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 |
|---|---|---|---|---|
2,820,796 | <p>In How many ways can a 25 Identical books can be placed in 5 identical boxes. </p>
<p>I know the process by counting but that is too lengthy .
I want different approach by which I can easily calculate required number in Exam hall in few minutes. </p>
<p>Process of Counting :
This problem can be taken partitions of... | Sri-Amirthan Theivendran | 302,692 | <p>In general, you can use the following recursive formula. Let $P_k(n)$ be the set of partitions of $n$ with exactly $k$ parts. Let $p_{k}(n)$ be the number of partitions of $n$ into $k$ parts i.e. $p_{k}(n)=|P_k(n)|$. Then
$$
p_{k}(n)=p_{k-1}(n-1)+p_{k}(n-k)
$$
by classifying $\lambda=(\lambda_1,\dots,\lambda_k)\in ... |
11,266 | <p>I have a list of time durations, which are strings of the form: <code>"hh:mm:ss"</code>. Here's a sample for you to play with:</p>
<pre><code>durations = {"00:09:54", "00:31:24", "00:40:07", "00:11:58", "00:13:51", "01:02:32"}
</code></pre>
<p>I want to convert all of these into numbers in seconds, so that I can a... | Mr.Wizard | 121 | <p>You didn't specify that performance is important but I decided to examine that aspect.</p>
<p>First I generate a list of 10,000 durations:</p>
<pre><code>durations =
ToString@Row[#, ":"] & /@
Map[IntegerString[#, 10, 2] &,
RandomInteger[{0, 24}, {10000, 3}], {2}];
</code></pre>
<p>I then time ... |
1,136,192 | <p>I need to solve this integral but I have no idea about how to procede, this is the integral:</p>
<p>$$\int \frac{x-1}{x+4x^3}\mathrm dx$$</p>
<p>This is how I solve the first part:</p>
<p>$$\int \frac{x}{x+4x^3}\mathrm dx - \int \frac{1}{x+4x^3}\mathrm dx$$</p>
<p>$$\int \frac{1}{1+4x^2}\mathrm dx - \int \frac{1... | Thomas | 103,537 | <p>Not sure if the question is about $1^n$ or $(-1)^n$ but in the latter case :</p>
<p>Hint: you have to group terms by two : $\sum \frac{1}{\ln(2n)} - \frac{1}{\ln(2n+1)}$.</p>
<p>Then use the $O()$ notation and express $\ln(2n+1)$ in terms of $\ln(2n)$ and $O(\frac{1}{n})$</p>
<p>This should lead you to $\sum O(\f... |
1,136,192 | <p>I need to solve this integral but I have no idea about how to procede, this is the integral:</p>
<p>$$\int \frac{x-1}{x+4x^3}\mathrm dx$$</p>
<p>This is how I solve the first part:</p>
<p>$$\int \frac{x}{x+4x^3}\mathrm dx - \int \frac{1}{x+4x^3}\mathrm dx$$</p>
<p>$$\int \frac{1}{1+4x^2}\mathrm dx - \int \frac{1... | Marco Cantarini | 171,547 | <p>First of all the series must start from $n=1$. If you have $(-1)^{n}$ at numerator the series converges for the alternating series test <a href="http://en.wikipedia.org/wiki/Alternating_series_test" rel="nofollow">http://en.wikipedia.org/wiki/Alternating_series_test</a>, and its sum is $-0.92429...$ If you have $1^{... |
3,864,729 | <p>I am studying for an exam and I am almost grasping compactness. However, some examples are still unclear. E.g. for <span class="math-container">$A = (a, b)$</span> with <span class="math-container">$a < b$</span> and <span class="math-container">$a, b \in \mathbb{R}$</span>.
How can <span class="math-container">... | fleablood | 280,126 | <p>"how is this set still covering (a,b) when b−1k is smaller than b"</p>
<p>Because <span class="math-container">$b$</span> is <em>not</em> in <span class="math-container">$(a,b)$</span>. so it never needs to be covered.</p>
<p><em>Every</em> point in <span class="math-container">$(a,b)$</span> is <em>also</... |
1,344,464 | <p>Consider <span class="math-container">$$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$</span></p>
<p>It's a power series with a radius, <span class="math-container">$R=1$</span>. at <span class="math-container">$x=1$</span> it converges. Hence, by Abel's thorem:</p>
<p><span class="math-container">$$\lim_... | Loreno Heer | 92,018 | <p>Note the following corollary:</p>
<blockquote>
<p>$$\int_a^b f(t)\, dt = F(b)-F(a)$$</p>
</blockquote>
<p>You are free to choose the points $a$ and $b$ as long the function $f$ is defined in $[a,b]$. But choosing $a$ such that $F(a)=0$ simplifies things.</p>
|
1,435,269 | <p>Let a sequence $x_n$ be defined inductively by $x_{n+1}=F(x_n)$. Suppose that $x_n\to x$ as $n\to \infty$ and $F'(x)=0$. Show that $x_{n+2}-x_{n+1}=o(x_{n+1}-x_n)$.</p>
<p>I'm not sure how to do this. Any solutions are greatly appreciated. I think The Mean-Value Theorem will be useful and we can assume that $F$ is ... | Paramanand Singh | 72,031 | <p>This is a direct consequence of the mean value theorem and the assumption that $F$ is continuously differentiable (without this assumption the result may not hold).</p>
<p>We have via mean value theorem $$x_{n+2}-x_{n+1}=F(x_{n+1})-F(x_{n})=(x_{n+1}-x_{n})F'(c)$$ for some $c$ between $x_n$ and $x_{n+1}$. As $n\to\i... |
588,802 | <p>The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$</p>
<p>The first thing I did was use the divergence test which didn't help since the result of the limit was 0.</p>
<p>If I multiply it through, the result is $\sum_{n=1}^{\infty} \frac{1}{n^2+3n}$</p>
<p>I'm wondering if I can consider this as a p-series and... | Hagen von Eitzen | 39,174 | <p>Note that $\frac1{n(n+3)}=\frac13\left(\frac1n-\frac1{n+3}\right)$ so this is a telescoping sum
$$\sum_{n=1}^m \frac1{n(n+3)}=\frac13\left(1+\frac 12+\frac13-\frac1{m+1}-\frac1{m+2}-\frac1{m+3}\right)\to \frac{11}{18}.$$</p>
|
249,908 | <p>This seems like a very inefficient way of doing what I want. I generate all permutations of (for example) <code>{1, 1, 2, 2}</code>, then for each permutation I generate its rotations, select the first one in <code>Sort</code> order, and use that for <code>GatherBy</code>.</p>
<pre><code>list = {1, 1, 2, 2};
display... | kglr | 125 | <pre><code>GroupOrbits[CyclicGroup[Length @ #], Permutations @ #, Permute] & @ list
</code></pre>
<blockquote>
<pre><code>{{{1, 1, 2, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 2, 1, 1}},
{{1, 2, 1, 2}, {2, 1, 2, 1}}}
</code></pre>
</blockquote>
|
1,865,062 | <p>I have seen some examples of inner automorphisms of Lie algebras. Can anyone please give me an example of an automorphism of Lie algebras that is not inner (with proof).
Note - An automorphism is said to be inner if it is of the form $exp(adx)$ for $adx$ nilpotent where $adx(y)$=$[x,y]$.
