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 |
|---|---|---|---|---|
2,869,261 | <blockquote>
<p>For positive real numbers $x,y,z$, Prove that,</p>
<p>$$\bigg(\frac{x^2+y^2+z^2}{x+y+z}\bigg)^{(x+y+z)}\geq x^xy^yz^z \geq
\bigg(\frac{x+y+z}{3}\bigg)^{(x+y+z)}$$</p>
</blockquote>
<p>I have no idea how to begin this? I am getting a hint for A.M. - G.M. equality but numbers don't seem to fit.<... | Michael Rozenberg | 190,319 | <p>Since $\ln$ is a concave function we obtain:
$$\sum_{cyc}\frac{x}{x+y+z}\ln{x}\leq\ln\left(\sum_{cyc}\frac{x}{x+y+z}\cdot x\right)=\ln\frac{x^2+y^2+z^2}{x+y+z},$$
which gives a left inequality.</p>
<p>The right inequality it's just Jensen for the convex function $f(x)=x\ln{x}:$
$$\frac{\sum\limits_{cyc}x\ln{x}}{3}\... |
4,115,343 | <p>I read about the construction of a function
<span class="math-container">$G : D_n → ∂D_n$</span> for an <span class="math-container">$n$</span>-dimensional disk <span class="math-container">$D_n$</span> using <span class="math-container">$F(p)$</span> and <span class="math-container">$p$</span> in the proof of Brouw... | Community | -1 | <p>Are you referring to the proof attributed to Hirsch, in which we map the point <span class="math-container">$p$</span> to <span class="math-container">$G(p)\in\partial D$</span> given by extending the segment through <span class="math-container">$p$</span> and <span class="math-container">$F(p)$</span> until it hits... |
1,301,773 | <p>decompose $\frac{x^2-2x+3}{(x-1)^2(x^2+4)}$</p>
<p>the way my teacher wants us to solve is by substitution values for x,</p>
<p>I set it up like this:</p>
<p>(after setting the variables to the common denominator and getting rid of the denominator in the original equation)</p>
<p>$x^2-2x+3= \frac{A}{x-1}+\frac{B... | Peter | 82,961 | <p>Hint : Start with $$\frac{x^2-2x+3}{(x-1)^2(x^2+4)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{CX+D}{x^2+4}$$ </p>
<p>Multiply with $(x-1)^2(x^2+4)$ and then insert the special values.</p>
|
129,854 | <p>I have a four points in plane and need to test (based on point coordinates) whether they are able to form a convex quadrilateral:</p>
<p><img src="https://i.stack.imgur.com/qMiCH.png" alt="convex quadrilateral"></p>
<p>Of course, the test should avoid configurations like these:</p>
<p><img src="https://i.stack.im... | lhf | 589 | <p>You want to know whether all four points are vertices of their convex hull. So find the convex hull using say <a href="http://en.wikipedia.org/wiki/Jarvis%27s_march" rel="nofollow">Jarvis's march</a> and check whether it has four vertices. You'll have to decide what to do when three points are collinear.</p>
|
129,854 | <p>I have a four points in plane and need to test (based on point coordinates) whether they are able to form a convex quadrilateral:</p>
<p><img src="https://i.stack.imgur.com/qMiCH.png" alt="convex quadrilateral"></p>
<p>Of course, the test should avoid configurations like these:</p>
<p><img src="https://i.stack.im... | Gene | 262,152 | <p>If no points are coincident and only one angle between the vertices rays is greater than 180, then it is concave. Concave quadrilaterals can never form a bow tie. If all the angles between vertices rays are less than 90 degrees, then you have a bow tie. To order the points of a convex quadrilateral, average the vert... |
2,949,936 | <p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are different complex numbers. Prove that the points <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> form an equi... | Ng Chung Tak | 299,599 | <p><strong>Hint:</strong></p>
<p>By equating sides and angles,
<span class="math-container">$$\frac{a-b}{b-c}=\frac{b-c}{c-a}=\frac{c-a}{a-b}=\omega$$</span></p>
<p>where <span class="math-container">$\omega=e^{\pm 2i\pi/3}$</span></p>
<blockquote>
<p>Observe that <span class="math-container">$a=b+(b-c)\omega$</sp... |
2,584,798 | <p>An usual recurrence relation can be written as:</p>
<p>$$a_{n} = \sum_k c_ka_{n-k}$$
or alternatively:
$$a_{n} - \sum_k c_ka_{n-k} = 0$$</p>
<p>for example, the famous Fibonacci sequence is the solution to the following recurrence relation:</p>
<p>$$a_n = 1\cdot a_{n-1} + 1 \cdot a_{n-2}$$</p>
<p>There exist ver... | Elias Costa | 19,266 | <p>By <a href="https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_definite_integrals" rel="nofollow noreferrer">mean value theorem for definite integrals</a> we have to
if $f : [a, b] \to \mathbb{R}$ is continuous and $g$ is an integrable function that does not change sign on $[a, b]$, then ther... |
3,491,692 | <p>A can do 50% of the job in 16 days, B can do 1/4th of the job
in 24 days. In how many days can they do 3/4 th of the job
working together?</p>
<pre><code>My approach.
A can complete the job in 32 days alone
B can complete the job in 96 days alone.
Together they can complete the job in ?
</code></pre>
<p>A rate... | Angina Seng | 436,618 | <p>In your second approach, note that <span class="math-container">$\cos(iy)=\cosh y$</span> and <span class="math-container">$\sin(iy)=i\sinh y$</span>.
You get
<span class="math-container">$$\cos x\cosh y-i\sin x\sinh y=0.$$</span>
So both the real and imaginary parts are zero: <span class="math-container">$\cos x\co... |
2,373,731 | <p>Can anyone think of a nice example of a homomorphism whose image is not normal? I think there must be one but am having no luck.</p>
<p>Any help would be appreciated. </p>
| carmichael561 | 314,708 | <p>Consider the homomorphism $\phi:\mathbb{Z}/2\mathbb{Z}\to S_3$ defined by $\phi(1)=(12)$.</p>
|
1,892,762 | <p>f(x)=1, x belongs to rational numbers and
=0, x belongs to irrational numbers
Prove that the above function is discontinuous over all real x.</p>
<p>I could not figure out any approach.</p>
| Kelenner | 159,886 | <p>I suppose that $f$ is not $0$. </p>
<p>1) I put first $a=f(0)$. Then $x=y=0$ gives $f(a)=0$. Putting $y=a$ gives $f(a^2)=0$, and replacing $x$ by $a$ gives $$f(-f(y))=a-2a^2f(y)+f(y^2)$$ </p>
<p>2) Let $b$ such that $f(b)=0$. Replacing $x$ by $b$ gives
$$f(-f(y))=a-2b^2f(y)+f(y^2)$$. By the above, this is true $... |
1,471,392 | <p>There is issue I have with this question, specifically part b.</p>
<p>a) Show that $(\sin A - \cos A)^2 = 1-\sin(2A)$</p>
<p>b) Hence, find the exact value of $\sin 15-\cos 15$.</p>
<p>I used what I worked out in part a. </p>
<p>$(\sin 15-\cos 15)^2 = 1-\sin 30 = 1 - \frac 1 2 = \frac 1 2$</p>
<p>and I thought ... | lab bhattacharjee | 33,337 | <p>$$\sin15^\circ-\cos15^\circ=\sqrt2\sin(15^\circ-45^\circ)=-\sqrt2\sin30^\circ<0$$</p>
|
1,471,392 | <p>There is issue I have with this question, specifically part b.</p>
<p>a) Show that $(\sin A - \cos A)^2 = 1-\sin(2A)$</p>
<p>b) Hence, find the exact value of $\sin 15-\cos 15$.</p>
<p>I used what I worked out in part a. </p>
<p>$(\sin 15-\cos 15)^2 = 1-\sin 30 = 1 - \frac 1 2 = \frac 1 2$</p>
<p>and I thought ... | Jean-François Gagnon | 271,452 | <p>You get $|ans|=1/\sqrt{2}$. This does not imply that $ans$ necessarily has both positive and negative value. It can be both or either. Here $ans$ is not a variable its an unknown (at first) quantity.</p>
|
842,437 | <p>What is the coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3)^6$ ? </p>
| vonbrand | 43,946 | <p>Use the multinomial theorem, here:
$$
(a+b+c+d)^n
= \sum_{k_1 + k_2 + k_3 + k_4 = n}
\binom{n}{k_1, k_2, k_3, k_4} a^{k_1} b^{k_2} c^{k_3} d^{k_4}
$$
The multinomial coefficient is here:
$$
\binom{n}{k_1, k_2, k_3, k_4}
= \frac{n!}{k_1! \, k_2! \, k_3! \, k_4!}
$$
Now you need to figure out which combin... |
842,437 | <p>What is the coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3)^6$ ? </p>
| Marc van Leeuwen | 18,880 | <p>I think the <em>fastest</em> way to compute this, after observing that $1$ and $-1$ are both obviously roots of $1-x-x^2+x^3$, and that $(1-x)(1+x)=1-x^2$, is to write $1-x-x^2+x^3=(1-x^2)(1-x)$ so that
$$
(1-x-x^2+x^3)^6 = (1-x^2)^6 (1-x)^6
