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 |
|---|---|---|---|---|
1,368,815 | <p>Let $U\subset\mathbb{R}^n$ be an open set. I proved that </p>
<p>$$
H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \}
$$</p>
<p>I have to show that $\dim H^0_{DR}=\text{number of connected components of}\,\, U$.</p>
<p>Here is my incomplete proof... | Andrew D. Hwang | 86,418 | <p>Hint: Suppose $X$ is a non-empty connected open set, and that $f:X \to \mathbf{R}$ is locally constant. Pick an arbitrary point $x_{0}$ in $X$, and let $U = \{x \in X : f(x) = f(x_{0})\}$.</p>
<p>Use local constancy of $f$ to prove $U$ is open and closed in $X$.</p>
|
52,194 | <p>Assuming you have a set of nodes, how do you determine how many connections are needed to connect every node to every other node in the set?</p>
<p>Example input and output:</p>
<pre><code>In Out
<=1 0
2 1
3 3
4 6
5 10
6 15
</code></pre>
| Jonas Meyer | 1,424 | <p>If there are $n$ nodes, then this is called "$n$ choose $2$", and is equal to the number of $2$-element subsets of a set of $n$ elements. The Wikipedia article on <a href="http://en.wikipedia.org/wiki/Binomial_coefficient">binomial coefficients</a> includes this and generalizations. </p>
<p>Since I started writin... |
2,679,525 | <p>Find the $\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$ over $\mathbb{Q}[x]$. Then find two polynomials $a(x),b(x) \in \mathbb{Q}[x]$ such that, $$a(x)(x^3-6x^2+14x-15) + b(x)(x^3-8x^2+21x-18)=\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$$ </p>
<p>I have managed to find,
$$x^3-6x^2+14x-15=(x-3)(x^2-3x+5)$$
$$x^3-8x^2+21x-18=... | Will Jagy | 10,400 | <p>This is just the Extended Euclidean Algorithm. Instead of back-substitution, I have always preferred to write the construction steps in the style of continued fractions. Furthermore, I have always depended on the kindness of strangers.</p>
<p>$$ \left( x^{3} - 6 x^{2} + 14 x - 15 \right) $$ </p>
<p>$$ \lef... |
2,866,389 | <p>Let $\gamma\colon\left[0,1\right]\to\mathbb{R}^{2}$ be a curve such
that $\gamma\in\mathcal{C}^{1}$ (Continuously Differentiable). I
need to show that $\gamma\left(\left[0,1\right]\right)$ has mesure
zero, that is to show there are generalized rectangles $R_{1},R_{2},\ldots$
such that $\gamma\left(\left[0,1\right]\r... | Adayah | 149,178 | <p>Hint: A $\mathcal{C}^1$ curve $\gamma : [0, 1] \to \mathbb{R}^2$ is Lipschitz with some constant $L$. For each $n \in \mathbb{N}$ and $1 \leqslant j \leqslant n$ let $I_j = \left[ \frac{j-1}{n}, \frac{j}{n} \right]$ so that $I_j$ are of length $\frac{1}{n}$ and cover $[0, 1]$. Now let $m_j$ be the center of $I_j$ so... |
13,478 | <p>This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?</p>
| Dmitri Panov | 943 | <p>Consider for simplicity smooth projective varieties defined over $\mathbb C$. In this case, the minimal size equals $n+1$ where $n$ is the dimension of the variety. </p>
<p>Proof. Let $M^n$ be such a variety. Take $n+1$ generic very ample divisors $D_0,...,D_n$. Then such divisors don't have a common intersection. ... |
52,338 | <p>I am interested in finding explicit formulae for (better yet characterizing) conformal functions from various domains onto the open unit disc $\mathbb{D}\subset\mathbb{C}$, and in understanding the key ideas necessary to establish such functions.</p>
<p>Specifically, what can $f$ look like when $f:G\to\mathbb{D}$ i... | t.b. | 5,363 | <p>Since you didn't show too many own thoughts, here are some hints only. By <em>conformal</em> I understand <em>biholomorphic</em>.</p>
<ol>
<li><p>First take $f(z) = z^2$ to map the quadrant biholomorphically onto the upper half-plane, then compose with the <a href="http://en.wikipedia.org/wiki/Cayley_transform#Conf... |