Thanks for any help.</p>
| Hanno | 81,567 | <p>The simplest example I know is ${\mathfrak s}{\mathfrak l}_n({\mathbb C})$ for $n\geq 3$, which has the non-inner automorphism $A\mapsto -A^t$. This is not inner because is doesn't preserve conjugacy classes of matrices (look at the eigenvalues) which however any inner automorphism $\text{exp}(\text{ad}(X))=\text{co... |
392,442 | <p>What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or disprove) the primality of every N-th number. I know this is a very large and subjective answer, however, I would li... | Community | -1 | <p>As others have mentioned there are many formulas for primes. </p>
<p>I can't pass up the opportunity to mention my favorite:</p>
<p>$$p_n=1+\sum^{2^n}_{m=1}\left\lfloor \sqrt[n]n \left( \sum^{m}_{x=1}\left\lfloor \cos^2\left( \pi \frac{(x-1)!+1}{x}\right) \right\rfloor \right)^{-1/n} \right\rfloor$$</p>
<p>Maybe ... |
2,802,959 | <p>If I write
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?</p>
<p>Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$? </p>
<p>Q2:</p>
<p>If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?</p>
| user061703 | 515,578 | <p>$x\in[0;1]$ means $0\le x\le 1$, any real number $x$ that satisfies this is true. For example:</p>
<p>$x^2-x\le 0\Leftrightarrow x(x-1)\le 0\Leftrightarrow x-1\le 0\le x\Leftrightarrow 0\le x\le 1$</p>
<p>$x\in\{0;1\}$ means $x=0$ or $x=1$ is true. For example:</p>
<p>$x^2-x=0\Leftrightarrow x(x-1)=0 \Leftrightar... |
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Arturo Magidin | 3,959 | <p>The <a href="http://en.wikipedia.org/wiki/Abc_conjecture%20%22ABC%20conjecture%22">ABC conjecture</a> and Vojta's conjectures come to mind.</p>
|
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Roy Maclean | 3,537 | <p>Clicking on Toolbox-What Link's Here in the wikipedia article <a href="http://en.wikipedia.org/wiki/Conditional_result">Conditional proof</a> brings up <a href="http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's hypothesis H</a> which the article says is used to prove conditional results in diophanti... |
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Gil Kalai | 1,532 | <p>In computational complexity there are several conjectures which are stronger than $NP \ne P$ which have important consequences. To mention a few</p>
<p>1) The conjecture that factoring is computationally hard is the basis to much theoretical and practical cryptography. </p>
<p>2) More broadly the conjecture that ... |
889,719 | <p>Example $5.9$ on page $103$ of John Lee's Smooth Manifolds says the following:</p>
<p>The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function
$$
x^i=f(x^1,\dots,x^{i-1},x^{i+1},\dots,x^{n+1})
$$
where $f\colon B^n\to\mathbb{R}$ is $f(u)=\sqrt{1-|u|^2}$. The intersection o... | Kelvin Lois | 322,139 | <p>I'm also confused by that. But after read Prof. Lee and koch comments i realized that maybe what it means is this :</p>
<p>$\mathbb{S}^n$ covered by the graphs of smooth function. The graph of smooth function is embedded submanifold. But an embedded submanifold satisfy local k-slice condition. So each graph that co... |
1,032,714 | <p>'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that $k_0 = i, k_n = j$ and $$U_{k_0} \cap U_{k_1} \neq \emptyset, U_{k_1} \cap U_{k_2} \neq \emptyset, ..., U_{k_{n-1}} \ca... | Community | -1 | <p>The first one is a special case of the second definition.
As far as I can see, the second definition <strong>does not</strong> follows from the first.</p>
<p>It is highly customary to accept the second as the definition.</p>
|
1,269,738 | <p>I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with. </p>
<p>Some examples that come to min... | zoli | 203,663 | <p>I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level.</p>
<p><a href="http://www.amazon.co.uk/Mathematical-Experience-Penguin-Press-Science/dp/0140134743/ref=sr_1_1?ie=UTF8&qid=1430917805&sr=8-1&keywords=Reuben+Hersh">Ph. Dav... |
25,917 | <p>$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ </p>
<p>$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$</p>
<p>Why they are different?</p>
| Kerry | 7,887 | <p>Proof:</p>
<p>By the fundamental theorem of algebra, every integer $N$ can be uniquely factored as $\prod^{n}_{i=1}p_{i}^{a_{i}}$. Now, mark $2=p_{1}$, note $a_{i}$ can take value of $0$. You got the theorem. </p>
<p>For the "inductive" proof, suppose for $n<k$ this is true. For $n+1$ its factors must be in pre... |
2,223,163 | <p>I don't have any idea on how to prove it, and I need it for one of my questions which is still unanswered: <a href="https://math.stackexchange.com/questions/2192947/what-is-the-largest-number-smaller-than-100-such-that-the-sum-of-its-divisors-is?noredirect=1#comment4521040_2192947">What is the largest number smaller... | Misha Lavrov | 383,078 | <p>It's always even, and it has digits $999\dots996$ so it's always divisible by $3$: it's $3$ times $333\dots332$.</p>
|
3,281,540 | <p>I wrote an algorithm by combining Fermat's Little Theorem and Euler's Method. However, I am experiencing a problem in Euler's method. </p>
<p>For instance, If I take <span class="math-container">$(A, B, M)$</span> such that <span class="math-container">$A^B mod(M)$</span>.</p>
<p>When the initial values are <span ... | Culver Kwan | 686,157 | <p>First, using modular arithmetic, we can reduce <span class="math-container">$12341$</span> to <span class="math-container">$12341-12313=28$</span>.</p>
<p>As <span class="math-container">$12313=7\times1759$</span>, we can first find the remainder of <span class="math-container">$28^{7113}$</span> divided by <span c... |
3,620,767 | <p><a href="https://imgur.com/a/i24lMmS" rel="nofollow noreferrer">https://imgur.com/a/i24lMmS</a></p>
<p>I tried solving this problem, but couldn't find an answer. Any suggestions? Thanks!</p>
| Quanto | 686,284 | <p>Let <span class="math-container">$x$</span> and <span class="math-container">$y$</span> be the side segments of the largest square segmented by the middle-square vertexes. Then,</p>
<p><span class="math-container">$$ x^2+y^2=37^2,\>\>\>\>\>16(x+y) =xy$$</span></p>
<p>where the second equation matche... |
2,435,505 | <p>It is a question from permutations and combination chapter and its ans is 48 as given in book! Please help me to do this. I am unable to figure out the solution! Please help!</p>
| Ross Millikan | 1,827 | <p>The answer is badly wrong. If you have to use all the digits you are asking for the number of permutations of your digits with the last one being even. You have $3$ choices for the units digit, then $5$ for the tens digit, $4$ for the hundreds and so on. You get $\frac {6!}2=360$ choices. If you don't have to us... |
723,707 | <p>I'm trying to understand what the relation is between the direct product and the quotient group. </p>
<p>If we let $H$ be a normal subgroup of a group $G$, then it is not too difficult to show that the set of all cosets of $H$ in $G$ forms a quotient group $G/H$:
\begin{equation}
G/H = \{ g H \mid g \in G \}
\end{e... | Manolito Pérez | 13,293 | <p>This statement is always true, since $H$ is a normal subgroup of $G \times H$ (i.e. $xH = Hx$ for all $x \in K$). To see why, consider two cases:</p>
<ol>
<li><p>The case $x \in H$</p></li>
<li><p>The case $x \notin H$</p></li>
</ol>
<p>Then, the statement follows from the fundamental theorem of homomorphisms. By... |
3,281,503 | <blockquote>
<p>For natural numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, what is the greatest value of <span class="math-container">$b$</span> so that <span class="math-container">$a^b$</span> has <span class="math-container">$b$</span> digits?</p>
</blockquote>
<p>I k... | Hagen von Eitzen | 39,174 | <p>A natural number <span class="math-container">$n$</span> has <span class="math-container">$b$</span> digits iff <span class="math-container">$10^{b-1}\le n<10^b$</span>. So for this to hold for <span class="math-container">$a^b$</span>, we clearly need <span class="math-container">$a<10$</span>, or: <span clas... |
813,715 | <p>Say I am asked to find, in expanded form without brackets, the equation of a circle with radius 6 and centre 2,3 - how would I go on about doing this?</p>
<p>I know the equation of a circle is $x^2 + y^2 = r^2$, but what do i do with this information?</p>
| Tunk-Fey | 123,277 | <p>The general solution for the circle with centre $(a,b)$ and radius $r$ is
$$
(x-a)^2+(y-b)^2=r^2.
$$
Now, we have the centre $(2,3)$ and the radius $6$, therefore the equation of the circle is
\begin{align}
(x-2)^2+(y-3)^2&=6^2\\
x^2-4x+4+y^2-6x+9&=36\\
x^2+y^2-4x-6y+4+9-36&=0\\
\large\color{blue}{x^2+y^... |
2,704,102 | <p>Let $X,Y,Z$ be topological spaces. Is the following statement true?