= \biggl(\sum_{i\geq0}(-1)^i\binom6ix^{2i}\biggr)
\biggl(\sum_{j\g... |
3,243,952 | <p>I'm trying to prove the following identity:</p>
<blockquote>
<p>For each <span class="math-container">$$i \in \mathbb{N}, \sum_{k=1}^{i} \binom{n-k}{n-i} = \frac{n \cdot \binom{n-1}{n-i}}{n+1-i}$$</span></p>
</blockquote>
<p>My attempt: I've tried to prove that <span class="math-container">$(n+1-i) \cdot \sum_{k... | Z Ahmed | 671,540 | <p><strong>Another Proof</strong>: Let <span class="math-container">$[x^j]$</span> denote the co-efficient of <span class="math-container">$x^j$</span> in the following expression. The the sum
<span class="math-container">$$S=\sum_{k=1}^i {n-k \choose n-i}=[x^{n-i}]~~\left ( \sum_{k=1}^{i} (1+x)^{n-k}\right) =[x^{n-i}... |
114,715 | <p>Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ? </p>
<p>The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory: </p>
<blockquote>
<p>Let $\mathfrak{o}$ be a noetherian integral domain. $\mathfrak{o}$ is a Dedekind domain if an... | Faisal | 430 | <p>I had the exact same question not too long ago. Apparently if you drop the noetherian precondition in Neukirch's definition of "Dedekind domain" then you get what some people call an "almost Dedekind domain". There are indeed examples of almost Dedekind domains that aren't Dedekind (i.e. aren't noetherian). The firs... |
114,715 | <p>Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ? </p>
<p>The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory: </p>
<blockquote>
<p>Let $\mathfrak{o}$ be a noetherian integral domain. $\mathfrak{o}$ is a Dedekind domain if an... | François Brunault | 6,506 | <p>The ring of integers $\mathcal{O}_{\mathbf{C}_p}$ of $\mathbf{C}_p$ is not noetherian, but its only nontrivial localization is $\mathbf{C}_p$, which is noetherian.</p>
<p><strong>EDIT</strong> This doesn't answer the question : the ring $\mathcal{O}$ is local, so its localization at the maximal ideal is $\mathcal{O... |
767,382 | <p>We are given a bag of marbles containing 6 blue marbles, 7 red marbles, and 5 yellow marbles. We select 4 marbles without replacement. How can we calculate the expected value of the number of blue, red and yellow marbles taken?</p>
<p>If we select 1 marble, then it becomes much easier: 6/18 blue marbles, 7/18 red m... | Juho Kokkala | 104,295 | <p>Hint: expectation is linear, i.e., $E(X_1 + X_2 + \ldots + X_n) = E(X_1) + E(X_2) + \ldots + E(X_n)$. Think about how this applies here.</p>
|
767,382 | <p>We are given a bag of marbles containing 6 blue marbles, 7 red marbles, and 5 yellow marbles. We select 4 marbles without replacement. How can we calculate the expected value of the number of blue, red and yellow marbles taken?</p>
<p>If we select 1 marble, then it becomes much easier: 6/18 blue marbles, 7/18 red m... | callculus42 | 144,421 | <p>just go step by step.</p>
<p><strong>Expected value for the blue marbles:</strong></p>
<p>four times blue marbles are selected: BBBB</p>
<p>calculate for this event the probability. Now multiply it with 4 (blue marbles).</p>
<p>Next possible selections: BBBR (variations)</p>
<p>Calculate for this event the prob... |
2,451,634 | <p>$$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$ I tried somthing,like go to the power of two , and change of variables... but it became more complicated . Is there an idea to solve this system of equation ?
Thanks in advance </p>
| DynamoBlaze | 395,207 | <p>Solving $1$:-</p>
<p>$\sqrt{x-1}+\sqrt{y-1}=4\sqrt3\\$</p>
<p>$\text{Squaring}$ </p>
<p>$\begin{align}x-1+y-1+2\sqrt{(x-1)(y-1)}&=48\\to\\50-x-y&=2\sqrt{(x-1)(y-1)}\\\end{align}$</p>
<p>$\text{Squaring}$ </p>
<p>$\begin{align}x^2 + y^2+ 2 x y- 100 x - 100 y + 2500&=4 x y - 4 x - 4 y + 4\\to\\x^2 ... |
957,891 | <p>if sets are created using this rule $a \times 2^m$ where $a$ is an odd number e.g</p>
<p>$a=1 {1,2,4,8,16,32,64}$</p>
<p>$a=3 {3,6,12,24,48,96}$</p>
<p>$a=99 {99}$</p>
<p>hence there are $50$ such sets. Will every number from $1-100$ inclusive appear in exactly one of these sets? If so why?</p>
| Shourja Chowdhury | 649,125 | <p>The function sgn<span class="math-container">$(x)$</span> is undefined at <span class="math-container">$x=0$</span>. A function can be continuous or discontinuous at the points of domain not at points outside of domain. Since <span class="math-container">$\{0\}$</span> does not belong to the domain of sgn<span class... |
27,439 | <p>A car has to pick up each person and take them to their destination. I am trying to find the shortest tour that will do this. How can I find the shortest path between any s and t.... visiting all vertices, and without coming back to the start?</p>
<p><img src="https://i.stack.imgur.com/LEJqM.png" alt="competition">... | Andreas Lauschke | 1,598 | <p>Finding the shortest paths between <em>any</em> s and t is usually called the "all-pairs shortest path" problem, as <em>any</em> is generally interpreted as <em>all</em>. You can do that with the <a href="http://www.jvmtools.net/moorebellmann.html" rel="nofollow">JFloyd</a> function in <a href="http://www.jvmtools.n... |
3,935,057 | <p>Since A is countable and B is uncountable then this would be undefined as B is larger than A right? Or am I missing something here?</p>
| Community | -1 | <p>It depends on the relation between <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. For example, if <span class="math-container">$A = \mathbb{N}$</span> and <span class="math-container">$B = \mathbb{R}$</span>, then <span class="math-container">$A \setminus B = \varnothing$</span>... |
1,746,213 | <p>I'm trying to prove that $$\lim_{x \rightarrow \infty} x^ne^{-x}=0$$</p>
<p>So far I have said:</p>
<p>Let $\epsilon>0.$ </p>
<p>Then $$|x^ne^{-x}-0|=|x^ne^{-x}| \leq |x^n...|$$</p>
<p>I'm not sure how to bound $e^{-x}$ though. How can I do this?</p>
| MathematicianByMistake | 237,785 | <p>Given that $n \in \Bbb{N}$, by applying consecutively ($n$-times) L'Hospital's rule we have
$$\frac{\lim_{n\rightarrow \infty}{(x^n)'}}{\lim_{n\rightarrow \infty}(e^x)'}=\frac{\lim_{n\rightarrow \infty}{(nx^{n-1})'}}{\lim_{n\rightarrow \infty}(e^x)'}=\cdots=\frac{\infty}{\infty}$$
but $$\frac{\lim_{n\rightarrow \in... |
1,838,738 | <p>A study that was carried on recently showed that even babies at the age of few months know that $1+1=2$. My question is : is this a fact that can be proved, or is it a just a postulate as those in geometry ? </p>
<p>Also what about the saying "A given statement can be either true or false." ? Why are there two and ... | marty cohen | 13,079 | <p>Because
(1) 2 is defined as (i.e., is the name of) the successor of 1
and
(2) the successor of 1 is 1+1.</p>
|
1,838,738 | <p>A study that was carried on recently showed that even babies at the age of few months know that $1+1=2$. My question is : is this a fact that can be proved, or is it a just a postulate as those in geometry ? </p>
<p>Also what about the saying "A given statement can be either true or false." ? Why are there two and ... | user21820 | 21,820 | <blockquote>
<p>1+1=2. My question is : is this a fact that can be proved, or is it a just a postulate as those in geometry ? </p>
</blockquote>
<p>For the question even to make sense, you'd have to precisely specify what you mean by "1", "2", "+" and "=". Otherwise no go. Usually "2" is defined as "1+1", and "x = y... |
1,410,498 | <p>How would I prove this using mathematical induction:
$\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$.</p>
<p>Here is what I have tried:</p>
<p>$\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$</p>
<p>Base case: $\dfrac{6^6}{3^6} < 6!$. Yes, this is true.</p>
<p>Assume for $k$ it's true so $\dfrac{k^k}{3^k} < k!$... | Daniel W. Farlow | 191,378 | <p>First recall the famous limit formula for the constant $e$:</p>
<blockquote>
<p>$$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e\approx 2.7182 \ldots<3.\tag{1}$$</p>
