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 |
|---|---|---|---|---|
463,353 | <p>How do I need to modify this in order for it to be correct?</p>
<p>The center of $S_n$ (for $n\geq$ 3) is the trivial identity. Proof: Assume the center of $S_n$ is $C = \{ id , \tau \}$ where $ \tau \in S_n$ and $\tau \neq \ id$. Then for some $n$ the factor group $S_n\backslash C$ is abelian and solvable, a contr... | Jared | 65,034 | <p>This formula should be helpful.</p>
<p>Suppose $\sigma,\tau\in S_n$, with $\tau=(a_1\ldots a_m)$ an $m$-cycle. Then:</p>
<p>$$\sigma\tau\sigma^{-1}=(\sigma(a_1)\ldots\sigma(a_m))$$</p>
|
109,528 | <p>Let S be a finite set of integers, do I can check with Gap that this set be a set of character degrees of small group?</p>
| Dima Pasechnik | 11,100 | <p>You most probably mean "the set of degrees of irreducible characters". Indeed, given a finite set of integers, it should not be hard to construct a finite group that has irreducible characters of that degrees (and many more other degrees).</p>
<p>Still, for "the set of degrees" the problem does not look easy, as de... |
2,710 | <p>The <em>Mandelbrot set</em> is the set of points of the complex plane whos orbits do not diverge. An point $c$'s <em>orbit</em> is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.</p>
<p>The shape of this set is well known, <strong>why is it that if you zoom into parts of the filaments you will find slight... | Sam Nead | 1,307 | <p>I really like this question! I can't yet upvote, so I'll offer an answer instead. This is only a partial answer, as I don't fully understand this material myself.</p>
<p>Suppose that $f$ is a quadratic polynomial. Suppose that there is an integer $n$ and a domain $U \subset \mathbb{C}$ so that the $n$-th iterate... |
2,043,132 | <p>I am having difficulty factorising the equation. </p>
| Michael Rozenberg | 190,319 | <p>Easy to see that $|U_1(x)|\leq\frac{\sqrt3}{2}$ and $|U_2(x)|\leq\frac{3}{4}$ for every $x$.</p>
<p>Assume that $|U_k(x)|\leq\left(\frac{\sqrt3}{2}\right)^k$ for every real $x$ and every $k\leq n$, for some $n\ge2$.</p>
<ul>
<li><p>If $|\sin{x}|\leq\frac{\sqrt3}{2}$, then $$|U_{n+1}(x)|=|\sin x|\cdot|U_n(2x)|\leq\... |
2,738,244 | <p>I'm trying to understand the growth of the term $\binom{n}{k}$ -
I saw <a href="https://math.stackexchange.com/questions/1265519/approximation-of-combination-n-choose-k-theta-left-nk-right">here</a> a proof that $\binom{n}{k} = O(n^k)$. However, if $k$ is quite large (say $k=n$) then this term is not polynomial. I ... | Pete L. | 516,021 | <p>For $n$ and $k$ large, it is helpful to think of this number in terms of Stirling's approximation.</p>
<p>Recall that </p>
<p>$${n \choose k}= \frac{n!}{k! (n-k)!}$$</p>
<p>and </p>
<p>$$n!\approx \sqrt{2\pi}\cdot \frac{n^{n+1/2}}{ e^{n} }.$$</p>
<p>Hence </p>
<p>$${n \choose k}\approx \frac{1}{\sqrt{2\pi}} \... |
3,604,877 | <p>Does a parabola eventually form a sort of ellipse when stretched to infinity along its axis? I am asking because I am trying to intuitively understand the following picture and the fact that the line at infinity is a tangent of parabola: </p>
<p><a href="https://i.stack.imgur.com/J64XZ.png" rel="nofollow noreferrer... | Jonas Linssen | 598,157 | <p>When talking about weird things like „line at infinity“ it is best to be clear on the spaces one is using.</p>
<blockquote>
<p><strong>Edit</strong> in a comment to the accepted answer the op has mentioned that the context is the projective plane. What follows is therefore irrelevant. However I feel like the view... |
581,605 | <p>Source: Miklos Bona, A Walk Through Combinatorics.</p>
<p>$$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$</p>
<p>The RHS is the upper bound of the Ramsey number $R(k,k)$.</p>
<p>How can I prove the inequality without using mathematical induction? I've merely expanded the LHS to obtain $\frac{(2k-2)!}{(k-1)!(k-... | Sugata Adhya | 36,242 | <p>$\dfrac{1}{3}+\dfrac{1}{4}\dfrac{1}{2!}+\dfrac{1}{5}\dfrac{1}{3!}+\dots\\=\displaystyle\sum_3^\infty\dfrac{1}{n}\times\dfrac{1}{(n-2)!}\\=\displaystyle\sum_3^\infty\dfrac{n-1}{n!}\\=\displaystyle\sum_3^\infty\left[\dfrac{1}{(n-1)!}-\dfrac{1}{n!}\right]\\=\dfrac{1}{2!}$</p>
|
165,492 | <p>Sorry for my English if there is any mistake. The exercice for which I need help is the following:</p>
<p>Compute using complex methods:
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$</p>
<p>i) Choose the complex function to integrate.</p>
<p>I guess it is $f(z)=1/(z^2+1)$</p>
<p>ii) Choose the contour.</p>
<p>I ... | Mike | 34,878 | <p>Well, I finally did it that way. Compute using complex methods: $$I=\int _1 ^\infty \frac{\mathrm{d}x}{x^2+1}$$</p>
<p>i) Choose the complex function to integrate.
$$f(z)=\log (z-1) \frac{1}{z^2+1}$$
Singularities: $z_0 =1$ (branch point), $z_1 =\mathrm{i}$ and $z_2=-\mathrm{i}$ (simple poles).</p>
<p>ii) Choose t... |
251,118 | <p>If I have a matrix that I know it can be written as (xi.xi)*KroneckerProduct(H,xi), where xi and H are vectors. Is there a way to obtain this expression from a given matrix?</p>
| Roman | 26,598 | <p>You can find <span class="math-container">$H$</span> as the only right eigenvector that is not in the nullspace, and <span class="math-container">$\xi$</span> as the only left eigenvector that is not in the nullspace:</p>
<p>Start with a random setup:</p>
<pre><code>SeedRandom[1234];
n = 5;
H = RandomVariate[NormalD... |
185,527 | <p>I would love to understand the famous formula <span class="math-container">$g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$</span>, which is valid in Riemannian normal coordinates and possibly more general situations.</p>
<p>I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's &quo... | Robert Bryant | 13,972 | <p>Perhaps the simplest way to understand this formula is to think about how you would go about deriving it: Try to find the 'best' coordinates you can centered on a given point and see what doesn't change in such coordinates.</p>
<p>Suppose $g$ is a Riemannian metric on $M$ and $p\in M$ is fixed. Start by choosing ... |
185,527 | <p>I would love to understand the famous formula <span class="math-container">$g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$</span>, which is valid in Riemannian normal coordinates and possibly more general situations.</p>
<p>I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's &quo... | Martin Gisser | 9,161 | <p>My question has been answered in comments by Liviu Nicolaescu:</p>
<p>The (almost) ultimate proof (for my taste) is via A. Gray's formula(e) for (symmetric) higher covariant derivative(s) of normal coordinate vector fields. <strong>Any exposition of normal coordinates lacking this formula is severely lacking.</stro... |
185,527 | <p>I would love to understand the famous formula <span class="math-container">$g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$</span>, which is valid in Riemannian normal coordinates and possibly more general situations.</p>