$X \times Z \cong Y \times Z \implies X \cong Y$?
how would you prove it? </p>
<p>and I know that if $A \cong B$, and $a \in A$ that there is a $b \in B$, such that $A\setminus{\{a\}} \cong B\setminus{\{b\}}.$ How would you prove the same for remov... | amir bahadory | 204,172 | <p>Let <span class="math-container">$ X=\mathbb{R^+} \cup \{0\} $</span> then <span class="math-container">$ \mathbb{R} \times X \cong X\times X $</span> with homeomorphism <span class="math-container">$(x,y) \to (x^2 -y^2,2xy)$</span> but <span class="math-container">$ \mathbb{R^+} \cup \{0\} \ncong \mathbb{R} $</span... |
4,398,873 | <p>Given f: (0,1) <span class="math-container">$\rightarrow$</span> <span class="math-container">$\mathbb R$</span>. f is a continuous function and improper integrable.</p>
<p>If <span class="math-container">$\int_{0}^{x}f(t)dt = 0$</span> <span class="math-container">$\forall x \in [0,1] $</span>,</p>
<p>Does <span cl... | Sarah | 1,018,684 | <p>As pointed out in the comments, using the fundamental theorem of calculus we have that <span class="math-container">$F(x)-F(0)=0$</span> where <span class="math-container">$F'(x)=f(x)$</span>. Hence <span class="math-container">$F(x)=F(0)$</span> for all <span class="math-container">$x$</span>, which means <span cla... |
4,398,873 | <p>Given f: (0,1) <span class="math-container">$\rightarrow$</span> <span class="math-container">$\mathbb R$</span>. f is a continuous function and improper integrable.</p>
<p>If <span class="math-container">$\int_{0}^{x}f(t)dt = 0$</span> <span class="math-container">$\forall x \in [0,1] $</span>,</p>
<p>Does <span cl... | B. S. Thomson | 281,004 | <p><strong>Can we appeal to the Fundamental Theorem of the Calculus?</strong></p>
<p>Certainly ... if you can find one in your textbook for <em>improper integrals</em>. Many textbooks make no such statement, assuming that you can use the regular version as needed.</p>
<p>If you cannot find one, then prove a simple one... |
587,878 | <p>Let $X$ be a reflexive Banach space of infinite dimension. </p>
<p>a) Prove that there exists a sequence $x_n$ such that $\| x_n \|=1$ and $x_n$ converges weakly to $0$.</p>
<p>b) Let $x_n$ be a sequence such that $\forall f \in X' \quad \exists \lim\limits_{n\to\infty} f(x_n)<\infty$ .Prove that $x_n$ converge... | gtoques | 638,316 | <p>What you said is not true in general, but there's a way around it. Find a closed subspace <span class="math-container">$S \subset X$</span> which is separable (and it is reflexive also by a common result). To do this, pick <span class="math-container">$\{x_{n}\}_{n \in \mathbb{N}}$</span> linearly independent vector... |
276,948 | <p>$a_n$ be a sequence of integers such that such that infinitely many terms are non zero, we need to show that either the power series $\sum a_n x^n$ converges for all $x$ or Radius of convergence is at most $1$. need some hint. thank you.</p>
| Davide Giraudo | 9,849 | <blockquote>
<p><strong>Claim:</strong> let <span class="math-container">$\{a_n\}$</span> a sequence of integers. The radius of convergence of <span class="math-container">$\sum_{n=1}^{+\infty}a_nx^n$</span> is</p>
<ul>
<li>infinite if <span class="math-container">$a_n=0$</span> when <span class="math-container">$n$</s... |
2,475,938 | <blockquote>
<p>How can I factor the polynomial <span class="math-container">$x^4-2x^3+x^2-1$</span>?</p>
</blockquote>
<p>This is an exercise in algebra. I have the solution showing that
<span class="math-container">$$
x^4-2x^3+x^2-1=(x^2-x-1)(x^2-x+1).
$$</span></p>
<p>But the solution does not show any details.... | Mr Pie | 477,343 | <p>Remember that for all values <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, <span class="math-container">$$(a + b)^2 = (a + b)(a + b)$$</span> which we then <a href="http://www.mathwords.com/f/foil_method.htm" rel="nofollow noreferrer"><em>FOIL</em></a> to work out that <span cl... |
4,065,797 | <p>Just to give a simple numerical example but in general the variables <span class="math-container">$x,y,z,u,v$</span> are not equal.</p>
<p><span class="math-container">$113= 2*4^2 + 2*4^2 +2*4^2 + 4^2 +1^2$</span></p>
<p>I am looking for a general method to solve this type of equation or a piece of software to do th... | giobrach | 332,594 | <p>As was hinted at in @Quanto's answer, we have the Wallis integral
<span class="math-container">$$W(\alpha) := \int_0^{\pi/2} \sin^\alpha(x)\ dx = \frac{\sqrt \pi}{2} \frac{\Gamma\left(\frac{\alpha+1}{2}\right)}{\Gamma\left(\frac \alpha 2 + 1\right)}, $$</span>
for all <span class="math-container">$\alpha \geqslant 0... |
849,583 | <p>I have several derivatives to find:</p>
<blockquote>
<p>For <span class="math-container">$g(s)=3s^3-s+4$</span>, <span class="math-container">$g'(s)=$</span></p>
<p>For <span class="math-container">$p(t)=\frac1t+t^2$</span>, <span class="math-container">$\frac{\mathrm dp}{\mathrm dt}=$</span></p>
<p>For <span class=... | Hakim | 85,969 | <p><strong>Hint:</strong> <br> Derivatives are linear, i.e. $$\dfrac{\mathrm d}{\mathrm dx}\Big[f(x)+g(x)\Big]=\dfrac{\mathrm d}{\mathrm dx}f(x)+\dfrac{\mathrm d}{\mathrm dx}g(x).$$
As an example: let $h(x)=x^2-x$, we then have: $$h'(x)=\dfrac{\mathrm d}{\mathrm dx}\Big[x^2-x\Big]=\dfrac{\mathrm d}{\mathrm dx}x^2+\dfra... |
1,687,868 | <p>This might be a really obvious question so I apologize in advance, but I'm having trouble seeing when matrices are commutative for general nxn matrices. For example, when proving tr(AB)=tr(BA), I can easily prove this in a 2x2 matrix but I'm getting confused for proving it in a nxn matrix.</p>
<p>I've searched onlin... | Salcio | 821,280 | <p>Of course one can prove this directly as above.