</blockquote>
<p>Now recall the binomial theorem:</p>
<blockquote>
<p><strong>Binomial theorem:</strong> $$(x+y)^n=\sum_{i=0}^n\binom{n}{i}x... |
35,757 | <p>I can't understand the intuition behind Pearson Product-moment correlation coefficient Formula<br />
for bivariate data. The formula is :<br />
<span class="math-container">$$\rho = \frac{\mathrm{cov}(X,Y)}{S_x \cdot S_y} $$</span><br />
where cov is covariance, <span class="math-container">$S_x$</span> and <span cl... | Gottfried Helms | 1,714 | <p>First the denominator can be removed, if the values in $X$ and $Y$ are standardized, call them $Z_x = X/S_x$ and $Z_y = Y/S_y$ , such that the standard-deviations of $S_{Z_x}=1$ and $S_{Z_y}=1$. </p>
<p>Then the correlation is simply the sum of the products of the individual values divided by <em>n</em>
$$ \rh... |
3,455,922 | <p>I have to find the sum of the following sequence </p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{x^n}{n-1},\quad x\geq0$$</span> and </p>
<p>How am I supposed to start?</p>
<p>Thanks.</p>
| Robert Israel | 8,508 | <p>Hint: Multiply by <span class="math-container">$x$</span> and take the derivative.</p>
|
1,339,390 | <p>I know that Szemerédi's theorem states that any set of integers with positive natural density contains arbitrary long arithmetic progressions. However, does this imply that such a set contains an <em>infinite</em> arithmetic progression? It seems to me that they are not exactly equivalent because the longer progre... | D Left Adjoint to U | 26,327 | <p>Try constructing a counterexample like follows. Try to construct the earliest (starting at $1$) sequence of integers that contains a growing pattern of arithmetic progressions yet no infinitely long one. Go by intuition for now. I come up with $$1, 2; \ \ 4,5,6; \ \ 8,9,10,11; \ \ 13, 14, 15, 16, 17; \ \ 19,20,21... |
3,317,887 | <p>domain and support are not the same thing.</p>
<p>Per <a href="http://mathworld.wolfram.com/Domain.html" rel="nofollow noreferrer">mathworld</a></p>
<blockquote>
<p>The term <code>domain</code> is most commonly used to describe the set of values
for which a function (map, transformation, etc.) is defined. For ... | pre-kidney | 34,662 | <p>The normal distribution is a distribution, not a function, so neither of the definitions you have given apply to it.</p>
<p>What is a distribution? There are <a href="https://en.wikipedia.org/wiki/Distribution_(mathematics)" rel="nofollow noreferrer">several</a> <a href="https://en.wikipedia.org/wiki/Measure_(mathe... |
1,293,387 | <p>I'm asked to show that the following argument is valid:</p>
<p>P1) <span class="math-container">$[E \lor (L \lor M)] \land (E \leftrightarrow F)$</span></p>
<p>P2) <span class="math-container">$L \rightarrow D$</span></p>
<p>P3) <span class="math-container">$D \rightarrow \neg L$</span></p>
<p>C) <span class="ma... | Frank Hubeny | 312,852 | <p>Here is a proof using the proof checker from the <em>forallx</em> text. The OP in a comment is looking for recommended books explaining logic. This text with the associated truth functional and first order logic proof checker may supplement what the OP is currently using.</p>
<p>Here is the proof:</p>
<p><a href=... |
12,394 | <p>There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.</p>
<p>Suppose $X$ is a set equipped with a metric $d$. I had initially assumed there must be an $n$ such that $X$ embeds isometrically into $\mathbb{R}^n$, but t... | Steve Huntsman | 1,847 | <p>I will just quote from the introduction of <a href="http://arxiv.org/abs/1203.2480" rel="nofollow">Johnson and Kambites' paper "Idempotent tropical matrices and finite metric spaces"</a> in giving a cheating answer that exploits the tropical--versus the usual--structure of $\mathbb{R}$:</p>
<blockquote>
<p>"every... |
1,067,382 | <p>Is there any way to find the <code>n-th</code> number in the series, by knowing it's genereting function.</p>
<p>For example, I found that the closed form solution for a generating function $\displaystyle\frac{(1+x)^2}{(1-x)^3}$ can be expressed as $2n^2 + 2n + 1$.</p>
<p>I am interested how can I derive similar f... | bof | 111,012 | <p>In other words, you want to find the coefficient of $x^n$ in the Maclaurin series for the function
$$f(x)=\frac{(1+x)^5}{(1-x)^6}=(1+x)^5(1-x)^{-6}.$$
To do that, first expand each factor using the binomial formula
$$(1+u)^m=\sum_{n=1}^\infty\binom mnu^n$$
and then multiply the two binomial series together using
$$\... |
683,780 | <p>How do i find the area enclosed within the $3$ points when the plane intersects the $x, y$ and $z$ axis and </p>
<p>The plane is $ax+by+cz=1$</p>
<p>Need help on solving this as it has been bothering me for some time</p>
| AnonSubmitter85 | 33,383 | <p>Find the points where the plane intersects the $x$, $y$, and $z$ axis. Then find the distances between these three points and use <a href="http://www.mathopenref.com/heronsformula.html" rel="nofollow">Heron's formula</a> to find the area of the trianlge defined by them.</p>
|
3,195,385 | <p>On p. 39 of James and Liebeck's book "Representations and Characters of Groups" The term FG module is defined for F a field, G a group V a vectors space and an action of G on V is defined. </p>
<p>Where is the module in all this? My understanding is that a module is a vector space with multiplication by a ring (i... | Alex Ortiz | 305,215 | <p>If <span class="math-container">$R$</span> is a ring, and <span class="math-container">$G$</span> is a group, then the <strong>group ring</strong> <span class="math-container">$\pmb{RG}$</span> is the ring of formal sums
<span class="math-container">$$
RG \stackrel{\text{def}}{=} \bigg\{\widetilde\sum_{g\in G}r_gg :... |
3,101,176 | <p>Suppose you have to pay <span class="math-container">$15$</span> to roll a dice repeatedly until the first <span class="math-container">$6$</span> comes up. Then you will get paid as many dollars as the square of the no of rolls <span class="math-container">$X^2$</span>.</p>
<p>I have a equation as follows:
<span c... | mannibu | 641,805 | <p>You are on the right track. Intuitively a large reward is plausible because of the squared increase in rewards. Very unlikely events have a huge reward which in this case outweigh the small losses. </p>
<p>Regarding your last question: Calculating the probability of rolling at least 4 times is correct, however, rat... |
427,794 | <p>The Euler and Weierstrass forms of the gamma function are :</p>
<p>$$\mathop{\mathrm{\Gamma}}\left(z\right)=\frac{1}{z}\prod^{\infty}_{n=1}\frac{\left(1+\frac{1}{n}\right)^{z}}{\left(1+\frac{z}{n}\right)}=\frac{e^{-\gamma z}}{z}\prod^{\infty}_{n=1}\frac{e^{\frac{z}{n}}}{\left(1+\frac{z}{n}\right)},\quad\mbox{for}\;... | Hagen von Eitzen | 39,174 | <p>I suggest, You could also simply say that the three expressions define the same meromorphic function on (all of) $\mathbb C$, since "meromorphic on $D$" already stands for "holomorphic on $D$ except for a set of isolated poles"</p>
|
970,578 | <p>Solve the following equation: $\arctan x + \arctan (x^2-1) = \frac{3\pi}{4}$.</p>
<h2>What I did</h2>
<p>Let $\arctan x = \alpha, \arctan(x^2-1) = \beta$, $\qquad\alpha+\beta = \frac{3\pi}{4}$</p>
<p>$\tan(\alpha+\beta) = \tan(\frac{3\pi}{4}) = -1$</p>
<p>$$\frac{\tan\alpha + tan\beta}{1-\tan\alpha\tan\beta} = ... | graydad | 166,967 | <p>Your three answers all look good to me. You did this in a pretty efficient manner, although I would've just jumped straight to the identity $$\arctan(A)+\arctan(B) = \arctan\left(\frac{A+B}{1-AB} \right)$$ without making the substitution $\arctan(x) = \alpha$, etc. </p>
<p>Anyway, you are confident that the equatio... |
4,224 | <p>Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?</p>
<p>I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers wi... | Chris Godsil | 1,266 | <p>If M is Hermitian then H + aI is positive semidefinite when a is large enough.