<p>I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's &quo... | Zurab Silagadze | 32,389 | <p>Still another approach to the Riemann normal coordinates expansion formula can be found in <a href="http://arxiv.org/abs/gr-qc/9712092" rel="noreferrer">http://arxiv.org/abs/gr-qc/9712092</a> (A Closed Formula for the Riemann Normal Coordinate Expansion, by U. Mueller, C. Schubert and A. van de Ven). This approach i... |
892,580 | <p>An object moving 12m/s passes north and hits an object. Due to the wind from a west direction, it is pushed sideways at 5m/s. Find the resultant velocity.</p>
<p>I don't know where to start with this one, I can do the other ones just fine. It involves vector addition and subtraction. Would anyone know? I have the a... | Mary Star | 80,708 | <p>Using the identity $$e^{a+b}=e^a e^b$$ we have the following:</p>
<p>$$\ln(\ln|x|) = 4 \ln(t) + C$$
$$ e^{\ln (\ln|x|)} = e^{4 \ln(t)+C} \Rightarrow \ln(|x|) = e^{\ln{(t^4)}} \cdot e^C \Rightarrow \ln(|x|) = t^4\cdot e^C$$</p>
<p>$C'=e^C$
$$ e^{\ln(|x|)} = e^{C' t^4} \Rightarrow x = e^{C't^4}$$ </p>
<p>$x(1) = 3... |
395,994 | <p>Suppose that $f_{x,y}(x,y) = \lambda^2 e^{\displaystyle-\lambda(x+y)}, 0\leq x , 0\leq y.$ Find $\operatorname{Var(X+Y)}$. </p>
<p>I'm having trouble with this problem the way to find $\operatorname{Var(X+Y)} = \operatorname{Var(X)}+\operatorname{Var(Y)}+2\operatorname{Cov(X,Y)}$, however if $X$ and $Y$ are indepen... | Seyhmus Güngören | 29,940 | <p>$$f_{xy}(x,y)\neq f_X(x)f_Y(y)$$</p>
|
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | chepner | 78,700 | <p>Assuming you consider</p>
<p>$$
\sum^n_{i = 1}i = \frac{n(n+1)}{2}
$$</p>
<p>to be a well-known fact, observe that your sum is just</p>
<p>$$
\begin{array}{rcl}
\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i & = & \sum^n_{i=1}i + \sum^n_{i=1}i - n \\
&=& 2\sum^n_{i=1}i - n\\
&=& 2\frac{n(n+1)}{2} - n... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Christiaan Kruger | 153,756 | <p>Ah man. I can't believe I'm late to this party. I discovered this as well a lot of years ago and came up with my own set of proofs.</p>
<p>I noticed that:
$$1 + 2 + .. + (n -1) + n + (n - 1) + ... + 2 + 1 = n^2$$
(which is the same thing that you have)</p>
<p><em>Proof by Induction</em>:</p>
<p>Base case:
For n =... |
2,421,896 | <p>I have the following integral to find:</p>
<p>$$\int 12x^2(3+2x)^5 dx$$</p>
<p>Now, I am aware of the integration by parts property - </p>
<p>$$\int \ u \frac{dv}{dx} = uv - \int v\frac{du}{dx}$$</p>
<p>Now, my question is the following - </p>
<p>When I make $u = 12x^2$, I find a different answer to when I make... | Apollo13 | 477,333 | <p>In addition to the answer already given (namely that that it does not matter which function you pick as "$ u $", as long as it's differentiable), may I suggest to make the substitution $ u = 3 + 2x $. Then
\begin{align*}
\int 12x^2(3+2x)^5 \: \mathrm{d}x &= \int 12 \left( \frac{u-3}{2} \right)^2 u^5 \: \frac{\ma... |
3,637,653 | <p>Suppose <span class="math-container">$A,B,C\in M_2(\mathbb{C})$</span>and they are linearly independent.Try to show that there exist <span class="math-container">$x_1,x_2,x_3\in \mathbb{C}$</span> such that the matrix <span class="math-container">$x_1A+x_2B+x_3C$</span>is invertible.</p>
<p>I know the span <span cl... | orangeskid | 168,051 | <p>Assume that we have a hyperplane inside the subset of singular matrices. It follows that there exist constants <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, <span class="math-container">$d$</span>, so that
<span class="math-container">$$ x... |
339,806 | <p>The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all dissections}). </p>
<p>Now, when I searched online of how to do the integral i... | ABC | 67,567 | <p>$$\int_0^a f(x)=Lt_{n\rightarrow\infty}^{h\rightarrow0} [1/n\left( f(0)+f(h)+f(2h)+........ +f(nh) \right)]$$
$$where, (nh)\rightarrow a \ as\ n\rightarrow \infty \ and \ h\rightarrow0$$
Now expand and use formula for sum using G.P,A.P. , sum of squares , sum of cubes etc.to get a single term (or more depending on ... |
339,806 | <p>The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all dissections}). </p>
<p>Now, when I searched online of how to do the integral i... | Ron Gordon | 53,268 | <p>An interesting sidelight which may give some insight here is the following. Consider an interval $[0,1]$ and $n$ points $x_k$ chosen completely at random, except that $0<x_1<x_2<\ldots<x_{n-1}<x_n<1$. Then for any continuous function $f$ defined on this interval, the expected value of the Rieman... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | MPW | 113,214 | <p>You should divide by the total number of voters, not the total of the scores:
$$\bar{x}=\frac{5\cdot1+3\cdot2+1\cdot3+17\cdot4+2\cdot5}{5+3+1+17+2}$$
$$=\frac{92}{28}\approx 3.29$$</p>
<p>To see why this is reasonable, suppose <em>everyone</em> voted for the same number (say $2$). Then you would certainly want the ... |
502,295 | <p>Use proof by contradiction to show that every integer greater than 11 is a sum of two composite numbers</p>
<p>My Solution: </p>
<p>Statement: For all integers $x$, if $x>11$, then $x = y + z$ whereby $y$ and $z$ are any composite numbers.</p>
<p>Proof by contradiction: There exists an integer $x$ such that $x... | njguliyev | 90,209 | <p>Hint for hint: One of the numbers $x-4$, $x-6$ and $x-8$ is divisible by $3$ and all of them are greater than $3$.</p>
<blockquote class="spoiler">
<p> Therefore $x = (x-4)+4$, $x = (x-6)+6$, or $x = (x-8)+8$.</p>
</blockquote>
|
4,060,755 | <p><a href="https://i.stack.imgur.com/xJH5E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xJH5E.png" alt="![enter image description here" /></a></p>
<p>I'm having trouble understanding how to do the step highlighted in red if anyone could help</p>
| Bobby Ocean | 76,965 | <p>The representation of a function may or may not provide all the values of where a function is definable. There are whole books dedicated to writing functions in multiple different ways in order to understand the values of the function on different subsets of the complex plane (integrals, dirichlet series, power seri... |
2,000,626 | <blockquote>
<p>Is it possible that $P(A\cap B)$ is greater than $P(A)$ or $P(B)$?</p>
</blockquote>
<p>I think not.<br>
Let's assume WLOG that $P(A\cap B) > P(A)$. Then,
$$P(B|A)=\frac{P(B\cap A)}{P(A)} > \frac{P(A)}{P(A)} = 1$$</p>
<p>Contradiction.</p>
<p>Is my proof valid?</p>
| jnyan | 365,230 | <p>If that were true, for $A =B, \, \,P(A) >P(A)$.And that can't happen, hence false.</p>
|
2,508,614 | <p>In $\mathbb{Z}_7[x]$ ,write $f(x)=2x^3+2x+3$ as product of irreducibles .
make all your coefficients either $0,1,2,3,4,5$ or $6$</p>
<p>My attempt: as $f(x)=2x^3+2x+3\rightarrow f(1)=2+2+3=7=0 $</p>
<p>so we can write $f(x)=(x-1)g(x)$</p>
<p>but how to find $g(x)?$</p>
| MCT | 92,774 | <p>When the degree is low we can just do some tricks to avoid long division. We have $f(1) = 7, f(3) = 63$, so these are both roots. So it splits into at least two linear polynomials, but by degree counting the other must be linear as well. So we have
$$2x^3 + 2x + 3 \equiv 2(x-1)(x-3)(x-a) \pmod 7$$
for some $a$. The ... |
2,508,614 | <p>In $\mathbb{Z}_7[x]$ ,write $f(x)=2x^3+2x+3$ as product of irreducibles .