Another proof is to assume first that <span class="math-container">$A$</span>, and <span class="math-container">$B$</span> are invertable (which is fine since in general case one can approximate any matrix by an invertable one) and check that <span class="math-containe... |
3,377,353 | <p>Given a purely real, rational integer <span class="math-container">$p$</span> that is prime in <span class="math-container">$\mathbb{Z}$</span>, we know very well that it ramifies in <span class="math-container">$\mathbb{Q}(\sqrt{pm})$</span> (where <span class="math-container">$m$</span> is a nonzero integer coprim... | Bob Happ | 215,060 | <p>The terminology of "ramify" and "split" carries over, but now needs the adjectives "completely" and [placeholder for others to add appropriate adjectives].</p>
<p>The ring of algebraic integers of <span class="math-container">$\mathbb{Q}(\sqrt{2} + \sqrt{3})$</span> provides richer examples, in my opinion. We see t... |
3,225,553 | <p>Show that <span class="math-container">$4x^2+6x+3$</span> is a unit in <span class="math-container">$\mathbb{Z}_8[x]$</span>.</p>
<p>Once you have found the inverse like <a href="https://math.stackexchange.com/questions/3172556/show-that-4x26x3-is-a-unit-in-mathbbz-8x">here</a>, the verification is trivial. But how... | Cornman | 439,383 | <p>To find an inverse polynomial for that holds <span class="math-container">$p(x)(4x^2+6x+3)=1$</span> so it has to be <span class="math-container">$3y=1\mod 8$</span> [edit: For more context on <span class="math-container">$y$</span>, see the comments below]. So <span class="math-container">$y=3$</span> and the polyn... |
3,225,553 | <p>Show that <span class="math-container">$4x^2+6x+3$</span> is a unit in <span class="math-container">$\mathbb{Z}_8[x]$</span>.</p>
<p>Once you have found the inverse like <a href="https://math.stackexchange.com/questions/3172556/show-that-4x26x3-is-a-unit-in-mathbbz-8x">here</a>, the verification is trivial. But how... | lhf | 589 | <p>Write
<span class="math-container">$$
4x^2+6x+3 = 3(4x^2+2x+1) = 3((2x)^2+(2x)+1) = 3 \frac{(2x)^3-1}{2x-1} = \frac{-3}{2x-1}
$$</span>
Therefore,
<span class="math-container">$$
\frac{1}{4x^2+6x+3} = \frac{2x-1}{-3} = 3(1-2x) = 3-6x = 2x+3
$$</span></p>
|
3,577,249 | <p>Let be <span class="math-container">$X$</span> a topological space and let be <span class="math-container">$Y\subseteq X$</span> such that <span class="math-container">$\mathscr{der}(Y)=\varnothing$</span>: so for any neighborhood <span class="math-container">$I_y$</span> of <span class="math-container">$y\in Y $</s... | Soma | 91,108 | <p>It is not possible. Let us consider <span class="math-container">$\mathbb{N}$</span> endowed with the discrete topology and let <span class="math-container">$X=\mathbb{N}\cup \{\infty\}$</span> as the one point compactification of such a space (<span class="math-container">$X$</span> is homeomorphic to a convergent ... |
2,378,508 | <p>I am reading about Arithmetic mean and Harmonic mean. From <a href="https://en.wikipedia.org/wiki/Harmonic_mean#In_physics" rel="nofollow noreferrer">wikipedia</a>
I got this comparision about them:</p>
<blockquote>
<p>In certain situations, especially many situations involving rates and ratios, the harmonic mea... | Vaas | 291,134 | <blockquote>
<p>Polynomials: These are functions which can be written in the form <span class="math-container">$f(x) = a_0 +a_1x+···+a_dx^d$</span> for some numbers <span class="math-container">$a_j, 0 ≤ j ≤ d, d ∈ N$</span>. If <span class="math-container">$a_d \neq 0$</span>, we say that f is a polynomial of degree d... |
3,536,135 | <blockquote>
<p>There are five multiple choice questions on a test, with four choices per question. A student was given 10 questions to study for the test and the teacher picked 5 out of 10 questions to put on the test. The student memorizes 7 of the 10 answers of the questions given. If the student encounters the th... | kccu | 255,727 | <p>Your answer to the first question is correct, but your notation is poor. You say <span class="math-container">$X1$</span> denotes question one, so what does it mean to write <span class="math-container">$X1=1$</span>? Instead I would define <span class="math-container">$B$</span> to be the event that question 1 is o... |
75,005 | <p>Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.</p>
<p>Then how do you check if his series is <... | cyborg | 17,917 | <p>Leaving aside the theoretical aspect of your question, there are also pragmatic answers to it because there are real world uses for high-quality <a href="http://en.wikipedia.org/wiki/Random_number_generation" rel="nofollow">random generators</a> (whether hardware or algorithmic). For statistical uses, <a href="http:... |
281,735 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/202452/why-is-predicate-all-as-in-allset-true-if-the-set-is-empty">Why is predicate “all” as in all(SET) true if the SET is empty?</a> </p>
</blockquote>
<p>In don't quite understand this quantification ov... | zrbecker | 19,536 | <p>Well if $\forall y \in \emptyset : Q(y)$ were false, we would be able to find some $y \in \emptyset$ such that $Q(y)$ were false. However, there are no $y \in \emptyset$. So $\forall y \in \emptyset : Q(y)$ should be true.</p>
|
281,735 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/202452/why-is-predicate-all-as-in-allset-true-if-the-set-is-empty">Why is predicate “all” as in all(SET) true if the SET is empty?</a> </p>
</blockquote>
<p>In don't quite understand this quantification ov... | Victor Jose Arana Rodriguez | 54,341 | <p>I think I got the idea, but it confuses me when I compare this reasoning with the existencial quantifier over the empty set
$ \exists x \in \emptyset : P(x) $</p>
<p>I think this is TRUE because I'm quantifying over the empty set too.</p>
|
438,490 | <p>I'm trying to find the eigenvalues and eigenvectors of the following <span class="math-container">$n\times n$</span> matrix, with <span class="math-container">$k$</span> blocks.
<span class="math-container">\begin{gather*}
X = \left( \begin{array}{cc}
A & B & \cdots & \\
B & A & B & \cdots \... | Toni Mhax | 121,643 | <p>The block matrix <span class="math-container">$X$</span> is a particular one. Set <span class="math-container">$\frac{n}{k}=m$</span>; you may diagonalize <span class="math-container">$A$</span> or <span class="math-container">$B$</span> by the same unitary <span class="math-container">$U$</span>, thus taking the di... |
271,489 | <p>I have a function for an ellipse that depends on time:</p>
<pre><code>elip = 1/2 (-((-0.476431 t + x) ((0.551344 (-0.476431 t + x))/t - (0.692939 (0.00786131 t + y))/t))
- (0.00786131 t + y) (-((0.692939 (-0.476431 t + x))/t) + (50.1052 (0.00786131 t + y))/t))
</code></pre>
<p>I want to plot this ellipse at a few d... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
elip = 1/
2 (-((-0.476431 t +
x) ((0.551344 (-0.476431 t + x))/
t - (0.692939 (0.00786131 t + y))/t)) - (0.00786131 t +
y) (-((0.692939 (-0.476431 t + x))/
t) + (50.1052 (0.00786131 t + y))/t)) //
Rationalize[#, 0] &... |
271,489 | <p>I have a function for an ellipse that depends on time:</p>
<pre><code>elip = 1/2 (-((-0.476431 t + x) ((0.551344 (-0.476431 t + x))/t - (0.692939 (0.00786131 t + y))/t))
- (0.00786131 t + y) (-((0.692939 (-0.476431 t + x))/t) + (50.1052 (0.00786131 t + y))/t))
</code></pre>
<p>I want to plot this ellipse at a few d... | Michael E2 | 4,999 | <p>Algebraically, <code>elip</code> is easy to analyze and much easier if the coefficients are exact than floating-point numbers (easier for Mma, that is):</p>
<pre><code>elipRat = Rationalize[Rationalize@elip, 0];
reg[t_] = ImplicitRegion[Log[1/2] == elipRat, {y, x}];
Show[
Module[{color = 0},
HighlightMesh[
D... |
3,080,124 | <p>Let <span class="math-container">$X$</span> be a topological space. Let <span class="math-container">$a\in X$</span>. Is it always true that <span class="math-container">$a$</span> is contained in a proper open set of <span class="math-container">$X$</span>? I don't know how to derive it directly by the axioms of a ... | jmerry | 619,637 | <p>No. Indeed, the "trivial topology" in which only the empty set and the full space are open, is a valid topology. It's not very interesting, but it does satisfy the axioms.</p>
|
3,881,029 | <p><span class="math-container">$R(A)$</span> is range of <span class="math-container">$A$</span> <br>
<span class="math-container">$N(A)$</span> is nullspace of <span class="math-container">$A$</span> <br>
<span class="math-container">$R(A^T)$</span> is range of <span class="math-container">$A^T$</span> <br>
<span cla... | Hagen von Eitzen | 39,174 | <p>With <span class="math-container">$y=Az$</span>,
<span class="math-container">$$ \langle x, y\rangle =\langle x, Az\rangle =\langle A^Tx,z\rangle=\langle 0,z\rangle =0.$$</span></p>
|
