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 |
|---|---|---|---|---|
44,200 | <p>Lets say I want to represent 20 Percent</p>
<p>What do you call/name the "0.2" and "20%" notations</p>
<p>I am calling them Fractional and WholeValue notations for the time being, but wondering if there was something better.</p>
| Macro | 11,636 | <p>I would call "0.2" a proportion and "20%" a percentage. </p>
|
3,330,625 | <p>An Affine transformation is a function <span class="math-container">$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$</span> such that <span class="math-container">$f(v) = Av+b $</span>, where <span class="math-container">$\det A \neq 0 $</span> and <span class="math-container">$b \in \mathbb{R}^2.$</span> </p>
<p>My profe... | Kavi Rama Murthy | 142,385 | <p>I think you have mis-interpreted the question. What is true is if <span class="math-container">$f(v)=Av+b$</span> for some square matrix <span class="math-container">$A$</span> and some vector <span class="math-container">$b$</span> then <span class="math-container">$f$</span> is one-to-one iff <span class="math-con... |
1,577,845 | <p>Find the volume of the solid inside the surface defined by the equation $\rho=8\sin \phi$ in spherical coordinates</p>
<p>So far I've set up an integral in spherical coordinates with $\rho$ from $0$ to $\rho=8\sin \phi$
$\theta$ from $0$ to $2\pi$,
$\phi$ from $0$ to $\pi$.</p>
<p>This is just a far off guess as I... | mickep | 97,236 | <p>Assuming the $a_k$ are all real we know that $\bar z$ is a solution to the characteristic equation given that $z$ is. </p>
<p>By the nature of solutions to homogeneous linear differential equations the term $x\sin x$ implies that $\pm i$ are solutions to the characteristic equation. Moreover, the $x$ in front of $\... |
3,638,741 | <p>Let <span class="math-container">$x_1, x_2, \ldots , x_n \in \{0, 1\}$</span>.</p>
<p>(a) Consider the equation <span class="math-container">$x_1 + x_2 + \ldots + x_n = 0 \pmod {2}$</span>. How many solutions does this equation have? Express your answer in terms of <span class="math-container">$n$</span>. For examp... | JMoravitz | 179,297 | <p>For (a) pick anything for <span class="math-container">$x_2,x_3,\dots,x_n$</span>. Given any specific choice of these there is a unique choice available for <span class="math-container">$x_1$</span> such that the overall sum is even. We were able to make <span class="math-container">$n-1$</span> choices freely and... |
124 | <p>Harold Williams, Pablo Solis, and I were chatting and the following question came up.</p>
<p>In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra <em>g</em>, you can find a faithful representation <em>g</em> → End(V) by Ado's theorem. Then you can take the group ... | Allen Knutson | 391 | <p>Let ${\mathbb R}$ act on ${\mathbb R}^2$ by rotation at unit speed, and on a second ${\mathbb R}^2$ by rotation at some irrational speed. Let $G$ be the (solvable) semidirect product ${\mathbb R} \ltimes {\mathbb R}^4$. Then the coadjoint orbits of $G$ may not be locally closed. I don't think that should happen in t... |
50,888 | <p>Why does </p>
<pre><code>Flatten[f[g[a, f[b]], f[c, d]]]
</code></pre>
<p>result in </p>
<blockquote>
<pre><code>f[g[a, f[b]], c, d]
</code></pre>
</blockquote>
<p>I expected the answer to be </p>
<blockquote>
<pre><code>f[g[a, b], c, d]
</code></pre>
</blockquote>
<p>What is wrong in this code?</p>
| ubpdqn | 1,997 | <p>This does not answer your question about <code>Flatten</code>. However, rasher's comment does. Here are some approaches to achieve your goal:</p>
<pre><code>f[f[g[a, f[b]], f[c, d]] /. f -> Sequence]
</code></pre>
<p>or</p>
<pre><code># /. f -> Sequence & /@ f[g[a, f[b]], f[c, d]]
</code></pre>
|
2,005,204 | <p>I am trying to prove this theorem: </p>
<blockquote>
<p>$A^o\cup A^{ext}\cup\ A^b=\mathbb{R}$</p>
</blockquote>
<p>I don't really know where to start. $A$ is a subset of $\mathbb{R}$.</p>
| drhab | 75,923 | <p>Let it be that $x\in\mathbb R$ and $A\subseteq\mathbb R$. </p>
<p>We discern the following statements:</p>
<ul>
<li><p>an open set $U$ exists with $x\in U\subseteq A$. </p></li>
<li><p>an open set $U$ exists with $x\in U\subseteq A^c$. </p></li>
<li><p>for every open set $U$ with $x\in U$ we have $U\cap A^c\neq\va... |
1,826,866 | <p>I can't understand why the projection of those two objects on plane $xy$ gives that the angle's range from -90 degress to 90 degrees. why not from 0 to 360?</p>
<p><a href="https://i.stack.imgur.com/rFDpf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rFDpf.png" alt="enter image description here... | Narasimham | 95,860 | <p>This is the famous Viviani scenario. Usually the unit sphere is centered at origin and cylinder axis displaced by unity. Here cylinder axis is z-axis and sphere is displaced by unity distance along x-axis.</p>
<p>By sketching them you note there is <em>no surface</em> outside the $\theta $ interval $ - \pi/2 , \pi... |
2,218,876 | <p>This is a question from Baltic way math competition (1995). </p>
<blockquote>
<blockquote>
<p>A polygon with $2n + 1$ vertices is given. Show that it is possible to label the vertices and midpoints of the
sides of the polygon, using all the numbers $1, 2, \dots , 4n + 2$, so that the sums of the three nu... | user1337 | 62,839 | <p>The natural domain of definiton of $\sqrt[3]{x}$ is indeed $\mathbb{R}$ as you've mentioned. However, the authors might have restricted the graph to the first quadrant in order to have this function differentiable as well (as $y=\sqrt[3]{x}$ has a vertical tangent at $x=0$). </p>
|
2,218,876 | <p>This is a question from Baltic way math competition (1995). </p>
<blockquote>
<blockquote>
<p>A polygon with $2n + 1$ vertices is given. Show that it is possible to label the vertices and midpoints of the
sides of the polygon, using all the numbers $1, 2, \dots , 4n + 2$, so that the sums of the three nu... | StackTD | 159,845 | <p>Technically, the domain is integral part of the definition of the function so you don't really <em>find</em> it, it is supposed to be <em>given</em>. However, in this context, when the domain isn't given but asked, it is usually assumed that they want you to give the <strong>maximal</strong> or <strong>natural domai... |
2,218,876 | <p>This is a question from Baltic way math competition (1995). </p>
<blockquote>
<blockquote>
<p>A polygon with $2n + 1$ vertices is given. Show that it is possible to label the vertices and midpoints of the
sides of the polygon, using all the numbers $1, 2, \dots , 4n + 2$, so that the sums of the three nu... | Community | -1 | <p>The cubic root of a real number $x$ is usually defined to be the real number $y$ such that</p>
<p>$$y^3=x.$$</p>
<p>By the observation that</p>
<p>$$(-y)^3=-y^3$$ the definition implies that</p>
<p>$$\sqrt[3]{-x}=-\sqrt[3]x$$ and the function is indeed defined for negatives (and is odd).</p>
<hr>
<p>Anyway, ne... |
222,730 | <p>This is a qualifying exam problem from Indiana University.</p>
<p>Prove or provide a counterexample to the following statement: </p>
<p>If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, then there exists a real number $L$ such that </p>
<p>$$\lim_{\epsilon \rightarrow 0} \int_{\epsilon \leq|x|\l... | Sangchul Lee | 9,340 | <p>To find a counterexample, it suffices to consider an odd function $f$ only. Rewriting the integral as</p>
<p>$$ 2 \int_{\epsilon}^{1} \frac{f(x)}{x} \, dx = 2 \int_{0}^{\log(1/\epsilon)} f(e^{-t}) \, dt,$$</p>
<p>it suffices to find a continuous function $g(t) = f(e^{-t})$ defined on $t \geq 0$ such that $\lim_{t\... |
497,118 | <p>find all the set of values for k such that $-2x^3+24x+8=k$ has more than one solution.</p>
<p>This problem can only be solved using basic Calculus 1 knowledge.</p>