make all your coefficients either $0,1,2,3,4,5$ or $6$</p>
<p>My attempt: as $f(x)=2x^3+2x+3\rightarrow f(1)=2+2+3=7=0 $</p>
<p>so we can write $f(x)=(x-1)g(x)$</p>
<p>but how to find $g(x)?$</p>
| ℋolo | 471,959 | <p>You have $2x^3+2x+3$ and you want to factor out $x-1$, so we need to find some expression in the form of $(x-1)(ax^2+bx+c)+d$ to solve it.</p>
<p>Let's expend this expression:$$(x-1)(ax^2+bx+c)+d=ax^3-ax^2+bx^2-bx+cx-c+d=ax^3+(b-a)x^2+(c-b)x+(d-c)$$</p>
<p>Now compare it to the original and you get:$$\begin{cases... |
3,807,296 | <p>Suppose you have a triangulated region in the plane, the triangulation consisting of <span class="math-container">$n$</span> triangles. Take an arbitrary triangle of this triangulation and call it <span class="math-container">$\Delta_i$</span> with <span class="math-container">$1\leq i\leq n$</span>.</p>
<p>The neig... | Ben Reiniger | 493,872 | <p>Your argument resembles a heuristic argument of another well-known historical case: can you bound the chromatic number of graphs with large girth (i.e. no short cycles)? If you consider graphs with no cycles shorter than, say, 1001, can you give an upper bound on the chromatic number? It's not unreasonable to thin... |
7,492 | <p>Inspired by <a href="https://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds">this thread</a>, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space underlying a complex varie... | Dmitri Panov | 943 | <p>Another good example are Brieskorn singularities
$z_1^2+z_2^2+z_3^2+z_4^3+z_5^{6k-1}=0$, $1\le k\le 28$, if you take a little sphere in
$C^5$ centered at zero, then its intersection with the hypersurface is $S^7$ with a non-standard smooth structue. So the hypersurface is homeomerphic to $R^8$ but does not have a ... |
3,691,993 | <p>According to the figure, CA is tangent to the circle, centre O, at A. ABT and POT are straight lines. </p>
<p><strong>Question:</strong></p>
<p>Given that BT is equal to the radius of the circle, prove that:</p>
<ol>
<li><span class="math-container">$\angle ABP = 3 \angle OBP$</span> </li>
<li><span class="math-c... | Michael Rozenberg | 190,319 | <ol>
<li><span class="math-container">$$\measuredangle ABP=\measuredangle OPB+\measuredangle OTB=\measuredangle OPB+\measuredangle TOB=$$</span>
<span class="math-container">$$=\measuredangle OPB+\measuredangle OPB+\measuredangle OBP=3\measuredangle OBP.$$</span></li>
<li><span class="math-container">$$\measuredangle P... |
1,490,051 | <p>As stated on the title, my question is: (a) represent the function $ f(x) = 1/x $ as a power series around $ x = 1 $. (b) represent the function $ f(x) = \ln (x) $ as a power series around $ x = 1 $. </p>
<p>Here's what I tried:</p>
<p>(a) We can rewrite $ 1/x $ as $ \frac{1}{1 - (1-x)} $ and thus using the series... | Daniel Muñoz | 1,073,098 | <p>For the function <span class="math-container">$f:(0,\infty)\to(0,\infty)$</span> defined as:</p>
<p><span class="math-container">$$f(x)=1/x$$</span></p>
<p>and given the general Taylor's expansion formula:</p>
<p><span class="math-container">$$\sum_{k=0}^{\infty}\left(D^kf(a)\right){(x-a)^k\over k!}$$</span></p>
<p>... |
3,103,639 | <p>Let's say I want to evaluate the following integral using complex methods - </p>
<p><span class="math-container">$$\displaystyle\int_0^{2\pi} \frac {1}{1+\cos\theta}d\theta$$</span></p>
<p>So I assume this is not very hard to be solved using real analysis methods, but let's transform the problem for the real plane... | jmerry | 619,637 | <p><span class="math-container">$1+\cos\theta$</span>? Are you sure that's the right problem? That's a second-order pole in the path of integration - or, equivalently, a <span class="math-container">$\frac{c}{(\theta-\pi)^2}$</span> singularity in the real form. By an elementary comparison, this one's <span class="math... |
2,696,400 | <p>The Peano axioms are intended to be able to prove very general statements about arithmetic, such as "all natural numbers can be written as the sum of two primes".</p>
<p>However, how can we use the peano axioms to mathematically derive all the rules that are being taught to primary school children, about how to add... | Ethan Bolker | 72,858 | <p>Yes, those rules do indeed follow from the Peano axioms. Your use of "$20$" suggests that those rules are about calculation (algorithms) in base $10$. You would have first to define and prove what you need about expressing numbers in any base.</p>
<p>I assume you meant your question literally - <em>can we prove?</e... |
2,964,020 | <p>Lets say you do A x B with A being an arbitrary set and B being the empty set. How could a cartesian point actually be constructed...</p>
| Fab's | 606,512 | <p>Suppose <span class="math-container">$A\times B\neq\varnothing$</span>. Hence, by AoC, there exists an element <span class="math-container">$(a,b)\in A\times B$</span>, so, by definition, <span class="math-container">$a\in A$</span> and <span class="math-container">$b\in B$</span>, but this is not true, because B is... |
2,964,020 | <p>Lets say you do A x B with A being an arbitrary set and B being the empty set. How could a cartesian point actually be constructed...</p>
| JMoravitz | 179,297 | <p>For all sets <span class="math-container">$A$</span> one has that <span class="math-container">$A\times \emptyset = \emptyset = \emptyset \times A$</span>.</p>
<p>This is seen directly as a result of the definition of cartesian product: <span class="math-container">$A\times B = \{(a,b)~:~a\in A,b\in B\}$</span> . I... |
1,712,502 | <p>How can I find the number of solutions of this equation in interval $[0,\pi]$:
$$ 3x + \tan x = \frac{5\pi}{2}$$
I have no clue how to proceed.</p>
| jim | 289,829 | <p>Depending on your ability, plot $\tan x$ and $\frac{5 \pi}{2} - 3x$ on the same graph and look at the number of intersections.</p>
|
2,240,781 | <p>Suppose $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{f}(\mathbf{x})=\mathbf{L}$ and $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{g}(\mathbf{x})=\mathbf{K}$. I want to prove that $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{f}(\mathbf{x})\bullet \mathbf{g}(\mathbf{x})=\mathbf{L}\bullet \mathbf{K}$, where $\bullet$ denotes the... | Community | -1 | <p>The upper bound you want is $\lVert f(x)\rVert\le \lVert f(x)-L\rVert+\lVert L\rVert$</p>
|
3,073,640 | <p>I think that I know how to do this task but I need know that I have right. This is my idea: I create matrix with vectors from lin and after elementary matrix I have: <span class="math-container">$$\begin{bmatrix}
0 & 0 & 1 & 0 \\
1 & -0,5 & 0 & -1,5
\end{bmatrix}$$</span> Then I think that th... | Peter Szilas | 408,605 | <p>Consider <span class="math-container">$V_n =(-1/n,1/n)$</span>, <span class="math-container">$n \in \mathbb{Z^+}$</span>, open and dense subsets.</p>
<p><span class="math-container">$A:=\displaystyle{\cap}_{n \in \mathbb {Z^+}} V_n= $</span>{<span class="math-container">$0 $</span>}.</p>
<p>Hence <span class="math... |
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| Eric Stucky | 31,888 | <p>I think it's more typical for computers to consider the <em>mod</em> operation as meaning "add/subtract three a bunch until the number gets near zero, then output". This would explain why Google gives you $\frac12 \text{mod } 3 = \frac12$.</p>
<p>But I agree with Daniel and 162520 that the 'right' answer is $2$.</p... |
1,710,762 | <p>I understand a counterexample is when $a=4, b=-3$ and $x=y=2$.</p>
<p>Yet, I get confused because according to this Bezout's identity it states that $ax+by=(a,b)$ Therefore shouldn't the above statement be true or am I misunderstanding the identity. </p>
| Bernard | 202,857 | <p>$ax+by=c$ only means $c$ is a multiple of $\gcd(a,b)$.</p>
|
756,111 | <p>Does anyone have any intuition on remembering or very quickly deriving that</p>
<p>$$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$$</p>
<p>holds for the Laplacian in spherical coordinates? Doing the IBP is too long and slow... | hyportnex | 117,814 | <p>for small $x$ we have $e^x = 1+x + x^2/2+.....$ therefore $e^{-1/n} \approx 1-\frac{1}{n}$ and $n^2 e^{-1/n} + n e^{-1/n} - n^2 \approx n^2(1-\frac{1}{n} + \frac {1}{2n^2}...)+ n(1-\frac{1}{n}+ \frac {1}{2n^2}...) -n^2
= n^2 -n +\frac{1}{2}-... n -1 +\frac{1}{2n}....- n^2 =\frac{1}{2}-1=-\frac{1}{2}$</p>
|
