pretend to have a constant acceleration such that, in a time <span class="math-container">$t$</span> you reach <span class="math-container">$0$</span> speed, then this acceleration must be:
... |
3,009,543 | <p>I am having great problems in solving this:</p>
<p><span class="math-container">$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$</span></p>
<p>I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I concluded that I h... | user | 505,767 | <p>By first order binomial expansion <span class="math-container">$(1+x)^r=1+rx + o(x)$</span>, we have</p>
<p><span class="math-container">$$\sqrt[3]{n+\sqrt{n}}=\sqrt[3]{n}\, \left(1+\frac1{\sqrt n}\right)^\frac13=\sqrt[3]{n}+\frac{\sqrt[3]{n}}{3\sqrt n}+o\left(\frac{\sqrt[3]{n}}{\sqrt n}\right)=\sqrt[3]{n}+\frac{1}... |
3,009,543 | <p>I am having great problems in solving this:</p>
<p><span class="math-container">$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$</span></p>
<p>I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I concluded that I h... | User | 125,635 | <p>Consider the function <span class="math-container">$f(x)=x^{1/3}$</span>. By the mean value theorem there's a number <span class="math-container">$y\in (n, n+\sqrt n)$</span> such that
<span class="math-container">$$
f(n+\sqrt n) - f(n) = f'(y)(n+\sqrt n - n)= \frac{y^{-2/3}}{3}\sqrt n<n^{-2/3}\sqrt n=n^{-1/6}\t... |
1,355,901 | <p>Let $A$ be the set of all integers $x$ such that $x = 2k$ for some integer $k$</p>
<p>Let $B$ be the set of all integers $x$ such that $x = 2k+2$ for some integer $k$</p>
<p>Give a formal proof that $A = B$.</p>
| user2034716 | 135,232 | <p>It suffices to show that both $A\subset B$ and $B\subset A$.</p>
<p>Fix $x \in A$. Then we can write $x=2k$ for some $k$, which means we could also write $x=2k=2k-2+2=2(k-1)+2$. Thus $x\in B$, because for $k'=k-1$ we can write $x=2k'+2$.</p>
<p>This proves one direction, that $A\subset B$. The other direction is n... |
555,955 | <p>Suppose I have two doors. One of them has a probability of $1/9$ to contain X, the other has a probability of $2/3$ to contain X. Then, supposing I pick randomly one of the two doors, what is the probability that it contains X?</p>
<p>(If one contains X, the other can also contain X. They are independent but not mu... | Tom | 103,715 | <p><em>Hint:</em> Let $E$ be the event that the door you open contains $X$. Assuming that you must choose either door $1$ or door $2$, but not both:</p>
<p>$$
P(E) = P(E~|~\text{choose door } 1)P(\text{choose door} 1) + P(E~|~\text{choose door } 2)P(\text{choose door} 2)
$$</p>
|
3,074,668 | <p>Good evening,</p>
<p>Could someone please demonstrate why this property is valid?</p>
<blockquote>
<p>Given <span class="math-container">$\sigma\in S_n$</span></p>
<p><span class="math-container">$$\left|\prod_{i<j} \frac{\sigma(j)-\sigma(i)}{j-i}\right|=1$$</span></p>
</blockquote>
| darij grinberg | 586 | <p>Detailed proof: See Exercise 5.13 <strong>(a)</strong> in <a href="https://github.com/darijgr/detnotes/releases/2019-01-10" rel="nofollow noreferrer">my <em>Notes on the combinatorial fundamentals of algebra</em>, 10th of January 2019</a>. The claim I prove there is more general: I show that if <span class="math-con... |
860,247 | <p>Simplify $$\frac{3x}{x+2} - \frac{4x}{2-x} - \frac{2x-1}{x^2-4}$$</p>
<ol>
<li><p>First I expanded $x²-4$ into $(x+2)(x-2)$. There are 3 denominators. </p></li>
<li><p>So I multiplied the numerators into: $$\frac{3x(x+2)(2-x)}{(x+2)(x-2)(2-x)} - \frac{4x(x+2)(x-2)}{(x+2)(x-2)(2-x)} - \frac{2x-1(2-x)}{(x+2)(x-2)(2-x... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $$x^2+x=\frac{(2x+1)^2-1^2}4$$ set $$2x+1=\sec\theta$$</p>
<p>For $x=0,\sec\theta=1\implies\theta=0$ and for $x=1,\sec\theta=3\implies \theta=\arccos\frac13$</p>
|
860,247 | <p>Simplify $$\frac{3x}{x+2} - \frac{4x}{2-x} - \frac{2x-1}{x^2-4}$$</p>
<ol>
<li><p>First I expanded $x²-4$ into $(x+2)(x-2)$. There are 3 denominators. </p></li>
<li><p>So I multiplied the numerators into: $$\frac{3x(x+2)(2-x)}{(x+2)(x-2)(2-x)} - \frac{4x(x+2)(x-2)}{(x+2)(x-2)(2-x)} - \frac{2x-1(2-x)}{(x+2)(x-2)(2-x... | medicu | 65,848 | <p>To calculate such integrals, it is often very useful change of variable of "type Euler".