<p>I thought about setting the equation to zero and then computing the x value that can then be rewritten in terms of k but I am not sure that is the c... | juantheron | 14,311 | <p>Let $f(x) = -2x^3+24x+8 = k$</p>
<p>Now Let $f(x) = -2x^3+24x+8$ and we will find the nature of cure and plot the curve in $X-Y$ plane</p>
<p>So $f^{'}(x) = -6x^2+24$ and for Max. and Min. $f^{'}(x) = 0$</p>
<p>we get $-6(x^2-4) = 0 \Rightarrow x = \pm 2$</p>
<p>Now $f^{''}(x) = -12x$</p>
<p>So at $x = -2$ , we... |
319,705 | <p>$$\lim_{x\rightarrow 0^-} \frac{\operatorname{arccot}(x) - \frac{\pi}{2}}{x}$$</p>
<p>The title says everything. I already know the limit is $+\infty$, I just want to see how it can be calculated. (Please don't use L'Hôpital's rule, I haven't covered it yet at school)</p>
| Dominic Michaelis | 62,278 | <p>At first a plot to show the limit is not -1.</p>
<p><img src="https://i.stack.imgur.com/xyTnR.png" alt="enter image description here"></p>
<p>The limit of $\operatorname{arccot}(x)$ as $x\to 0$ and $x<0$ is $-\frac{\pi}{2}$.</p>
<p>To see that, we use that $\operatorname{arccot(x)}=\frac{1}{\arctan\left(\frac{... |
3,112,196 | <p>Let T : V <span class="math-container">$\rightarrow$</span> W. Then T is injective if and only if null T = {0}.</p>
<p>What does the null T = {0} intuitively mean here?</p>
| farruhota | 425,072 | <p>Referring to the graph:</p>
<p><span class="math-container">$\hspace{3cm}$</span><a href="https://i.stack.imgur.com/ZXshe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZXshe.png" alt="![enter image description here"></a></p>
<p>You need to find <span class="math-container">$DF$</span> (black o... |
5,381 | <p>How-to compute the <a href="http://en.wikipedia.org/wiki/Genus_%28mathematics%29#Orientable_surface">genus</a> of an algebraic <a href="http://en.wikipedia.org/wiki/Algebraic_curve">curve</a> in Mathematica ?</p>
<p>In my case the algebraic curve is explicitly defined by a polynomial.</p>
| Dominic | 47,466 | <p>Don't understand why I'm being flagged for this response: Knowing the monodromy (ramified coverings) around each singular point enables us to easily compute the genus using the Riemann-Hurwitz formula:</p>
<p><span class="math-container">$$G=1/2\sum^N (e_p-1)-D+1$$</span></p>
<p>where the set <span class="math-co... |
4,568,768 | <p>I was reading about <strong>Reservoir Sampling</strong> and in the article it was mentioned</p>
<blockquote>
<p>To prove that this solution works perfectly, we must prove that the probability that any item stream[i] where <span class="math-container">$0 \le i < n$</span> will be in final reservoir[] is <span clas... | Lourrran | 1,104,122 | <p>You split a group of N elements into 2 groups, K and N-K elements.
What is the probability of a specific element to be in group 1 if it is not K/N ?</p>
|
2,118,676 | <p>Let $u,v,w$ be 3 pairwise coprime integers. Then $$u^3+v^3+3^{5}w^3=2\cdot3^{2}uvw$$ has no non-trivial solutions. How can I prove this? </p>
<p>I have tried to consider many individual cases such as $uvw>0$,$uvw<0$, $max(u,v,w)=u$ etc. a pretty tedious approach. I am certain there must be simpler ones. Any h... | S.C.B. | 310,930 | <p>Let us rearrange $p,q,r$ without any particular order into $a,b,c$. </p>
<p>Assume that $a=19$. Then we have to solve $$bc-b-c=19 \implies (b-1)(c-1)=20$$
So $(b,c)=(2, 21), (3, 11), (5, 6), (6,5), (11,3), (21, 2)$. </p>
<p>As $b$ and $c$ are both prime, we have $\{a,b,c\}=\{19,3,11\}=\{p,q,r\}$.
Since $p<q<... |
1,309,680 | <p><strong>EDIT: Starting Over</strong></p>
<p>I would like to see a formal proof in the language of ZFC of:</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\neg\exists S: \forall x:[x\in S \iff x\notin x]$</span></p>
</li>
<li><p><span class="math-container">$\forall S: \exists x: x\notin S$</span> (added ... | letsmakemuffinstogether | 131,029 | <p>Two Proofs in ZFC that the Russell ``Set" is not a set.</p>
<p>(1) The axiom of regularity states: $\forall x ( x \neq \emptyset \rightarrow \exists y \in x\ ( x \cap y = \emptyset) )$. If we had a set $R = \{x: x \notin x\}$ then consider now $\{R\}$. We see that there must be an element of $\{R\}$ which is disjoi... |
685,081 | <p>Tried <a href="http://www.proofwiki.org/wiki/Quotient_Group_of_Direct_Products" rel="nofollow noreferrer">http://www.proofwiki.org/wiki/Quotient_Group_of_Direct_Products</a>
<img src="https://i.stack.imgur.com/GZVFe.png" alt="enter image description here" /></p>
<blockquote>
<p><a href="http://jupiter.math.nctu.edu.... | Nicky Hekster | 9,605 | <p>In general, if $N$ is a normal subgroup of the group $G$, then there is always a homomorphism from $G$ onto (it is surjective) $G/N$, namely $\pi: G \rightarrow G/N$, defined by $\pi(g)=gN$ (also written as $\bar g$). Here $G/N$ is the set of cosets of $N$ in $G$, and this set of cosets can be endowed with a natural... |
3,136,298 | <blockquote>
<p>Calculate the number of ways to sort the numbers <span class="math-container">$1,2,...,n$</span> with
<span class="math-container">$n\ge 4$</span> with the condition:</p>
<p><span class="math-container">$i)\space1$</span> goes before <span class="math-container">$2$</span>, and <span class="mat... | N. F. Taussig | 173,070 | <blockquote>
<p>Calculate the number of ways to sort the numbers <span class="math-container">$1, 2, 3, \ldots, n$</span> with <span class="math-container">$n \geq 4$</span> if <span class="math-container">$1$</span> goes before <span class="math-container">$2$</span>, and <span class="math-container">$3$</span> goes... |
4,071,323 | <p>In his paper, "Euler and the Partial Sums of the Prime Harmonic Series" (<a href="http://pollack.uga.edu/eulerprime.pdf" rel="nofollow noreferrer">pdf</a>), Paul Pollacks makes a small statement at the top of page 4:</p>
<p><span class="math-container">$$ \log \frac{3}{2} < (\frac{3}{2} - 1) = \frac12 $... | fleablood | 280,126 | <p>If <span class="math-container">$x = 1$</span> then <span class="math-container">$0 =\ln x = x-1$</span>.</p>
<p>Would you accept that as <span class="math-container">$x, x-1$</span> and <span class="math-container">$\ln x$</span> get larger than <span class="math-container">$x$</span> and <span class="math-containe... |
241,367 | <p>I want to create a FormFunction that takes arbitrary many integer numbers and calculates the average. You want to use only one input field, but for more convenince allow users to write their numbers in different formats:</p>
<ol>
<li>input1: "1; 2; 3; 4; 5 "</li>
<li>input2: "1, 2, 3, 4, 5 "</li>... | Daniel Huber | 46,318 | <p>You may use <code>StringReplace</code>to transform the different inputs into WL. The only problematic case is when two numbers are only separated by a space, but we can treat this by using a regular expression.</p>
<pre><code>mean[in_String] := Module[{st = in},
st = StringReplace[st, {";" -> ",&... |
241,367 | <p>I want to create a FormFunction that takes arbitrary many integer numbers and calculates the average. You want to use only one input field, but for more convenince allow users to write their numbers in different formats:</p>
<ol>
<li>input1: "1; 2; 3; 4; 5 "</li>
<li>input2: "1, 2, 3, 4, 5 "</li>... | Roman | 26,598 | <p>The tricky part is the number extraction:</p>
<pre><code>extract[s_String] :=
ToExpression /@ StringSplit[s, (" " | "," | ";" | "/") ..]
extract /@ {"1; 2; 3; 4; 5 ",
"1, 2, 3, 4, 5 ",
"1/ 2/ 3/ 4/ 5 ",
&quo... |
241,367 | <p>I want to create a FormFunction that takes arbitrary many integer numbers and calculates the average. You want to use only one input field, but for more convenince allow users to write their numbers in different formats:</p>
<ol>
<li>input1: "1; 2; 3; 4; 5 "</li>
<li>input2: "1, 2, 3, 4, 5 "</li>... | Alexsom23814 | 77,724 | <p>that tasks seems like a tough one. As in the previous solutions mentioned, the hard part for me was adding a space a separator but eventually I tried RegularExpression[].
