qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,127,468 | <p>Suppose we have two functions <span class="math-container">$f,g:\Bbb R\rightarrow \Bbb R$</span>. The chain rule states the following about the derivative of the composition of these functions, namely that
<span class="math-container">$$
(f \circ g)'(x) = f′(g(x))\cdot g′(x).
$$</span>
However, the equivalent expre... | Jackozee Hakkiuz | 497,717 | <p><span class="math-container">$f'(g(x))$</span> means the derivative of <span class="math-container">$f$</span> evaluated at <span class="math-container">$g(x)$</span>.
Really the ambiguous one is Leibniz notation, because it makes you think the function <span class="math-container">$f$</span> <em>"cares"</... |
91,252 | <p>Let $v_1, \ldots, v_n$ be a set of vectors in a vector space $V$. Show that $v_1, \ldots, v_n$ is a basis of $V$ if and only if for any non-zero linear function $f$ on $V$ there is a vector $v$ in $\operatorname{span}(v_1, \ldots, v_n)$ such that $f(v) \neq 0$. </p>
<p>Suppose that $v_1, \ldots, v_n$ is not a basis... | lhf | 589 | <p>Every linear transformation is determined by its values on a basis. These values may be chosen arbitrarily. So, your $f$ is a linear functional on $V$ if you require that $f(x)=1$ for a basis of $W$, not for the whole of $W$. A constant function cannot be linear unless it is zero.</p>
|
635,893 | <p>I am trying to prove following statement:</p>
<blockquote>
<p>$[m,n]$ is a set of functions defined as $f \in [m,n] \leftrightarrow f: \{1,...,m\} \rightarrow \{1,...,n\}$. The size of $[m,n]$ is $n^m$ for $m,n \in \mathbb{N}_{\gt0}$.</p>
</blockquote>
<p>I have tried to prove it but I am not entirely sure about... | Hagen von Eitzen | 39,174 | <p>Assume $a:=f(x_0)\ne0$ for some $x_0\in(0,1)$. Then $\frac1a{f(x)}>\frac12$ on some open neighbourhood $x_0\in (x_0-r,x_0+r)\subset(0,1)$ because $f$ is continous. Hence
$$ 0=\frac1a\int_0^{x_0+r}f(t)\,\mathrm dt-\frac1a\int_0^{x_0-r}f(t)\,\mathrm dt=\int_{x_0-r}^{x_0+r}\frac1af(t)\,\mathrm dt\ge \int_{x_0-r}^{x_... |
2,251,240 | <p>What is the first derivative and nth derivative of the following function $ y = \sqrt {2 +\sqrt {3 + \sqrt {x}}}$ </p>
<p>I think taking the ln for both sides will remove the first square root only?
Could anyone give me a hint ? </p>
| gandalf61 | 424,513 | <p>Alternative approach (same answer) ...</p>
<p>$y^2 = 2+ \sqrt{3+\sqrt{x}}$</p>
<p>$\Rightarrow y^2-2=\sqrt{3 + \sqrt{x}}$</p>
<p>$\Rightarrow y^4-4y^2+4=3+\sqrt{x}$</p>
<p>$\Rightarrow 4y^3\frac{dy}{dx} - 8y\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$</p>
<p>$\Rightarrow \frac{dy}{dx} = \frac{1}{8y(y^2-2)\sqrt{x}}$</p>... |
2,734,109 | <p>This is what my lecture notes have but I cannot find anything like it online and there is no explanation in the notes. The example given is for 903 and 444.
<a href="https://i.stack.imgur.com/9NXYk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NXYk.png" alt="enter image description here"></a>
... | fleablood | 280,126 | <p>Let $a = 903$ and $b= 444$.</p>
<p>$903 - 2*444 = 15$ So $a - 2b = 15$.</p>
<p>$444- 29*15 = 444-435 = 9$ or in other words $b - 29(a-2b) = 59b-29a = 9$.</p>
<p>$15 - 9=6$ or in other words $(a-2b) -(59b-29a) = 30a -61b = 6$.</p>
<p>$9 - 6 = 3$ of in other words $(59b-29a) - (30a -61b) = 120b - 59a = 3$.</p>
<... |
2,734,109 | <p>This is what my lecture notes have but I cannot find anything like it online and there is no explanation in the notes. The example given is for 903 and 444.
<a href="https://i.stack.imgur.com/9NXYk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NXYk.png" alt="enter image description here"></a>
... | Will Jagy | 10,400 | <p>I like to write these as (simple) continued fractions. </p>
<p>$$ \gcd( 903, 444 ) = ??? $$ </p>
<p>$$ \frac{ 903 }{ 444 } = 2 + \frac{ 15 }{ 444 } $$
$$ \frac{ 444 }{ 15 } = 29 + \frac{ 9 }{ 15 } $$
$$ \frac{ 15 }{ 9 } = 1 + \frac{ 6 }{ 9 } $$
$$ \frac{ 9 }{ 6 } = 1 + \frac{ 3 }{ 6 } $$
$$ \frac{... |
1,264,353 | <blockquote>
<p>Evaluate the determinants given that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-6.$</p>
</blockquote>
<ol>
<li>$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}$ </li>
<li>$\begin{vmatrix} a & b &am... | Upax | 157,068 | <p>I'm not sure I understand. The determinant
\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=
$a e i-a f h-b d i+b f g+c d h-c e g$=-6.
While
\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}=
$-a e i+a f h+b d i-b f g-c d h+c ... |
129,693 | <p><img src="https://i.stack.imgur.com/tFha8.png" alt="enter image description here"></p>
<p>I'm lost on what's happening here. This is regarding MinML( "an idealized programming language" ) . More pics below: Thank You Very Much</p>
<p><img src="https://i.stack.imgur.com/BYjTF.png" alt="enter image description here"... | LeeJordan | 776,157 | <p>If you take
<span class="math-container">$$f(z) = \frac{e^{iz}}{1+z^2}$$</span>
and apply the residue theorem over the path <span class="math-container">$\Gamma = \gamma_1 + \gamma_2$</span>:
<a href="https://i.stack.imgur.com/Y6nFb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Y6nFb.png" alt="p... |
1,471,415 | <p>I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one:</p>
<p>$$
v = 1 - e^{-t/\tau}
$$</p>
<p>The derivative is</p>
<p>$$
\frac{dv}{dt} = \frac{1-v}{\tau}
$$</p>
<p>Can someone walk me through this? If this is explained somewhere else, I'd love to k... | zahbaz | 176,922 | <p>$$\frac{dv}{dt}= \frac{d}{dt} (1 - e^{-t/\tau})$$</p>
<p>The derivative of $1$ is zero, so</p>
<p>$$= -\frac{d}{dt}e^{-t/\tau}$$</p>
<p>Using the chain rule,</p>
<p>$$= -e^{-t/\tau}\cdot\frac{d}{dt}\frac{-t}{\tau}$$
$$= -e^{-t/\tau}\cdot\frac{-1}{\tau}$$
$$= \frac{e^{-t/\tau}}{\tau}$$</p>
<p>Now, we know that $... |
423,938 | <p>I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.</p>
<p>$$\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$$</p>
| mick | 39,261 | <p>Perhaps not an elegant proof but I considered this.</p>
<p>Use a Taylor series for $\sin$.</p>
<p>$\sin(x)=x+ a_1x^3 + a_2x^5 + ...$</p>
<p>Let $A$ be the value of the limit.</p>
<p>Then we get
$A=\dfrac{x^2+\frac{1}{x}-\frac{1}{x}+a_1(x^2+\frac{1}{x})^3-a_1(\frac{1}{x})^3+\ldots}{x} = \dfrac{x^2+a_1(x^2+\frac{... |
976,392 | <p>I would need a proof that <span class="math-container">$n \left(1-p^{\frac{1}{n}}\right)$</span> is increasing in <span class="math-container">$n \in \mathbf{N}$</span> for any <span class="math-container">$p \in (0,1)$</span>.</p>
<h3>Context</h3>
<p>I am working on a larger question and this is the last missing pi... | mvggz | 167,171 | <p>0 < p < 1 : let f : x-> x*(1 - $p^\frac{1}{x}$)</p>
<p>$f'(x) = (1-p^\frac{1}{x}) +\frac{1}{x}*ln(p)*p^\frac{1}{x} = (1-p^\frac{1}{x}) -\frac{1}{x}*ln(\frac{1}{p})*p^\frac{1}{x} $ = $1 - [1+\frac{1}{x}*ln(\frac{1}{p})]*p^\frac{1}{x}$</p>
<p>Let $ h : u-> 1 - [1 + u*ln(\frac{1}{p})]*p^u = f'(\frac{1}{u}) ... |
4,625,921 | <p><a href="https://i.stack.imgur.com/mIKMW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mIKMW.png" alt="Two polar curves." /></a></p>
<p>I am looking for the area outside the circle <span class="math-container">$r = 3\cos\Theta$</span> and inside the limaçon <span class="math-container">$r = 1 + ... | Sammy Black | 6,509 | <p><strong>EDIT:</strong> Original answer didn't address radial function taking on negative values.</p>
<hr />
<p>The positive direction for the radial polar variable <span class="math-container">$r$</span> is <em>outward</em> so if you want the area of the region <em>outside</em> the polar curve <span class="math-cont... |
2,802,035 | <p>(a) All roots of $f(x)$ are real.<br>
(b) $f(x)$ has one real root and $2$ complex roots.<br>
(c) $f(x)$ has two roots in $(-1,1).$<br>
(d) $f(x)$ has at least one negative root.</p>
<p>I thought of solving this question using Descartes Rule. 'c' is negative and 'a' is turning out to be positive. 'b' should be nega... | nonuser | 463,553 | <p>So $f(1) >0$ and $f(-1)>0$ and $f(0)<0$: </p>
<p>\begin{array}{cccc}
x & -1 & 0 & 1 \\
f(x) & + & - & + \\
\end{array}</p>
<p>So $a)$ and $c)$ and $d)$ are true.</p>
|
265,037 | <p>Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.</p>
<blockquote>
<p><strong>Question.</strong> Is it true there is always a $\pi\in\mathfrak{S}_k$ such that the following are pair-wise distinct in $\mathbb{F}_p$?