$$\sqrt{x+x^2} = x-t $$ with $ x= \frac{t^2}{2t+1}$ and $dx= \frac{2t(t+1)}{2t+1} $</p>
<p>Integral reduces to an integral rational calculation:</p>
<p>$$\int_0^1 x^{2}\sqrt{x+x^2}dx =2\int_{1-\sqrt{2}}^0 \frac{t^6(t+1)^2}{(2... |
1,378,536 | <p>Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known <em>nice real analysis tools</em> for calculating them (<em>including here all possible sources<br>
in literature that are publicaly available</em>). At some point I'll add my <em>real analy... | Random Variable | 16,033 | <p>After making the substitution $u = \text{arctanh}(x)$, we could use the Laplace transform $$\int_{0}^{\infty} \cos(ax) \, e^{-bx} \, dx = \frac{b}{a^{2}+b^{2}} \, , \, \text{Re} (b) >0 $$</p>
<p>and then switch the order of integration.</p>
<p>Specifically,</p>
<p>$$ \begin{align} \int_{-\infty}^{\infty} \frac... |
1,378,536 | <p>Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known <em>nice real analysis tools</em> for calculating them (<em>including here all possible sources<br>
in literature that are publicaly available</em>). At some point I'll add my <em>real analy... | robjohn | 13,854 | <p>$\newcommand{\sech}{\operatorname{sech}}\newcommand{\arctanh}{\operatorname{arctanh}}\newcommand{\Res}{\operatorname*{Res}}$
$\Res\limits_{z=\frac\pi2i}\left(\frac{\sech^2(z)}{\pi^2+4z^2}\right)=-i\frac{3+\pi^2}{12\pi^3}$ and for $k\ge1$, $\Res\limits_{z=\frac{(2k+1)\pi}2i}\left(\frac{\sech^2(z)}{\pi^2+4z^2}\right)=... |
3,190,594 | <p>From Rick Durrett's book <em>Probability: Theory and Examples</em>:</p>
<blockquote>
<p>We define the conditional expectation of <span class="math-container">$X$</span> given <span class="math-container">$\mathcal{G}$</span>, <span class="math-container">$E(X | \mathcal{G})$</span> to be any random variable <span cla... | kccu | 255,727 | <p>Definition 1 <span class="math-container">$\Rightarrow$</span> Definition 2 follows from Theorem 4.1.14 in <a href="https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf" rel="nofollow noreferrer">Durrett (5th edition)</a> and by propert (2) with <span class="math-container">$A=\Omega$</span>. </p>
<p>Definition ... |
1,285,177 | <p>I know this is very simple and I'm missing something trivial here...</p>
<p>I'm having trouble converting this set of equations to polar form:</p>
<p>$$
\dot{x_1}=x_2-x_1 (x_1^2+x_2^2-1)\\
\dot{x_2}=-x_1-x_2 (x_1^2+x_2^2-1)
$$</p>
<p>where</p>
<p>$$
r= (x_1^2+x_2^2)^{1/2}\\
\theta=\arctan\left(\frac{x_2}{x_1}\ri... | Brian Fitzpatrick | 56,960 | <p>Note that
$$
A=\begin{bmatrix}a&b\\c&d\end{bmatrix}
$$
satisfies $A^\top=-A$ if and only if
$$
\begin{bmatrix}
a&c\\b&d
\end{bmatrix}
=
\begin{bmatrix}
-a&-b\\-c&-d
\end{bmatrix}
$$
That is, $A^\top=-A$ if and only if