However, I am not sure how to implement my idea in the function Interpreter[] as it is needed by your request.</p>
|
2,426,726 | <p>So I know that </p>
<p>$$\bar{x}y + x\bar{y} = x \oplus y$$ </p>
<p>Can this be applied to something like</p>
<p>$$ \bar{a}\bar{b}cd + ab\bar{c}\bar{d} $$</p>
<p>to get
$$ab \oplus cd$$</p>
| Community | -1 | <p>Using the first formula,</p>
<p>$$ab\oplus cd=\overline{ab}cd+ab\overline{cd}=(\bar a+\bar b)cd+ab(\bar c+\bar d)=\bar acd+\bar bcd+ab\bar c+ab\bar d.$$</p>
|
706,461 | <p>Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,
$$\lim_{n\to\infty} \ln((n!)^{1/n}) = \lim_{n\to\infty} (1/n)\ln(n!) = \lim_{n\to\infty} (\ln(n!)/n)$$
Applying L'hopital's r... | SK_ | 967,760 | <p>Let a <span class="math-container">$n\in \Bbb N$</span>. By definition <span class="math-container">$$[\frac n2]\leq \frac n2<[\frac n2]+1.$$</span>
Then <span class="math-container">$n!=1\cdot 2\cdot ...\cdot[\frac n2]\cdot ([\frac n2]+1)\cdot...\cdot n>(\frac n2)^{n-[\frac n2]+a}>(\frac n2)^{\frac n2 +a}$... |
2,903,178 | <p>If X is a G-set, an automorphism $G$-invariant is a map $f:X \to X$ such that $f(g*x)=g*f(x)$</p>
<p>I need to prove that $N_G(H)/H$ is isomorphic to $G$-automorphism invariants of $G/H$</p>
<p>I tried with two natural maps:</p>
<ol>
<li><p>The map such that $\phi(gH)=f_{gH}$ where $f_{gH}(g'H)=(g'g)H$, but this ... | Ben West | 37,097 | <p>The map $\phi\colon N_G(H)\to\operatorname{Aut}_G(G/H)$ defined by $\phi(g)(g'H)=g'g^{-1}H$ is well defined as $g\in N_G(H)$, and
$$
\phi(g)(g'H)=g'g^{-1}H=g'\phi(g)(H)
$$
and
$$
\phi(gk)(g'H)=g'(gk)^{-1}H=g'k^{-1}g^{-1}H=\phi(g)(g'k^{-1}H)=\phi(g)\circ\phi(k)(g'H).
$$
This implies $\phi(g)\phi(g^{-1})=\phi(e)=1_{\o... |
3,253,368 | <p>I saw this property in an exercise about metric spaces: Let <span class="math-container">$(E,d)$</span> be a metric space and <span class="math-container">$A$</span>, <span class="math-container">$B$</span> be two non empty subsets of <span class="math-container">$E$</span>. If <span class="math-container">$A$</span... | Matt Samuel | 187,867 | <p>Let
<span class="math-container">$$A=\bigcup_{i=1}^\infty [i-1+1/(10i),i-1/(10i)]$$</span>
<span class="math-container">$$B=\mathbb Z$$</span>
Then <span class="math-container">$d(A, B) =0$</span>, but the two sets do not intersect. </p>
|
3,253,368 | <p>I saw this property in an exercise about metric spaces: Let <span class="math-container">$(E,d)$</span> be a metric space and <span class="math-container">$A$</span>, <span class="math-container">$B$</span> be two non empty subsets of <span class="math-container">$E$</span>. If <span class="math-container">$A$</span... | Mitchell Spector | 350,214 | <p>Let <span class="math-container">$A$</span> be the set of all integers greater than <span class="math-container">$1,$</span> and let <span class="math-container">$B$</span> be <span class="math-container">$\lbrace n + \frac1 n \vert n \in A \rbrace.$</span></p>
|
1,590,039 | <p>In programming, often the value of True is also 1, and False is 0.</p>
<p>This means that:</p>
<pre><code>(x>5)*4
</code></pre>
<p>will return 4 if x is greater than 5 (because <code>(x>5)==1</code>), else 0.</p>
<p>I need to accomplish a similar thing using mathematical operators (no piecewise functions, ... | Brevan Ellefsen | 269,764 | <p>To simplify Hardy's answer, a function I commonly use is
$$\frac{1}{2} + \frac{n-x}{2|n-x|}$$
If you want to make calculations less taxing by working only with integers during calculations, this is obviously the same as<br>
$$\frac{1+\frac{n-x}{|n-x|}}{2}$$<br>
This solution immediately arises from the fact that $\... |
1,392,273 | <p>I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them.</p>
<p>What are the actual differences between algebra and analysis, and how do I tell which is for me?</p>
| Community | -1 | <p>You need to know both. Honestly at higher levels of math things aren't broken down into analysis or algebra. There's functional analysis. Algebraic topology etc. </p>
<p>Both are important to know.</p>
<p>The real question however, is how do you eat your corn?</p>
|
1,829,082 | <p>I was wondering how many base cases are needed and when to stop (in general).
For example, I have 4-cent and 11-cent stamps and I need to determine the amount of postage I can make, the cases I have that work are:
$4,8,11,12,15,16,17,19,20,22,23,24,26,27,28,30 \geq$</p>
<p>Now what I'm stuck on is: what stops me fr... | Daniel W. Farlow | 191,378 | <p>Nothing at all is stopping you from "adding 31 to that pile." In fact, I believe your goal is to show that you can form $n$ cents of postage whenever $n\geq30$ (when you have just 4-cent and 11-cent stamps at your disposal). </p>
<p>Let $P(n)$ be the statement that you can form $n\geq30$ cents of postage using excl... |
3,146,277 | <blockquote>
<p>Let <span class="math-container">$\langle a_n\rangle$</span> be a sequence where <span class="math-container">$a_n = \sum\limits_{j=1}^{n}(-1)^j2/n$</span></p>
<ol>
<li><p>is oscillatory</p>
</li>
<li><p>diverges to <span class="math-container">$+\infty$</span></p>
</li>
<li><p>diverges to <span class="... | Peter Szilas | 408,605 | <p>1)<span class="math-container">$a_n= (-2/n)$</span>, <span class="math-container">$n$</span> is odd.</p>
<p>2) <span class="math-container">$a_n=0$</span>, <span class="math-container">$n$</span> is even.</p>
<p>1')<span class="math-container">$\lim_{ k \rightarrow \infty } a_{2k+1} =0$</span>, <span class="math-c... |
4,387,701 | <p>I have been trying to find a proof I can follow (I am not familiar with metric spaces, for instance) for the fact that any given function only has a countable number of jump discontinuities.</p>
<p>I have been looking at <a href="https://math.stackexchange.com/questions/263606/prove-that-the-number-of-jump-discontin... | CyclotomicField | 464,974 | <p>We will explicitly construct a mapping between a countable set and the jump discontinuities of a function. To being we know that the left and right hand limit exist there left-continuous and right-continuous neighborhoods near each jump discontinuity. Because the rational numbers are dense in the reals and we can ma... |
3,471,096 | <p><span class="math-container">$\frac{d}{dx} xy^2 e^{-x^2}$</span></p>
<p>I am not sure where to get started and what method I can use to deal with the <span class="math-container">$x^2$</span> power</p>
| Gabe | 449,093 | <p>Depends on if <span class="math-container">$y$</span> is a constant or a function of <span class="math-container">$x$</span>, which you haven’t clarified yet.</p>
<p>If it’s a constant, this is just simply product rule,
<span class="math-container">$$\frac{d}{dx}xy^2 2e^{-x^2}=2y^2\left( e^{-x^2}+x\frac{d}{dx}e^{-x... |
327,239 | <p>I wanted to show my nephew(16) a simple approach to calculate the distance between two geo-locations. The mathematical knowledge of a 16-year old boy is limited to simple geometrical shapes like triangle, circles, etc.</p>