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | Rodrigo A. Pérez | 13,923 | <p>It is probably too much to ask from one editor to find space for two 50 page papers on the same topic, so it is natural to send the second part elsewhere. The exceptions I know involve solutions to big problems and/or big names.</p>
<p>About your second problem... include paper one as a helpful reference for the re... |
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | Noah Snyder | 22 | <p>One option to consider, which we did with the small index subfactor papers, is to submit to the same editor but at <em>different</em> journals. This retains the logistical advantage of having the same referees, but doesn't require as much space in one journal. Furthermore you can put the better half in a better jour... |
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | Stefan Kohl | 28,104 | <p>First of all, I wonder how you can be sure that a 100-pages-proof is correct before you even have typed it in completely -- at least unless it consists mostly of routine computations!</p>
<p>As to your question, I think one proof should ideally be given in one paper, or a least in two papers in the same journal. --... |
2,930,003 | <p>Can there be a relation which is reflexive, symmetric, transitive, and antisymmetric at the same time? I tried to find so.</p>
<p>If <span class="math-container">$A = \{ a,b,c \}$</span>. Let <span class="math-container">$R$</span> be a relation which is reflexive, symmetric, transitive, and antisymmetric.</p>
<p>... | 5xum | 112,884 | <p>For any set <span class="math-container">$A$</span>, there exists only one relation which is both reflexive, symmetric and assymetric, and that is the relation <span class="math-container">$R=\{(a,a)| a\in A\}$</span>.</p>
<p>You can easily see that any reflexive relation must include all elements of <span class="m... |
2,137,530 | <blockquote>
<p>Given $N$ trials of a die roll, where we have defined $D$ as the number of distinct outcomes, what would be the mean and standard deviation of $D$?</p>
</blockquote>
<p>If we have defined $I(k)$ as an indicator random variable which equals 1 if outcome $k$ (such as 6) appears at least once, and 0 oth... | Graham Kemp | 135,106 | <p>Good tactic. The distributions of the indicators are identical, though <em>not</em> independent, as you noticed. Fortunately that is not a critical issue.</p>
<p>A really useful though counterintuitive, property of Expectation is that it is Linear and this holds whether the random variables are indepe... |
2,137,530 | <blockquote>
<p>Given $N$ trials of a die roll, where we have defined $D$ as the number of distinct outcomes, what would be the mean and standard deviation of $D$?</p>
</blockquote>
<p>If we have defined $I(k)$ as an indicator random variable which equals 1 if outcome $k$ (such as 6) appears at least once, and 0 oth... | Parcly Taxel | 357,390 | <p>We approach this problem from a combinatorial perspective. The number of $n$-roll sequences with $k$ distinct values ($1\le k\le6$), out of $6^n$ sequences total, is
$$D(n,k)=\binom6kk!\left\{n\atop k\right\}=\binom6k\sum_{j=0}^k(-1)^{k-j}\binom kjj^n$$
where $\left\{n\atop k\right\}$ is the <a href="https://en.wiki... |
88,511 | <p>In version <a href="http://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn102.html">10.2</a> there is a new experimental function: <a href="http://reference.wolfram.com/language/ref/FindFormula.html"><code>FindFormula[]</code></a>.</p>
<p>I suspect that a <a href="https://en.wikipedia.org/wiki/Symbolic_... | TheDoctor | 2,263 | <p>I doubt that this is very robust. Consider a simple change in the DE example in the Documentation:</p>
<pre><code>sol = y /. NDSolve[{y'[x] == y[x] Cos[x], y[0] == 2}, y, {x, -5, 300}][[1]];
times = N[Range[-5, 600]/9];
data = Transpose[{times, sol[times] + RandomReal[0.05, Length[times]]}];
lp = ListPlot[data, Plo... |
1,991,950 | <p>Let $m$ and $n$ be relatively prime integers, with $n \ne 0,1$, so that $a=m/n$ is a non-integer rational fraction. Let $p$ be an odd prime.</p>
<blockquote>
<p><strong>QUESTION 1</strong>: Can Fermat's Little Theorem be applied, <em>i.e.</em>, can one say $a^p \equiv a\!\pmod{p}$, without any further considerati... | Jyrki Lahtonen | 11,619 | <p>It can be applied in the following sense.</p>
<ul>
<li>You need to assume that <span class="math-container">$p\nmid n$</span> (division by <span class="math-container">$p$</span> amounts to division by zero when doing arithmetic <span class="math-container">$\pmod p$</span>, so is a no-no).</li>
<li>We can work in ... |
400,838 | <p>I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$</p>
<p>I tried and tried... friends of mine tried as well and we don't know how to get out of:</p>
<p>$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$</p>
<p>(this is what we get after multiplying by the conjugate of $2 + \sqrt{3+x}$)</p>
<p>How to proce... | amWhy | 9,003 | <p>You had the right idea: the issue is in your simplification of the numerator:</p>
<p>$$\begin{align}
(2 - \sqrt{3 + x})(2 + \sqrt{3 + x}) & = 2^2 - \left(\sqrt{(3 + x)}\right)^2 \\ \\
& = 4 - (3 + x) \\ \\
& = 4 - 3 - x \\ \\
& = 1 - x = -(x - 1)
\end{align}$$</p>
<p>That gives you $$\begin{align}... |
1,811,443 | <p>Let $(a_n)$ be a sequence of rational numbers, where <strong>all rational numbers are terms</strong>. (<em>i.e. enumeration of rational numbers</em>)</p>
<p>Then, is there any convergent sub-sequence of $(a_n)$?</p>
| Aloizio Macedo | 59,234 | <p>The argument given by Elio Joseph can be refined in order to prove that for any given real number, there always exists a subsequence which converges to it.</p>
<p>For that, let $x \in \mathbb{R}$ be a real number. Consider the interval $[x-1,x+1]$. It contains infinitely many rationals, but in particular it contain... |
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | James Garrett | 457,432 | <p>Noticing that $x-1=-(1-\sqrt{x})(1+\sqrt{x})$ then $$\lim_{x \to 1} \frac{\sin(1-\sqrt{x})}{-(1-\sqrt{x})(1+\sqrt{x})}=\lim_{x \to 1} \frac{\sin(1-\sqrt{x})}{1-\sqrt{x}}\cdot \lim_{x \to 1} \frac{-1}{1+\sqrt{x}}=\lim_{x\to 1} \frac{-1}{1+\sqrt{x}}=\frac{-1}{2},$$ using the fact that $$\lim_{u \to 0} \frac{\sin(u)}{u... |
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | Michael Rybkin | 350,247 | <p>In order to find this limit, you're going to have to employ the following well-known fact from elementary limit theory:
$$
\lim \limits_{x \to 0} \frac{\sin{x}}{x}=1
$$</p>
<p>What it says is that as $x$ approaches zero, sine of $x$ over $x$ approaches $1$. Here's <a href="https://www.youtube.com/watch?v=7Dy8ASSE8o... |
3,657,026 | <p>I need to prove expression using mathematical induction <span class="math-container">$P(1)$</span> and <span class="math-container">$P(k+1)$</span>, that:</p>
<p><span class="math-container">$$
1^2 + 2^2 + \dots + n^2 = \frac{1}{6}n(n + 1)(2n + 1)
$$</span></p>
<p>Proving <span class="math-container">$P(1)$</span>... | fleablood | 280,126 | <p><span class="math-container">$\frac{1}{6}k(k+1)(2k+1) + (k+1)^2=$</span></p>
<p><span class="math-container">$\frac{1}{6}k(k+1)(2k+1) + \frac 16\cdot 6(k+1)^2=$</span> </p>
<p><span class="math-container">$\color{green}{\frac{1}{6}}k\color{blue}{(k+1)}(2k+1) + \color{green}{\frac 16}\cdot 6\color{blue}{(k+1)}^2=$<... |
3,718,347 | <p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers and <span class="math-container">$$x^2+y^2-4x-6y-1=0$$</span>what is the minimum value of <span class="math-container">$x + y$</span>?</p>
<p>I tried to factor it but I couldn't. Then I tried to make squares so al... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>Set <span class="math-container">$x+y=c$</span></p>
<p>Replace <span class="math-container">$y$</span> with <span class="math-container">$c-x$</span> to form a quadratic equation in <span class="math-container">$x$</span></p>
<p>As <span class="math-container">$x$</span> is real, the discriminant must be... |
3,718,347 | <p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers and <span class="math-container">$$x^2+y^2-4x-6y-1=0$$</span>what is the minimum value of <span class="math-container">$x + y$</span>?</p>
<p>I tried to factor it but I couldn't. Then I tried to make squares so al... | Angelo | 771,461 | <p><span class="math-container">$x^2+y^2-4x-6y-1=$</span></p>
<p><span class="math-container">$=x^2-4x+y^2-6y-1=$</span></p>
<p><span class="math-container">$=(x^2-4x+4)-4+(y^2-6y+9)-9-1=$</span></p>
<p><span class="math-container">$=(x-2)^2-4+(y-3)^2-9-1=$</span></p>
<p><span class="math-container">$=(x-2)^2+(y-3)^2-1... |
2,482,669 | <p>Find sum of the expression,
$$x^n+x^{n-1}y+x^{n-2}y^2+x^{n-3}y^3+\dots+xy^{n-1}+y^n$$
where $x,y$ are real numbers and $n$ is a natural number.</p>
| Macavity | 58,320 | <p><strong>Hint:</strong> It’s a geometric progression with common ratio $\dfrac{y}x$.</p>
|
211,623 | <p>Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} s_{n} \text{ for $n \geq 0$}.