... | Salvo Tringali | 16,537 | <p>I first learned about this problem from Éric Balandraud (in 2013). So I wrote to him a couple of days ago, and he has just sent me an e-mail explaining that the question dates back (at least) to 1971. Éric attributes it to Erdős and Graham: He hasn't provided me with a reference (he is in the desert these days), but... |
840,445 | <p>Stirling number of the second kind is the number of ways to partition a set of n objects into k non-empty subsets - S(n,k). I want to restrict/constrain this partition so I can count the ways to partition a set of n objects into k non-empty subsets while at least p subsets out of k will have size r. When p=k the ans... | epi163sqrt | 132,007 | <p>Here is an approach based upon generating functions. The following can be found in section II.3.1 in <em><a href="http://algo.inria.fr/flajolet/Publications/book.pdf" rel="nofollow noreferrer">Analytic combinatorics</a></em> by P. Flajolet and R. Sedgewick:</p>
<blockquote>
<p>The class $S^{(A,B)}$ of set partit... |
2,471,546 | <p>In order to find a shorter proof for this thread : </p>
<p><a href="https://math.stackexchange.com/questions/2470262/find-all-functions-fmfn2-1-fnfmn-fm-n/2470409?noredirect=1#comment5106478_2470409">Find all functions $f(m)[(f(n))^2-1]=f(n)[f(m+n)-f(m-n)]$</a></p>
<p><em>Rem: just to be clear, $\mathbb N^*$ stand... | stochasticboy321 | 269,063 | <p>Note that $a \alpha = F_1 + \frac{\beta}{a},$ which implies that $$F_n = F_1 a^{n-1} + \beta \left( \frac{a^{2n-2} +(-1)^n}{a^n} \right).$$ Since both $F_n$ and $F_1 a^{n-1}$ are naturals for every $n$, we must have that for every $n$, $$\beta \left( \frac{a^{2n-2} + (-1)^n}{a^n} \right) \in \mathbb{Z}.$$ In partic... |
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | Community | -1 | <p>Before doing anything, you know that <span class="math-container">$x\ge0$</span> because the real square root function is non-negative.</p>
<p>By inspection <span class="math-container">$x=1$</span>, which is the only solution.</p>
|
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | mlchristians | 681,917 | <p>In your title you ask, is <span class="math-container">$x = -2$</span> a solution of <span class="math-container">$\sqrt{2-x} = x$</span>?</p>
<p>To answer this question, you simple need to check the answer by substituting <span class="math-container">$x=-2$</span> into the original question and see if a true state... |
647,757 | <blockquote>
<p>How many integers $n$ are there such that $\sqrt{n}+\sqrt{n+7259}$ is an integer? </p>
</blockquote>
<p>No idea on this one.</p>
| dmk | 88,878 | <p>(This assumes $n \in \mathbb{Z}$.)</p>
<p>First, note that $k^2 + (2k + 1) = (k+1)^2$; that is, the difference between two consecutive sqaures is an odd number. This implies that the difference between <em>any</em> two perfect squares is a sum of consecutive odd numbers.</p>
<p>We start by finding the largest $n$... |
3,887,526 | <p>I need to prove <span class="math-container">$\frac{|x+y+z|}{1+|x+y+z|} \le \frac{|x|}{1+|y|+|z|}+\frac{|y|}{1+|x|+|z|}+\frac{|z|}{1+|x|+|y|}$</span>. I've tried to use triangle inequality or to explore the form of <span class="math-container">$(a+b+c)^2$</span> but it won't get me anywhere. I would be grateful for ... | Albus Dumbledore | 769,226 | <p>as <span class="math-container">$|x+y+z|\le |x|+|y|+|z|$</span></p>
<p>let <span class="math-container">$|x|=a,|y|=b,|z|=c$</span>
it suffices to prove <span class="math-container">$$\sum \frac{a}{1+b+c}\ge \sum \frac{a+b+c}{1+a+b+c}$$</span><br />
indeed by C-S/titu's lemma; <span class="math-container">$$\sum \fra... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Sunil B N | 71,150 | <p>To explain it more precisely, $n!$ grows very fast when compared to a power $n$. Because the greater number is multiplied with the product each time: $$(n+1)!=1 \cdot 2 \cdots n \cdot (n+1).$$ But in case of exponential function, $$a^{n+1} = a \cdot a \cdots a,$$ the term $a$ remains constant.</p>
|
2,028,594 | <p>Given a simple undirected connected graph. What is the minimum number of edge-disjoint paths needed to cover all edges of the graph? I have only found information about the vertex-covering one.</p>
<p>My friend suggests that such value equals <em>u/2</em>, with <em>u</em> being number of vertices with odd degree, o... | user392700 | 392,700 | <p>In a directed graph, we can transform this into a flow problem with each edge having the lower-bound capacity of $1$ and all link to a source and a sink with edges with lower-bound capacity $0$. The least number of paths will be the minimum flow value. Is this idea correct and if so can it be generalized to 'undirec... |
2,123,926 | <p>I have a triangle with the coordinates (0, 20), (15, -10), (-15, -10). So the centre of the triangle is 0,0. </p>
<p>I want to rotate this by one degree. Im using the formula:</p>
<pre><code>x' = cos(theta)*x - sin(theta)*y
y' = sin(theta)*x + cos(theta)*y
</code></pre>
<p>Yet when I work out the coordinate for a... | S.C.B. | 310,930 | <p>$$y''-2y'+y=0 \iff y''-y'-y'+y=0 \iff (y'-y)'-(y'-y)=0$$</p>
<p>Note that $$(y'-y)'-(y'-y)=0 \iff y'-y=ce^{x} \iff (y'-y)e^{-x}=c$$
For some constant $c$. Now note $$(y'-y)e^{-x}=(ye^{-x})'$$
So we have that $$(ye^{-x})'=c \iff ye^{-x}=cx+b$$
For some constant $c,b$. So your answer is incorrect. The answer should b... |
88,284 | <p>If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$
under usual addition and multiplication, then what are the left and right ideals of $R$?</p>
| rschwieb | 29,335 | <p>This is covered in full on page 17 of Lam's <em>First course in noncommutative rings</em>. In general the "triangular ring" where $R$ and $S$ are rings and $M$ is an $R-S$ bimodule looks like: </p>
<p>$$T= \begin{pmatrix} R &M\\ 0 & S \end{pmatrix}
$$</p>
<p>You can also visualize the ring as $R\oplus M\o... |
1,842,779 | <p>This is Q28 from Australian Maths Competition 2014.</p>
<p>A circle is surrounded by 6 other circles,in a hexagonal formation.The leftmost circle is 0,which the rightmost circle is 1000.Each of the five missing numbers is the average of its neighbours. What is the largest of the 5 missing numbers?</p>
<p>What I tr... | Théophile | 26,091 | <p>Starting off with @heropup's approach, we can speed up the calculations by appealing to symmetry:</p>
<p>\begin{array}{ccccc} & a & & b & \\ 0 & & c & & 1000 \\ & d & & e & \end{array}</p>
<p>Observe that we must have $a=d, b=e$, and $a = 1000-b$. This eliminates all... |
910,020 | <p>Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$.</p>
<p><strong>Attempt:</strong> $|G|=168=2^3.3.7$</p>
<p>Then number of sylow $7$ subgroups in $G = n_7 = 1$ or $8$.</p>
<p>Given that $H$ is a normal subgroup of order $4$ in $G$. <... | Andreas Caranti | 58,401 | <p>If $K$ is a normal subgroup of $G$ of order $4$, you may well argue in $H = G/K$, and reduce to show, via the correspondence theorem, that in a group of order $2 \cdot 3 \cdot 7 = 42$ there must be a normal subgroup of order $7$.</p>
<p>To do this, just consider that the number of $7$-Sylow subgroups in $H$ must di... |
910,020 | <p>Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$.</p>