$$
A=
\begin{bmatrix}
a&b\\c&d
\end{bmatrix}
=
\begin{bmatrix}
0&am... |
1,285,177 | <p>I know this is very simple and I'm missing something trivial here...</p>
<p>I'm having trouble converting this set of equations to polar form:</p>
<p>$$
\dot{x_1}=x_2-x_1 (x_1^2+x_2^2-1)\\
\dot{x_2}=-x_1-x_2 (x_1^2+x_2^2-1)
$$</p>
<p>where</p>
<p>$$
r= (x_1^2+x_2^2)^{1/2}\\
\theta=\arctan\left(\frac{x_2}{x_1}\ri... | math.n00b | 135,233 | <p>More generally, even if $A$ and $B$ are $n \times n$ matrices, they still form a subspace because:
$(A+\lambda B)^T = A^T + \lambda B^T$</p>
<p>So, if $A^T = -A$ and $B^T = -B$ we get $(A+\lambda B)^T = -(A + \lambda B)$ which proves that $A + \lambda B$ belongs to the set of skew symmetric $n \times n$ matrices.</... |
362,895 | <p>I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one please check this proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field. Thank you.</p>
<p>Proof: Notice that ... | Math Gems | 75,092 | <p>One must also prove that the quotient ring is $\ne \{0\}.\:$ Below is a complete proof.
$\rm\quad \Bbb Z\stackrel{h}{\to}\, \Bbb Z[{\it i}\,]/(1\!-\!{\it i}\,)\:$ is $\rm\,\color{#0b0}{\bf onto,\:}$ by $\rm\:mod\,\ 1\!-\!{\it i}\,:\ {\it i}\,\equiv 1\phantom{\dfrac{|}{|}}\!\!\!\Rightarrow\:a\!+\!b\,{\it i}\,\equiv ... |
3,632,097 | <p>Given a sheaf <span class="math-container">$F$</span> on a topological space <span class="math-container">$X$</span> and <span class="math-container">$U$</span> is an open subset of <span class="math-container">$X$</span>. Denote <span class="math-container">$F|_U$</span> be the restricted sheaf of <span class="math... | diracdeltafunk | 19,006 | <p>@Bueggi has the right answer from first principles. I'd like to mention a slightly more abstract way to think about it, which I hope will be helpful to see if you're starting to learn about sheaves.</p>
<p>If <span class="math-container">$*$</span> is a topological space with one point, the category sheaves on <spa... |
874,946 | <p>What is the remainder when the below number is divided by $100$?
$$
1^{1} + 111^{111}+11111^{11111}+1111111^{1111111}+111111111^{111111111}\\+5^{1}+555^{111}+55555^{11111}+5555555^{1111111}+55555555^{111111111}
$$
How to approach this type of question? I tried to brute force using Python, but it took very long time.... | Ben Grossmann | 81,360 | <p>Two facts help here:</p>
<ol>
<li>if $a \equiv b \pmod m$, then $a^n \equiv b^n \pmod m$</li>
<li>For any $a$ relatively prime to $100$, $a^{40} \equiv 1 \pmod {100}$</li>
</ol>
<p>So, for example,
$$
111^{111} \equiv 11^{111} \equiv (11^{40})^2 11^{31} \equiv 11^{31} \pmod{100}
$$</p>
|
874,946 | <p>What is the remainder when the below number is divided by $100$?