<p>My approach bases on the assumption that you can draw the two locations as two points (A a... | Dan Shved | 47,560 | <p>The first bells started ringing around the word "trapezoid". Trapezoids are objects on a Euclidean plane, and we are on a sphere. So, technically, $ABCD$ is not a trapezoid but something else. I am not just being picky here, this is exactly where the first pitfall is hidden.</p>
<p>You see, in your case $ABCD$ is s... |
821,747 | <p>Ok so far what i understand is this lets say...Having to draw a card from 52card-deck its probability is of course 1/52.Now the probability to say that i will keep drawing this same card 10 times of course it is 1/(52)^10 because now i have alot more possibilities.Now my question is this.I also know that drawing a ... | Ben Grossmann | 81,360 | <p>The question it seems you are asking about can be phrased as follows:</p>
<blockquote>
<p>Given that I have selected the ace of spades 9 times in a row (with replacement), what is the probability that, upon drawing another card at random, I will draw the ace of spades again?</p>
</blockquote>
<p>The traditional ... |
2,539,855 | <p>I'm not convinced this is possible, as soon as you have $2$ distinct elements mapping to the same number, the function is no longer $1$-$1$ and therefore not a bijection. </p>
| zwim | 399,263 | <p><a href="https://i.stack.imgur.com/BSrms.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BSrms.jpg" alt="enter image description here"></a></p>
<p>Remark that if a bijection is monotonous on $[0,1]$ you can build piecewise non-monotonous function based on the restriction of this bijection scaled ... |
359,196 | <p>Let <span class="math-container">$d \gg 1$</span>. Let <span class="math-container">$G:=(U, V, E)$</span> be some bipartite graph such that deg<span class="math-container">$(u) \le d$</span> for all <span class="math-container">$u\in U$</span> and deg<span class="math-container">$(v) \le 3$</span> for all <span clas... | Tony Huynh | 2,233 | <p>Let <span class="math-container">$G=(U,V,E)$</span> be a bipartite graph where <span class="math-container">$U=[n], V=\binom{[n]}{3}$</span>, and there is an edge between <span class="math-container">$u \in U$</span> and <span class="math-container">$v \in V$</span> if and only if <span class="math-container">$u \in... |
1,753,313 | <p>I am being asked to find the sum of the following convergent series :</p>
<p>$$\sum_{n=0}^∞ \frac{2^n}{6^n} + \frac{3^n}{6^n}$$</p>
<p>Attempting to generalize from partial sums yields nothing of interest:</p>
<p>$s_1 = \frac{5}{6}$</p>
<p>$s_2 = \frac{5}{6} + \frac{13}{36} = \frac{43}{36}$</p>
<p>$s_3 = \frac{... | zz20s | 213,842 | <p>$\frac{2^n}{6^n}=(\frac{1}{3})^n$ and $\frac{3^n}{6^n}=(\frac{1}{2})^n$</p>
<p>Now, use the well known result that for $|x|<1$,</p>
<p>$$\sum \limits_{n=0}^{\infty}x^n=\frac{1}{1-x}$$</p>
|
1,753,313 | <p>I am being asked to find the sum of the following convergent series :</p>
<p>$$\sum_{n=0}^∞ \frac{2^n}{6^n} + \frac{3^n}{6^n}$$</p>
<p>Attempting to generalize from partial sums yields nothing of interest:</p>
<p>$s_1 = \frac{5}{6}$</p>
<p>$s_2 = \frac{5}{6} + \frac{13}{36} = \frac{43}{36}$</p>
<p>$s_3 = \frac{... | user315531 | 315,531 | <p>Note that the sum $\sum_{n=0}^{\infty}r^{n}=\frac{1}{1-r}$ for $|r|<1$. According to this fact $\sum_{n=0}^{\infty}\frac{2^{n}+3^{n}}{6^{n}}=\sum_{n=0}^{\infty}\frac{2^{n}}{6^{n}}+\sum_{n=0}^{\infty}\frac{3^{n}}{6^{n}}=\sum_{n=0}^{\infty}(\frac{2}{6})^{n}+\sum_{n=0}^{\infty}(\frac{3}{6})^{n}=\frac{1}{1-1/3}+\frac... |
3,354,815 | <p>Let <span class="math-container">$ f(x)$</span> has the convergent power series representation of the form <span class="math-container">$$\sum_{n=0}^{\infty}a_nx^n$$</span>. Then What is the power series representation of the function <span class="math-container">$\left[f(x)\right]^m$</span> for some integer <span c... | Saketh Malyala | 250,220 | <p>Well, in the case of <span class="math-container">$\cos(x)$</span>, notice that <span class="math-container">$\cos(0)=1$</span>. So, obviously, near <span class="math-container">$x=0$</span>, <span class="math-container">$\displaystyle \frac{\cos(x)}{x}$</span> does not converge.</p>
<p>Whereas, in the case of a <s... |
306,487 | <p>$$y' = y^2 - 1$$</p>
<p>I think this cannot be solved using the integration factor method because of the square power of the y.</p>
<p>Attempted:
$$ \frac{y'}{y^2} = -1 $$
$$ y^{-1} = t + C $$
$$ \frac{1}{y} = t + C$$</p>
<p>It's not the right answer.</p>
<p>How do I go about solving this?
Thanks.</p>
| André Nicolas | 6,312 | <p>There was an algebra error at the beginning. From $y'=y^2-1$ it is not correct to conclude that $\frac{y'}{y^2}=-1$. We <em>could</em> conclude that $\frac{y'}{y^2}=-\frac{1}{y^2}$, but that is not helpful. Our DE is <em>separable</em>, and there is a standard method for dealing with separable equations. </p>
<p>... |
93,096 | <p>Hi, what are the "best" values for lenght of needle (l) and distance between paralles (d) for an accurate approximation of pi? Does it have to be l-d-1.0 or ld? Thanx</p>
| Arthur B | 8,737 | <p>The naive estimator is biased. If there are $N$ trials and $i$ success, a Rao-Blackwellisation of the naive estimator gives the unbiased estimator $\frac{2l}{d}\left(\frac{n}{i}+\frac{1}{i}\right)$ (to be fair this hides an assumption for a uniform prior for the probability of crossing, which induces a weird prior o... |
782,812 | <p>I'm trying to solve this recurrence relation: </p>
<p>$$a_m = 8 \cdot a_{m-1} + 10^{m-1}, a_1=1$$</p>
<p>By the change of variable $\displaystyle b_m = \frac{a_m}{10^m}$ I obtained this linear non-homogenous relation:</p>
<p>$$b_m = \frac{8}{10}b_{m-1} + \frac{1}{10}, b_1 = \frac{1}{10}$$</p>
<p>Now I'm trying t... | Community | -1 | <p>Your method fails because </p>
<p>$$ A \left[ \begin{matrix} b_m \\ b_{m-1} \end{matrix} \right]$$</p>
<p>isn't anything useful.</p>
<p>Instead, try setting up a matrix such that</p>
<p>$$ \left[ \begin{matrix} 1 \\ b_{m} \end{matrix} \right] = A \left[ \begin{matrix} 1 \\ b_{m-1} \end{matrix} \right]$$</p>
<p>... |
4,343,810 | <p>We must to calculate <span class="math-container">$$\lim_{x\to \pi/2}\frac{\sin x-1}{2x-\pi}.$$</span>
It is very simple with l'Hospital's rule that the limit is zero. But what would happen if I didn't use l'Hospital's rule?</p>
<p>In fact if <span class="math-container">$t=x-\frac{\pi}2$</span> we have <span class=... | Chee Han | 242,589 | <p>Another way to rewrite it is<br />
<span class="math-container">$$ \frac{\sin(t + \pi/2) - 1}{2t} = \frac{\sin t\cos(\pi/2) + \sin(\pi/2)\cos t - 1}{2t} = \frac{\cos t - 1}{2t}. $$</span></p>
|
2,288,983 | <p>A card is drawn and replaced in an ordinary pack of 52 playing cards. Minimum number of times must a card be drawn so that there is at least an <strong>even chance</strong> of drawing a heart?</p>
<p>What does the second sentence of the question mean?