\end{eqnarray*}
This sequence is what we get when replace the letters in the algae L-s... | Alireza Abdollahi | 19,075 | <p>You may start with the PhD thesis of P. P. Campbell </p>
<p><a href="http://www-circa.mcs.st-and.ac.uk/Theses/ppcphd.pdf" rel="nofollow">http://www-circa.mcs.st-and.ac.uk/Theses/ppcphd.pdf</a></p>
|
118,742 | <p>The PDF for $Y$ is
$$f_Y(y) = \begin{cases}
0 & |y|> 1 \\
1-|y| & |y|\leq 1
\end{cases}$$</p>
<p>How do I find the corresponding CDF $F_Y(y)$? I integrated the above piecewise function to get
$$F_Y(y)=\begin{cases}
1/2 -y/2-y^2/2 & [-1,0] \\
1/2-y/2+y^2/2 & [0,1]
\en... | Community | -1 | <p>We have that $F(y) = \displaystyle \int_{-\infty}^y f(x) dx$. In your case, we are given that $$f(x) = \begin{cases} 0 & x <-1\\ 1 + x & x \in[-1,0]\\ 1-x & x \in [0,1]\\ 0 & x > 1\end{cases}$$</p>
<ul>
<li>If $y < -1$, then we have $F(y) = \displaystyle \int_{-\infty}^y f(x) dx = \displays... |
118,742 | <p>The PDF for $Y$ is
$$f_Y(y) = \begin{cases}
0 & |y|> 1 \\
1-|y| & |y|\leq 1
\end{cases}$$</p>
<p>How do I find the corresponding CDF $F_Y(y)$? I integrated the above piecewise function to get
$$F_Y(y)=\begin{cases}
1/2 -y/2-y^2/2 & [-1,0] \\
1/2-y/2+y^2/2 & [0,1]
\en... | Dilip Sarwate | 15,941 | <p>This is the kind of problem that gives integration a bad name among students.</p>
<ul>
<li><p><strong>Draw</strong> a graph of the density function. It looks like an isoceles
right triangle with hypotenuse $2$ and apex at $(0,1)$ and very obviously has area $1$ (useful as a check on one's work.)</p></li>
<li><p>Fo... |
2,352,527 | <p><em>This is rather a simple problem that I'm posting ; looking forward not for the solution of it but the different ways it could be solved.</em></p>
<p>What is the value of </p>
<p>$\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}$ </p>
<p><em>Do I ju... | Fred | 380,717 | <p>For $z \ne 0$ let $g(z)=\frac{f(z)}{z}$. Then $g$ is holomorphic on $ \mathbb C \setminus \{0\}$ and in $0$ the function $g$ has an isolated singularity.</p>
<p>From $|f(z)|\leq M|z|$, we see that $|g(z)| \le M$ for $ z \in\mathbb C \setminus \{0\}$.</p>
<p>Riemann says: $0$ is a removable singularity of $g$. Hen... |
756,662 | <p>I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with with indicator functions but I'm not sure how to write this.
Thanks!</p>
| user139388 | 139,388 | <p>Well, <em>any</em> indicator function won't do. You can see that there is no way to write $(0,25,0.5)$ as an element of the $\sigma$-algebra generated by those sets, and so the indicator of that set won't be measurable. But the set of simple functions which are measurable in that space is most likely dense with resp... |
272,468 | <p>I wrote the <a href="https://www.researchgate.net/publication/334884635_On_a_Bivariate_Frechet_Distribution" rel="nofollow noreferrer">Frechet Distribution</a> as follows:</p>
<pre><code>dist1 = ProbabilityDistribution[{"PDF", \[Lambda]1/\[Alpha]1 (x/\[Alpha]1)^(-\[Lambda]1 - 1)E^-((x/\[Alpha]1)^-\[Lambda]... | Michael Seifert | 27,813 | <p>The two problems seem to be related, or at least can be cured together:</p>
<pre><code>obj2[a_?NumericQ] :=
Chop[NIntegrate[(E^(1/2 (-x[1]^2 - x[2]^2)) Log[
Abs[1 - a (x[1]^2 + x[2]^2)]] x[1]^2)/(\[Pi] (x[1]^2 +
x[2]^2)), {x[1], -\[Infinity], \[Infinity]}, {x[
2], -\[Infinity], \[Infinit... |
80,966 | <p>I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological condition on this morphism).</p>
<p>For compact Lie groups, using the exponential map, the answers is no, but in ge... | Andreas Thom | 8,176 | <p>The answer is no in general, but this is a rather deep fact.</p>
<blockquote>
<p><strong>Theorem:</strong> (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely generated (abstract) quotient of $G$ is finite.</p>
</blockquote>
<p>N. Nikolov and D. Segal, <em>Generators and commu... |
80,966 | <p>I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological condition on this morphism).</p>
<p>For compact Lie groups, using the exponential map, the answers is no, but in ge... | Alain Valette | 14,497 | <p>Andreas shot first, but I still encourage everybody to have a look at the lemma on p.263 of R. Alperin,Locally compact groups acting on trees and property $T$. Monatsh. Math. 93 (1982), no. 4, 261–265: any homomorphism from a locally compact group to $\mathbb{Z}$, is continuous. This answers Florent's question.</p>... |
456,581 | <p>If one requires simply the existence of partial derivatives of first order rather than all orders, then a standard example is the function</p>
<p>$$ f(x,y) = \left\{\begin{array}{l l}
\frac{2xy}{x^2+y^2} & \quad \text{if $(x,y)\neq(0,0),$}\\
0 & \quad \text{if $(x,y)=(0,0).$}
\end{array} \... | user | 89,499 | <p>For ease of reading, I TeXified a part of <a href="http://mathforum.org/kb/message.jspa?messageID=5449291">sci.math post</a> by Dave L. Renfro (this post is a CW). </p>
<hr>
<p>What follows is from The American Mathematical Monthly 67 #8 (October 1960), 813-814.</p>
<p><strong>Discontinuous Function with Partial... |
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | DonAntonio | 31,254 | <p>An idea for you:</p>
<p>$$d\,\mid\,n\implies n=qd\;,\;\;q\in\Bbb Z\;,\;\;\text{and from here}: $$</p>
<p>$$x^n-1=\left(x^d\right)^q-1=\left(x^d-1\right)\left(\left(x^d\right)^{q-1}+\left(x^d\right)^{q-2}+\ldots+x^d+1\right)$$</p>
<p>The above uses the basic relation from geometric series:</p>
<p>$$x^a-1=(x-1)(x^... |
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | Xiao | 131,137 | <p>Hint from Dummit and Foote: </p>
<p>Suppose $n= kd+r$, where $0 \leq r < d$ we have
$$x^n -1 = x^{kd+r} - x^r + x^r -1 = x^r (x^{kd}-1) + (x^r -1)$$
then
can you conclude $d|n$ if and only if $x^d-1 | x^n-1$.</p>
|
2,532,156 | <p>As the title suggests I am confused between what arguments will qualify a explanation as a proof and when does the intuition betrays us. Here is the question that made me think about this:</p>
<p>On a certain planet Tau Cetus more than half of its land is dry. Prove that a tunnel can be dug straight through center ... | neptun | 192,385 | <p>You are asking one of the hardest questions of mathematics. Much about mathematics is seeing the reason for something to be true and then carefully writing it out, because very often the sheer complexity of a problem makes you overlook certain things that could go wrong. There are many statements that are obvious in... |