1,029,489 | <p>I am studying the book "introduction to set theory", by Donald Monk, and I am having difficulties to solve some exercises about proper classes, could anybody help me?</p>
<p>here they are:</p>
<p>Prove that:
there are proper classes A, B such that $A \cap B= 0$<br>
there are proper classes A, B such that $A \subs... | Joel | 85,072 | <p>If you instead consider $$\left( \frac{x}{2} + \frac{y}{2} \right)^4$$ we know that the function $(\cdot)^4$ is convex. This leads to: $$\left( \frac{x}{2} + \frac{y}{2} \right)^4 \le \frac12 x^4 + \frac12 y^4$$</p>
<p>Multiply both sides by $16$ and we have: $$(x+y)^4 \le 8x^4 + 8y^4.$$</p>
<p>This process works ... |
1,141,357 | <p>When deriving the integration by parts formula, you can use the product rule to do so,<br>
i.e. $\{uv\}' = uv' + vu'$</p>
<p>$\Rightarrow \int \{uv\}' = \int udv + \int vdu$</p>
<p>hence $uv = \int udv + \int vdu$. </p>
<p>If $uv$ is the integral of $\{uv\}'$ then why is the formula rearranged to read</p>
<p>$... | randomgirl | 209,647 | <p>$\text{ Say you want to evaluate } \\ I=\int \ln(x) dx \\ \text{ we will first write } \\ I \text{ as an integral of a product so we can apply the integrate by parts formula }\\ I=\int 1 \cdot \ln(x) dx \\ \text{ so this is in the form } 1 \cdot \ln(x) dx \\ \text{ is in the form } u \cdot dv \\ \text{ we just need... |
2,720,539 | <p>Prove that for every $x \in(0,\frac{\pi}{2})$, the following inequality:</p>
<p>$\frac{2\ln(\cos{x})}{x^2}\lt \frac{x^2}{12}-1$</p>
<p>holds</p>
<p>I don't see room to use derivatives, since it seems a little messy to calculate the $\lim_{x\to 0}$ of $\frac{2\ln(\cos{x})}{x^2}$
(which, I think, is necessary in o... | user | 505,767 | <p>Note that since $e^x$ is strictly increasing</p>
<p>$$\frac{2\ln(\cos{x})}{x^2}\lt \frac{x^2}{12}-1\iff\ln \cos x< \frac{x^4}{24}- \frac{x^2}{2}\iff \cos x<e^{\frac{x^4}{24}- \frac{x^2}{2}}$$</p>
<p>and since</p>
<ul>
<li>$e^x>1+x$</li>
<li>$\cos x < 1-\frac{x^2}{2}+\frac{x^4}{24}$ </li>
</ul>
<p>(re... |
2,720,539 | <p>Prove that for every $x \in(0,\frac{\pi}{2})$, the following inequality:</p>
<p>$\frac{2\ln(\cos{x})}{x^2}\lt \frac{x^2}{12}-1$</p>
<p>holds</p>
<p>I don't see room to use derivatives, since it seems a little messy to calculate the $\lim_{x\to 0}$ of $\frac{2\ln(\cos{x})}{x^2}$
(which, I think, is necessary in o... | Crostul | 160,300 | <p>You need the following two inequalities:
$$\cos x = 1- \frac{x^2}{2}+ \frac{x^4}{24}-\frac{x^6}{6!} + \cdots < 1- \frac{x^2}{2}+ \frac{x^4}{24}$$
and
$$\ln (1+y) = y -\frac{y^2}{2}+ \cdots\le y$$</p>
<p>(you can formally prove them using derivatives, if you want).</p>
<p>Combining them substituting $y= - \frac... |
2,720,539 | <p>Prove that for every $x \in(0,\frac{\pi}{2})$, the following inequality:</p>
<p>$\frac{2\ln(\cos{x})}{x^2}\lt \frac{x^2}{12}-1$</p>
<p>holds</p>
<p>I don't see room to use derivatives, since it seems a little messy to calculate the $\lim_{x\to 0}$ of $\frac{2\ln(\cos{x})}{x^2}$
(which, I think, is necessary in o... | Umberto P. | 67,536 | <p>First note that $\sec^2 x > 1$ if $x \in (0,\frac \pi 2)$, so that $$\tan x = \int_0^x \sec^2 t \, dt > \int_0^x 1 \, dt = x$$ for all $x \in (0,\frac \pi 2)$. Similarly,
$$ - \ln (\cos x) = \int_0^x \tan t \, dt > \int_0^x t \, dt = \frac{x^2}{2}$$ for all $x \in (0,\frac \pi 2)$. Consequently
$$\frac{2 \... |
4,569,871 | <p>I have a question on the equality of the ring of regular functions <span class="math-container">$\mathcal{O}_X(X)$</span> on <span class="math-container">$X\subset\mathbb{A}^n$</span> and the affine coordinate ring <span class="math-container">$A(X)$</span>. Some sources state it is as an isomorphism (Harthshorne, c... | cigar | 1,070,376 | <p>If <span class="math-container">$a$</span> and <span class="math-container">$b$</span> commute, then we immediately get <span class="math-container">$\lvert ab\rvert \mid \rm{lcm}(\lvert a\rvert, \lvert b\rvert)$</span>. <em>Without</em> equality however. This is easy to see.</p>
<p>If <span class="math-container"... |
3,991,914 | <p>I've been thinking about this one for a while, at first I thought it only to be valid if <span class="math-container">$U,W$</span> were open (the union of open sets in <span class="math-container">$\mathbb{R}^{n}$</span> is open and <span class="math-container">$\mathbb{R}^{n}$</span> is an open set), then I remembe... | Duncan Ramage | 405,912 | <p>No, of course not. Consider <span class="math-container">$n = 1$</span> and, say, <span class="math-container">$U = \mathbb{Q}$</span> and <span class="math-container">$W = \mathbb{R} - \mathbb{Q}$</span>.</p>
|
177,102 | <p>I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find a... | Neil Strickland | 10,366 | <p>In Maple you can just do </p>
<pre><code>with(Optimization):
g := (your function):
Minimize(g,iterationlimit = 200);
</code></pre>
<p>On my machine this takes only about 1.5 seconds to return the following:</p>
<pre><code>[2.35579022955789696*10^(-9),
[x0 = .696531801759957, x1 = .286105658731833, x10 = .3429734... |
1,360,394 | <p>For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.</p>
| Community | -1 | <p><strong>Hint</strong>:</p>
<p>$$=|======\ ==|=====\ ===|====$$</p>
<p>Think of a cursor that you can move. How many positions are allowed ?</p>
|
3,546,184 | <p>Take three numbers <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, and <span class="math-container">$x_3$</span> and form the successive running averages <span class="math-container">$x_n = (x_{n-3} + x_{n-2} + x_{n-1})/3$</span> starting from <span class="math-container">$x_4$<... | emacs drives me nuts | 746,312 | <p>With
<span class="math-container">$$
v_n:=
\begin{pmatrix}
x_{n}\\
x_{n-1}\\
x_{n-2}\\
\end{pmatrix}
\qquad\text{and}\qquad
M:=\frac13
\begin{pmatrix}
1&1&1\\
3&0&0\\
0&3&0\\
\end{pmatrix}
$$</span>
we can write:
<span class="math-container">$$v_n = M v_{n-1} = M^{n-3}v_3
\;\text{ for }\; n \... |
1,212,433 | <p>Problem statement:</p>
<p>Let $ f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ be defined as $ f(m, n) = (3m + 7n, 2m + 5n) $. Is $f$ a bijection, i.e., one- to-one and onto? If yes then give a formal proof, based on the definitions of one-to-one and onto, and derive a formula for $ f^-1 $. If ... | Fermat | 83,272 | <p>The function is onto since for any given $(a,b)\in \mathbb Z\times \mathbb Z$ we have $$f(-7b+5a,3b-2a)=(a,b)$$
To see why the function is one-to-one, assume that $f(m_1 ,n_1)=f(m_2,n_2)$ then
$$3(m_1-m_2)=7(n_2-n_1)$$ and $$2(m_1 -m_2)=5(n_2-n_1)$$ multiplying both side of these two equation by $5$ and $7$ respect... |
4,143,585 | <p>Recently, I came across <a href="https://www.youtube.com/watch?v=joewDkmpvxo" rel="noreferrer">this</a> video, the method shown seemed good for a few logarithms. Then I tried to plot the equation <span class="math-container">$$ \frac{x^{\frac{1}{2^{15}}}-1}{0.000070271} $$</span> and it looks <em>exactly</em> like t... | Jair Taylor | 28,545 | <p>This can be derived using the approximation <span class="math-container">$e^x \approx 1+x$</span> for small <span class="math-container">$x$</span>. We have</p>
<p><span class="math-container">\begin{align*}
x^{2^{-15}} &= e^{2^{-15} \ln x } \\
&\approx 1 + 2^{-15} \ln x
\end{align*}</span></p>
<p>and so
<s... |
711,122 | <p>I'm trying to derive the equation </p>
<p>$$y = (2x-6)^4$$</p>
<p>I thought that it would be</p>
<p>$$\frac{dy}{dx} = 8(2x-6)^3$$</p>
<p>Wolframalpha says $dy/dx = 64(x-3)^3$</p>
<p>Who's correct? I thought it would be a simple calc-1 chain rule. </p>
| Fly by Night | 38,495 | <p>There is no error. There is a common fact of $2$ in $(2x-6)$. That means we can write
\begin{eqnarray*}
(2x-6)^4 &\equiv& \left[2(x-3)\right]^4 \\ \\
&\equiv& 2^4\cdot(x-3)^4 \\ \\
&\equiv& 16(x-3)^4
\end{eqnarray*}
If you differentiate this then you will get what $64(x-3)^3$ as Wolfram Alpha... |
1,158,666 | <p>I know that every closed and bounded set in $\mathbb{R}$ is compact (like $[a,b]$)</p>
<p>so i can conclude that every bounded set in $\mathbb{R}$ is relatively compact, by contradiction i say that let $A\subset\mathbb{R}$ be a bounded set but not relatively compact it means that $\overline{A}$ is not compact, but ... | Haha | 94,689 | <p>Suppose $A$ is bounded so $|x-y|\leq M$ for $x,y \in A$. If $\overline A$ is not bounded then there is a $z\in \overline A$ such that $dist(z,A)>0$. But $z\in \overline A$ means $dist(z,A)=0$. Contradiction.</p>