<p><strong>Attempt:</strong> $|G|=168=2^3.3.7$</p>
<p>Then number of sylow $7$ subgroups in $G = n_7 = 1$ or $8$.</p>
<p>Given that $H$ is a normal subgroup of order $4$ in $G$. <... | aaa | 186,150 | <p>If K is a normal subgroup of G of order 4 , you may well argue in $H=\frac{G}{K}$ , and reduce to show, via the correspondence theorem, that in a group of order $2\cdot 3\cdot 7=42$ there must be a normal subgroup of order 7 .</p>
<p>To do this, just consider that the number of 7 -Sylow subgroups in H must divi... |
507,133 | <p>I came across a problem in a book that asked us to find the first number $n$ such that $\phi(n)\geqslant 1,000$ it turns out that the answer is 1009, which is a prime number. There were several questions of this type, and our professor conjectured that it will always be the next prime. However, no one has been able ... | mjqxxxx | 5,546 | <p>If $n$ is not prime, then it has a prime factor $\le \sqrt{n}$, and so
$$
\varphi(n) = n \prod_{p\;|\;n}\left(1-\frac{1}{p}\right)\le n\left(1-\frac{1}{\sqrt{n}}\right)=n-\sqrt{n};
$$
while $\varphi(n)=n-1$ if $n$ is prime.</p>
<p>So, fix $M$. Let $p$ be the next prime number after $M$, so $\varphi(p)=p-1\ge M$; a... |
3,929,952 | <p><strong>Question:</strong> How do I estimate the following integral? <span class="math-container">$$\int_{2}^{\infty} \frac{|\log \log t|}{t\, (\log t)^2} \, dt$$</span></p>
<p><strong>Attempt:</strong> Setting <span class="math-container">$u=\log t$</span>, we get</p>
<p><span class="math-container">$$\int_{2}^{\in... | Jacky Chong | 369,395 | <p>Observe that
<span class="math-container">\begin{align}
\int^\infty_2 \frac{|\log\log t|}{t(\log t)^2}\ dt = \int^\infty_{\log\log 2} |u|e^{-u}\ du.
\end{align}</span></p>
<p>You can even evaluate this by hand.</p>
|
44,082 | <p>I have basic questions about elliptic curves over finite fields. </p>
<ol>
<li><p>Where to find general references? Hartshorne for instance restricts to algebraically closed ground fields.</p></li>
<li><p>Over an arbitrary field $K$, is the right definition of an elliptic curve a smooth proper
curve of genus 1 with... | Yuri Zarhin | 9,658 | <p>Elliptic curves $E$ and $E'$ over a finite field $K$ are $K$-isogenous if and only if the orders of $E(K)$ and $E'(K)$ coincide. However, it may happen that the groups $E(K)$ and $E'(K)$ have the same order (and even isomorphic) but $E$ and $E'$ are not isomorphic over $K$. Even worse, there exist such a $K$ and n... |
2,531,538 | <p>I wish to show the following in equality:</p>
<p>$$\dfrac {n!}{(n-k)!} \leq n^{k}$$</p>
<p>Attempt:</p>
<p>$$\dfrac {n!}{(n-k)!} = \prod\limits_{i = 0}^{k -1} (n-i) = n\times (n-1)\times \cdots \times(n-({k-1})) $$</p>
<p>I am not sure how to make the argument that $n\times (n-1)\cdots \times (n-({k-1})) \leq n^... | Sri-Amirthan Theivendran | 302,692 | <p>Each of the $k$ terms in the product is less than or equal to $n$. </p>
<p>ALternatively, the LHS counts the number of injective maps from a set with $k$ elements to a set with $n$ elements while the RHS counts all the maps from a set with $k$ elements to a set with $n$ elements. The result follows.</p>
|
981,181 | <p>I'm stuck with a logic problem like this</p>
<blockquote>
<p>I eat ice cream if I am sad.</p>
<p>I am not sad.</p>
<p>Therefore I am not eating ice cream.</p>
</blockquote>
<p>Is this conclusion logical? The first sentence can be understood both like "ice cream <span class="math-container">$\implies$</span> sad... | Bruno Bentzen | 174,538 | <p><strong>Argument:</strong></p>
<blockquote>
<p>"I eat ice cream if I am sad. I am not sad.Therefore I am not eating ice cream."</p>
</blockquote>
<p>Whenever you are stuck in an argument, try represent it in proper symbolism:</p>
<p><strong>Glossary</strong>:</p>
<p>$E \equiv $ 'I eat ice cream'</p>
<p>$S \eq... |
2,568,544 | <p>The exercise is to construct a cube (inscribed in the unit spher) with one of the corners at:
$$ \vec{v} = \frac{1}{\sqrt{10}} (3,1,0) \in S^2 $$</p>
<p>I'm a bit stuck constructing the other seven vertices. My first guess was to use the eight reflections available to me:</p>
<p>$$ \square \stackrel{?}{=}\left... | hmakholm left over Monica | 14,366 | <p>A possible brute-force approach:</p>
<p>Suppose you have orthogonal transformations $T$ and $U$ such that
$$ T(\mathbf e_1) = \frac1{\sqrt{10}}(3,1,0) \\ U(\mathbf e_1)=\frac1{\sqrt3}(1,1,1)$$</p>
<p>Then $T(U^{-1}(\frac1{\sqrt3}(\pm1,\pm1,\pm1)))$ will be a possible solution.</p>
<p>You can construct matrix repr... |
2,246,889 | <p>$\displaystyle \lim_{\substack{x \rightarrow 4 \\ y \rightarrow \pi}} x^2 sin \frac{y}{x}$</p>
<p>I don't know in which way should I try to solve this example, can maybe the single variable limit of $sin \frac{1}{x}$ help me ?</p>
| hamam_Abdallah | 369,188 | <p>Your limit is simply</p>
<p>$$(4)^2\sin (\frac {\pi}{4} )$$
$$=8\sqrt {2} .$$</p>
|
2,452,135 | <p>I need to prove the set
$A=\left \{ \left ( r\cos t,r\sin t \right ) \mid 0\leq t\leq \theta ,r\geq 0\right \}$ is closed. I have tried to use the theorems for continuous functions, but I always into troubles cause the domain is not bounded. </p>
| Math1000 | 38,584 | <p>Let $\{(x_n,y_n)\}$ be a sequence of points in $A$ such that $\lim_{n\to\infty}(x_n,y_n)=(x,y)$ for some $(x,y)\in\mathbb R^2$. We can assume that $(x,y)\ne(0,0)$ since clearly $0=0\cdot\cos t=0\cdot\sin t$ for any $t$ and so $(0,0)\in A$. Set $\rho = \sqrt{x^2+y^2}$ and $\theta = \mathrm{atan2}(y,x)$, where
$$\mat... |
4,278,202 | <p>Suppose, I am working on <span class="math-container">$Eq(X)$</span> = the set of all equivalence relations on <span class="math-container">$X$</span>. Let <span class="math-container">$\theta_1, \theta_2$</span> denote arbitrary equivalence relations.</p>
<p>Then how does <span class="math-container">$\theta_1 \lor... | Tereza Tizkova | 943,366 | <p>If we take two arbitrary elements <span class="math-container">$a, b$</span>, how does <span class="math-container">$a (\theta_1 \lor \theta_2) b$</span> look like?</p>
<p>If this holds, there must exist an element <span class="math-container">$c$</span> such that</p>
<p><strong><span class="math-container">$a \: \t... |
2,258,720 | <p>Define a set X of integers recursively as follows:</p>
<p>Base: 5 is in X</p>
<p>Rule 1: If x is in X and x>0, then x+3 is in X</p>
<p>Rule2: If x is in X and x>0, then x+5 is in X</p>
<p>Show that every integer n>7 is in X</p>
<p>I am pretty new to this stuff and I am very lost on this. Please help!</p>
| fleablood | 280,126 | <p>Let $n \ge 13$. Divide $n$ by $5$ and take the remainder to get $n = k*5 + r$ and $k$ is at least $3$.</p>
<p>Case 0: if $r = 0$ then $n = 5+5+5+5 ..... = 5*k + 0 = n$ and $n $ is in $X$.</p>
<p>Case 3: if $r = 3$ then $n = 5+ 5+ 5+ .... + 3 = 5+ 3 + 5(k-1)$ is in $X$.</p>
<p>Case 1: if $r = 1$ then $n = 5+5+5+5... |
1,722,256 | <p>I'm trying to calculate the percentage difference $\Bbb{P}$ between two numbers, $x$ and $y$. I'm not given much context about how these numbers are different (for example, if one is "expected outcome" and one is "observed outcome").</p>
<p>When is each of this formulas relevant?</p>