733,908 | <p>How do i start off with integrating the below function? i tried applying trig substitution and U substitution. how do i go about solving this function? should i split them up further into 2 separate functions ? need some help in this as i can't seem to figure out how to continue on with it </p>
<p>$$\int\fra... | André Nicolas | 6,312 | <p>Rewrite the integrand as $\frac{x(x^2-1)}{(x^2-1)^{1/2}}+\frac{x}{(x^2-1)}$, which is $x(x^2-1)^{1/2}+\frac{x}{(x^2-1)^{1/2}}$. Then $u=x^2-1$ finishes things. </p>
|
163,816 | <p>I'm looking for a good exercise book for probability theory, preferably at least partially with solutions to it. I want it to be detailed, not trivial, providing me solid fundamentals in the topic to be developed in the future. I'd wish it to be more of a "applied" technical university approach than the highly "abst... | Dilip Sarwate | 15,941 | <p><em>One Thousand Exercises in Probability</em> by Grimmett and Stirzaker, Oxford Press, 2001 might suit your needs.</p>
|
3,717,243 | <p>I am reading Introduction to automata theory, languages, and computation 3ed, by John E. Hopcroft, et al. The wikipedia article (<a href="https://en.wikipedia.org/wiki/Turing_machine#Formal_definition" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Turing_machine#Formal_definition</a>) for Turing machine al... | Mostafa Ayaz | 518,023 | <p>I don't think the equality generally holds, specially by differentiating both sides we must have <span class="math-container">$$\sum_{i=1}^{\infty} {1\over (i+x)\cdot i}={1\over x}$$</span>which for <span class="math-container">$x\in\Bbb N-\{1\}$</span> means that <span class="math-container">$$\sum_{i=1}^{x}{1\over... |
2,816,965 | <p>Hey so I was wondering how to differentiate $(e^2)^x$ without using the chain rule.</p>
<p>I tried but I always end up using the chain rule in this case.</p>
<p>Would appreciate some help! (No hints please).</p>
| Rhys Hughes | 487,658 | <p>Indeed, using the chain rule is certainly the easiest: $$f(x)=e^{2x}\to f'(x)=2e^{2x}$$</p>
<hr>
<p>You could however, use the product rule:
$$f(x)=(e^2)^x=e^xe^x$$
Then $p=q=e^x$, and $p'=q'=e^x$</p>
<p>Product rule is $f'(x)=p'q+q'p\to e^xe^x+e^xe^x=2e^{2x}$</p>
<p>Or the quotient rule can also be used:
$$f(x)... |
118,311 | <p>Let $A$ be an $n\times n$ matrix with entries in an arbitrary field $k$. </p>
<p>Is the characteristic polynomial $\det(tI_n-A)$ dependent only on the trace and determinant of $A$?</p>
| Noah Snyder | 8 | <p>By column orthogonality you can immediately work out the entries in the last two columns. For the first two missing columns, the key observation is that x and $x^{-1}$ are not conjugate. This means that the first two missing columns are complex conjugates of each other. You then have to split into cases based on ... |
26,062 | <p>Given a <code>TabView</code> panel like this one</p>
<pre><code>TabView[{
DynamicModule[{x = False}, {Checkbox[Dynamic[x]], Dynamic[x]}],
"foo"}]
</code></pre>
<p>I would like to reset the value of <code>x</code> to its initial value (<code>False</code>) every time the first tab gets selected - that is, I basi... | Albert Retey | 169 | <p>Here is a version which fully localizes the variables. Other than that it does the same as Kubas answer so I'd suggest to accept his answer:</p>
<pre><code>With[{xDefault = False},
DynamicModule[{x = xDefault, visibleTab = 1},
TabView[{
Row[{Checkbox[Dynamic[x]], " ", Dynamic[x]}],
Dynamic[x]
}, Dyna... |
1,774,084 | <p>I think it is convergent to $1$ because as $n$ tends to $\infty$ , $1/\sqrt(n)$ tends to $0$. Is it true?</p>
<p>Thanks!</p>
| Piquito | 219,998 | <p>Let $y_n=n^{\frac{1}{\sqrt n}}$ so we have $\log y_n=\frac{\log n}{\sqrt n}$.</p>
<p>Applying now the Hospital rule you have $$\lim_{n\to \infty}y_n=\lim_{n\to \infty}\frac{\log n}{\sqrt n}=\lim_{n\to \infty}\frac{\frac 1n}{-2\sqrt{n^3}}=0$$</p>
<p>Thus your limit is equal to $1$ so your sequence is convergent to... |
448,694 | <p>Let $k$ be a field, and $A$ be a finitely generated $k$-algebra. Then does $k$ also have to be a finitely generated field?</p>
<p>Motivation: Let $A$ be generated by $\{a_1,a_2,\dots,a_n\}$, and $k$ be generated by $\{k_1,k_2,k_3,\dots\}$. Then the element $k_1k_2k_3\dots a_1\in A$ is not generated by a finite numb... | Chris Eagle | 5,203 | <p>No. For example, $\Bbb{C}$ is a finitely generated $\Bbb{C}$-algebra but not a finitely generated field (since finitely generated fields are countable).</p>
|
3,546,661 | <p>The following is from <em>Elementary Differential Geometry</em> by A.N. Pressley, page 102.</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/KkYsL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KkYsL.png" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/t2... | Emilio Novati | 187,568 | <p>Because the first curve, with <span class="math-container">$v=v_0$</span> has <span class="math-container">$\dot v =0$</span>,
the second, with <span class="math-container">$u=u_0$</span> has <span class="math-container">$\dot u=0$</span> so are used to compare only the coefficients <span class="math-container">$E_i... |
2,231,391 | <p>I'd like a low-discrepancy sequence of points over a 3D-hypercube <span class="math-container">$[-1,1]^3$</span>, but don't want to have to commit to a fixed number <span class="math-container">$n$</span> of points beforehand, that is just see how the numerical integration estimates develop with increasing numbers ... | user32038 | 438,631 | <p>another good solution to get an open-ended sequence is using the Halton method. It is also very easy to implement, even for any dimension! For d<8 it has usually good properties, beyond this more difficult will typically outperform Halton.</p>
|
2,323,223 | <p>I am having a hard time with this question for some reason. </p>
<p>You and a friend play a game where you each toss a balanced coin. If the upper faces on
the coins are both tails, you win \$1; if the faces are both heads, you win \$2; if the coins
do not match (one shows head and the other tail), you lose \$1.
C... | robjohn | 13,854 | <p>Variance is the mean of the squares minus the square of the mean.
$$
0.25\cdot1^2+0.25\cdot2^2+0.5\cdot(-1)^2-0.25^2=1.6875
$$
For independent events, the variance of a sum is the sum of the variances, so the variance for $50$ events is
$$
50\cdot1.6875=84.375
$$</p>
|
2,323,223 | <p>I am having a hard time with this question for some reason. </p>
<p>You and a friend play a game where you each toss a balanced coin. If the upper faces on
the coins are both tails, you win \$1; if the faces are both heads, you win \$2; if the coins
do not match (one shows head and the other tail), you lose \$1.