52,338 | <p>I am interested in finding explicit formulae for (better yet characterizing) conformal functions from various domains onto the open unit disc $\mathbb{D}\subset\mathbb{C}$, and in understanding the key ideas necessary to establish such functions.</p>
<p>Specifically, what can $f$ look like when $f:G\to\mathbb{D}$ i... | user786 | 786 | <p>Here is a sketch:</p>
<p>For (1) we want to map the open first quadrant onto the unit disk. What we can do is first map the open first quadrant onto the upper half plane then the upper half plane onto the unit disk. the mapping $z \mapsto z^2$ maps the first quadrant to the upper half plane. Then the upper half pla... |
1,576,235 | <p>Take, for example: $3x + y + 2z = 6$.</p>
<p>Parameterized as: $ui + (6 -3u - 2v)j + vk$
restricted to the first octant</p>
| Slade | 33,433 | <p>No, everywhere it meets $x=0$, $y=0$, or $z=0$, is a boundary point.</p>
<p>One way to see this: in a manifold $M$ with boundary, $x\in M$ is a boundary point if and only if $x$ has a neighborhood $U$ such that $U\setminus \{x\}$ is contractible. But in your example, $M\setminus \{x\}$ is convex, hence contractibl... |
342,684 | <p>It would be useful to me to have a result of the following kind (which I would need to generalize, but this case is already interesting). Let <span class="math-container">$r<n$</span> be positive integers and let <span class="math-container">$\delta>0$</span> be a fixed constant such as 1/100. Does there exist... | Ilya Bogdanov | 17,581 | <p>Any maximal <span class="math-container">$r$</span>-separated set is an <span class="math-container">$r$</span>-net, otherwise you can augment it with a non-covered point.</p>
<p>But the same holds for linear codes! If a subspace <span class="math-container">$V$</span> is <span class="math-container">$r$</span>-sep... |
2,181,633 | <p>Given a real polynomial $P(x)$, find another real polynomial $Q(x)$ such that the powers of $x$ in $P(x)Q(x)$ are multiples of an integer $n$. Is it always possible to find such a polynomial?</p>
<p>For e.g if $P(x) = 1+3x+2x^2$ I can find $Q(x)=(2 x - 1) (x - 1) (4 x^2 + 1) (x^2 + 1)$ such that $P(x)Q(x) = 16 x^... | Community | -1 | <p>Here are three lemmas that underlie one approach to the problem: the first underlies the main idea, and the other two in dealing with the complication of real coefficients instead of complex ones.</p>
<p>Let $\zeta$ be a primitive $n$-th root of unity, and let Let $\bar{R}(x)$ be the polynomial whose coefficients a... |
1,324,824 | <p>Compute the flux integral with vector field,
$\vec F(x,y,z)= <2x,y,3z>$
and a sphere of radius 36 centered at the point $(1,2,-1)$</p>
<p>so what I did was I used divergence theorem which is</p>
<p>$\int\int \vec F \cdot \vec n dS = \int \int \int (div \vec F) dV$</p>
<p>I used the right hand side and so</p... | Raman Singh | 166,287 | <p>I read about flux in my electostatics class . So I may be wrong but :- </p>
<p>Consider flow of a liquid with velocity <strong>v</strong>, through a small
flat surface <strong>dS</strong>, in a direction normal to the surface. The
rate of flow of liquid is given by the volume crossing the
area per unit time <strong... |
3,653,016 | <p>I am solving this equation: </p>
<p><img src="https://i.stack.imgur.com/WN7ce.png" alt="Image of the equation"><br>
My issue arises on line 2 where we have (n + 1 - 2 + 1)(n + 1 + 2)/2. This is what i understand. We have the formula n! = n(n+1)/2. Subbing values into the equation yields us with 10[(n+1)(n+1 + 1)/2]... | JMP | 210,189 | <p><span class="math-container">$n+1-2+1$</span> gives the number of terms, which is the number of integers in the inclusive range <span class="math-container">$[2,n+1]$</span>.</p>
<p><span class="math-container">$\dfrac{n+1+2}{2}$</span> gives the average term, which is the largest term (<span class="math-container"... |
723,355 | <p>To win the lottery you must pick the one winning ticket.