Why should there be any change in the chance of drawing a heart,... | Dr. Sonnhard Graubner | 175,066 | <p>i have got $$f(0)=0,f(-1)=2,g(0)=-3,g(-1)=-5$$
$$x=-3,x=2$$
$$p=-1,p=4$$
$$z=2$$
$$z=1$$
we have to solve $$z^2-z=2z-3+1$$ and this is $$z^2-3z+2=0$$
$$z_{1,2}=\frac{3}{2}\pm\sqrt{\frac{9}{4}-\frac{8}{4}}$$</p>
|
3,506,267 | <p>In the last few days, I have been wondering about the following equation in <span class="math-container">$x,y \in N$</span>: <span class="math-container">$$x^2+y^2 = M$$</span> with <span class="math-container">$M \in N$</span>. More precisley, I don't understand how can we compute the number of solutions that this ... | Somos | 438,089 | <p>This is in general a very hard problem because any
method which finds solutions to
<span class="math-container">$\, n = a^2+b^2 = c^2+d^2\,$</span> allows us to
factor <span class="math-container">$\,n.\,$</span>
However, factoring large numbers is a very
hard problem according to Wikipedia
<a href="https://en.wikip... |
2,194,461 | <p><a href="https://i.stack.imgur.com/Q73yr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Q73yr.jpg" alt="enter image description here"></a></p>
<p>in this I tries to substitute x by $\pi -t$ </p>
<p>and thought about it </p>
<p>but after that got stuck.</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Considering the problem of the antiderivative first, making $x+\frac \pi 4=t$ leads to $$\int e^{2 x} \sin \left(x+\frac{\pi }{4}\right)\,dx=e^{-\frac{\pi }{2}} \int e^{2 t} \sin (t)\,dt$$ Now consider $$\sin(t)=\frac{e^{it}-e^{it}}{2i}$$ which makes $$e^{2 t} \sin (t)=\frac i 2(e^{(2-i... |
265,101 | <p>Let</p>
<pre><code>L1={1,2,4,6,8,9,10,14,15,17,21,22,72,76,80,96,106,116,117}
</code></pre>
<p><strong>Question1)</strong> Group from 10-th to 14-th elements in <code>L1</code>.</p>
<p>So that the output becomes</p>
<pre><code>L2={1,2,4,6,8,9,10,14,15,{17,21,22,72,76},80,96,106,116,117}
</code></pre>
<p>A possible s... | imida k | 34,532 | <p><strong>-1></strong> Today (2022.3.19) I was able to find even faster and intuitive way to achieve it. Thanks for Edmund's introduction of <code>TakeList</code> command. The prerequisite for <code>L</code> and <code>P</code> are similar to <strong>0></strong>.</p>
<pre><code>L = Range[3300000]; P = Partition[P... |
1,213,371 | <blockquote>
<p>How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ?</p>
</blockquote>
<hr>
<p>Inspired by <a href="https://math.stackexchange.com/questions/1200624">this question</a>, I sou... | epi163sqrt | 132,007 | <p><em>Note:</em> Here is an outline of an alternate proof, which could be helpful.</p>
<p>In section $21$ in <em><a href="https://books.google.de/books?id=qy1GNv2ovHQC%5C&pg=PA74#v=onepage%5C&q%5C&f=false" rel="noreferrer">Bessel Functions and Their Applications</a></em> B.G. Korenev introduces the so-cal... |
48,366 | <p>Does the Laplace equation on a rectangle with Dirichlet boundary conditions at two opposing sides and Neumann boundary conditions at the other two, always have a solution? If it does, is it unique? Is the same true for the discrete Laplace equation (with the standard five-point laplacian)?</p>
| Anand | 36,814 | <p>I am not familiar with the discrete case. As for continuous case, I think you can use the method of images to solve this problem. It should always have a solution and it should be unique, due to the fact that the underlying <a href="http://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equat... |
3,264,994 | <p><a href="https://i.stack.imgur.com/AvHVZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AvHVZ.png" alt="enter image description here"></a></p>
<p>This is question regarding solving for a percentile of the cumulative distribution function of a certain pdf. What I am wondering is how I am meant to... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Let <span class="math-container">$(\pi_{0.75}+6)^2=x$</span> and solve for <span class="math-container">$x$</span>
<span class="math-container">$$\frac 3 4=1-\left(1-\frac x {60} +\frac 4 {60}\right)^2$$</span> that is to say
<span class="math-container">$$\frac 1 4=\left(\frac{16}{15}-... |
3,264,994 | <p><a href="https://i.stack.imgur.com/AvHVZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AvHVZ.png" alt="enter image description here"></a></p>
<p>This is question regarding solving for a percentile of the cumulative distribution function of a certain pdf. What I am wondering is how I am meant to... | Eric Towers | 123,905 | <p><span class="math-container">\begin{align*}
\dfrac{3}{4} &= 1 - \left[1-\dfrac{1}{60}(\pi_{0.75}+6)^2 +\frac{4}{60} \right]^2 \\
\dfrac{-1}{4} &= - \left[1-\dfrac{1}{60}(\pi_{0.75}+6)^2 +\frac{4}{60} \right]^2 \\
\dfrac{1}{4} &= \left[1-\dfrac{1}{60}(\pi_{0.75}+6)^2 +\frac{4}{60} \right]^2 \\
\pm \dfr... |
1,369,805 | <p>How do I calculate the follwing Limit:</p>
<p><span class="math-container">$$ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $$</span></p>
<p>I have been solving an exam from my University's collection of previous exams, and in one of them, there wasn't an official solution, b... | Chuks | 217,178 | <p>Already typing up the solution before I saw the solution of user1337.</p>
<p>So from where he stopped, I continue...</p>
<p>$\lim_{x \to 0} \frac{e^{x^2}-1}{2x\sin(x^2)}=
\lim_{x \to 0}\frac{2xe^{x^2}}{4x^2\cos(x^2)+2\sin(x^2)}=\lim_{x \to 0}\frac{4x^2e^{x^2}+2e^{x^2}}{8x\cos(x^2)-8x^3\sin{x^2}+4x\cos(x^2)}=\frac{... |
594,724 | <p>my question is:
Consider the Taylor polynomial of $e^x$ of order $n$ at $x=0$, or $P_n(x)=e^0 + e'^{(0)}x +...+\frac{e^{(n)(0)}x^n}{n!}$.
Prove that the sequence {$P_n(x)$} converges uniformly to $e^x$ on any bounded interval.
I tackled the first part of the question by proving, through Taylor's theorem, that $e^x =... | D Wiggles | 103,836 | <p>The key to this proof is that for a fixed $x$, $x^n\leq n!$ when $n$ is sufficiently large. Let's consider the ratio
$$\frac{x^n}{n!}=\frac{x\cdot x\cdot\ldots\cdot x}{1\cdot 2\cdot 3\cdot\ldots\cdot (n-1)\cdot n}$$
If $x$ is large, then for small $n$, the numerator is clearly larger. However, once $n\geq x$, each ... |
3,396,103 | <p>Is <span class="math-container">$S$</span> is a bounded, non-empty set (<span class="math-container">$S \subset \mathbb{R}^n$</span>), then I know it has the least upper bound property. Let <span class="math-container">$x = \sup S$</span>. </p>
<p>Is <span class="math-container">$x \in S$</span>? If no, counterexam... | Hagen von Eitzen | 39,174 | <p>Let <span class="math-container">$A=\Bbb R^N$</span>, <span class="math-container">$K_n=\{\,x\in\Bbb R^N:\frac 1n\le |x|\le n\,\}\cup\{0\}$</span>, <span class="math-container">$K=\{\,x\in\Bbb R^N: |x|\le 1\,\}$</span>.</p>
|
3,396,103 | <p>Is <span class="math-container">$S$</span> is a bounded, non-empty set (<span class="math-container">$S \subset \mathbb{R}^n$</span>), then I know it has the least upper bound property. Let <span class="math-container">$x = \sup S$</span>. </p>
<p>Is <span class="math-container">$x \in S$</span>? If no, counterexam... | DanielWainfleet | 254,665 | <p>After the edit of the Q. If <span class="math-container">$K_n\subset int(K_{n+1})$</span> then, rather obviously, <span class="math-container">$A=\cup_n\,K_n\subset \cup_n \, int(K_{n+1})\subset \cup_n\,K_{n+1}\subset \cup_n\,K_n=A,$</span> so <span class="math-container">$A=\cup_n \,int(K_{n+1}).$</span></p>
<p>If... |
2,196,347 | <p><strong>Problem:</strong> What will be the number of distinct real solution of <span class="math-container">$|x|^2-5|x|+6=0$</span> ?</p>