So the Hermitian + positive semidefinite case is essentially the same as the Hermitian case</p>
|
1,054,169 | <p>Find all real functions $f:\Bbb R\rightarrow\Bbb R$ so that $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$, for all real numbers $x$ and $y$.</p>
<p>$f(x)=x^2$ is the only solution I think.</p>
<p>So far I have got:</p>
<p>$f(x^2)=[f(x)]^2+2f(x)f(0)+[f(0)]^2-f(0)$</p>
<p>$f(2y^2)=4[f(y)^2]-f(0)$</p>
<p>And by putt... | Kelenner | 159,886 | <p>Very partial solution.
I show that $f$ is even when $f(0)=0$. I use a comment of @soulEater: in this case we have
$$(f(x+y)+f(x-y))(f(y)-f(-y))=0$$
Now suppose that there exists an $y_0$ such that $f(y_0)-f(-y_0)\not =0$, put $L=2y_0\not = 0$. We have $f(x+y_0)=-f(x-y_0)$ for all $x$, hence $f(x+L)=-f(x)$ and $f(x+... |
204,384 | <p>Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that are kind of central and popular right now ? Thanks!</p>
| Kimball | 6,518 | <p>I believe more work is being done on nonabelian CFT than abelian CFT nowadays (see the link in Stopple's comment), but from the comments I guess you are mainly interested in abelian CFT. I'm no expert in abelian CFT, but my impression is most recent work in CFT proper is studying things like extensions with ramific... |
1,243,414 | <p>I'm have a lot of trouble trying to wrap my head around how exactly to use the central limit theorem in this question.</p>
<p>Image a gambling casino consisting of 100 busy roulette tables. Suppose that each table brings in an
average hourly profit of $50 with a standard deviation of 25.</p>
<p>(a) What is the... | Nescrio | 230,074 | <p>So we have a hundred tables $X_1, \dots, X_{100}$. Because we have more than lets say $30$ tables, or stochastic processes, we can use the CLT and we know that $\frac{1}{100}(X_1 + \dots + X_{100}) \sim N(50,\frac{25^2}{100})$. All these tables, or $X_i$'s are stochastic with $E(X_i) = 50$ and $V(X_i) = 25^2$.</p>
... |
1,243,414 | <p>I'm have a lot of trouble trying to wrap my head around how exactly to use the central limit theorem in this question.</p>
<p>Image a gambling casino consisting of 100 busy roulette tables. Suppose that each table brings in an
average hourly profit of $50 with a standard deviation of 25.</p>
<p>(a) What is the... | Community | -1 | <p>Hint: By the CLT, the distribution of the profit for whole casino follows a normal law of mean $5000\$/$hour and standard deviation $250\$/$hour. $4500$ is $2\sigma$ from the mean.</p>
|
4,551,007 | <blockquote>
<p>Let <span class="math-container">$A$</span> be a <span class="math-container">$2 \times 2$</span> matrix with <span class="math-container">$-3$</span> and <span class="math-container">$-1$</span> as eigenvalues. The eigenvectors are <span class="math-container">$v_1=[-1,1]$</span> and <span class="math-... | Rócherz | 451,007 | <p>Where you claim you “integrated each eqn.”, you actually <strong>differenciated them instead:</strong></p>
<p><span class="math-container">\begin{align}
x(t) &= \frac13 e^{-3t}-e^{-t} \\
y(t) &=-\frac13 e^{-3t}-e^{-t}
\end{align}</span></p>
<p>The system for the constants after applying the initial condition... |
2,944,889 | <p><a href="https://i.stack.imgur.com/JZETM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JZETM.jpg" alt="Why does f(x) belongsto closure of F(A)"></a></p>
<p>looking at the image above, the line <span class="math-container">$3 \to 4$</span>, I can't seem to understand why <span class="math-contai... | Henno Brandsma | 4,280 | <p>You started with an arbitary point of <span class="math-container">$\overline{A}$</span>. You then show </p>
<blockquote>
<p>Every neighbourhood of <span class="math-container">$f(x)$</span> intersects <span class="math-container">$f[A]$</span>.</p>
</blockquote>
<p>This is the definition of <span class="math-co... |
4,081,780 | <p>I have seen the statement that if <span class="math-container">$R$</span> is a commutative ring and <span class="math-container">$R$</span> is a semisimple ring, then <span class="math-container">$R$</span> is Noetherian several times while reading through some algebra resources; however, I have never seen an explan... | lhf | 589 | <p>One interpretation of "same geometrical object" is "isometric".</p>
<p>Let <span class="math-container">$F$</span> be the graph of <span class="math-container">$f$</span> and <span class="math-container">$G$</span> be the graph of <span class="math-container">$g$</span>.</p>
<p>Then <span class="... |
264,887 | <p>I have a question about the integral of <span class="math-container">$\ln x$</span>.</p>
<p>When I try to calculate the integral of <span class="math-container">$\ln x$</span> from 0 to 1, I always get the following result.</p>
<ul>
<li><span class="math-container">$\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1)... | Koushik | 35,112 | <p>Suppose a person takes a bow and starts shooting arrows at a target. Let Xn be his score in n-th shot. Initially he will be very likely to score zeros, but as the time goes and his archery skill increases, he will become more and more likely to hit the bullseye and score 10 points. After the years of practice the pr... |
264,887 | <p>I have a question about the integral of <span class="math-container">$\ln x$</span>.</p>
<p>When I try to calculate the integral of <span class="math-container">$\ln x$</span> from 0 to 1, I always get the following result.</p>
<ul>
<li><span class="math-container">$\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1)... | T. Eskin | 22,446 | <ol>
<li><p>The thing is that even though $X_{n}\to X$, we might have $X_{n}(\omega)\neq X(\omega)$ for all $\omega\in\Omega$ and $n\in\mathbb{N}$. Since the sequence $(X_{n})$ <strong>converges</strong> to $X$, it is not necessarily the case that it ever <strong>reaches</strong> it. It just comes arbitrarily close to ... |
2,690,659 | <p>I can prove other inequalities like $\|x\|_{1}\leq \sqrt{n}\|x\|_{2}$ and $\|x\|_{1}\leq {n}\|x\|_{\infty}$ using cauchy-schwarz and holder's inequality, but I don't know how to prove this one $\left\|x\right\|_2\leq\sqrt{n}\left\|x\right\|_\infty$. Can anybody tell me how to prove this one?</p>
<p>Also, how do I p... | Fred | 380,717 | <p>If $x=(x_1,...,x_n)$ , then there is $j_0 \in \{1,2,...,n\}$ such that $|x_j|=||x||_{\infty}.$</p>
<p>Thus $||x||_2 \le (\sum_{k=1}^n x_{j_0}^2)^{1/2}=\sqrt{n}\left\|x\right\|_\infty$.</p>
|
72,220 | <p>I am a graduate student thinking about some stuff.</p>
<p>1) If I have an idea for a result but it is not yet complete, is it advisable to mention it to others (especially more experienced people)? Is it possible that he/she will take it, develop it quicker than I do and publish it? How should one behave in these s... | Donu Arapura | 4,144 | <p>To be honest, I don't like the question very much, but I will answer it by
plagiarizing, I mean quoting, Hailong's succinct answer:</p>
<blockquote>
<p>"There is a nonzero risk.