Given the option of drawing 1 out of 10 tickets or 10 out of 100 tickets to win which is the better option? Are they both 1 out of 10 or is it better to pick 1 out of 100 first, then 1 out of 99, then 1 out of 98 (assuming you don't pick the winner) up to you... | Thanos Darkadakis | 105,049 | <p>Why do you think it's not the same?</p>
<p>It's easier to count the probability that you will NOT win.</p>
<p>In the first case it is $P_1=\frac9{10}$</p>
<p>In the second case it is: </p>
<p>$P_2=\frac{99}{100}\times\frac{98}{99}\times\frac{97}{98}\times\frac{96}{97}\times\frac{95}{96}\times\frac{94}{95}\times\... |
247,315 | <p>Please see my Edited version at the end of the post.</p>
<p>================================</p>
<p><a href="http://en.m.wikipedia.org/wiki/Cantor_set" rel="nofollow">http://en.m.wikipedia.org/wiki/Cantor_set</a></p>
<p>My definition of Cantor Set is just like that of wikipedia.</p>
<p>That is, $C=[0,1]\setminus... | Stephen Nand-Lal | 78,546 | <p>In $\mathbb{R}^2$, for any two Cantor sets $A$ and $B$ we can find a homeomorphism $h \colon \mathbb{R}^2 \to \mathbb{R}^2$ which carries $h(A)$ onto $B$. This essentially says they're equivalently embedded. I'm not sure this directly answers your question but in $\mathbb{R}^2$, talk of `the' Cantor set isn't necess... |
4,253,485 | <p>I understand that if you roll <span class="math-container">$10$</span> <span class="math-container">$6$</span>-faced dice (D6) the different outcomes should be <span class="math-container">$60,466,176$</span> right? (meaning <span class="math-container">$1/60,466,176$</span> probability)</p>
<p>but if you are playi... | Rezha Adrian Tanuharja | 751,970 | <p>A simple argument is you can roll a <span class="math-container">$1$</span> using one <span class="math-container">$60$</span> sided die but you cannot using ten <span class="math-container">$6$</span> sided dice.</p>
|
4,253,485 | <p>I understand that if you roll <span class="math-container">$10$</span> <span class="math-container">$6$</span>-faced dice (D6) the different outcomes should be <span class="math-container">$60,466,176$</span> right? (meaning <span class="math-container">$1/60,466,176$</span> probability)</p>
<p>but if you are playi... | Adam Rubinson | 29,156 | <p>Adding onto other answers…</p>
<p>Rolling ten six-sided dice, the most likely sum you are to get is <span class="math-container">$\frac{1+6}{2}*10=35.$</span> Then, getting a sum of <span class="math-container">$34$</span> is slightly less likely than getting a sum of <span class="math-container">$35$</span>. Etc… a... |
2,198,462 | <p>Let $\mathbf{f} : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by:
$$\mathbf{f}(u, v) = (u − v,\sin(u) \sin(v)).$$</p>
<p>Let $(y, z) = (0, 3/4).$ Find the point $(a, b)$ with $0 < a < \frac{\pi}{2}$ such that $\mathbf{f}(a, b)=(y, z)$.</p>
<p>Then prove that the function $\mathbf{f}^{−1}$ such that $\mat... | Jaideep Khare | 421,580 | <p><strong>Answer to Part I</strong></p>
<p>You want $$\mathbf{f}(a, b)=(y, z)=(0, 3/4)$$</p>
<p>Using definition :</p>
<p>$$\mathbf{f}(a, b) = (a − b,\sin(a) \sin(b))=(0,3/4)$$</p>
<p>i.e. $a-b=0 \implies a=b$ </p>
<p>and ,