$$
1^{1} + 111^{111}+11111^{11111}+1111111^{1111111}+111111111^{111111111}\\+5^{1}+555^{111}+55555^{11111}+5555555^{1111111}+55555555^{111111111}
$$
How to approach this type of question? I tried to brute force using Python, but it took very long time.... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$(1+10n)^{1+10n}=1+\binom{1+10n}1(10n)\pmod{100}\equiv1+10n$$</p>
<p>and $$(5+50n)^{1+10n}=5^{1+10n}+\binom{1+10n}1(50n)5^{10n}\pmod{100}$$</p>
<p>Now, $$5^{m+2}-5^2=5^2(5^m-1)\equiv0\pmod{100}\implies5^{m+2}\equiv25\pmod{100}$$ for integer $m\ge0$</p>
<p>$$\implies5^{1+10n}+\binom{1+10n}1(50n)5^{1... |
1,067,131 | <p>I'm reading a analysis book for fun and I got stuck on a problem.</p>
<p>The task is to find the function $f$ if
$$f(x-y,x+y) = \frac{x^2 + y^2}{2xy}$$</p>
<p>Since I can see the solution $\frac{x^2 + y^2}{y^2 - x^2}$ from the book (it's given in the back), I can backwards engineer the solution:</p>
<p>$$ \frac{(... | GPerez | 118,574 | <p>If you've studied the cyclotomic polynomial the answer becomes quite simple! Since $\omega_7=e^{2\pi i/7}$ is a root of the seventh cyclotomic polynomial $\Phi_7$, and $\Phi_7$ can be shown to be irreducible, the degree of the extension is $\deg\Phi_7 = \varphi(7) = 6$, because $7$ is prime ($\varphi$ denotes the Eu... |
1,067,131 | <p>I'm reading a analysis book for fun and I got stuck on a problem.</p>
<p>The task is to find the function $f$ if
$$f(x-y,x+y) = \frac{x^2 + y^2}{2xy}$$</p>
<p>Since I can see the solution $\frac{x^2 + y^2}{y^2 - x^2}$ from the book (it's given in the back), I can backwards engineer the solution:</p>
<p>$$ \frac{(... | Jyrki Lahtonen | 11,619 | <p>Extended hint:</p>
<p>The field $\Bbb{Q}(\omega_7+\omega_7^5)$ is contained in the 7th cyclotomic field $\Bbb{Q}(\omega_7)$. That field is an abelian extension of $\Bbb{Q}$, so all intermediate fields are themselves Galois extensions of $\Bbb{Q}$. This follows from Galois correspondence as all subgroups of an abeli... |
902,522 | <p>How would I simplify a fraction that has a radical in it? For example:</p>
<p>$$\frac{\sqrt{2a^7b^2}}{{\sqrt{32b^3}}}$$</p>
| Wouter Stekelenburg | 27,375 | <p>The right way to build such a category is a philosophical question. There are different approaches in the mathematical literature. One thing is clear though: the objects should be propositions, not just theorems.</p>
<p>The problem is to define equality of proofs in a sensible way. For example, let $\Pi$ be Pythago... |
813,395 | <p>I have read that linear independence occurs when:</p>
<p>$$\sum_{i=1}^n a_i v_i =0$$</p>
<p>Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean that such a vector set is not linearly independent?</p>
<p>What if I have:</p>
<p>Let $\{c_0,c_1,c_2... | Andreas Caranti | 58,401 | <p>Any set of vectors containing zero is linearly dependent, that is, <em>not</em> linearly independent.</p>
<p>This is simply because, as you have said, if $v_{1} = 0$, say, then
$$
1 \cdot v_{1} + 0 \cdot v_{2} + \dots + 0 \cdot v_{n} = 0,
$$
and not all coefficients are zero.</p>
|
3,011,862 | <p>Test the convergence <span class="math-container">$$\int_0^1 \frac{x^n}{1+x}dx$$</span></p>
<p>I have used comparison test for improper integrals..by comparing with <span class="math-container">$1/(1+x)$</span>...
so I found it convergent ..