But the rewards ... far outweigh the risk". </p>
</blockquote>
<p>To be more explicit, talking to an expert might save you a great d... |
72,220 | <p>I am a graduate student thinking about some stuff.</p>
<p>1) If I have an idea for a result but it is not yet complete, is it advisable to mention it to others (especially more experienced people)? Is it possible that he/she will take it, develop it quicker than I do and publish it? How should one behave in these s... | Thierry Zell | 8,212 | <p>Ask yourself: do you want to be part of a community, or do you want to do math by yourself in the woods?</p>
<p>There is no wrong answer: doing either is a fine and potentially rewarding lifestyle, but giving up on the community means giving up on both the positives and negatives that it brings, and most answers he... |
236,711 | <p>It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal Form computation. By "solved" I mean either "decide, for a given $A,b$, whether a solution exists" or "decide whether a ... | Peter Franek | 10,072 | <p>Ok, apparently, professor Storjohann sent me the reference I was looking for: Theorem 19 in <a href="http://e-collection.library.ethz.ch/eserv/eth:3348/eth-3348-01.pdf" rel="nofollow">A Fast Practical Deterministic Algorithm for Triangularizing Integer Matrices</a>. (Thanks)</p>
|
212,681 | <p>I am currently working on problem that I think could be expressed as an integer lattice problem.</p>
<p>Given $u \in \mathbb{R}^n$ and a <em>bounded</em> integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like to find an integer vector $v \in L$ that minimizes the <em>angle</em> between $u$ and $v$. That is, ... | O. S. Dawg | 73,680 | <p>Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. In the original LLL article, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions (in polynomial time) to very similar problems at th... |
1,066,761 | <p>I have this problem I can't figure out. </p>
<p>Suppose R is a relation on A, and let S be the reflexive closure of R. Prove
that if R is symmetric, also is S.</p>
<p>Could you suggest me how to do it?</p>
<p>Thanks</p>
| quid | 85,306 | <p>Consider $(x,y)\in S$. You need to show that $(y,x) \in S$. </p>
<ul>
<li><p>Case $x=y$. Trivial.</p></li>
<li><p>Case $x \neq y$. Then we observe $(x,y) \in R$ and finish the proof based on the symmetry of $R$. </p></li>
</ul>
|
3,226,962 | <p>I know that if <span class="math-container">$f$</span> is a polynomial and <span class="math-container">$f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$</span>, then <span class="math-container">$a$</span> is a root of multiplicity <span class="math-container">$k$</span>. Does this work for a differentiable function that is not... | PierreCarre | 639,238 | <p>You say that <span class="math-container">$f(x)$</span> has a double root on <span class="math-container">$x=x_0$</span> if <span class="math-container">$f(x)=(x-x_0)^2 \varphi(x)$</span>, with <span class="math-container">$\varphi(x)\ne 0$</span> in a neighbourhood of <span class="math-container">$x_0$</span>. When... |
1,722,670 | <p>In a 52-card deck, Jack is worth $1$ point, Queen $2$ points, King $3$ points, Ace $4$, and everything else is worth $0$.</p>
<p>How to calculate the probability you have a hand with sum $C\geq13$ points? where one hand is $13$ cards.</p>
| Aloizio Macedo | 59,234 | <p>As commented, $\frac{n-1}{n^2}=\frac{1}{n}-\frac{1}{n^2}$. Therefore,</p>
<p>$$\sum_{n=1}^N\frac{n-1}{n^2}+\sum_{n=1}^N\frac{1}{n^2}=\sum_{n=1}^N\frac{1}{n}.$$</p>
<p>If your proposed series were convergent, then the left side would converge, and hence the right side would too, a contradiction.</p>
|
1,722,670 | <p>In a 52-card deck, Jack is worth $1$ point, Queen $2$ points, King $3$ points, Ace $4$, and everything else is worth $0$.</p>
<p>How to calculate the probability you have a hand with sum $C\geq13$ points? where one hand is $13$ cards.</p>
| Doug M | 317,162 | <p>It seems like a few people have pointed you in the right direction, but you still need help.</p>
<p>The limit comparison test. If you have two series $\sum a_n$ and $\sum b_n$ and you know what whether a_n converges or diverges.</p>
<p>And,
$\lim_\limits{n\to \infty} \dfrac{b_n}{a_n} = M$ for some bounded M not e... |
1,722,670 | <p>In a 52-card deck, Jack is worth $1$ point, Queen $2$ points, King $3$ points, Ace $4$, and everything else is worth $0$.</p>
<p>How to calculate the probability you have a hand with sum $C\geq13$ points? where one hand is $13$ cards.</p>
| mweiss | 124,095 | <p>Try it this way: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ definitely converges, right?</p>
<p>Now if your series $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ converges, then the sum $\sum_{n=1}^{\infty} \frac{n-1}{n^2} + \sum_{n=1}^{\infty} \frac{1}{n^2}$ would converge as well.</p>
<p>But that sum is $\sum_{n=1}^{\infty} \f... |
1,351,381 | <p>I tried to figured it out if the follwing series converges or not</p>
<p>$$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\ln^2n}\ (-1)^{n}$$</p>
<p>I tried to show that $\sin(\frac{1}{n})$ is a monotonic but I'm not sure its in the right direction.</p>
| Simon S | 21,495 | <p><em>Hint:</em> By the alternating series test, if the sequence $a_n$ is monotonically decreasing and $a_n \to 0$ then $$\sum_{n=1}^\infty (-1)^n a_n$$ converges.</p>
<p>$\displaystyle a_n = \frac{\sin(1/n)}{\ln^2 n}$ meets those hypotheses as </p>
<ul>
<li>$\displaystyle \sin\left(1 \over n+1\right) < \sin\lef... |
2,410,736 | <p>If we have the product of 3 primes,
A<em>B</em>s
two of them large: A, B
and one small: s
We don t know the product of A<em>B,
(Given that if we knew A</em>B we couldn t factorize it, because of the size and the computational complexity of the problem.)
can we find the small number s through factorization?