|
225,813 | <p>Let M be a finite dimensional von Neumann algebras with a normal faithful trace. Let e and f be two projections with rank 1. I want to know if e and f have identical traces. (This is obviously true if M is a factor.)</p>
<p>I guess it is false, while i have no counterexample.</p>
| Sam Nead | 1,650 | <p>This follows, fairly easily, from the hypothesis of irreducibility and from the "annulus theorem" (see page 130 of Jaco-Shalen's book "Seifert Fibered Spaces in 3-Manifolds"). You can remove the hypothesis of irreducibility if you are willing to use the Poincaré conjecture. </p>
|
2,860,013 | <p>I'm trying to prove that the following system of congruence equations has a solution:</p>
<p>$X \equiv 2 $ (mod $5^N$)</p>
<p>$X \equiv 1 $ (mod $7^N$)</p>
<p>$X \equiv 4 $ (mod $6^N-4$)</p>
<p>being $N$ an integer number, $N\geq 2$</p>
<p>I guess that this may be answered using the <a href="https://en.wikipedi... | JJacquelin | 108,514 | <p>$$(y(x)-x)y'(x)=1$$
$$y-x(y)=x'(y)$$
This first order linear ODE is easy to solve :
$$x(y)=c\,e^{-y}+y-1$$
The condition $x(0)=0$ implies $c=1$
$$x(y)=e^{-y}+y-1$$
The condition $x(1)=a$ implies $x(1)=a=e^{-1}+1-1$
$$a=\frac{1}{e}$$</p>
|
2,720,922 | <blockquote>
<p>With $0<a,b,c<1$,
\begin{align}
\begin{bmatrix}
X_n \\
Y_n
\end{bmatrix}=\begin{bmatrix}
a^2 & (1-b)^2 \\
(1-a)^2 & b^2
\end{bmatrix}^n \begin{bmatrix}
c^2 \\
(1-c)^2
\end{bmatrix}
\end{align}
How fast does $X_n+Y_n$ go to zero?</p>
</blockquote>
<p>This is coming from a Marko... | lulu | 252,071 | <p>There are no such functions. To see that, suppose we had such a function. We will derive a contradiction.</p>
<p>First note that the constant function $0$ does not work, hence there must be some value $y_0$ for which $f(y_0)\neq 0$. </p>
<p>Taking $x=0, y=y_0$ then yields $$f(y_0)=f(0)\,f(y_0)\implies f(0)=1$$<... |
912,030 | <p><img src="https://i.stack.imgur.com/FTQSg.png" alt=""></p>
<p>Hallo everybody,</p>
<p>I have the following problem regarding shortest paths in $R^2$.</p>
<p>Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture.
I am looking for a path from $p$ to $q$ through a point $c_1$ in the firs... | David | 119,775 | <p>Solving your equation gives
$$\sin3y=3\cos^2x$$
and hence
$$3y=\arcsin(3\cos^2x)+2k\pi\quad\hbox{or}\quad 3y=-\arcsin(3\cos^2x)+(2k+1)\pi$$
for some integer $k$. Checking the initial condition, these give respectively
$$\pi=2k\pi\quad\hbox{or}\quad\pi=(2k+1)\pi\ .$$
The first is impossible, the second gives $k=0$ a... |
3,767,656 | <p>Using L'hopital rule: <span class="math-container">$\lim_{x \to x_i, y \to y_i} \frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}} = \lim_{x \to x_i, y \to y_i} \frac{\frac{d(x-x_i)}{dx}}{\frac{d\sqrt{(x-x_i)^2 + (y-y_i)^2}}{dx}} = \lim_{x \to x_i, y \to y_i} \frac{1}{\frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}}} = \lim_{x \t... | boreal | 811,202 | <p>L'hopital rule is for one variable not multiple variables,
you have a problem lim (x->x0) lim(y->y0) f(x,y) =/= lim(y->y0) lim (x->x0) f(x,y)</p>
<p>in the first case the limit is 1 and in the second case lim(y->y0) sqrt((y-y0)^2)/0 which is infinite.</p>
<p>So the answer is no, the limit for multiple... |
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Pandelis Dodos | 3,096 | <p>This is not an answer to your question about the Erdos-Rado theorem, but a remark concerning another example of a forcing proof of a ZFC result (a result which is actually quite useful and observed by several authors). For every infnite subset $L$ of $\mathbb{N}$ denote by $[L]^{\infty}$ the set of all infinite subs... |
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Timothy Chow | 3,106 | <p>Matthew Wiener once explained to me that because of the close connections between forcing and the Baire category theorem, forcing could be used to prove certain results in analysis that are more commonly proved via BCT. Unfortunately, all I've been able to find is <a href="https://groups.google.com/forum/#!msg/sci.... |
3,722,842 | <p>Kevin wants to fence a rectangular garden using <span class="math-container">$14$</span> rails of
<span class="math-container">$8$</span>-foot rail, which cannot cut. What are the dimensions of the rectangle that will maximize the fenced area?</p>
<p>So the number of rails in each dimension the rectangle could be ei... | poetasis | 546,655 | <p>The closest we can come to a square is a rectangle with <span class="math-container">$2$</span> sets of adjacent sides, each with a semi-perimeter of <span class="math-container">$\frac{14}{2}=7$</span> rails. The best we can do from here is to make each of the <span class="math-container">$2$</span> "corners&q... |
75,355 | <p>A groupoid is a category in which all morphisms are invertible.(*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits both left and right (weak) adjoints. So you can localize (or <em>complete</em>) a category to a groupoid. If E denote... | Andreas Blass | 6,794 | <p>Here's a different (as far as I can see) proof for the same example Todd gave, the monoid of endofunctions of $\mathbb N$. Consider the following elements of this monoid: </p>
<ul>
<li><p>$b$ is the bijection that interchanges $2n$ with $2n+1$ for all $n$.</p></li>
<li><p>$e:n\mapsto 2n$ and $f:n:\mapsto 2n+1$.</p... |
2,059,571 | <blockquote>
<p>Let $$V=\{(x_1,x_2,x_3,\dots,x_{100})\in\mathbb{R}^{100}\,|\, x_1=x_2=x_3 \text{ and } x_{51}=x_{52}=x_{53}= \dots=x_{100}\}$$ What is $\dim V$?</p>
</blockquote>
<p>If W is a subspace of vector space $V$ then $$\dim W = \dim V - \text{number of linearly independent restrictions}$$
In our case $\dim ... | Fimpellizzeri | 173,410 | <p>Each 'equal sign' is a different, linearly independent restriction. So you have $2 + 49 = 51$ restrictions.</p>
<p>Another way to think about this is in terms of degrees of freedom. Choosing a value for $x_1$ fills in $3$ of the coordinates, and choosing a value for $x_{51}$ fills in another $50$ coordinates. For t... |
124,530 | <p>I want to have a function value of an expression where some variables are solutions to some set of equations, with some values of parameters. I had an idea to use pure functions for that. </p>
<p>However, since both the expression and the equations are lengthy I'd like to place them in a separate expressions define... | Alexey Popkov | 280 | <p>Your issue is quite simple and fundamental: you try to use <em>lexical scoping</em> (<code>SetDelayed</code> and <code>Function</code>) as it would be <em>dynamic scoping</em> (<code>Block</code>). So your problem can be solved easily by outsourcing the dynamic scoping to <code>Block</code> with minimal modification... |
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| John Joy | 140,156 | <p>Try this as an exercise in geometry. Show that if you fit a sphere into a cylinder just big enough to contain it, that the cylinder and the sphere have equal area. In other words, the projection of a sphere onto its containing cylinder is area preserving. I think that Archimedes did something similar.</p>
<p><img s... |
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| Ant | 66,711 | <p>Without integrals, albeit a little less rigorous:</p>
<p>Take half a sphere. Imagine to press it on a plane until it has become a circle.</p>
<p>Of course the surface area is still the same; what's the radius of the circle?</p>
<p>it is clearly $r\sqrt{2}$, where $r$ is the radius of the sphere (basically the poi... |
3,812,773 | <p>Please some hint on how to solve in the set of natural numbers
<span class="math-container">$$x^{100} − y^{100} = 100!$$</span>
The question comes from the <a href="https://dms.rs/wp-content/uploads/2020/08/JSMO2020.pdf" rel="nofollow noreferrer">Serbian Junior Mathematical Olympiad 2020</a>.</p>
<p>I have tried wit... | tkf | 117,974 | <p>You have shown that a solution requires <span class="math-container">$x=101k$</span>. Therefore <span class="math-container">$$x^{100}-y^{100}\geq101^{100}-100^{100}\geq100\times100^{99}=100^{100}>100!$$</span></p>
<p>The first inequality follows from <span class="math-container">$(101k)^{100}-y^{100}$</span> ta... |
2,264,252 | <p>If a function $f: M \rightarrow N$ is surjective, does it also mean it has an inverse? or is it the same thing?</p>
| Martin Argerami | 22,857 | <p>$$
0=\ln1=\ln\left(x\times\frac1x\right)=\ln x+\ln\frac1x.