C... | Graham Kemp | 135,106 | <p>You are confusing the distribution of $50X_1$ and $\sum_{k=1}^{50}X_k$ when ${(X_k)}_{k=1}^n$ is a sequence of independent and identically distributed random variables.</p>
<p>It is true that $\mathsf E(50X_1)=50\mathsf E(X_1)$ and $\mathsf{Var}(50X_1)=2500\mathsf {Var}(X_1)$. However, that is not what you a... |
35,220 | <p>It is a basic result of group cohomology that the extensions with a given abelian normal subgroup <em>A</em> and a given quotient <em>G</em> acting on it via an action $\varphi$ are given by the second cohomology group $H^2_\varphi(G,A)$. In particular, when the action is trivial (so the extension is a central exten... | Richard Borcherds | 51 | <p>There is probably no easy answer to this question. Even the special case when all groups are of order 2 seems hopeless. While it is not the same as the problem of classifying all finite groups of order a power of 2, it is similar, and probably of about the same difficulty. Classifying groups of order 2<sup>n</sup> i... |
160,165 | <p>For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that
$$
0 \rightarrow E(\Bbb{Q})/2E(\Bbb{Q}) \rightarrow S^{(2)}(E/\Bbb{Q}) \rightarrow Ш(E/\Bbb{Q})[2] \rightarrow 0
$$
is an exact seque... | Alex B. | 3,212 | <p>There is no "natural" structure. The point is that if $E(\mathbb{Q})\cong \Delta\oplus \mathbb{Z}^r$, where $\Delta$ is a finite group, then for any positive integer $n$,
$$
E(\mathbb{Q})/nE(\mathbb{Q}) \cong \Delta/n\Delta\oplus (\mathbb{Z}/n\mathbb{Z})^r.$$
Note that this is a finite group. Since we can basically... |
2,856,180 | <p>I've been learning some about counting and basic combinatorics. But some scenarios were not explained in my class...</p>
<p><strong>Example problem:</strong> You are given 6 tiles. 1 is labeled "1", 2 are labeled "2", and 3 are labeled "3".</p>
<p><strong>Problem 1:</strong> How many different ways can you arrange... | Shrey Joshi | 572,044 | <p>Question 1:
This is the same as finding the number of permutations of the 'word' $ABBCCC$. Let us distinguish the same letters by subscripts so that we now have $A_1B_1B_2C_1C_2C_3$. The number of of permutations of this is just $6!$. But for each valid permutation, there are $2!3!$ repeats due to permutations of th... |
2,856,180 | <p>I've been learning some about counting and basic combinatorics. But some scenarios were not explained in my class...</p>
<p><strong>Example problem:</strong> You are given 6 tiles. 1 is labeled "1", 2 are labeled "2", and 3 are labeled "3".</p>
<p><strong>Problem 1:</strong> How many different ways can you arrange... | Karn Watcharasupat | 501,685 | <p><strong>For the case where order does not matter.</strong></p>
<p>Suppose you have $x$, $y$, and $z$ objects A, B and C respectively. You are choosing $k$ tiles. </p>
<p>If $k-(y+z)>0$, then you need to pick at least $(k-(y+z))$ of object A. Otherwise, at least zero.</p>
<p>For $i$ of object A chosen, you must... |
35,585 | <p>Let H be an infinite dimensional and separable Hilbert space. Let C be a closed and
connected subset of H containing more than one point. Can C ever be the countable union
of closed and totally disconnected subsets of H?</p>
| BS. | 6,451 | <p>Let $E_c$ be the <em>complete Erdös space</em> (Erdös, Annals of Math vol 41 1940), defined as the subspace of $\ell^2(\mathbb{N})$ where all coordinates are irrationals. It is <a href="http://en.wikipedia.org/wiki/Polish_space" rel="nofollow">polish</a> (separable and completely metrizable) and totally disconnected... |
3,710,804 | <p><span class="math-container">$$f(x) = \int \frac{\cos{x}(1+4\cos{2x})}{\sin{x}(1+4\cos^2{x})}dx$$</span></p>
<p>I have been up on this problem for an hour, but without any clues. </p>
<p>Can someone please help me solving this?</p>
| Vishu | 751,311 | <p>The integral is <span class="math-container">$$ \int \frac{\cos x(5-8\sin^2x)}{\sin x(5-4\sin^2x)} dx$$</span>
Substitute <span class="math-container">$t=\sin x \implies dt =\cos x dx$</span> to get <span class="math-container">$$\int \frac{5-8t^2}{t(5-4t^2)} dt$$</span> Now, use partial fraction decomposition: <spa... |
2,466,947 | <p>In class we are beginning discrete probability and we are being introduced to counting. He said in an example today that the word "anagram" (which has $7$ letters) can be rearranged to have $7!/3!$ possible words when we don't count the order of the a's. To me this seemed like a combination but this result reduces t... | JMoravitz | 179,297 | <p>This can be explained as the following using multiplication principle:</p>
<ul>
<li>Pick which of the seven available spaces is occupied by the <code>n</code></li>
<li>Pick which of the six remaining available spaces is occupied by the <code>g</code></li>
<li>$\vdots$</li>
<li>Pick which of the four remaining space... |
1,003,096 | <p>Let $G=(\mathbb{Q}-\{0\},*)$ and $H=\{\frac{a}{b}\mid a,b\text{ are odd integers}\}$.</p>
<ol>
<li>Show $H$ is a normal subgroup of $G$.</li>
<li>Show that $G/H \cong (\mathbb{Z},+)$</li>
</ol>
<p>I know that there are multiple definitions for normal subgroup and I am having a hard time to develop the proof for th... | Mike Earnest | 177,399 | <p>Subgroups of abelian groups are always normal. One way to show normality is to show that for any $g\in G$, and $h\in H$, that $ghg^{-1}\in H$. But this follows immediately , since $ghg^{-1}=gg^{-1}h=h\in H$, since multiplication is commutative.</p>
|
57,131 | <p>Evaluating the following lines in my computer takes near six seconds in Mathematica 10 and near 5 in Mathematica 9. I consider this very slow as only quantiles are being calculated from data to create this simple set of charts. I think that Mathematica should do much better than this. Do you agree? Thoughts?</p>
... | Verbeia | 8 | <p>It is calculating the quantiles that takes the time. Your code takes a bit under 4 seconds on my machine in version 9, but if I cut the sample sizes to 10,000 instead of 100,000, it's just under 0.4 seconds. Clearly calculating quantiles is roughly linear in performance with respect to the sample size. </p>
<pre><c... |
9,168 | <p>I'm having a doubt about how should we users encourage the participation of new members. So far I have only presented MSE to three of my fellow colleagues in grad school. In an overall way I feel like if MSE becomes too open and wide known, some of the high-rank researchers and top-class grad and undergrads users wi... | Community | -1 | <p>As the faq has,</p>
<p><code>Mathematics - Stack Exchange is for people studying mathematics at any level and professionals in related fields.</code></p>
<p>So I would suggest and strongly enourage in spreading the word around (I certainly spread the word among my co-grad students and Profs).</p>
<p>That said, I ... |
65,582 | <p>Let $G$ be a locally compact group on which there exists a Haar measure, etc..</p>
<p>Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$ such that $d(x,y) = d(gx,gy)$ for all $x,y,g \in G$. How to... | Lucas Kaufmann | 12,501 | <p>What if you take the original metric $d_0$ and define the new metric by<br>
$d(x,y) = \int_G d_0(gx,gy) d\mu$</p>
<p>where $d\mu$ is the Haar measure?<br></p>
<p>The invariance of the measure implies the invariance of the integral, and hence of $d$.</p>
<p>EDIT: As Theo pointed out, this works only when $G$ is co... |
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