<p><strong>Solution:</strong></p>
<blockquote>
<p>Case1: When <span class="math-container">$x<0$</span>, then <span class="math-container">$x=-2,-3$</span></p>
<p>Case2: When <... | StackTD | 159,845 | <p>You are correct.</p>
<p>It is a quadratic equation in $|x|$. Or you could set $y=|x|$, then:
$$y^2-5y+6=0 \iff y=2 \vee y = 3$$
So with $y=|x|$, you get:
$$|x|=2 \implies x = \pm 2 \quad\mbox{and}\quad |x|=3 \implies x = \pm 3$$</p>
|
3,229,978 | <p>I just questioned myself this:</p>
<p>Given that I have on a reddit post:</p>
<ul>
<li><span class="math-container">$5$</span> net upvotes</li>
<li><span class="math-container">$65\%$</span> upvote ratio.</li>
</ul>
<p>How many upvotes and downvotes do I have in total?.</p>
<p>I think its solvable, but after som... | cryptograthor | 614,938 | <p><span class="math-container">$x$</span> up, <span class="math-container">$y$</span> down, <span class="math-container">$\frac x {x+y} =.65$</span>, and <span class="math-container">$x-y=5$</span> solve two equations in two unknowns.</p>
|
3,854,931 | <p>If <span class="math-container">$f(x + iy) = u(x,y) + i v(x,y)$</span>, then <span class="math-container">$\frac{\partial f}{\partial x}$</span> describes how <span class="math-container">$f$</span> changes if we take take small step in the <span class="math-container">$x$</span>-direction in the input space, and <s... | Community | -1 | <ol>
<li>Use continuity of the function <span class="math-container">$f(x):=x^n$</span>.</li>
</ol>
<blockquote class="spoiler">
<p> <span class="math-container">$f(0)\le a$</span> and <span class="math-container">$f(\max(a,1))\ge a$</span> so by the intermediate value theorem there must be an <span class="math-contain... |
3,802,674 | <p><strong>Origin.</strong> Yesterday I posted a somewhat similar question asking about the number of subsets of <span class="math-container">$\{1,2,...,n\}$</span> which do not contain three consecutive integers. That one was solved completely, and I have had a fairly well understanding. However, I have been intrigued... | IncredibleSimon | 812,406 | <p>Thanks to lulu's hint. I am trying to write out the complete solution below.<br/></p>
<p>Let <span class="math-container">$a_{i,j}$</span> denote the number of <span class="math-container">$i$</span>-element subsets of <span class="math-container">$\{1,2,...,j\}$</span>, where each of such subset does not involve th... |
3,339,064 | <p>Let <span class="math-container">$R$</span> be a commutative ring with identity and let <span class="math-container">$f\colon R \to R[X]$</span> be the natural ring homomorphism. Let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span>. What I have shown is that <span class=... | Community | -1 | <p>I guess that you mean that <span class="math-container">$f$</span> is the inclusion. My approach would be the following:</p>
<p>First, show that <span class="math-container">$I^e = I[X]$</span>, the ideal of polynomials in <span class="math-container">$R[X]$</span> with coefficients in <span class="math-container">... |
355,150 | <p>A. I wonder if every generator of <span class="math-container">$Z({\rm Spin}_n^{\epsilon}(q))$</span> is a square element in <span class="math-container">${\rm Spin}_n^{\epsilon}(q)$</span>?</p>
<p>B. When <span class="math-container">$Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$</span>, is the unique element of order tw... | Derek Holt | 35,840 | <p>Since nobody has answered Question A, I computed a few examples with small dimensions (up to <span class="math-container">$14$</span>), and small finite fields (up to order 9, depending on the dimension) in Magma.</p>
<p>The results were consistent and can be summed up as follows. All generators of <span class="mat... |
3,118,987 | <p>I consider the application <span class="math-container">$f$</span> from <span class="math-container">$H$</span> to <span class="math-container">$\mathbb{Z}_{n/m}$</span> how <span class="math-container">$[m]_{n}$</span> to <span class="math-container">$[1]_{n/m}$</span> , where <span class="math-container">$\langle ... | Community | -1 | <p>Any subgroup of a cyclic group is cyclic. By Lagrange's theorem the order of <span class="math-container">$H$</span> divides <span class="math-container">$n$</span>. <span class="math-container">$\therefore n=m\mid H\mid\implies \mid H\mid=\frac nm\implies H\cong\Bbb Z_{\frac nm}$</span>.</p>
<p>Alternatively, <... |
1,132,024 | <p>Let $a,b,c \in \Bbb Z$. Show that the Diophantine equation $ax+by = b+c$ is solvable in integer $x , y$ iff $ax+by =c$ is solvable.</p>
<p>We know that a Diophantine equation $ax+by =c$ is solvable iff $d|c$ where $d=gcd(a,b)$.
From here how can we do the above proof??</p>
| User8976 | 98,414 | <p>As you have said we use the following theorem :</p>
<p>A Diophantine equation $ax+by =c$ is solvable iff $d|c$ where $d=gcd(a,b)$.</p>
<p>$\Rightarrow$ Let us assume that $ax+by =b+c$ is solvable in integer $x,y$. Thus $d|b+c$ where $d=gcd(a,b)$ and since $d|b$ so $d|c$ thus $ax+by =c$ is solvable in integer $x,y$... |
1,931,335 | <p>I am trying to find the greatest value of $"c"$ for which $7x + 9y = c$ has exactly six solutions over positive integers.</p>
<p>My approach: Let one of the solution = $(a, b)$</p>
<p>General solution is $(a + 9t, b - 7t)$</p>
<p>For values t = 0, 1, 2, 3, 4, 5 we get 6 solutions.</p>
<p>Also $b - 7t$ should be ... | Piquito | 219,998 | <p>When you don´t restraint the values $(x,y)$ the general solution is given by $(x,y)=(a+9t,b-7t)$ and there are infinitely many solutions hence, in order to have exactly six solutions we take positive or non-negative ones. We take positive solutions. </p>
<p>It follows one can take the values $t=0,1,2,3,4,5$ so $7t\... |
1,931,335 | <p>I am trying to find the greatest value of $"c"$ for which $7x + 9y = c$ has exactly six solutions over positive integers.</p>
<p>My approach: Let one of the solution = $(a, b)$</p>
<p>General solution is $(a + 9t, b - 7t)$</p>
<p>For values t = 0, 1, 2, 3, 4, 5 we get 6 solutions.</p>
<p>Also $b - 7t$ should be ... | Brian Tung | 224,454 | <p>Imagine a line of slope $-\frac{7}{9}$, initially extending through the origin and making its way "northeastward"—that is, sliding in the direction of positive $x$ and positive $y$. As it does so, it traverses a lattice of points with non-negative integer coordinates. (It will be simpler, computationally, to consi... |
354,661 | <p>Can any one help me to answer this question:</p>
<blockquote>
<p>Assuming <span class="math-container">$$f(x)=\begin{cases} x &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$</span></p>
<p>show that <span class="math-container">$f$</span> is continuous only at <span... | Did | 6,179 | <p>By definition of the radius of convergence, $a_nx^n\to0$ for every $|x|\lt R$ and $a_nx^n\not\to0$ for every $|x|\gt R$. Thus, $a_nx^{2n}\to0$ for every $|x|\lt\sqrt{R}$ and $a_nx^{2n}\not\to0$ for every $|x|\gt\sqrt{R}$. Using the definition again, this proves that the radius of convergence of the series $\sum\li... |
3,100,406 | <p>Let <span class="math-container">$G$</span> be some group, and let <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> be some subgroups. Then <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> are said to be commensurable iff <span class="math... | Mike Earnest | 177,399 | <p>For the second identity, start with the binomial theorem:
<span class="math-container">$$
(1-t)^n=\sum_{k=0}^n\binom{n}k(-1)^kt^k.