Is t... | Carmeister | 290,127 | <p>Consider the rationals with the standard topology. Then $(-\infty,a)\cap\mathbb Q$ is a clopen subset of $\mathbb Q$ for any irrational $a$.</p>
<p>Now take the set
$$S=\{(-\infty,a)\cap\mathbb Q:a\text{ is irrational and }a<0\}.$$</p>
<p>In the lattice of clopen sets, the meet of that subset is the smallest cl... |
321,032 | <p>Take any convex set <span class="math-container">$A\subset\mathbb{R}^n$</span> which contains a neighborhood of the origin and let <span class="math-container">$f_A$</span> be the associated <em>Minkowski functional</em>
<span class="math-container">$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$</span>
which... | Michael Renardy | 12,120 | <p>I note that the result quoted from Giusti's book a priori limits the Lipschitz constant to a specific value. Without this limitation, a minimizer may not exist. Consider, for instance, the annulus <span class="math-container">$1<r<2$</span> in two dimensions, and the minimization of <span class="math-container... |
918,430 | <p>I am reading a paper (in biology) which performs a clustering algorithm.</p>
<p>At one point in the paper, it is stated that:</p>
<p>$$
\arg \max_{\lVert X\rVert=1} X^T S\, X
$$</p>
<p>can be computed as the normalized eigenvector of $S$ with largest eigenvalue...</p>
<p>How is this possible? I am guessing that ... | mvw | 86,776 | <p>If $S$ had the decomposition $S = A^H A$, then the square of the spectral norm of $A$ would be
$$
\lVert A\rVert_2^2
=
\max_{\lVert x\rVert_2 = 1} \lVert A x\rVert_2^2
=
\max_{\lVert x\rVert_2 = 1} Ax \cdot Ax
=
\max_{\lVert x\rVert_2 = 1} A^H Ax \cdot x
=
\max_{\lVert x\rVert_2 = 1} x^T A^H Ax
= \lambda_1
$$
whe... |
1,021,830 | <p>I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here.</p>
<p>The integral is:</p>
<p>$$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$</p>
<p>I don't know a way or I know it but I can't see which way or method I have to use.</p>
<p>If you ... | Venus | 146,687 | <p>You may use this formula:
\begin{equation}\int_0^{2\pi}\frac{\cos mx}{p-q\cos x}\, dx=\frac{2\pi}{\sqrt{p^2-q^2}}\left(\frac{p-\sqrt{p^2-q^2}}{q}\right)^m\qquad\hbox{for}\qquad |q|<p
\end{equation}
The complete proof can be seen <a href="https://math.stackexchange.com/a/955655/146687">here</a>.</p>
<p>Your integ... |
1,021,830 | <p>I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here.</p>
<p>The integral is:</p>
<p>$$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$</p>
<p>I don't know a way or I know it but I can't see which way or method I have to use.</p>
<p>If you ... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
621,604 | <p>I am reading <a href="http://www.researchgate.net/publication/238887959_On_Subsets_of_Finite_Abelian_Groups_with_No_3Term_Arithmetic_Progressions" rel="nofollow">this</a> paper. It is about finite abelian groups of odd order. I need to find maximal subset which doesn't contain 3-term arithmetic progression. I don't ... | Dietrich Burde | 83,966 | <p>Indeed, the result for odd order abelian groups has been generalized here (Theorem 1 implies the result of Meshulam):
<a href="http://www.math.ksu.edu/~cvs/liu_spencer-roth_group.pdf" rel="nofollow">http://www.math.ksu.edu/~cvs/liu_spencer-roth_group.pdf</a>. Reading the proof there, it seems vital that $k$ is odd (... |
621,604 | <p>I am reading <a href="http://www.researchgate.net/publication/238887959_On_Subsets_of_Finite_Abelian_Groups_with_No_3Term_Arithmetic_Progressions" rel="nofollow">this</a> paper. It is about finite abelian groups of odd order. I need to find maximal subset which doesn't contain 3-term arithmetic progression. I don't ... | Ashot | 51,491 | <p>I think even order abelian groups are not considered, because there exists 3-term nontrivial arithmetic progressions which consists of 2 numbers.</p>
<p>For example in $\mathbb Z_{10}$, set $\{1,6\}$ contains arithmetic progression. $1 + 1 = 2 * 6$.</p>
|
2,045,154 | <p>Given the equivalence relation R on $\mathbb{Z}$ </p>
<p>by $aRb$ iff $3\vert 5a^2+b^2$</p>
<p>Determine the distinct equivalence classes of R </p>
<p>$[0]=\left\{x \in \mathbb{Z}: xR0\right\}=\left\{x \in \mathbb{Z}:3|5x^2+0\right\}$
Would this only give $0$?
$[1]=\left\{x \in \mathbb{Z}: xR1\right\}=\left\{x ... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ {\rm mod}\,\ 3\!:\,\ 0 \equiv b^2\!+5a^2\equiv b^2\!-a^2\equiv (b\!-\!a)(b\!+a\!)\iff \,b\equiv \pm a$</p>
<hr>
<p>It is not clear what you mean by "since this is equivalent to a mod function it would have $3$ distinct equivalence classes "? Note that equivalence relations on $\{0,1,2\}$ ... |
402,655 | <p>I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to industry after my PhD? I'm also based in the USA.</p>
<p>I apologize if this question is kind of "bare", I'm a Ph... | Tomcat | 354,290 | <p>Just answering to your question in the title: computer algebra is definitely an active area of research! For example, there is a huge research program across several German universities funded by the DFG (German version of the NSF): <a href="https://www.computeralgebra.de/sfb/" rel="noreferrer">https://www.computera... |
948,306 | <p>Assuming a standard English 26-letter alphabet, and $k<26$, what's the number of $k$-letter words where the letters don't repeat? So for example 'house' has no repeat letters, while 'rolls' has a repeat. </p>
<p>The way I solved this was to first select the $k$ letters which will make up the word, and then count... | Jack D'Aurizio | 44,121 | <p>Let $S_N$ be the set of the strings over the alphabet $\Sigma=\{0,1\}$ with length $N$, avoiding $4$ consecutive $1$'s, and $T_N=|S_N|$. The only possible prefixes of an element of $S_N$ can be:
$$0,\quad 10,\quad 110, \quad 1110$$
hence we have:
$$ T_N = T_{N-1}+T_{N-2}+T_{N-3}+T_{N-4} $$
and:
$$ T_1=2,\quad T_2=4,... |
948,306 | <p>Assuming a standard English 26-letter alphabet, and $k<26$, what's the number of $k$-letter words where the letters don't repeat? So for example 'house' has no repeat letters, while 'rolls' has a repeat. </p>
<p>The way I solved this was to first select the $k$ letters which will make up the word, and then count... | David | 175,745 | <p>Just count them up using placeholders. Notice I start with the heads in the leftmost slots and then gradually work them to the right one placeholder at a time (case $8$ is a simple example to illustrate this).</p>
<p>Let H = Head, T = Tail, - = don't care (could be Head or Tail), (nH) = n consecutive Heads.</p>
... |
2,467,608 | <p>For a vector-valued function $f:c\in E\subseteq \Bbb R^n\to\Bbb R^m$, I want to prove that the linear transformation $T$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T(h)}{|h|}=0$ is unique.</p>
<p>My attempt:
Suppose $T_1,T_2:\Bbb R^n\to\Bbb R^m$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T_... | mfl | 148,513 | <p><strong>Hint</strong></p>
<p>Note that
$$\lim_{t\to 0^+}\frac{T_2(tv)-T_1(tv)}{|tv|}=\lim_{t\to 0^+}\frac{t(T_2(v)-T_1(v))}{|tv|}=\lim_{t\to 0^+}\frac{T_2(v)-T_1(v)}{|v|}.$$</p>
|
4,072,364 | <p><span class="math-container">$$
\exists x (P(x) \lor Q(x)) ≡ \exists x P(x) \lor \exists x Q(x)
$$</span></p>
<p>I have to prove this. I'm extremely new and yet to grasp the whole tautology logics. I believe, that <span class="math-container">$$ \exists x P(x) \lor \exists x Q(x) $$</span> requirement is that all in... | Will Jagy | 10,400 | <p>Thought I would throw in the 3 by 3 case, which is worth remembering:</p>
<p>The eigenvalues are the roots of
<span class="math-container">$$ \lambda^3 - T \lambda^2 + P \lambda - D $$</span>
where trace
<span class="math-container">$$T = a_{11} +a_{22} +a_{22} $$</span>
and the sum of the two by two principal min... |
868,643 | <p>How do you integrate the following by using Euler's formula, without using integration by parts? <span class="math-container">$$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}d\theta$$</span></p>
<p>I did integrate it by parts, by writing the <span class="math-container">$3$</span> in the numerat... | xpaul | 66,420 | <p>Hint:
$$ \frac{d}{dx}\frac{\sin x}{3\cos x+4}=\frac{3+4\cos x}{(3\cos x+4)^2}.$$</p>
|
1,048,379 | <p>The equation is $f(x) = 2^{x-3}$</p>
<p>I know you switch the $x$ and $y$ and then solve for $y$, giving $x = 2^{y-3}$, but I'm not sure how to solve the equation...?</p>
<p>Thanks in advance!</p>
<p>And yes, I'm familiar with logarithms and I know that you would take the log of each side, giving $\ln x = \ln 2^{... | Ilya | 5,887 | <p>Just use logarithms:
$$
x = 2^y-3\iff x+3 = 2^y\iff y = \log_2(x+3)
$$</p>
|