<p>$$\Bbb{P} = \frac{|x-y|}{x}... | peter.petrov | 116,591 | <p>All of these can be relevant in a certain context. The 1st one is the percentage difference as a percentage of $x$, the 2nd one is the percentage difference as a percentage of $y$ the 3rd one is the percentage difference as a percentage of the max of $x, y$ and so on. </p>
|
134,213 | <p>Is there a way to simplify the following expression</p>
<p>$$A = \frac{\left (a^6-b^6 \right )^2}{c^6-4a^3b^3}$$</p>
<p>assuming</p>
<p>\begin{eqnarray}
a = &\ (a-b)^2+b\ (a+1) \cr
b = &\ (b-c)^2+c\ (b+1) \cr
c = &\ (c-a)^2+a\ (c+1)
\end{eqnarray}</p>
<h3>Edit</h3>
<p>I would like to determi... | Bob Hanlon | 9,362 | <pre><code>Solve[Eliminate[{A == (a^6 - b^6)^2/(c^6 - 4 a^3 b^3),
a == (a - b)^2 + b (a + 1), b == (b - c)^2 + c (b + 1),
c == (c - a)^2 + a (c + 1)}, {a, b, c}], A] // Simplify
(* {{A -> -6 - 3*((1/2)*(123 -
55*Sqrt[5]))^(1/3) -
3*((1/2)*(123 + 55*Sqrt[5]))^
(1... |
895,325 | <p>I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is:</p>
<p>$$ \mathbb{C}[X,XY,XY^2] \cong \frac{\mathbb{C}[U,V,W]}{\langle V^2 - UW \rangle}. $$</p>
<p>I would like a nice way to write the Kahler differentials of such rings. Fo... | Robert Israel | 8,508 | <p>If $a$ and $b$ are coprime, every odd divisor of $ab$ can be written in exactly one way as the product of an odd divisor of $a$ and and odd divisor of $b$, and every such product is an odd divisor of $ab$. So $f(ab) = f(a) f(b)$.</p>
|
3,072,198 | <p>How can one explain that <span class="math-container">$$\frac{d}{dx}\left(\int_0^x{\cos(t^2+t)dt}\right) = \cos(x^2+x)$$</span>
Without solving the integral?</p>
<p>I know it's related to the fundamental theorem of calculus, but here we have a derivative with respect to <span class="math-container">$x$</span>, whil... | Paramanand Singh | 72,031 | <p>Perhaps you are mixing two parts of the Fundamental Theorem of Calculus (henceforth referred to as FTC). First part of the theorem deals exactly with finding derivative of things of the form <span class="math-container">$\int_{a} ^{x} f(t) \, dt$</span>.</p>
<p>More formally, let the function <span class="math-cont... |
3,690,260 | <blockquote>
<p>Find two vectors <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span>, whose sum is <span class="math-container">$\langle 0,-2\rangle$</span> where <span class="math-container">$v_1$</span> is parallel to <span class="math-container">$\langle 4,-3\rangle $</span>, wh... | Community | -1 | <p>Let <span class="math-container">$v_1 = \langle 4t, -3t \rangle, v_2 = \langle 3s,4s \rangle\implies v_1 \parallel \langle 4,-3\rangle, v_2 \perp \langle 4,-3\rangle$</span>.We solve for <span class="math-container">$s,t$</span>: <span class="math-container">$v_1+v_2=u\implies \langle 4t+3s, -3t+4s\rangle=\langle 0,... |
1,396,045 | <p>$$\int{\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx}$$</p>
<p>$$\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx=\int \frac{(1-\sin x)(2-\sin x)}{\sqrt{(1-\sin x)(2-\sin x)(1+\sin x)(2+\sin x)}}dx$$</p>
<p>I am stuck. Please help me....</p>
| juantheron | 14,311 | <p>Let $$\displaystyle I = \int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$$</p>
<p>We can write $$\displaystyle \sqrt{\frac{1-\sin x}{1+\sin x}} = \sqrt{\frac{1-\sin x}{1+\sin x}\times \frac{1+\sin x}{1+\sin x}} = \frac{\cos x}{1+\sin x}$$</p>
<p>So we get $$\displaystyle I = \int\frac{\cos x}{1+\sin ... |
3,552,264 | <p>How to find function <span class="math-container">$f(x)$</span> that has continuous derivative on <span class="math-container">$[0,2]$</span> satisfies the following conditions:</p>
<ol>
<li><span class="math-container">$f(2) = 3$</span></li>
<li><span class="math-container">$\displaystyle \int_0^2 [f'(x)]^2 dx = 4... | Riemann | 27,899 | <p><span class="math-container">$$ \int_0^2 x^3f'(x)\, dx= \int_0^2 x^3 \,df(x)=x^3 f(x)|^2_0-3\int_0^2 x^2f(x) \,dx=8\cdot f(2)-3\cdot\frac{1}{3}=24-1=23.$$</span></p>
|
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | John_dydx | 82,134 | <p>I can certainly recommend <em><a href="http://www.springer.com/gb/book/9783540761976" rel="nofollow">Elementary Number Theory</a></em> by Gareth A Jones et al. It will get you started and then you can move onto more advanced texts. It's a very short book (about 300 pages) which means you can easily read through the ... |
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | Ethan Bolker | 72,858 | <p>Dover publishes many number theory titles. At \$10-\$15 each they're a bargain - no need even to look for the Amazon discount. You can get several and jump back and forth among them to get different perspectives on each topic. You can write yourself notes in the margins. Take them to the library to read.</p>
<p>Thi... |
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | Community | -1 | <p>In my opinion Hardy &Wright's book on Number Theory is not the best possible book for someone "who has no prior training in Number Theory", I would suggest the following books.</p>
<blockquote>
<ul>
<li><p><em>Elementary Number theory</em> by David M. Burton.</p>
</li>
<li><p><em>Number Theory A Histor... |
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | J2 student | 306,580 | <p>For the first rope, label one end 0 and the other end A. Starting from end 0, let derivative of time passed w.r.t. fire position x be f(x). The corresponding function starting from end A is f'(x). </p>
<p>Since rope A takes 1h to burn from either end, integrating f(x) from 0 to A, and integrating f'(x) from A to 0 ... |
4,308,155 | <p>Given an operator <span class="math-container">$T \in B(H)$</span>, denote its spectrum by <span class="math-container">$\sigma(T)$</span>. We write <span class="math-container">$\Omega := \sigma(T)\setminus \{0\}$</span>. The following result is well-known and <em><strong>can be assumed without proof</strong></em>:... | Community | -1 | <p>Here is the proof how I wrote it down:</p>
<p><strong>Theorem</strong>: Let <span class="math-container">$T \in B_0(H)$</span> is a self-adjoint operator compact operator. If <span class="math-container">$\mathcal{F}_\lambda$</span> is an orthonormal basis for <span class="math-container">$\mathcal{E}_\lambda$</span... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Laie | 6,013 | <p>One possible way to think about this midnight question might be to ask what smaller results have been important or interesting to you, and then to appreciate the existence of the mathematicians who have discovered/invented it. My guess would be that everyone can compile a list of such results by players who are not ... |
4,038,151 | <p>In Bernt Oksendal's <em>Stochastic Differential Equations</em>, Chapter 4, one has the following stochastic differential equation (whose solution is geometric Brownian motion):
<span class="math-container">$$dN_t=rN_tdt+\alpha N_tdB_t\;\;\;\text{ ie } \;\;\; N_t-N_0=r\int_0^t N_sds+\alpha\int_0^tN_sdB_s,$$</span>
wh... | nullUser | 17,459 | <p>Typically associativity of the integral is proved early on. If <span class="math-container">$X$</span> is a semimartingale and the integral <span class="math-container">$K \cdot X = \int K dX$</span> makes sense then the integral <span class="math-container">$(HK) \cdot X = \int HK dX$</span> makes sense if and only... |