$$</span>
Integrating both sides from <span class="math-container">$0$</span> to <span class="math-container">$x$</span> with respect to <span class="math-container">$dt$</span>,
<span c... |
2,315,400 | <p>My questions are: </p>
<p>If the $\mathscr{C}^{1}$ mapping $f:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{m+n}$
defined by $f(\mathbf{x},\mathbf{y})=\{\mathbf{x},G(\mathbf{x},\mathbf{y})\}$
has an inverse such that $f^{-1}(\mathbf{x},G(\mathbf{x},\mathbf{y}))=\{\mathbf{x},\mathbf{y}\}$
does that imply </p>
<p>$\mathit{... | Mirko | 188,367 | <p>Hint.<br>
First prove that if $K$ is any component of $(\mathbb{R}^n-S)$, then $S\cup K$ is connected. </p>
<p>Some details: </p>
<blockquote class="spoiler">
<p> It is easily seen that $K$ is closed. It cannot also be open as this would partition $\mathbb{R}^n$. If $p$ is any boundary point of $K$ then $p\in\ov... |
2,022,062 | <p>Does anybody know a function $f$ <strong>that is continuous on $[0, \infty)$</strong>, so that $\int_{0}^{\infty} f(x)dx$ converges but $\int_{0}^{\infty} f(x)^2dx$ diverges? Thanks</p>
<p>EDIT: Sorry! I meant that $f$ is also continuous there.</p>
| Simply Beautiful Art | 272,831 | <p>Notice that <a href="https://math.stackexchange.com/questions/105107/prove-int-0-infty-sin-x2-dx-converges">the following integral converges</a>, and to the particular value.</p>
<p>$$\int_0^\infty\sin(x^2+2x+1)dx\approx3.16389$$</p>
<p>On the other hand,</p>
<p>$$\int_0^\infty\sin^2(x^2+2x+1)dx$$</p>
<p>does no... |
3,294,874 | <p>How should I solve <span class="math-container">$$y'+2y=x\sin(x)$$</span>
I tried this calculation using integrating factor, but it costs me a lot of calculation. Is there better way to solve this kind of problem?</p>
| Jimmy | 564,828 | <p><span class="math-container">$y'+2y=x\sin x$</span> Integrating factor is <span class="math-container">$e^{\int 2\mathrm dx}=e^{2x}$</span>.</p>
<p>Multiplying by the Integrating factor both side of the equation,</p>
<p><span class="math-container">$e^{2x}y'+2e^{2x}y=e^{2x}x\sin x$</span></p>
<p><span class="math... |
2,525,145 | <p>How to prove: $Cl_{\mathbb{R}}(\mathbb{IQ})=\mathbb{R}$ ($\mathbb{IQ}$-set of irrational numbers)?</p>
<p>I know how to prove: $Cl_{\mathbb{R}}(\mathbb{Q})=\mathbb{R}$ ($\ast$)</p>
<p><strong>Proof of ($\ast$)</strong>: Let: $r \in \mathbb{R}$ , $q \in \mathbb{Q}$, $\epsilon \in \mathbb{R}_{+}$. Then: $\exists n \... | nonuser | 463,553 | <p>We can assume $a=0$ (otherwise reduce all numbers for $a$)</p>
<p>$$|c|+|b-d|<|d|+|b-c| \;\;\;/ ^2$$
$$c^2+b^2+d^2-2bd+2|c||b-d| < d^2+b^2+c^2-2bc +2|d||b-c|$$
$$-bd+|c||b-d| < -bc +|d||b-c|$$
$$|c||b-d|-|d||b-c| < b(d-c)$$</p>
<p>Last one is true since we have, by triangle inequality ($|x|-|y|\leq|x-y... |
3,678,402 | <p>I have verified using the eigenvalue method that around <span class="math-container">$(0,0)$</span> the system </p>
<p><span class="math-container">\begin{align}\dot x&=y - 3x - x^3 \\ \dot y &= 6x - 2y \end{align}</span></p>
<p>is stable. However, I have been trying to find a suitable Lyapunov function <s... | user577215664 | 475,762 | <p>You can try this:
<span class="math-container">$$V(x,y)=\dfrac {y^2}2+ 3 x^2 \ge 0$$</span>
Hence we have:
<span class="math-container">$$V'(x,y)=\left(\dfrac {y^2}2+ 3 x^2\right)'$$</span>
<span class="math-container">$$V'(x,y)=\left( {y'y}+ 6x'x\right)$$</span>
<span class="math-container">$$V'(x,y)= {y(6x-2y)}+ 6... |
173,781 | <p>This question arises from one of the Berkeley Problems in Mathematics, fall 1981. It asks me to find a counterexample to the following claim:</p>
<p>Suppose $\lim\limits_{x\rightarrow x_{0}}f(x)=a$, $\lim\limits_{t\rightarrow a}g(t)=b$, prove or find a counterexample for the claim $\lim\limits_{x\rightarrow x_{0}}g... | William Vickery | 334,699 | <p>Another counter-example that would be more familiar to a college or high-school calculus student would be the following.
consider $f,g: \mathbb{R} \to \mathbb{R}$ defined </p>
<p>$$
\begin{array}{ll}
g(x) = \frac{x^2 -1}{x-1} = \frac{(x+1)(x-1)}{x-1} \text{and}\\
f(x) = 1.
\end{array}
$$</p>
<p>Then
$$
\lim_{x... |
2,842,589 | <p>It has been a while since my undergraduate introductory analysis course, so I am a little rusty. I know that to show two sets $B, C$ are equal, you first show that $B\subseteq C$ and then $C\subseteq B$ . To show $B\subseteq C$ you take any element of $B$, say $b \in B$ and show that $b \in C$. </p>
<p>The main is... | nonuser | 463,553 | <p>Do you know when two sets are equal? </p>
<p>Say $B=(a,b)$ and $$A=\bigcup^\infty_{n=1}(a,b-\frac{1}{n}]$$ You must prove $A=B$.</p>
<p>So take any $x\in A$ and prove that it is also in $B$. Then do vice versa.</p>
|
885,335 | <blockquote>
<p>There is convex body $T$ (with the area is $1$), show that
there is a triangle $\Delta ABC$, such $A,B,C\in T$, and
$$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/nkuDz.jpg" alt="enter image description here"></p>
<p>This problem is from Ch... | Anna | 89,100 | <p>I can't add comments since my reputation, so I use this to comment.
I just list something easy to notice about:
If $T$ is the circle, then the triangle with the maximal area in it is the equilateral triangle, so the area of it is exactly $\frac{3\sqrt 3}{4\pi}$. Thus if $T$ is the ellipse with the area equals to $1$... |
885,335 | <blockquote>
<p>There is convex body $T$ (with the area is $1$), show that
there is a triangle $\Delta ABC$, such $A,B,C\in T$, and
$$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/nkuDz.jpg" alt="enter image description here"></p>
<p>This problem is from Ch... | Christian Blatter | 1,303 | <p>Yesterday I went to the ETH Zurich library and got hold of Wilhelm Blaschke's 1917 paper </p>
<p><em>Über affine Geometrie III: Eine Minimumeigenschaft der Ellipse.</em> Leipziger Berichte <strong>69</strong> (1917), pages 3–12. </p>
<p>In this paper he proves that any convex body $B$ in the plane contains a trian... |
45,172 | <blockquote>
<p>If <span class="math-container">$\displaystyle t = \frac{1}{x}$</span> then</p>
<p>a) Explain why <span class="math-container">$\displaystyle\lim_{x \to 0^-}f(x)$</span> is equivalent to <span class="math-container">$\displaystyle\lim_{t \to -\infty}f\left(\frac{1}{t}\right)$</span></p>
<p>b) Using that... | Arturo Magidin | 742 | <p>(a) does not have to do with $f$; rather, the point is that if $t=\frac{1}{x}$, then $x=\frac{1}{t}$, and $\lim\limits_{t\to-\infty}x = 0^-$ (that is, as $t$ approaches $-\infty$, the value of $\frac{1}{t}$ approaches $0$ from the left; and conversely, as $\frac{1}{t}=x$ approaches $0$ from the left, the value of $t... |
4,318,277 | <p>I've been reading up on Bayes Theorem and thought I'd try to apply it to a hypothetical medical test, but I'm not sure I'm applying it correctly.</p>
<p>I contrived this scenario:</p>
<ul>
<li>A medical test has a sensitivity of 60%. In other words, the false
negative rate is 40%.</li>
<li>The test also has a specif... | ryang | 21,813 | <blockquote>
<p>Now my only confusion is how the <code>P(False Negative Result)</code> differs from the <code>Negative Predictive Value</code>.</p>
</blockquote>
<p>Consider this probability tree:</p>
<p><a href="https://i.stack.imgur.com/XzHrl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XzHrl.pn... |
3,329,650 | <p>Let <span class="math-container">$f$</span> be a complex function <span class="math-container">$f(z) = \frac{z^2 +1}{z(z-i)^3}$</span>. It needs to be put jnto Laurent's series, so that it will converge in a set <span class="math-container">$D = \{ g(z) \mid Re(z) > 0 \}$</span>, where <span class="math-containe... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>The function <span class="math-container">$f(x)$</span> diverges only in its singularities. So do any of those singularities belong to the inside or on the boundary of <span class="math-container">$D$</span>?</p>
|
122,415 | <p>Suppose G is a Grassmannian variety. What are cohomology groups $H^i(G,T_G)$ $(i\geq 0)$ of the tangent bundle ?
thanks.</p>
| David E Speyer | 297 | <p>This answer is in characteristic zero so that I can use Borel-Bott-Weil; I'm not sure if it's still right in finite characteristic. As Serge says, $H^0(G(k,V), T) = \mathrm{End}(V)/\langle \mathrm{Id} \rangle$. All the other cohomology groups are zero.</p>
<p>Proof sketch: Let $Fl(V)$ be the variety of complete fla... |
4,072,810 | <p>Currently trying to prove that <span class="math-container">$\log(n+2)/\log(n+1)$</span> converges to <span class="math-container">$1$</span> with analysis tools.</p>
<p>if I can prove that <span class="math-container">$\forall \varepsilon >0$</span>, <span class="math-container">$\exists N$</span> such that
<sp... | user158293 | 158,293 | <p>to MOishe : I had to put it as an answer because the following was too long for comment. Ok so you are considering the <span class="math-container">$\{\pm I\}$</span> subgroup, being finite and discrete is considered not connected I may guess? as far as that internet wikpedia article goes yes i did read that prior a... |
86,800 | <p>lets take this small example code:</p>
<pre><code>Manipulate[Plot[Sin[x (1 + a* x)], {x, 0, 6}], {a, 0, 100},
Button["a=90", ToExpression["a=90"]]]
</code></pre>
<p>By clicking the button I want to set the manipulate variable a to 90. An important thing is, that the variable name comes out of an imported sting in ... | Karsten 7. | 18,476 | <p><strong>Update:</strong></p>
<p>Based on your updated question I guess you want something like</p>
<pre><code>Manipulate[{a, b}, {a, 1, 10}, {b, 1, 10},
Button["export", Export[SystemDialogInput["FileSave",".txt"], {{"a", a}, {"b", b}}, "Table"],
Method -> "Queued"],
Button["import", {a, b} = Import[System... |
64,605 | <p>If I print the following <code>BarChart</code> in black and white, then it is very difficult to tell which bar is "a" and which is "b".</p>
<pre><code>BarChart[{{1, 2}}, ChartLegends -> {"a", "b"}]
</code></pre>
<p><img src="https://i.stack.imgur.com/uzYiB.png" alt="enter i... | Bob Hanlon | 9,362 | <p>You can use both <code>ChartLabels</code> and <code>ChartLegends</code> for when the colors may not be easily differentiated.</p>
<pre><code>BarChart[{{1, 2, 3}, {1, 3, 2}, {5, 2, 1}},
ChartLabels -> {"a", "b", "c"},
ChartLegends -> {"a", "b", "c"}]
</code></pre>
<p><img src="https://i.stack.imgur.com/d83c... |
1,286,720 | <blockquote>
<p>to get a license, one's need to pass a 3 stages exam.