1,566,275 | <p>I'm familiar with the "construction" of non-measurable sets on $\mathbb{R}$. But of interest to me is if there is a way to construct a countably additive probability measure $\mu$ on $\mathbb{N}$ such that we can't extend $\mu$ to $2^{\mathbb{N}}$. Apparently there's no way to do it when $\mu$ is defined on all sing... | Thomas Andrews | 7,933 | <p>Let $\sim$ be any equivalence relation on $\mathbb N$. Define a measure on $\mathbb N/\sim$. Then $\mathbb N$ inherits this measure, and the measurable sets are exactly unions of equivalence classes. In particular, if $\sim$ is not the equality relation, then there are non-measurable sets.</p>
<p>For example, we ca... |
2,474,279 | <p>I am asked to find $$\int_{-\infty}^{\infty}\frac{\cos(x)}{e^{x} + e^{-x}}dx$$</p>
<p>My attempt so far:</p>
<p>I've defined a function $f(z) = \frac{e^{iz}}{e^{z} + e^{-z}}$, the idea being that since $\cos(x)=\Re(e^{ix})$, we can work with $e^{ix}$ over the upper half plane. The poles of such a function occur at... | Ian | 83,396 | <p>Main idea: the denominator decays when the real part is large, uniformly in the imaginary part, while the numerator decays when the imaginary part is large and positive, uniformly in the real part. So if you want a piece of the contour integral to decay, it should be a vertical or horizontal segment with either a la... |
2,474,279 | <p>I am asked to find $$\int_{-\infty}^{\infty}\frac{\cos(x)}{e^{x} + e^{-x}}dx$$</p>
<p>My attempt so far:</p>
<p>I've defined a function $f(z) = \frac{e^{iz}}{e^{z} + e^{-z}}$, the idea being that since $\cos(x)=\Re(e^{ix})$, we can work with $e^{ix}$ over the upper half plane. The poles of such a function occur at... | Jack D'Aurizio | 44,121 | <p>Alternative way: the hyperbolic secant is almost a fixed point the Fourier transform, while the Fourier transform of $\cos(x)$ is a linear combination of Dirac delta<em>s</em>. From
$$ \mathfrak{F}^{-1}\left(\frac{1}{2\cosh x}\right)=\sqrt{\frac{\pi}{2}}\cdot\frac{1}{2\cosh\frac{\pi s}{2}},\qquad \mathfrak{F}\left(\... |
253,835 | <p>Whether any even number can be written as sum of odd number of primes?(3,5,7.. primes) I know that Goldbach's conjecture state that a even number can be written as sum of two primes.</p>
<p>D=A+B+C+...+n such that where no of elements in the equation is a odd number and where A,B,C are prime numbers and D is even n... | Henry | 6,460 | <p>All positive integers except $1$ and $4$ can be written as the sum of an odd number of primes. </p>
<p>$6=2+2+2$ can be written as the sum of an odd number of primes.</p>
<p>$7=2+2+3$ can be written as the sum of an odd number of primes.</p>
<p>$8=2+3+3$ can be written as the sum of an odd number of primes.</p>
... |
253,835 | <p>Whether any even number can be written as sum of odd number of primes?(3,5,7.. primes) I know that Goldbach's conjecture state that a even number can be written as sum of two primes.</p>
<p>D=A+B+C+...+n such that where no of elements in the equation is a odd number and where A,B,C are prime numbers and D is even n... | André Nicolas | 6,312 | <p>We need a $2$. It turns out that for even $n \ge 8$, one $2$ is enough, the rest can be odd primes. This is easily verified for the first few even numbers $\ge 8$. After that, we can use induction, since if $n-6$ is a sum of an even number of odd primes, then so is $n$. </p>
|
98,527 | <p>Let $I\subset\mathbb{R}$ be a compact interval and let $J$ denote its interior.<br>
Consider $f:J\to\mathbb{R}$ being continuous.</p>
<ol>
<li>Under which conditions does the following statement hold?<br>
$$
\text{There exists a continuous extension $g:I\to\mathbb{R}$ of $f$.}\tag{A}
$$</li>
<li>Is boundedness of $... | Robert Israel | 8,508 | <p>Hint for 1: think about limits.</p>
<p>Hint for 2: try $f(x) = \sin(h(x))$ for suitable functions $h$.</p>
|
754,277 | <p>I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube. </p>
<p>For example:</p>
<p>2^2 -- a 2 by 2 square</p>
<p>2^3 -- a 2 by 2 by 2 cube</p>
<p>I've been puzzling over how something would look when you make it to the power of a fraction.... | Logan Tatham | 71,844 | <p>Another problem is that because just because a set exists doesn't mean you can find it.</p>
|
754,277 | <p>I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube. </p>
<p>For example:</p>
<p>2^2 -- a 2 by 2 square</p>
<p>2^3 -- a 2 by 2 by 2 cube</p>
<p>I've been puzzling over how something would look when you make it to the power of a fraction.... | Travis Nell | 62,249 | <blockquote>
<p>Clearly this is either true or false - there either exists such a set, or there does not exist such a set.</p>
</blockquote>
<p>This is the crux of the issue here. Gödel's completeness theorem tells us that a theory T proves a sentence $\varphi(\overline{x})$ if and only if $\varphi(\overline{x})$ ho... |
301,259 | <p>Problem : How many ways are there to make $n$ by adding $k$ non-negative integers, where order matters. Suppose $n=4$ and $k=3$. There are 15 solutions using $0, 1, 2, 3, 4$:</p>
<p>$(0,0,4), (0,1,3), (0,2,2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, ... | Elchanan Solomon | 647 | <p>Here's the proof. Instead of thinking of this problem as adding together $k$ non-negative integers to make $n$, let us think of the following problem instead. We have $k$ boxes and $n$ coins. How many ways can we distribute the coins? This is in fact the same problem, for if we have $10$ coins and $3$ boxes, we can ... |
42,099 | <p>Given the following time series data where row is date and column is time intervals.</p>
<pre><code> rawdata = {{"Date", "08:00", "08:30", "09:00", "09:30", "10:00", "10:30",
"11:00", "11:30", "12:00", "12:30",
"13:00"}, {{2011, 1, 4, 0, 0, 0.}, 47., 59., 71., 77., 115., 111.,
103., 75., 101.... | ciao | 11,467 | <pre><code>With[{data = #},
MapThread[{#1, #2} &, {(#1 + DateList[#2]*{0, 0, 0, 1, 1, 1}) & @@@
Tuples[{data[[2 ;;, 1]], Rest[data[[1]]]}],
Flatten[data[[2 ;;, 2 ;;]]]}]] &[rawdata]
</code></pre>
<p>All those excess characters... Here's the Twitterized version:</p>
<pre><code>Transpose@{(... |
195,464 | <p>Random variable <strong>X</strong> (as a measureable function from some probability space to the reals) has support <em>S</em> and image <em>T</em>. True or False: </p>
<p>i.) <em>S</em> is always a subset of <em>T</em>.</p>
<p>ii.) <em>(S,E,dF)</em> is the induced probability space with: sample space <em>S</em>, ... | Zhiyong Wang | 34,197 | <p>We only need to prove the second claim for the following reason.</p>
<p>If $|x_k|\rightarrow (\gamma (1+\alpha)/2)^{1/(1-\alpha)}$,
$$\frac{x_{k+1}}{x_k}=\left(1-\frac{\gamma(1+\alpha)}{|x_k|^{1-\alpha}}\right) \rightarrow-1.$$
Thus ${x_{k}}$ will be an oscillating sequence.</p>
<p>I will write it later about ho... |
2,351,690 | <p><strong>Background Information:</strong></p>
<p>I am using the book Partial Differential Equations by Walter A. Strauss. In chapter $5$ section $6$ the author solves the diffusion equation with sources at the endpoints.
$$u_t = k u_{xx} \ \ 0<x<l, t>0$$
$$u(0,t) = h(t) \ \ \ \ \ u(l,t) = j(t)$$
$$u(x,0) =... | justanewb | 215,215 | <p>I guess this is super difficult based on the lack of comments or answers. Here is an answer that I have provided from a general solution I found online for solving these types of problems.</p>
<p>Given the system
$$\begin{cases}
u_t = u_{xx} \ \ \text{for} \ \ 0 < x < 1, t > 0\\
u(0,t) = e^{t} = T_1 \ \ \... |
2,282,738 | <p>I was pondering the PMI and there's something bothering me about proofs that I've seen that the well-ordering principle (WOP) and the PMI imply each other. It seems to me that you could use the PMI on a set $S \subseteq \Bbb{Z}$ that has a maximum element $a$. Let $p:S\to\{T,F\}$ be a propositional function on S. Th... | Bram28 | 256,001 | <p>Yes, what you do is perfectly ok, and you made a good insight about induction, well done!</p>
<p>Please note though that the WOP may not hold relative to the $<$ relation as normally defined for $\mathbb{Z}$ and $\mathbb{Q}$, but if you have a countable set $S$ then that means that the WOP <em>does</em> hold ...... |
993,304 | <p>thank you for your time at first!</p>
<p>It's my homework, so I don't expect answer with result, only some hint.</p>
<p>With given iteration method
$$x_{n+1} = \frac{x_n(x_n^2 + 3U)}{3x_n^2 + U} $$
show cubic convergence to $\sqrt U$</p>
<p>My first consider is lack of $x_0$ or $x_1$ (I will assume that $x_1$ is... | Empy2 | 81,790 | <p>Can you show that $$\frac{x_{n+1}-\sqrt{U}}{(x_n-\sqrt{U})^3}$$ converges?</p>
|
112,048 | <p>This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giant-component" and "fully connected"?