$$
It follows that
$$
\ln\frac1x=-\ln x.
$$</p>
|
1,284,820 | <blockquote>
<p>How would this differential equation be solved?
$$y{\partial z\over \partial x}+z{\partial z\over \partial y}={y \over x}$$</p>
</blockquote>
<p>I was taught to solve them like : $${dx \over y}={dy \over z}={dz \over {y \over x}}$$ Then find the constants $c_1$ and $c_2$ and the answer being $F(c_1... | Crostul | 160,300 | <p>$$\frac{1}{2}\sum_{i=0}^k (x-i)(x-i-1) = \frac{1}{2} \left( \sum_{i=0}^k (x^2-2xi+i^2 + x-i) \right) =$$
$$=\frac{1}{2} \left( \sum_{i=0}^k (x^2+ x) + \sum_{i=0}^k (-2xi-i) + \sum_{i=0}^k i^2\right)= \frac{1}{2} \left( (k+1)(x^2+x) -(2x+1)\left(\frac{k(k+1)}{2} \right) + \frac{k(k+1)(2k+1)}{6}\right)$$</p>
|
1,771,075 | <p>I have a large number of data sets that have either a unimodal normal distribution or a bimodal normal distribution. I'm not a statistician by any means, so I'm quite limited in my experience.</p>
<p>For the bimodal data sets, I have implemented (through a library) the Expectation-Maximization method for identifyin... | BruceET | 221,800 | <p>The bimodal data you have may be a mixture of normal components,
but that mixture is not normal. Thus it may be enough for you
to use ordinary tests of normality. Most software packages incorporate such tests. I will show you briefly how R statistical software can
be used for this purpose.</p>
<p>First, some bimoda... |
1,663,244 | <p>How to represent a graph in a function?</p>
<p>For example, I used 3 functions : </p>
<p>$$f(x)=x^2$$
$$g(x)=x$$
$$h(x)=3$$</p>
<p>These 3 functions were plotted on the same graph and the result (after edit) is as given below</p>
<p>How would you represent the below graph in a function, lets say $k(x)$ ?</p>
<p... | Étienne Bézout | 92,466 | <p>Depending on the context, it might also be useful to represent the function using Heaviside's step function $H(x)$, which equals $0$ for $x < 0$ and $1$ for $x > 0$. You could also define $H(0)=0$ or $H(0) = 1$ if you want, but it might not be so important, again depending on the context.</p>
<p>With this not... |
3,486,036 | <p>Given a complete, weighted graph as the input to TSP, is the edge from <span class="math-container">$i$</span> to <span class="math-container">$j$</span> with minimum weight always in the solution?</p>
| bof | 111,012 | <p>Consider the four points <span class="math-container">$P(-2,0)$</span>, <span class="math-container">$Q(0,1)$</span>, <span class="math-container">$R(2,0)$</span>, <span class="math-container">$S(0,-1)$</span> with edges weighted by the Euclidean distance. The shortest edge is <span class="math-container">$QS$</span... |
2,917,858 | <p>Consider the vector space of all functions $f: \mathbb{R} \rightarrow \mathbb{C}$ over $\mathbb{C}$. If $W$ is a subspace spanned by $\beta$ = $\{1, e^{ix}, e^{-ix}\}$, show that $\beta$ is a basis for $W$.</p>
<p>I think I am very confused - I know I just have to show that $\beta$ is linearly independent, which me... | Neal | 20,569 | <p>You need to show that
$$ a + be^{ix} + ce^{-ix} = 0 \Rightarrow a, b, c = 0 $$
<em>for all $x$</em>.</p>
<p>The vector space you're considering is linear combinations of <em>functions</em>, so the zero on the right of the equation is the zero function, not the number zero.</p>
|
2,016,540 | <p>Can someone help me prove this transition? </p>
<p><a href="https://i.stack.imgur.com/u4YVA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/u4YVA.png" alt="enter image description here"></a></p>
| Olivier Oloa | 118,798 | <p>By the change of index $k=n-j$ in the first sum, one gets the second sum.</p>
|
553,571 | <p>We <a href="http://www.maa.org/programs/maa-awards/writing-awards/the-circle-square-problem-decomposed" rel="noreferrer">know</a> that there is no paper-and-scissors solution to <a href="http://en.wikipedia.org/wiki/Tarski%27s_circle-squaring_problem" rel="noreferrer">Tarski's circle-squaring problem</a> (my six-yea... | tim_hutton | 11,400 | <p>Not really an answer but there are some fantastic dissections on <a href="http://mathworld.wolfram.com/Dissection.html" rel="noreferrer">this page</a>, including these two:</p>
<p>Dissecting an octagon into a square with five pieces:</p>
<p><img src="https://i.stack.imgur.com/wv24h.png" alt="enter image descriptio... |
553,571 | <p>We <a href="http://www.maa.org/programs/maa-awards/writing-awards/the-circle-square-problem-decomposed" rel="noreferrer">know</a> that there is no paper-and-scissors solution to <a href="http://en.wikipedia.org/wiki/Tarski%27s_circle-squaring_problem" rel="noreferrer">Tarski's circle-squaring problem</a> (my six-yea... | Zachary Krueger | 451,195 | <p>I believe I have found the optimal method for n=6 and n≥10.</p>
<p>Above you state that an inscribed dodecagon yields coverage of 95.49%, which you demonstrate is not the best answer. You were on the right track, but a better approach is to overlay the circle with a dodecagon of <em>equal area</em>, and then dissec... |
1,723,331 | <blockquote>
<p>$$\frac{1}{1-x^2}$$</p>
</blockquote>
<p>$$\frac{1}{1-x^2}=\frac{a}{1-x}+\frac{b}{1+x}$$</p>
<p>$$1=a+ax+b-bx$$</p>
<p>$$1=a+b+x(a-b)$$</p>
<p>$a+b=1$ and $x(a-b)=0\Rightarrow a-b=0\Rightarrow a=b$</p>
<p>$$2a=1\Rightarrow a=\frac{1}{2}$$</p>
<p>$b=\frac{1}{2}$</p>
<p>$$\frac{1}{1-x^2}=\frac{1}... | Jan Eerland | 226,665 | <blockquote>
<p>Notice:</p>
<ul>
<li><span class="math-container">$$\frac{1}{1-x^2}=\frac{1}{-(x-1)(x+1)}=-\frac{1}{(x-1)(x+1)}$$</span></li>
</ul>
</blockquote>
<p>So, we get:</p>
<p><span class="math-container">$$-\frac{1}{(x-1)(x+1)}=-\frac{a}{x-1}-\frac{b}{x+1}$$</span></p>
<p>And, now we can see that:</p>
<p><span... |
32,797 | <p>Let $\phi\colon F\to G$ be a homomorphism of finitely generated abelian groups. If $F$ is free, then $\ker(\phi)$ is also free and thus admits a basis.</p>
<p>Question: <em>Is there a general procedure to find a basis for $\ker(\phi)$?</em></p>
<hr>
<p>As an example, consider the homomorphism $\phi\colon\mathbb Z... | Arturo Magidin | 742 | <p>One way is to proceed recursively, by considering subgroups of $\mathbb{Z}^k$ as $k$ increases.</p>
<p>For $k=1$, you are considering subgroups of $\mathbb{Z}$, which are straightforward: they are generated by the least positive element in the subgroup (if there is one; otherwise, they are the zero subgroup).</p>
... |
643,505 | <p>I am struggling to solve the following equation numerically: $x'^2 - xx''=W(t)$, where W(t) is a sinusoidal function, only known by its samples (i.e. no analytic form is known).