1,961,614 | <p>I can't say I've gotten very far. You can show $3 + \sqrt{11}$ is irrational, call it $a$. Then I tried supposing it's rational, i.e.:</p>
<p>$a^{1/3}$ = $\frac{m}{n}$ for $m$ and $n$ integers.</p>
<p>You can write $m$ and $n$ in their canonical factorizations, then cube both sides of the equation...but I can't s... | Deepak | 151,732 | <p>If you denote $x = (3 + \sqrt{11})^{\frac 13}$, you can show that the equation $x^6 - 6x^3 - 2 = 0$ holds and appeal to Rational Root Theorem.</p>
<p><a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">Rational Root Theorem</a></p>
|
1,961,614 | <p>I can't say I've gotten very far. You can show $3 + \sqrt{11}$ is irrational, call it $a$. Then I tried supposing it's rational, i.e.:</p>
<p>$a^{1/3}$ = $\frac{m}{n}$ for $m$ and $n$ integers.</p>
<p>You can write $m$ and $n$ in their canonical factorizations, then cube both sides of the equation...but I can't s... | Steven Alexis Gregory | 75,410 | <p><strong>Rational Root Theorem</strong></p>
<blockquote>
<p>If $P(x) = a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_1x + a_0$, where all of the $a_i$ are integers, and $P\left( \dfrac uv \right)=0$ where $u$ and $v$ are integers, then $v \mid a_n$ and $u \mid a_0$.</p>
</blockquote>
<p>Let $x = (3 + \sqr... |
2,419,922 | <p>An involution is a permutation $P$ which is its own inverse: $P\cdot P = \text{id}$.</p>
<p>Every permutation can be written (in various ways) as a sequence of single-element swaps (transpositions). Sometimes these sequences are palindromes, in that the reversal of the sequence is the same sequence.</p>
<p>If $T$ ... | user326210 | 326,210 | <p>It's not possible in general to express an involution as a palindrome of transpositions. In fact, because of cancellation properties, the only involutions that can be expressed as palindromes are (1) a single transposition, and (2) the identity.</p>
<p>Indeed, any even-length palindrome of transpositions cancels its... |
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| Nasser | 70 | <pre><code>a = RandomReal[{1, 10}, 10];
b = RandomReal[{1, 10}, 10];
lst = a/b;
Min[lst]
(*0.442821447015283*)
Position[lst, Min[lst]]
(* {{4}} *)
</code></pre>
<p><strong>Update to answer comment below</strong></p>
<p><code>How can I implement the b[i]>0 condition</code></p>
<p>One way can be to use <a href="h... |
541,541 | <blockquote>
<p>Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product $f_1\otimes f_2\colon V_1\otimes V_2\to W_1\otimes W_2$ is defined, and it's obvious that $\ker f_1\otimes V_2+ V... | Jendrik Stelzner | 300,783 | <h1>When <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span> are surjective</h1>
<h2>Constructing an isomorphism</h2>
<p>Suppose first that <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span> are surjective, and that we are working over an arbitrar... |
2,326,551 | <p>Prove that $\varphi(n^2)=n\cdot\varphi(n)$ for $n\in \Bbb{N}$, where $\varphi$ is Euler's totient function.</p>
| Nemanja Beric | 405,086 | <p>$Hint:$ Use multiplicity of Euler's function and fundamental theorem of arithmetics.</p>
|
544,763 | <p>Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: $m(A)>\frac{2n-1}{2n}(b-a)$. I need to show that $A$ contains an arithmetic sequence with <em>n</em> numbers ($a_1,a_1+d,...,a_1+(n-1)*d$ for some d).</p>
<p>I thought about dividing [a,b] into n equal parts, and show that if I put one part on top of the o... | George | 54,962 | <p>** Fact 1.** <em>Let $p$ be a linear normalized Lebesgue
measure on a circle $C$ centered at the origin of the plane $R^2$.
Suppose that $p(A)> (n-1)/n$. Then there exist $n$ points in $A$
which are vertices of a regular $n$-sided polygon.</em></p>
<p><strong>Proof.</strong>
Let us consider a rotation defined ... |
544,763 | <p>Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: $m(A)>\frac{2n-1}{2n}(b-a)$. I need to show that $A$ contains an arithmetic sequence with <em>n</em> numbers ($a_1,a_1+d,...,a_1+(n-1)*d$ for some d).</p>
<p>I thought about dividing [a,b] into n equal parts, and show that if I put one part on top of the o... | George | 54,962 | <p>Fact 2 implies that we need no a restriction that $m(A)>\frac{2n-1}{2n}(b-a)$. It is sufficient to require that $m(A)>0$. </p>
<p>It is subject of interest that there is a Lebesgue null set $X$ in $\mathbb{R}$ which
satisfies the following conditions:</p>
<p>(i) $\mbox{card}(X)=2^{\aleph_0}$;</p>
<p>(ii) f... |
1,895,422 | <p>I would like to know if it is possible to prove the identity $\mathbb E[e^{4W_t-8t}(W_t-4t)^4]=3t^2$ without the use of a measure change and Girsanov Theorem? I only know the solution by using the exponential martingale for the first term and then simplify terms under the new measure. </p>
| Mark Fischler | 150,362 | <p>For arbitrary real $a$,
$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}e^{4x-2a\sigma^2}(x-a\sigma^2)^4 \, dx = e^{(8-2a)\sigma^2}\sigma^4 \left( (a-4)^4\sigma^4 + 6(a-4)^2 \sigma^2 + 3\right)
$$
The relation you want to prove is the special case of $a=4$.</p>
|
1,989,532 | <p>The equation I am considering is </p>
<p>$$x^{3}-63x=162$$</p>
<p>Here is how far I got:</p>
<p>$$3st=-63$$</p>
<p>$$s=\frac{-63}{3t}=-21t$$</p>
<p>$$s^{3}t^{3}=-162$$</p>
<p>$$(-21t)^{3}-t^{3}=-162$$</p>
<p>$$-9261t^{3}-t^{3}=-162$$</p>
<p>Where would I go from here?</p>
| Dietrich Burde | 83,966 | <p>The roots are given by the factorization
$$
x^3-63x-162= (x + 6)(x + 3)(x - 9),
$$
and this is much easier than to use Cardano's formula for it (because it is not so obvious from the formulas that the roots are indeed integral).</p>
|
2,021,324 | <blockquote>
<p>Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$ with $p>1$, and $x,y,z$ positive </p>
</blockquote>
<p>By Jensen I got taking $p_1=p_2=p_3=\frac13$, (say $x=x_1,y=x_2,z=x_3$)</p>
<p>$\left(\sum p_kx_k\right)^p\le\sum p_kx^p$ which means;</p>
<p>$3^{1-p}(x... | Michael Rozenberg | 190,319 | <p>Indeed, Jensen works.</p>
<p>Let $x+y+z=3$.</p>
<p>Hence, we need to prove that $\sum\limits_{cyc}f(x)\geq0$, where $f(x)=\frac{x^p}{3-x}-\frac{1}{2}$.</p>
<p>$f''(x)=\frac{x^{p-2}\left((p-2)(p-1)x^2-6p(p-2)x+9p(p-1)\right)}{(3-x)^3}$.</p>
<p>If $p=2$ so $f''(x)>0$.</p>
<p>If $p>2$ so since $f''(0)>0$ ... |
969,871 | <p>First, by definition of odd functions I have $-f(x) = f (-x)$. It would follow that $|–f(x)|= |f(-x)|$. Then given $\epsilon > 0$ there exists $\delta > 0$ s.t. when $0 < |x| < \delta$, it follows that $|f(x)-M| < \epsilon$. With $M = 0$, then $|f(x)| < \epsilon$. (approaching $0$ from the left... | Community | -1 | <p>Since $f$ is odd then $f(0)=0$ and since $f$ is continuous then
$$\lim_{x\to0}f(x)=f(0)=0$$</p>
|
2,530,221 | <p>Let $X$ be a topological space, and equip its homeomorphism group $\text{Homeo}(X)$ with the compact-open topology. I probably need to restrict $X$ to be locally compact for it to play off with $\text{Homeo}(X)$ nicely.</p>
<p>The usual definition of $X$ being <b>homogeneous</b> is that given $x,y \in X$ there exis... | Alex Ravsky | 71,850 | <p>One of such natural classes are topological groups $G=X$ with $\theta(x,y)(t)=yx^{-1}(t)$ for each $t\in G$. Indeed, clearly that $\theta(x,y)(x)=y$ for each $x,y\in G$. To prove the continuity of the map $\theta: G\times G\to \operatorname{Homeo}(X)$ consider a subbase $\mathcal B$ of the space $\operatorname{Homeo... |