<br>
If one's fail in one stage he can not continue to the next stage.
<br> The probability he will pass the first stage is $0.9$, if he passed the first stage, the probability he will pass the second test is $0.8$.
if he passed the first... | Adelafif | 229,367 | <p>just apply Chinese remainder theorem (See Burton)</p>
|
1,286,720 | <blockquote>
<p>to get a license, one's need to pass a 3 stages exam.
<br>
If one's fail in one stage he can not continue to the next stage.
<br> The probability he will pass the first stage is $0.9$, if he passed the first stage, the probability he will pass the second test is $0.8$.
if he passed the first... | Thomas Andrews | 7,933 | <p>The quick solution to the Chinese Remainder Theorem exercise is to find integers $w,x,y,z$ such that $105w + 70x+42y + 30z=1$. Then:</p>
<p>$$x\equiv 0\cdot 105w + 1\cdot 70x + 4\cdot 42y + 1\cdot 30z\pmod {210}$$</p>
|
4,451,267 | <p>In Chapter 2 of <em>Numerical Optimization</em> by Nocedal and Wright, the authors have written the following equation:</p>
<p><span class="math-container">$$
\nabla f(\overrightarrow x + \overrightarrow p) = \nabla f(\overrightarrow x) + \int_0^1 \nabla^2 f(\overrightarrow x + t\overrightarrow p)\overrightarrow pdt... | Tryst with Freedom | 688,539 | <p>Picture:</p>
<p><a href="https://i.stack.imgur.com/bgDsB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bgDsB.png" alt="enter image description here" /></a></p>
<p>The idea is to first consider a line from <span class="math-container">$x$</span> to <span class="math-container">$ x +p$</span> , an... |
3,551,494 | <p>Can someone help me out with <span class="math-container">$$\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})?$$</span> I tried Taylor's expansion to no avail.</p>
| Community | -1 | <p><span class="math-container">$$\lim_{x\rightarrow 0}\ln\left(1+\frac{1}{x}\right)^{x}=\lim_{x\rightarrow 0}\frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}=\frac{\frac{-\frac{1}{x^{2}}}{1+\frac{1}{x}}}{-\frac{1}{x^{2}}}=\frac{x+1}{x}=1$$</span></p>
|
3,350,551 | <p>Given two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. We know that <span class="math-container">$B \setminus A = B$</span> holds. What can we say about the relation between <span class="math-container">$A$</span> and <span class="math-container">$B$</span>?</p>
<p>I wou... | Peter Szilas | 408,605 | <p>A bit formal:</p>
<p>Let <span class="math-container">$A,B \subset X$</span>.</p>
<p><span class="math-container">$B$</span> \ <span class="math-container">$A= B\cap A^c$</span> , where <span class="math-container">$A^c$</span> is the complement of <span class="math-container">$A$</span> in <span class="math-conta... |
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Community | -1 | <p><a href="/questions/tagged/status-completed" class="post-tag moderator-tag" title="show questions tagged 'status-completed'" rel="tag">status-completed</a></p>
<p>I propose the following word</p>
<p><strong>urgent</strong></p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with th... |
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | bof | 111,012 | <p>I propose the word </p>
<p><strong>interesting</strong></p>
<p>for inclusion in the HNQ block-list. Please up-vote if you agree with the inclusion, and down-vote if you disagree with the inclusion.</p>
|
1,626,579 | <p>Is there an exact, workable description of sets of positive integers closed under the lcm or gcd operations? In other words, a set of ideals of Z which is closed under intersections or sums. My motivation for asking this is the fact that the order of the product of two elements in an abelian group is the lcm of thei... | Ross Millikan | 1,827 | <p>The powers of any number of the form $a^k$, with $a,k$ natural satisfies this. You can stop at any power you want. You can also take the union of powers of $a^k$ and $b^m$ as long as $\gcd(a,b)=1$ and their products, stopping each at any power you want. This extends to as many pairwise coprime numbers as you want. ... |
2,890,207 | <p>Consider the following system of equations where $x,y,z$ are variables and for a constant $\mathrm C$, $y \times z = \mathrm C \neq 0$</p>
<p>\begin{equation}
\left\{
\begin{array}{lcl}
a_1x + b_1y + c_1z &= 0\\
a_2x + b_2y + c_2z &= 0\\
\end{array}
\right.
\end{equation}</p>
<p>The most straightforward so... | Jack D'Aurizio | 44,121 | <p>Technical recommendations:</p>
<ol>
<li><em>Generatingfunctionology</em> by Wilf and <em>Analytic Combinatorics</em> by Flajolet-Sedgewick. They cover the huge analogy between combinatorial classes and power series, the former with a focus on combinatorics, the latter with a focus on asymptotics (Laplace and saddl... |
28,463 | <p>The <a href="http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations" rel="nofollow noreferrer">Wikipedia article on numerical differentiation</a> mentions the formula</p>
<p>$$
h=\sqrt \epsilon \times x
$$</p>
<p>where $\epsilon$ is the machine epsilon (approx. $2.2\times 10^{-16}$ for 64-... | Gerry Myerson | 3,684 | <p>If you click through to the reference given for the Wikipedia piece, you'll find an answer. The formula given there is to take $h$ to be roughly $\sqrt{\epsilon_f}x_c$, where $\epsilon_f$ isn't necessarily "machine epsilon," but more to the point, where $x_c$ isn't necessarily $x$. </p>
|
4,075,642 | <p>I recently found this problem, it was part of a regional qualifier in Southern America (Venezuela- I believe) in January 2021. As I can’t find the solution anywhere and it is very different from the problems I have tackled so far I thought I might ask you all for some help. I have tried finding cases where <span cla... | Robert Shore | 640,080 | <p>The given operation preserves the parity of the number of odd numbers on the board -- that number always drops either by <span class="math-container">$2$</span> or <span class="math-container">$0$</span>. Since you started with an odd number of odd numbers, it's impossible for <span class="math-container">$2020$</s... |
2,721,285 | <p>I would like to prove the following statement.</p>
<blockquote>
<p>If $n$ is a prime number, then the factorization of $x^n-1$ over
$\Bbb{Z}$ contains only $2$ factors, and those factors are:
$$x^n-1=(x-1)(x^{n-1}+x^{n-2}+\cdots+x+1).$$</p>
</blockquote>
<p>I have already proved that the factorization abov... | wendy.krieger | 78,024 | <p>The polynomial $x^n-1$ contains as many algebraic factors, as $n$ contains divisors. Each of these algebraic factors contain a first term $x^a$, where $a$ is the Euler totative of the divisor.</p>
<p>These equations can be demonstrated and evaluated from the decimal. This works at least for all values where the e... |
1,019,191 | <p>I'm negating this proposition: "If you study you will not fail." </p>
<p>I'm using proposition P: "You study" and proposition Q: "You will fail." </p>
<p>The original statement can be written as "$P → ¬Q.$"</p>
<p>My instructor has the negation of this statement like this:</p>
<p>$¬(P → ¬Q) = ¬(¬P \lor ¬Q) = P\l... | Dan Christensen | 3,515 | <p>I find it more convenient to use: $$[A\implies B]\equiv \neg[A \land \neg B]$$</p>
<p>Then your statement could be restated as: It cannot be that you both study and fail.
$$ [P\implies \neg Q]\equiv \neg[P\land Q]$$ </p>
<p>The negation is that you both study and fail.<br>
$$P\land Q$$</p>
<p>Or equivalently, by ... |
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