Up until now I tried to write it as two first-order ODEs: $\begin{cases}v'=u \\ u'=\displaystyle\frac{u^2-W(t)}{v}\end{cases}$ and find a s... | Chris Rackauckas | 244,186 | <p>ode45 would not be the way to go for this type of problem since you're going to have oscillations around zero. These oscillations translate to large eigenvalues in the Jacobian, meaning you will need a highly-A-stable method. You can give ode15s a try, though you may need to use an implicit solver like Implicit Eule... |
679,807 | <p>I am trying to find the value of $2^{-1-i}$. </p>
<p>I rewrite it like this, $2^{-1-i}=e^{\ln(2)(-1-i)}={1\over{e^{\ln2}e^{i\ln2}}}=1/2$</p>
<p>Since $e^{i\ln2}=e^{Re(i\ln2)}=e^0=1$. </p>
<p>This looks way nicer than it should be, I think, can anyone tell me where I go wrong, and maybe a good way to do this probl... | viplov_jain | 129,167 | <p>$e^{iln2} \neq e^{Re(iln2)}$
$$e^{iln2}=cos(ln2)+isin(ln2)$$
and, $$2^{-1-i}=\frac{(cos(ln2)-isin(ln2))}{2}$$
As u expected this doesnt look good.</p>
|
507,657 | <p>I tried many approaches but none of them really worked I treated $p^b-1$ as a Geometric progression but it didn't work and that is about as far as I have been able to go I have no clue how to move forward</p>
| Community | -1 | <p>In the group $(\mathbb Z_a^*,\times)$ since $p$ isn't a multiple of $a$ then $\gcd(p,a)=1$ and then $\overline p\in \mathbb Z_a^*$ and let $b=o(\overline p)$ then
$$\overline p^b=\overline 1\iff p^b-1\equiv 0 \mod a$$</p>
|
4,322,437 | <p>I'm attempting some questions from Abstract Algebra Theory and Applications by Judson (found <a href="http://abstract.ups.edu/index.html" rel="nofollow noreferrer">here</a>) and this one is a bit problematic.</p>
<p>Let <span class="math-container">$S = R\setminus \{−1\}$</span> and define a binary operation on <spa... | Shaun | 104,041 | <p>Here <span class="math-container">$-1$</span> is an obstacle, essentially, because of the following.</p>
<p><strong>Lemma:</strong> Let <span class="math-container">$e$</span> be the identity of a group <span class="math-container">$G$</span>. Then the only idempotent of <span class="math-container">$G$</span> is <s... |
725,547 | <p>This is from the Chapter 15 text of Gourieroux and Monfort's Statistics and Econometric Models II:</p>
<p><strong>Set Up</strong>: Suppose that there are 2 possible parameter values $\theta_0$ and $\theta_1$ from which there are 2 density functions $l_{\theta_0}(y)$ and $l_{\theta_1}(y)$ on the data. Define
$$
F(k)... | William Ballinger | 79,615 | <p>At the beginning of the step you asked about, we have that $x = apq + r$. When we work modulo $a$, $a=0$, so $$\begin{align} x \pmod{a} &= \\ apq + r \pmod{a} &= \\apq\pmod{a} + r\pmod{a} &= \\0 + r \pmod{a} &= \\r\pmod{a}.\end{align}$$</p>
|
3,031,174 | <p><span class="math-container">$$\lim_{x\to2}\left(\frac{1}{x(x-2)^2}-\frac{1}{x^2-3x+2}\right)$$</span></p>
<p><strong>What I tried:</strong> Got the fraction to the same denominator
<span class="math-container">$$\begin{align}
\lim_{x\to2}{\frac{x^2-3x+2-x(x-2)^2}{x(x-2)^2(x^2-3x+2)}}&=\lim_{x\to2}{\frac{(x-2)(... | user | 505,767 | <p>By <span class="math-container">$y=x-2 \to 0$</span> we have that</p>
<p><span class="math-container">$$\lim_{x\to 2} \left({\frac{1}{x(x-2)^2}-\frac{1}{x^2-3x+2}}\right)=\lim_{y\to 0} \left({\frac{1}{y^2(y+2)}-\frac{1}{y(y+1)}}\right)$$</span></p>
<p>and</p>
<p><span class="math-container">$${\frac{1}{y^2(y+2)}-... |
1,258,199 | <p>The question I need help is:</p>
<blockquote>
<p>Prove that $U(\mathbb I_9) \cong \mathbb I_6$ and $U(\mathbb I_{15}) \cong \mathbb I_4 \times \mathbb I_2$.</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/2fi10.jpg" alt="enter image description here"></p>
<p>U() is the group of units in a ring</p>
<p>... | Martin Sleziak | 8,297 | <blockquote>
<p>I don't know what a group of units is and the like. </p>
</blockquote>
<p><a href="https://en.wikipedia.org/wiki/Unit_%28ring_theory%29" rel="nofollow">Unit</a> (also called invertible element) of a ring $R$ is an element $a\in R$ for which there exists exists $b\in R$ such that $ab=1$. I.e., it is a... |
2,130,062 | <p>I have 4 decks, each with five unique cards. If I select 3 from the first, 4 from the second, 2 from the third and 2 from the fourth. In case order matters and I do not put cards back, in how many ways can I arrange the 11 cards if the order matters for each selected card? </p>
<p>I think that I use the following f... | Mark Bennet | 2,906 | <p>Actually the "ugly way" is not too tricky, because there are some easy intermediate cancellations. Expanding using the first row gives $$(a^2+1)(b^2+c^2+1)-ab(ab)+ac(-ac)=a^2+b^2+c^2+1$$</p>
|
4,130,129 | <p>I was thinking that it might has to be <span class="math-container">$m$</span> and <span class="math-container">$n$</span> coprimes, but I don't have a consolidated idea of how I can prove it. Incidentally, how could I prove that it doesn't work for any integers? (is there any counterexample? I was thinking about <s... | Narasimham | 95,860 | <p>HINTS:</p>
<p>The polar relation</p>
<p><span class="math-container">$$( r/a)^2= \sin \theta , \text{ which wlog can be written as :} $$</span></p>
<p><span class="math-container">$$( r_A= \sqrt{\sin \gamma \;E_B},\; r_B= \sqrt{\sin \delta \;E_A} ) $$</span></p>
<p>represent a drop / flattened circle shapes of maxi... |
95,709 | <p>Let $F$ be a fixed free group of finite rank. If $H$ is a finitely generated subgroup of $F$ and $A$ is a basis for $F$, then we can form the Stallings graph $\Gamma_A(H)$ for $H$. It is the unique smallest ($|A|$-labelled) subgraph of the covering space of a bouquet of $A$-circles corresponding to $H$ that contains... | Ian Agol | 1,345 | <p>If you take a free group on two generators to be $F=\langle a,b\rangle$, and take an index 2 subgroup to be $K$, then the subgroup $H$ generated by $[a,b]$ is not a geometric free factor in $K$ for any choice of generators for $F$ (but is a free factor). One may verify this for all three 2-generator subgroups of $F$... |
2,782,489 | <p>The solutions to a tutorial question I am working on are as follows:</p>
<p><span class="math-container">$$\cos\left(\frac{p}{\sqrt{\eta}}x\right) = \cos\left(\frac{p}{\sqrt{\eta}}x\right)\cos\left(\frac{p}{\sqrt{\eta}}2L\right)-\sin\left(\frac{p}{\sqrt{\eta}}x\right)\sin\left(\frac{p}{\sqrt{\eta}}2L\right)$$</span>... | amWhy | 9,003 | <p>Hint: The RHS gives $$\cos\left(\frac p{\eta} x +\frac p{\eta}2L\right)^{(\dagger)}$$</p>
<p>So what can you conclude if $$\cos\left(\frac p\eta x\right) = \cos\left(\frac p{\eta} x +\color{blue}{\frac p{\eta}2L}\right)\;?$$</p>