3,115,566 | <p>Convergence of the series for <span class="math-container">$a \in \mathbb R$</span> <span class="math-container">$$\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$$</span> </p>
<p>I saw this problem in a calculus book and it gave a hint that says </p>
<p><strong>HINT</strong> First show that <span class="mat... | Peter Foreman | 631,494 | <p>For small <span class="math-container">$x$</span> we have that <span class="math-container">$\sin{(x)} \approx x$</span>. One can prove this from the Taylor series expansion of <span class="math-container">$\sin{(x)}$</span>. As
<span class="math-container">$$\frac{\pi a^2}{\sqrt{n^2+a^2}+n} \approx \frac{\pi a^2}{... |
2,592,999 | <p>Prove that $$\lim_{x \to 2} x^3 = 8$$</p>
<p>My attempt, </p>
<p>Given $\epsilon>0$, $\exists \space \delta>0$ such that if $$|x^3-8|<\epsilon \space \text{if} \space 0<|x-2|<\delta$$ </p>
<p>$$|(x-2)(x^2+2x+4)|<\epsilon$$</p>
<p>I'm stuck here. Hope someone could continue the solution and exp... | Shashi | 349,501 | <p>$\newcommand{\Res}{\operatorname{Res}}\newcommand{\Arg}{\operatorname{Arg}}$I understand that in Complex Analysis class, if the problem is to find a definite integral, then very likely it is expected from you to use the Residue Theorem on that specific integral. </p>
<p>I provided you a <a href="https://math.stack... |
255,215 | <p>I am very new to Mathematica and already spent a lot of time trying to do this but failed.</p>
<p>I am trying to solve an ODE:</p>
<pre><code>solution = DSolve[{-((m (1 + m) + 4/(9 (-2/3 + t) t)) y[t]) +
2 (-1/3 + t) y'[t] + (-2/3 + t) t y''[
t] == (-4 (1 + C/2))/(9 (-2/3 + t) t), y[1] == 1, y'[1] == C}... | Cesareo | 62,129 | <p>We can have an insight about the solutions behavior for each <code>m</code> by doing</p>
<pre><code>tmax = 1000;
solution = ParametricNDSolve[{-((m (1 + m) + 4/(9 (-2/3 + t) t)) y[t]) + 2 (-1/3 + t) y'[t] + (-2/3 + t) t y''[t] == (-4 (1 + c/2))/(9 (-2/3 + t) t), y[1] == 1, y'[1] == c}, y, {t, 1, tmax}, {m, c}]
m = ... |
2,326,072 | <p>We define 3 sequences $(a_n),(b_n),(c_n)$ with positive terms so that
$$ a_{n+1}\leq\frac{b_n+c_n}{3}\ ,\ b_{n+1}\leq\dfrac{a_n+c_n}{3}\ ,\ c_{n+1}\leq\dfrac{a_n+b_n}{3} $$
Check if any of $(a_n),(b_n),(c_n)$ converge, and if they do find their limit.</p>
<p>PROOF</p>
<p>My part of the proof is this: By adding the... | CY Aries | 268,334 | <p>Since $\{x_n\}$ is decreasing and bounded from below, it is convergent. But this does not guarantee that the limit is zero. Indeed, since</p>
<p>$$0< x_{n+1}\le\frac{2}{3}x_n$$</p>
<p>we have</p>
<p>$$0< x_n\le\left(\frac{2}{3}\right)^{n-1}x_1$$</p>
<p>This implies that $x_n\to0$ as $n\to\infty$.</p>
<p>W... |
1,690,248 | <p>Hello I am hoping to find some direction on solving this try it yourself problem in my textbook.</p>
<p>Let S be an arbitrary set of symbols and let $\Phi = \{v_0 \equiv t | t \in T^S\} \cup \{\exists v_0 \exists v_1 \neg v_0 \equiv v_1\}$. </p>
<p><em>Note: $T^S$ is the set of all S-terms.</em></p>
<p>Show that ... | aventurin | 308,622 | <p>You can use a ring buffer of appropriate size. Then the average speed is the sum of the numbers in the ring buffer divided by the count of numbers currently in the buffer. The sum of the numbers at time $t+1$ is the sum at time $t$ plus the number added to the ring buffer minus the number that leaves the ring buffer... |
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | DenDenDo | 166,977 | <p>You already found that z has to be real, now just write out $|z| = \sqrt{Re(z)^2 + Im(z)^2}$<br>
Note that $i+1 = \sqrt{2} e^{i\frac{\pi}{4}} = \sqrt{2} \cos(\frac{\pi}{4}) + i\sqrt{2} \sin(\frac{\pi}{4})$<br>
Then you need to find when $a^2 =|a-(1+i)^n|^2 = a^2 + 2 \cdot 2^\frac{n}{2} a \cos(\frac{\pi}{4}n) + 2^n$<... |
2,029,279 | <blockquote>
<p>Let <span class="math-container">$T:\Bbb R^2\to \Bbb R^2$</span> be a linear transformation such that <span class="math-container">$T (1,1)=(9,2)$</span> and <span class="math-container">$T(2,-3)=(4,-1)$</span>.</p>
<p>A) Determine if the vectors <span class="math-container">$(1,1)$</span> and <span cla... | copper.hat | 27,978 | <p>If two vectors $x_1,x_2$ are linearly dependent, the either
$x_1 = \lambda x_2$ or $x_2=\lambda x_1$ for some $\lambda$, in other words
they lie on the same line.</p>
|
986,875 | <p>A service center consists of two servers, each working at an exponential rate
of two services per hour. If customers arrive at a Poisson rate of three per hour, then,
assuming a system capacity of at most three customers,
What fraction of potential customers enter the system?</p>
<p>I was thinking if I could thi... | cyberboy | 395,432 | <p>fraction of potential customers enter the system = 1 - πN<br>
here N is system capacity
= 1 - π3<br>
= 1-27/143<br>
= 0.811188</p>
<p>similar question with solution {see question 3}<br>
<a href="https://www2.isye.gatech.edu/~ashapiro/publications/MT2-solutions.pdf" rel="nofollow noreferrer">https://www2.isye.ga... |
3,305,861 | <p>This is an answer from Slader for <em>Understanding Analysis</em> by Abbott. The book suggests to use <span class="math-container">$a = a-b+b$</span> but the steps for this answer doesn't make any sense to me. For one, why does s/he write use <span class="math-container">$a=a-b+b$</span> when it looks like they used... | Alejandro Nasif Salum | 481,187 | <p>First of all, it is real because it converges and because <span class="math-container">$\arctan\left(\frac1{1+x^2}\right)$</span> is a real number for every <span class="math-container">$x\in \mathbb R$</span>.</p>
<p>On the other hand, that some number can be represented with an expression involving <span class="m... |
3,305,861 | <p>This is an answer from Slader for <em>Understanding Analysis</em> by Abbott. The book suggests to use <span class="math-container">$a = a-b+b$</span> but the steps for this answer doesn't make any sense to me. For one, why does s/he write use <span class="math-container">$a=a-b+b$</span> when it looks like they used... | Yuriy S | 269,624 | <p>A pretty integral, and I'm not sure I would be able to guess the primitive right away.</p>
<p>Still, substitution is always a good way to simplify things:</p>
<p><span class="math-container">$$I=\int_a^b \arctan\left(\dfrac{1}{1+x^2}\right)dx, \\ 0 \leq a<b$$</span></p>
<hr>
<p><span class="math-container">$$... |
3,990,717 | <p>I'm currently a self-learner on Abstract Algebra and Group theory. The book that I'm currently reading is <em>A First Course In Abstract Algebra</em> by <em>John B. Fraleigh</em>. So I know this question might truly be a basic question which every student would be taught in class and there is confusion about it. How... | Mike | 544,150 | <p>Write</p>
<p><span class="math-container">$$(ab)^n = \underbrace{ababab \ldots ab}_{n {\text{ times}}}$$</span></p>
<p>Then</p>
<p><span class="math-container">$$(ba)^n = \underbrace{bababab \ldots ba}_{n {\text{ times}}} = (\underbrace{bababab \ldots ba}_{n {\text{ times}}})bb^{-1} = b(\underbrace{ababab \ldots ab... |