e.g. a detailed picture for say, $p_n=\frac{(\log n)^\gamma}{n}$, with $0<\gamma<1$</p>
| Carl Feynman | 7,759 | <p>"Random Graphs" by Bollobas has an extensive discussion of the thresholds at which various properties appear in the Erdos-Renyi model, for example the appearance of the first cycles. It's quite well-written, too. I'm familiar with the first edition, which is a little behind the times, but I see a new edition was p... |
3,341,926 | <p>Seen in martycohen's answer to <a href="https://math.stackexchange.com/questions/2394033">How to prove that $\frac{\text{d}}{\text{d}x} x^n = nx^{n-1}$ without using the Binomial Theorem?</a>.</p>
| J.G. | 56,861 | <p>More generally, <span class="math-container">$(uv)^\prime=uv^\prime+u^\prime v$</span>. Now use <span class="math-container">$x^\prime=1$</span>.</p>
|
2,768,590 | <p>If f is uniformly differentialable on $(a,b)$ then $f'$ is continuous on $(a,b)$? </p>
<p>I just knew the definition </p>
<p>A function $f:(a,b)\to R$ is said to be uniformly differentialable iff f is differntiable on $(a,b)$, and for each $\epsilon>0$ there is $\delta>0$ such that $0<|x-y|<\delta$ an... | 97DL | 554,614 | <p>Given $x \in (a,b)$
\begin{align*}
|f'(y)-f'(x)| & \leq |f'(x) - \frac{f(x)-f(y)}{x-y} | + |\frac{f(x)-f(y)}{x-y}-f'(y)|
\end{align*}
Given that $f$ is differentiable at $x$, the first member of the sum is small for $y$ sufficiently close to $x$. The second member is small for $y$ sufficiently close to $x$, for... |
2,363,381 | <p>I tried to prove two directions. </p>
<p>The ($\Leftarrow$) direction is not so hard, it's just in $\mathbb R$, since $\mathbb R$ is complete, Heine-Borel Theorem holds. However, I have a hard time proving ($\Rightarrow$) direction. Which is using Heine-Borel Theorem we can prove the Completeness Axiom. </p>
<p>Th... | Vassilis Markos | 460,287 | <p>Let <span class="math-container">$A\subseteq\mathbb{R}$</span> be a non-empty bounded set - actually it is needed to be bounded from above only. Then, the set <span class="math-container">$B=\overline{A}$</span>, which is the closure of <span class="math-container">$A$</span>, is a closed and bounded set which, by H... |
985,930 | <p>There are 10 distinct sticks of length 1,..,10. How many triangles can be formed? I do not know whether there are some counting tricks for this one.</p>
| Asinomás | 33,907 | <p>given the triangle inequality and a hint of euclidean geometry, a triangle(possibly degenerate) can have sides $a,b,c$ with $a\leq b\leq c$ if and only if $c< a+b$</p>
<p>There are $\binom{10}{3}$ ways to select three integers between $1$ and $10$</p>
<p>It should be clear there are $k-1$ ways to write the numb... |
1,947,154 | <p>So im not being able to understand a statement in the proof of the following theorem:</p>
<blockquote>
<p>Theorem: If <span class="math-container">$S,T$</span> are subgroups of a finite group <span class="math-container">$G$</span> then</p>
<p><span class="math-container">$$|ST||S\cap T|=|S||T|$$</span></p>
</blockq... | user21820 | 21,820 | <p>You should not be afraid of quantifiers. Think of them simply as specifying a context. When you say "everyone likes candy", for example, you are restricting to the context where you are given any arbitrary person P, and in that context you claim that P likes candy. Conditional statements are also the same,... |
1,907,997 | <p>Prove $3^n \geq 2n^2 +1$ for $n = 1,2,\ldots$ using induction</p>
<p>This is what I have so far</p>
<p>Base case - $n=1,3^1 \geq 2^1 + 1 = 3$ true</p>
<p>Induction step - Assume true for some n, then,</p>
<p>$ 3*3^n \geq 3*(2n^2 +1)=6n^2 +3 $ </p>
<p>I have to somehow manipulate and show its $\geq 2*(n+1)^2 +1$... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$3^1\geq2\cdot1^2+1$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$3^n\geq2n^2+1$</p>
<p><strong>Third, prove that this is true for $n+1$:</strong></p>
<p>$3^{n+1}=$</p>
<p>$3\cdot\color\red{3^n}\geq$</p>
<p>$3\cdot(\... |
4,405,311 | <p>I am trying to solve the below problem.</p>
<blockquote>
<p>Suppose that five <span class="math-container">$1$</span>s and four <span class="math-container">$0$</span>s are arranged around a circle. Form a new circle by placing a <span class="math-container">$0$</span> between any two unequal adjacent numbers and a ... | Paul Sinclair | 258,282 | <p>Yet another argument: Swap the roles of <span class="math-container">$1$</span> and <span class="math-container">$0$</span> completely in the problem. At each stage you will get a circle that is just the complement of the one in this problem.</p>
<p>But now, working modulo <span class="math-container">$2$</span>, ea... |
1,787,867 | <p>Prove that $\dfrac{2\cos x}{\cos 2x + 1 }= \sec x$. </p>
<p>So far I have:
$\dfrac{2\cos x}{\cos 2x + 1 }= \dfrac 1 {\cos x}$</p>
<p>Where do I go from here?</p>
| Jan Eerland | 226,665 | <p>$$\frac{2\cos(x)}{1+\cos(2x)}=\sec(x)\Longleftrightarrow$$</p>
<hr>
<p>Use $\sec(x):=\frac{1}{\cos(x)}$</p>
<hr>
<p>$$\frac{2\cos(x)}{1+\cos(2x)}=\frac{1}{\cos(x)}\Longleftrightarrow$$
$$2\cos^2(x)=1+\cos(2x)\Longleftrightarrow$$</p>
<hr>
<p>Use $\cos(2x)=2\cos^2(x)-1$</p>
<hr>
<p>$$2\cos^2(x)=1+2\cos^2(x)-1... |
151,134 | <p>I am having trouble with notations like 1 cm$^{-3}$, especially since I am converting them between compound units.</p>
<p>Is there a way to express 1 cmcm$^{-3}$ without writing the negative exponent?</p>
<p>The particular queston I was asked was:</p>
<blockquote>
<p>The EC standard for lead in the atmosphere i... | Gere | 15,588 | <p>You can treat units like ordinary factors.</p>
<p>For example</p>
<p>$2 \mu g/m^{−3}=2 \cdot \mu g/m^{−3}=2\cdot(0.000001g)/(100cm)^{-3}=2\cdot 0.000001/100^{-3}\cdot g/(cm)^{-3}$</p>
<p>Just do plain algebra and replace $m\to 100\cdot cm$ with no further thinking.</p>
<p>EDIT: there is no need to visualize this... |
2,002,099 | <p>Some things I know:
omega_1 is the first uncountable ordinal, made up of all of the countable ordinals<br>
f is not necessarily continuous </p>
<p>For this proof, I currently have x as a condensation point in my range, and I know that (x - (1/n) , x + (1/n) ) is uncountable and everything to the left and right of... | Brian M. Scott | 12,042 | <p>HINT: There is an order-isomorphism $h:\Bbb R\to(0,1)$. If you had a strictly increasing $f:\omega_1\to\Bbb R$, you could compose it with $h$ to get a strictly increasing function from $\omega_1$ to $(0,1)$, so there is no harm in assuming that $f$ has bounded range. Let $a=\sup_{\xi\in\omega_1}f(\xi)$, and for $n\i... |
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