But the solution set says that it is convergent if <span class="math-conta... | fleablood | 280,126 | <p>Pick the last digit <em>first</em>. You have <span class="math-container">$3$</span> choices (<span class="math-container">$2,6$</span> or <span class="math-container">$8$</span>).</p>
<p>Then pick the first digit. As you have already picked a digit you have only <span class="math-container">$4$</span> choices re... |
3,546,615 | <p>Why do we take thickness be differential of distance apart of elemental mass when calculating volume and be differential length of arc when calculating area of the sphere when integrating in terms of angle.</p>
<p><a href="https://i.imgur.com/Mw8oW85m.jpg" rel="nofollow noreferrer"><img src="https://i.imgur.com/Mw8... | Emilio Novati | 187,568 | <p>This is because <span class="math-container">$\theta \to 0$</span> so
<span class="math-container">$$
\frac{d \sin \theta}{d\theta}=\lim_{\theta \to 0}\frac{\sin \theta}{\theta}=1
$$</span>
In other words if <span class="math-container">$\theta$</span> is small the difference with <span class="math-container">$\sin ... |
1,483,489 | <p>What I have trying is:</p>
<p>Suppose that $f(x)$ has at least one zero $\alpha$ such that $f(x) = (x - \alpha)^sq(x)$, $s > 1$ in some extension. Then I guess that $(x-\alpha)^{s-1} \mid f(x)$.
So, $f(x)$ is not irreducible, where $f(x) = (x-a)^{s-1}h(x)$.
But is seems wrong once I neither used the hypothesis $... | André Nicolas | 6,312 | <p>Outline: We can assume that $f$ has degree $n\gt 1$. Then $f'(x)$ has degree $n-1$. (This is where we use characteristic $0$. In characteristic $p$, this part can fail. For example, the derivative of $x^p+1$ is the $0$ polynomial.) </p>
<p>Since $f$ is irreducible, $f(x)$ and $f'(x)$ are relatively prime over $F$. ... |
1,012,158 | <p>$$ y\in R$$
Prove: <br>
if for every positive number $b$:
$$ \left\lvert y \right\rvert \leq b $$
so $y=0$</p>
<p>I tried seperating into cases where</p>
<p>$$ -b\leq y\leq 0 $$ and $$ 0\leq y\leq b $$</p>
<p>But I can't see where it helps me, any ideas? thanks</p>
| Empiricist | 189,188 | <p>Assume on the contrary $y \neq 0$.</p>
<p>Then $\frac{|y|}{2} > 0$ and hence $\frac{|y|}{2} \geq |y|$ and thus $|y| \leq 0$. Contradiction.</p>
|
3,805,286 | <p>This is a question on the convergence of a sequence of real, convex, analytic functions (it does not get better than that!):</p>
<p>Let <span class="math-container">$(f_n)_{n\in \mathbb N}$</span> be a sequence of convex analytic functions on <span class="math-container">$\mathbb R$</span>.</p>
<p>Suppose that <span... | Greg Martin | 16,078 | <p>No—not even necessarily differentiable! The function <span class="math-container">$f_n(x) = \frac1n\log(1+e^{nx})$</span> is convex and analytic on <span class="math-container">$\Bbb R$</span>, but
<span class="math-container">$$
\lim_{n\to\infty} \frac1n\log(1+e^{nx}) = \begin{cases} 0, &\text{if } x\le 0, \\ x... |
3,322,492 | <p><strong>Prove:</strong> <span class="math-container">$A \cap (B - C) = (A \cap B) − (A \cap C)$</span></p>
<p>I can understand this using Venn Diagrams, however I am struggling to translate this into a formal proof. </p>
| Henno Brandsma | 4,280 | <p><span class="math-container">$$A \cap (B-C) = A \cap (B \cap C^\complement)$$</span> while</p>
<p><span class="math-container">$$(A\cap B) - (A \cap C) = (A \cap B) \cap (A \cap C)^\complement = \\
(A \cap B) \cap (A^\complement \cup C^\complement) (\text{ de Morgan) } =\\
(A \cap B \cap A^\complement) \cup (A \cap... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Michael Hoffman | 429 | <p>Whenever I've heard the term "elementary number theory", the speaker seems to mean "analytic number theory"</p>