3,243,440 | <p>Find the number k such that:</p>
<p><span class="math-container">$$det\begin{bmatrix}
3a_1 & 2a_1 + a_2 - a_3 & a_3\\\
3b_1 & 2b_1 + b_2 - b_3 & b_3\\\
3c_1 & 2c_1 + c_2 - c_3 & c_3\end{bmatrix}$$</span></p>
<p><span class="math-container">$$ = k \bullet det\begin{bmatrix}
a_1 & a_2 &... | Mohammad Riazi-Kermani | 514,496 | <p>Performing elementary column operations, <span class="math-container">$$det\begin{bmatrix}
3a_1 & 2a_1 + a_2 - a_3 & a_3\\\
3b_1 & 2b_1 + b_2 - b_3 & b_3\\\
3c_1 & 2c_1 + c_2 - c_3 & c_3\end{bmatrix}=$$</span></p>
<p><span class="math-container">$$det\begin{bmatrix}
3a_1 & 2a_1 + a_2 ... |
137,253 | <p>We have the following: $A,B,C$ are sets. </p>
<p>$C = \{ab: a \in A, b \in B\}$. </p>
<p>What is the relationship between $\sup(C),\sup(A)$, and $\sup(B)$?. </p>
<p>Is it: $$\sup(C) \le \sup(A) \sup(B)\;,$$ and why?.</p>
| Brian M. Scott | 12,042 | <p>If $A$ and $B$ are sets of non-negative real numbers, your conjecture is correct. Let $x=\sup A$ and $y=\sup B$. Then for any $c\in C$ there are $a\in A$ and $b\in B$ such that $c=ab$, and since $a,b\ge 0$, $ab\le xy$. That is, $c\le xy$ for every $c\in C$, so $\sup C\le xy=\sup A\sup B$.</p>
<p>But we can say more... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | LCarvalho | 37,895 | <p>Try this:</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -> 200];
fLine[p1_, p2_, case_] :=
If[TrueQ[case == v], {p1, {p1[[1]], p2[[2]]}}, {... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | Sumit | 8,070 | <p>Quick Draw (for an alternative way to do the job using some unknown MMA functionality)</p>
<pre><code>n=10;
pts= RandomReal[1, {n, 2}];
ListLinePlot[pts, InterpolationOrder -> 0,
Mesh -> Full, MeshStyle -> Purple, PlotStyle -> {Red, Dashed}, Frame -> True]
</code></pre>
<p><a href="https://i.stack... |
3,588,068 | <p>I have the following system of 3 equations and 3 unknowns:
<span class="math-container">$$c_{0} = \frac{x_0}{x_0 + x_1},\ \ c_{1} = \frac{x_1}{x_1 + x_2},\ \ \ c_{2} = \frac{x_2}{x_2 + x_0},$$</span>
where all <span class="math-container">$c_i\!\in\!(0,1)$</span> are known and all <span class="math-container">$x_i &... | Dietrich Burde | 83,966 | <p>We rewrite the equations as a system of linear equations <span class="math-container">$Ax=0$</span> with
<span class="math-container">$$
A=\begin{pmatrix} c_0-1 & c_0 & 0 \\ 0 & c_1-1 & c_1 \\
c_2 & 0 & c_2-1& \end{pmatrix}
$$</span>
and <span class="math-container">$x=(x_0,x_1,x_2)^t$</s... |
200,063 | <p>I am looking to evaluate</p>
<p>$$\int_0^1 x \sinh (x) \ \mathrm{dx}$$</p>
| user 1591719 | 32,016 | <p>If you use Taylor's series representation of the integrand and then interchange the sum and integral, you simply get</p>
<p>$$\sum_{k=0}^{\infty} \left(\frac{1}{(2k+2)!}-\frac{1}{(2k+3)!}\right) = \frac{1}{e}.$$ </p>
|
2,186,560 | <p>I was hoping someone can help me with the following.</p>
<p>I had to solve the following ODE and get the implicit form. </p>
<p>$$\frac{dy}{dx}=\frac{1+3y}{y} * x^2$$</p>
<p>this give by separating and integrating the following:
$1/9*(3y-ln(3y+1))=1/3*x^3 +c $
which can also be written as (c is an arbitrary cons... | Emilio Novati | 187,568 | <p>By definition of the <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow noreferrer">Lambert W</a> function: if $ze^z=A$ than $z=W(A)$.</p>
<p>so, if we have:
$$
ue^{-u}=e^{-3x^2+c}
$$
that is :
$$
-ue^{-u}=-e^{-3x^2+c}
$$
we have:
$$
-u=W\left(-e^{-3x^2+c} \right)\quad \iff \quad u=-W\left(-e... |
301,905 | <p>I'm having troubles to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$</p>
<p>I've tried a lot using so many techniques without success. I found $\int_{-1}^1 \frac{x^{2012}}{1+e^x}dx=1/2013$, but I couldn't solve from 0 to 1.</p>
<p>Thanks a lot.</p>
| mrf | 19,440 | <p>I doubt there is a simple expression for your integral.</p>
<p>One approach would be to write</p>
<p>$$\frac{1}{1+e^x} = \frac{1}{e^x(1+e^{-x})} = e^{-x} \sum_{k=0}^\infty (-1)^k e^{-kx} = \sum_{k=0}^\infty (-1)^k e^{-(k+1)x}$$</p>
<p>with uniform convergence on $x \ge c$ for every $c > 0$. Hence</p>
<p>\begi... |
3,523,491 | <p>Consider a <span class="math-container">$m \times n$</span> matrix <span class="math-container">$M$</span> such that each cell in <span class="math-container">$M$</span> is equal to the sum of its adjacent cells (sharing either an edge or a corner with this cell). What are the values in this matrix. </p>
<p>I am tr... | Jaroslaw Matlak | 389,592 | <p>For sure such matrix could be found for <span class="math-container">$m \in 2\mathbb{N}+1, n \in 3\mathbb{N}+2$</span>:</p>
<p><span class="math-container">$$\left[\begin{matrix}
a&a&0&-a&-a&0&a&a&0&\cdots & 0 & a &a & 0 &-a &-a\\
0&0&0&0&... |
2,520,669 | <p>How are these equal? </p>
<blockquote>
<p>$$|\sqrt{x} - \sqrt{c}| = \frac{|x-c|}{|\sqrt{x} + \sqrt{c}|},$$ </p>
</blockquote>
| Nosrati | 108,128 | <p>Simpler $$\frac{\cos(x+h)-\cos(x)}{(x+h)^{1/2} - x^{1/2}}\times\frac{(x+h)^{1/2} + x^{1/2}}{(x+h)^{1/2} + x^{1/2}}=\frac{\cos(x+h)-\cos(x)}{h}[(x+h)^{1/2} + x^{1/2}]$$</p>
|
2,520,669 | <p>How are these equal? </p>
<blockquote>
<p>$$|\sqrt{x} - \sqrt{c}| = \frac{|x-c|}{|\sqrt{x} + \sqrt{c}|},$$ </p>
</blockquote>
| lab bhattacharjee | 33,337 | <p>Hint : </p>
<p>$$\dfrac1{\sqrt{x+h}-\sqrt x}=\dfrac{x+h-x}{\sqrt{x+h}+\sqrt x}=?$$</p>
<p>and Use <a href="http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html" rel="nofollow noreferrer">Prosthaphaeresis Formulas</a> $$\cos(x+h)-\cos x=-2\sin\dfrac h2\sin\dfrac{2x+h}2$$</p>
<p>Finally, $\lim_{u\to0}\dfrac... |
549,014 | <p>If $y=\sin x$, then find the value of $$\frac{d^2(\cos^7 x)}{dy^2}$$ </p>
<p>I have no idea on how to proceed in this problem. Please help.</p>
| Claude Leibovici | 82,404 | <p>You can also start replacing x by ArcSin[y] and remember that Cos[ArcSin[z]]=Sqrt[1-z^2]. Can you continue with this ?</p>
|
2,861,406 | <p>I want to calculate the answer of the integral $$\int\frac{dx}{x\sqrt{x^2-1}}$$ I use the substitution $x=\cosh(t)$ ($t \ge 0$) which yields $dx=\sinh(t)\,dt$. By using the fact that $\cosh^2(t)-\sinh^2(t)=1$ we can write $x^2-1=\cosh^2(t)-1=\sinh^2(t)$. Since $t\ge 0$, $\sinh(t)\ge 0$, and we have $\sqrt{x^2-1}=\si... | Community | -1 | <p>Verify by differentiation.</p>
<p>$$(2\arctan(x+\sqrt{x^2-1}))'=2\frac{1+\dfrac x{\sqrt{x^2-1}}}{(x+\sqrt{x^2-1})^2+1}=2\frac{x+\sqrt{x^2-1}}{2(x^2+x\sqrt{x^2-1)}\sqrt{x^2-1}}$$</p>
<p>and</p>
<p>$$(\arctan\sqrt{x^2-1})'=\frac{\dfrac{x}{\sqrt{x^2-1}}}{(\sqrt{x^2-1})^2+1}.$$</p>
<p>Hence both answers are correct.... |
2,924,831 | <p>Suppose you have <span class="math-container">$100$</span> coins whose probabilities of obtaining the outcome "head" are <span class="math-container">$p_1,\ldots,\,p_{100}$</span>. These probabilities are not necessarily equal each other. Consider the following random experiment divided into two rounds.</p>
<ul>
<l... | Student1981 | 476,980 | <p>I found a solution to my problem. This solution builds on the paper </p>
<p>Nelsen, R. B. (1987). Discrete bivariate distributions with given marginals and correlation. <em>Communications in Statistics-Simulation and Computation</em>, 16(1), 199-208.</p>