<p>This equality only occurs when $\color{blue}{\dfrac {2Lp}{\eta}} = 2n\pi$, where $... |
2,030,841 | <p>Let $a,b \in \mathbb {Z} $ and let $m$ be an integer greater than $2$. I found a counterexample to the equation</p>
<p>$$(a+b)\mathrm {mod} m = a\mathrm {mod} m + b \mathrm {mod} m $$</p>
<p>where $m>2$. But that was only after I thought I had proven that the equation does hold, so I was wondering if someone co... | Bernard | 202,857 | <p>Actually, the formula should be
$$(a+b)\bmod m\equiv (a\bmod m+b\bmod m)\bmod m.$$
That's why computing in the ring $\mathbf Z/m\mathbf Z$ is simpler than computing with congruence classes in $\mathbf Z$.</p>
|
3,845,602 | <p>I was reading Dummit and Foote and encountered the following statement: any two elements in <span class="math-container">$S_n$</span> are conjugate if and only if they have the same cycle types.</p>
<p>However, I am able to produce a counter example:</p>
<p>Let <span class="math-container">$(1 2 3)$</span> and <span... | Community | -1 | <p>Of course a permutation is conjugate to itself. It has the same cycle type as itself, as well. You could just as well have conjugated by the identity.</p>
<hr />
<p>Conjugation takes <span class="math-container">$k$</span>-cycles to <span class="math-container">$k$</span>-cycles: <span class="math-container">$\pi... |
4,489,437 | <p>Is it possible to find the value of n (n belongs to Real number) such that it satisfies the equation: <span class="math-container">$2^n = n^8$</span> <strong>without any help of computer or graph generator</strong> (i.e. only manually). If possible, please explain and if it requires explanation of some large number ... | Thomas Andrews | 7,933 | <p>The closed form for the solutions can be expressed in terms of the <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow noreferrer">Lambert W function</a>, a non-elementary function.</p>
<p>You have: <span class="math-container">$$e^{n\log 2/8}=\pm n$$</span></p>
<p>Multiplying both sides by <spa... |
4,489,437 | <p>Is it possible to find the value of n (n belongs to Real number) such that it satisfies the equation: <span class="math-container">$2^n = n^8$</span> <strong>without any help of computer or graph generator</strong> (i.e. only manually). If possible, please explain and if it requires explanation of some large number ... | Claude Leibovici | 82,404 | <p><em>All done by hand</em></p>
<p>Suppose that we look for the zeros of function
<span class="math-container">$$f(x)=2^x-x^8$$</span>
A very quick expansion shows that there are roots close to <span class="math-container">$\pm 1$</span>.</p>
<p>Using Taylor expansion
<span class="math-container">$$f(x)=-\frac{1}{2}+... |
259,388 | <p>Let $A$ be a real $2\times 2$ matrix such that $\det A=1$, show that $\|A\|=\left\|A^{-1}\right\|$.</p>
<p>Any hint would be appreciated, thanks.</p>
<p>EDIT: $\|\cdot\|$ is the operator norm $\|A\|=\max_{\|x\|=1}\|Ax\|$, all vector norms are Euclidean norms.</p>
| Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Use the definition of the inverse matrix</p>
<blockquote>
<p>If A is invertible, then $$ \det(\mathbf{A}) \mathbf{A}^{-1}=\mathrm{adj}(\mathbf{A}) , $$
where $\mathrm{adj}(\mathbf{A})$ is the adjugate matrix. </p>
</blockquote>
<p>The adjugate of the 2 × 2 matrix</p>
<p>$$\mathbf{A} = ... |
2,757,743 | <p>It is clear that the Galois group of $\mathbb{Q}(\omega):\mathbb{Q}$ where $\omega$ is a primitive root of unity is a cyclic group of order 10. Let $\sigma$ be a primitive element then $\sigma^2$ has order 5. </p>
<p>I want to find an $\alpha$ such that $Fix(<\sigma^2>)=\mathbb{Q}(\alpha)$.</p>
<p>Using the ... | C Monsour | 552,399 | <p>Add up the five $11^{th}$ roots in an orbit of your group element $\sigma^2$ of order $5$, and that is the $\alpha$ you seek.</p>
<p>In other words $\omega +\omega^4 +\omega^5 +\omega^9 +\omega^3$.</p>
<p>You can then see that this has minimal polynomial $x^2+x+3$.</p>
|
140,117 | <p>Does anyone know how one can plot contour lines on a meshed surface? For example I would like to be able to plot contour lines indicating lines of equal distance from a given point such as data coming from (<a href="https://mathematica.stackexchange.com/questions/129207/how-to-estimate-geodesics-on-discrete-surfaces... | Jason B. | 9,490 | <p>You can feed this data almost as-is to <a href="http://reference.wolfram.com/language/ref/ListSliceContourPlot3D.html" rel="noreferrer"><code>ListSliceContourPlot3D</code></a>,</p>
<pre><code>surf = MeshRegion[vertices, Map[Polygon, tris]];
ListSliceContourPlot3D[MapThread[Append, {vertices, data}], surf]
</code></... |
320,194 | <p>I have 5 types of symptoms, I want to know all kind of combinations a patient could have:</p>
<p>The set is $(vomit, excrement, urine, dizzyness, convulsion)$</p>
<p>As patient can show only one, or even 5 of them I am listing them as:</p>
<p>So</p>
<h1>Possibilities with one symptom</h1>
<pre><code>vomit
excre... | milcak | 5,105 | <p>To find the number of combinations of $r$ symptoms, the answer sure is $\binom{5} {r}$. If you want all of the combinations, you sum through $r= 1, 2, \dots, 5$. As a special case to the <a href="http://en.wikipedia.org/wiki/Binomial_theorem" rel="nofollow">Binomial theorem</a> we have $\sum_{i=0}^n \binom{n}{i} = ... |
711,863 | <p>$$\int{\frac{3}{5y^2 + 4}}dy$$</p>
<p>$$\frac{3}{4}\int{\frac{1}{\left(\frac{\sqrt{5}y}{2}\right)^2 + 1}}dy$$</p>
<p>$$u = \frac{\sqrt{5}y}{2}$$</p>
<p>$$dy = \frac{2}{\sqrt{5}}du$$</p>
<p>My solution to this problem was</p>
<p>$$\frac{3}{2\sqrt{5}}\left(\frac{1}{\tan\left(\frac{\sqrt{5}y}{2}\right)} + c\right)... | Shashank | 134,380 | <p>Doesn't matter if c is inside or outside the bracket. It just denotes a constant value.
If you multiply the coefficient with the constant, you will just get another constant.</p>
|
711,863 | <p>$$\int{\frac{3}{5y^2 + 4}}dy$$</p>
<p>$$\frac{3}{4}\int{\frac{1}{\left(\frac{\sqrt{5}y}{2}\right)^2 + 1}}dy$$</p>
<p>$$u = \frac{\sqrt{5}y}{2}$$</p>
<p>$$dy = \frac{2}{\sqrt{5}}du$$</p>
<p>My solution to this problem was</p>
<p>$$\frac{3}{2\sqrt{5}}\left(\frac{1}{\tan\left(\frac{\sqrt{5}y}{2}\right)} + c\right)... | Community | -1 | <p>Remember that $c$ is a generic constant - it doesn't hold a particular value until it is assigned one. In this case, note that $\frac{3}{2\sqrt{5}}*c$ is still a constant.
$$\frac{3}{2\sqrt{5}}\left(\frac{1}{\tan(\frac{\sqrt{5}y}{2})} + c\right)=\frac{3}{2\sqrt{5}}\left(\frac{1}{\tan(\frac{\sqrt{5}y}{2})}\right)+\f... |
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