<p>I would imagine that the reason for not using complex numbers is at least partially related to the idea that they're going over old results, that were essentially developed before the development of co... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Community | -1 | <p>Your usage of "elementary" is correct; your definition is the one that most number theorists would use. You don't have to take my word for it however; just consider the first sentence of <a href="http://www.jstor.org/pss/1969455" rel="noreferrer">Selberg's Elementary Proof of the Prime Number Theorem</a>:</p>
<p><e... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Alexey Ustinov | 5,712 | <p>Probably there is no correct bounary between elementary and non-elementary number theory. There two possibilities: either we can apply limits or not. It is like the axiom of choice in the set theory.</p>
<p>1) If we have no <span class="math-container">$\lim$</span> then we have no <span class="math-container">$\pi... |
459,579 | <blockquote>
<p>Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$</p>
</blockquote>
<p>Doesn't this thing approaches 0 at the end? why does it approaches 1?</p>
| Harish Kayarohanam | 30,423 | <p>$ 3^{12} \times 3^{sum\ of\ geometric\ series }$ </p>
<p>geometric series is </p>
<p>$ 1 + 1/3 + 1/9 + .... $ </p>
<p>$ = 1 / (1-1/3) $ </p>
<p>$ = 3/2 $ </p>
<p>so </p>
<p>$= 3^ {12 + \frac{3}{2} } $ </p>
<p>$= 3 ^{ 27/2 } $</p>
|
192,394 | <p>I'm re-reading some material from Apostol's Calculus. He asks to prove that, if $f$ is such that, for any $x,y\in[a,b]$ we have</p>
<p>$$|f(x)-f(y)|\leq|x-y|$$</p>
<p>then:</p>
<p>$(i)$ $f$ is continuous in $[a,b]$</p>
<p>$(ii)$ For any $c$ in the interval,</p>
<p>$$\left|\int_a^b f(x)dx-(b-a)f(c)\right|\leq\fr... | Seirios | 36,434 | <p>Another way is to write $$(b-a)^2= ((b-c)-(a-c))^2= (b-c)^2+ (a-c)^2-2(b-c)(a-c) \geq (b-c)^2+(a-c)^2$$ because $(b-c)(a-c) \leq 0$.</p>
<p>For your second question, consider $\displaystyle \frac{1}{\lambda} f$ and apply your first result.</p>
|
261,031 | <p>i hope some of you can support to solve my problem, i need to work on data in the following way, where the length of each of the lists or sublists is equal. As an example i want to share the data-pattern with you:</p>
<pre><code>list1={a,b,c};
list2={{d,e,f},{g,h,i},......} (in reality the number of sublists in list... | user1066 | 106 | <pre><code>Inner[{#2,#1}&,list2,list1,List]
(* {{{a, d}, {b, e}, {c, f}}, {{a, g}, {b, h}, {c, i}}} *)
</code></pre>
<p><strong>A Slot-free version</strong></p>
<p><a href="https://mathematica.stackexchange.com/users/50/j-m-cant-deal-with-it">J. M. can't deal with it </a> (in a comment) gives the following neat mo... |
2,961,023 | <p>Is it allowed to solve this inequality <span class="math-container">$x|x-1|>-3$</span> by dividing each member with <span class="math-container">$x$</span>? What if <span class="math-container">$x$</span> is negative?</p>
<p>My textbook provides the following solution:</p>
<blockquote>
<p>Divide both sides b... | Will Jagy | 10,400 | <p><img src="https://i.stack.imgur.com/bKIyr.jpg" alt="">
<span class="math-container">${}{}{}{}{}{}{}{}{}{}{}{}{}{}$</span></p>
|
4,262,888 | <p>My task is to prove that if an atomic measure space is <span class="math-container">$\sigma$</span>-finite, then the set of atoms must be countable.</p>
<p>This is my given definition of an atomic measure space:</p>
<blockquote>
<p>Assume <span class="math-container">$(X,\mathcal{M},\mu)$</span> is a measure space w... | Lazy | 958,820 | <p>That does appear to be a bit sloppy. But you can mend that by simply specifying the sum over any countable selection and taking the supremum of that. You can easily show: If you have an uncountable family of positive values, then for any countable finite sum there exists an even larger one, so this sup must be <span... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.