<p>Since <span class="math-container">$\mathbb{P}(Y_1=y_1,\,... |
356,583 | <p>In the paper <a href="https://arxiv.org/abs/1811.02002" rel="nofollow noreferrer">"Finding Mixed Nash Equilibria of Generative Adversarial Networks"</a> the authors write in equation (1) on page 2:</p>
<blockquote>
<p>Consider the classical formulation of a two-player game with
<em>finitely</em> many strategies... | qwer1304 | 155,643 | <p>This <strong>non-standard</strong> formulation matches better the intended application - (Wasserstein) GAN. It can be mapped to the <strong>standard</strong> formulation <span class="math-container">$\langle \boldsymbol{q}, \tilde{A}\boldsymbol{p} \rangle$</span> through a new payoff matrix <span class="math-contain... |
2,013,459 | <p>The period of $\sin(x)$ is $2\pi$. Thus the period of $\sin(\pi x)$ will become $T_1=2$, similarly the period of $\sin(2\pi x)$ is $T_2=1$ and for $\sin(5\pi x)$, the period is $T_3=\frac{2}{5}$.</p>
<p>To find the period of $\sin(\pi x)-\sin(2\pi x)+\sin(5\pi x)$, we can have
$$\frac{T_1}{T_{2}}=\frac{2}{1}\Righta... | AsianDuckKing | 381,322 | <p>I think you are right, since <span class="math-container">$2$</span> divided by <span class="math-container">$\frac{1}{3}$</span> is an integer, as well as the <span class="math-container">$1$</span>. You can have a check by plotting</p>
<p><a href="https://i.stack.imgur.com/o3OOA.png" rel="nofollow noreferrer">The ... |
2,060,418 | <p>We have</p>
<p><span class="math-container">$$ \begin{cases} \dot{x} = - y - y^3 \\ \dot{y} = x \end{cases} $$</span></p>
<p>where <span class="math-container">$x,y \in \mathbb{R}$</span>. Show that the critical point for the linear system is a <span class="math-container">$\mathbf{center}$</span>. Prove that the ty... | MrYouMath | 262,304 | <p>You can use the following Lyapunov function candidate</p>
<p><span class="math-container">$$V(x,y)=1/2x^2+1/2y^2+1/4y^4,$$</span></p>
<p>which is positive definite and <span class="math-container">$V(x=0,y=0)=0$</span>.</p>
<p>The derivative is given by</p>
<p><span class="math-container">$$\dot{V}=x\dot{x}+y\do... |
116,417 | <p>To be honest, I don't really know, whether or not the following is a research level
question:</p>
<p>Let $M$ be a smooth manifold, $C^\infty(M)$ the smooth function ring on $M$ and
suppose $R\subset C^\infty(M)$ is a subring. What are conditions, such that
$R$ is the smooth function ring of a smooth manifold ?</p>... | David C | 27,816 | <p>You have to be careful: let $Y\subset V$ be a submanifold of $V$, you have a restriction map $$C^{\infty}(V)\rightarrow C^{\infty}(Y)$$
whose kernel is an ideal $p_Y$, and if $Y$ is a closed submanifold:
$$C^{\infty}(Y)\cong C^{\infty}(V)/p_Y$$ </p>
<p>Thus the question is rather what ideals of $C^{\infty}(V)$ are ... |
4,011,999 | <p>Let <span class="math-container">$n$</span> be a positive integer, and consider a sequence <span class="math-container">$a_1 , a_2 , \cdots , a_n $</span> of positive integers. Extend it periodically to an infinite sequence <span class="math-container">$a_1 , a_2 , \cdots $</span> by defining <span class="math-conta... | Calvin Lin | 54,563 | <p>This is a great question to showcase how you can interpret that statement. I suggest that you try to make sense of the condition, and understand how it restricts the values.</p>
<p>The main observation that you need is: If <span class="math-container">$a_i = k$</span>, then <span class="math-container">$a_k \leq n+i... |
3,243,906 | <p>I am doing a problem where I am differentiating from first principles, but I can't simplify the final expression:</p>
<p><span class="math-container">$\frac{-2xh - h^2}{x^4h + 2x^3h^2+x^2h^3}$</span></p>
<p>Could someone explain it in steps?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$\frac{-2x-h}{x^2(x^2+2xh+h^2)}=\frac{-2x-h}{x^2(x+h)^2}$$</span>
it must be <span class="math-container">$$x\ne 0,x\ne -h$$</span>
If <span class="math-container">$$h$$</span> tends to zero we obtain <span class="math-container">$$\frac{-2x}{x^2(x^2)}=\frac{-2x}{x^4}=\frac{-2}{x^... |
511,921 | <p>Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of <strong>similar</strong>: "We say that operators $S,T \in \mathcal{L}(V)$ are <strong>similar</strong> if there exists an isomorphism (in this context of linear maps, a bijecti... | Giuseppe Negro | 8,157 | <p>Write your linear map as an equation: $$y=Lx.$$
Now apply the following:
$$\tag{1}x=Px',\quad y=Py'.$$
Since $P$ is invertible, you get
$$y'=P^{-1}LPx'.$$
In a finite dimensional space, $(1)$ is what you call a <em>change of coordinates</em>.</p>
|
749,731 | <p>if $G' <H < G$ then $H$ is normal in $G$. ($G'$ is the commutator subgroup of $G$.)</p>
<p>This is what I do:</p>
<p>because $G' < H$ we have $\frac{H}{G'} \triangleleft \frac{G}{G'}$. because $\frac{G}{G'}$ is abelian then $\frac{\frac{G}{G'}}{\frac{H}{G'}} \approx \frac{G}{H} $ is abelian and it means ... | egreg | 62,967 | <p>The homomorphism theorems imply that when $N$ is a normal subgroup of $G$, there is a bijection between the subgroups of $G$ containing $N$ and the subgroups of $G/N$ given by
$$
H\mapsto H/N
$$
In this correspondence, normal subgroups correspond to normal subgroups.</p>
<p>When $N=G'$, the quotient $G/G'$ is abeli... |
62,621 | <p>There are three elements: x, y, z and a relation C:</p>
<p>x C y, y C z, z C x, x C x, y C y, z C z.</p>
<p>Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e. </p>
<p>x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z </p>
... | Alex 'qubeat' | 14,175 | <p>Just few hours ago I found, that the construction was used in talks of J. B. Nation
“How aliens do math”, and “Logic on other planets” (<a href="http://www.math.hawaii.edu/~jb/talks.html" rel="nofollow">here</a>).
Despite of such titles, the works look quite instructive.</p>
|
2,494,552 | <p>Give an example of metric spaces $X,Y,Z$ and a function $f:X\times Y \to Z$ such that :</p>
<p>(a)if $a\in X$, then $f(a,y)$ is continuous</p>
<p>(b)if $b\in Y$, then $f(x,b)$ is continuous</p>
<p>(c)$f:X\times Y \to Z$ is not continuous</p>
<p>Actually I think $f$ must be continuous if it satisfies (a) and (b) ... | user284331 | 284,331 | <p>I think the following works:</p>
<p>$f(z)=\exp\left(-\dfrac{1}{z^{4}}\right)$ for $z\ne 0$ and $f(0)=0$. Here we are dealing with complex plane.</p>
<p>$f$ is not continuous at $z=0$ because $f(h+ih)=\exp\left(\dfrac{1}{4h^{4}}\right)$ for real $h\ne 0$. </p>
<p>For the separate continuous, we have to consider on... |
2,494,552 | <p>Give an example of metric spaces $X,Y,Z$ and a function $f:X\times Y \to Z$ such that :</p>
<p>(a)if $a\in X$, then $f(a,y)$ is continuous</p>
<p>(b)if $b\in Y$, then $f(x,b)$ is continuous</p>
<p>(c)$f:X\times Y \to Z$ is not continuous</p>
<p>Actually I think $f$ must be continuous if it satisfies (a) and (b) ... | Math1000 | 38,584 | <p>Let $f:\mathbb R^2\to\mathbb R$ be defined by
$$
f(x,y) = \begin{cases}
\frac{xy}{x^2+y^2},& (x,y)\ne(0,0)\\
0,& (x,y)=(0,0).
\end{cases}
$$
Clearly $f$ is continuous on $\mathbb R^2\setminus\{(0,0)\}$. Now, for each $y\ne 0$ we have $f(0,y)=0$, so $\lim_{y\to 0}f(0,y)=0$, so the map $y\to f(0,y)$ is continu... |
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