$$\sin a \sin b =\sin^2a=3/4 \implies \sin a =+\frac{\sqrt{3}}{2}$$ (Since $0< a<... |
168,926 | <p>How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure? To be more precise, can anyone describe <em>how much</em> rigidity is imposed on the structure of rings by the distributive axiom which "connec... | William | 13,579 | <p>One example of where the distributivity axiom adds additional properties to ring-like structures is the following: </p>
<p>The ring axioms <em>with</em> the multiplicative identity axiom and <em>without</em> the commutativity of addition axiom can already prove the commutativity of addition. </p>
<p>This requires ... |
219,763 | <p><strong>The function $f$ is defined on $\mathbb{R}$ such that for every $\delta\gt0$, $|f(y)-f(x)|\lt\delta^2$ for all $x,y\in\mathbb{R}$ and $|y-x|\lt\delta$. Prove that $f$ is a constant function.</strong> </p>
<p>So, what I know, is that I need to show that $f(a)=f(b)$ for all points $a,b\in\mathbb{R}$. Or for ... | Berci | 41,488 | <p><strong>Hint:</strong> Show that $f$ is <em>differentiable</em> in each point and the same time that the derivative is $0$.</p>
|
219,763 | <p><strong>The function $f$ is defined on $\mathbb{R}$ such that for every $\delta\gt0$, $|f(y)-f(x)|\lt\delta^2$ for all $x,y\in\mathbb{R}$ and $|y-x|\lt\delta$. Prove that $f$ is a constant function.</strong> </p>
<p>So, what I know, is that I need to show that $f(a)=f(b)$ for all points $a,b\in\mathbb{R}$. Or for ... | Sangchul Lee | 9,340 | <p>It is natural to think that the given condition implies the inequality
$$\left| f(y) - f(x) \right| \leq C \left| y - x \right|^2. \tag{1}$$
But this implication requires some justification. So here is a proof.</p>
<p>Let $\epsilon > 0$ be any positive real number. To prove $(1)$, we decompose the set $\{ 0 <... |
396,363 | <p>I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to calculate both, the disitrbution functions $F_D(t), F_Z(t)$ and the the density functions $f_D(t),f_Z(t)$</p>
<p>To do... | wolfies | 74,360 | <p>My goodness ... what a lot of work. The nice thing about using a computer algebra system to solve this is: a) it's basically a one-liner, and b) if it is for a uni course, you will still need to know yourself how to get there, but at least you will know if your workings are correct. Here I am using the mathStatica a... |
396,363 | <p>I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to calculate both, the disitrbution functions $F_D(t), F_Z(t)$ and the the density functions $f_D(t),f_Z(t)$</p>
<p>To do... | Montaigne | 45,170 | <p>I thought it makes more sense to write an answer myself, but I still need someones help to complete the calculations. Here is my summary:</p>
<p>For $F_D(t)=\int_{\frac{1}{2}}^1\int_{-t+y}^{t+y}f_X(x) dx f_Y(y) dy=\int_{\frac{1}{2}}^1\int_{-t+y}^{t+y} 2*2 dx dy=4t$ for $x\in(1/2,1]$</p>
<p>For $f_D(t)=P(D=t)=\int_... |
2,709,532 | <p>(True/False) If $S$ is a non-empty subset of $\mathbb{N}$, then there exists an element $m \in S$ such that $m\ge k$ for all $k \in S$.</p>
<p>So my reasoning is that the above statement is false. Since $S$ is a non-empty subset of $\mathbb{N}$, it may also be the case that $S=\mathbb{N}$, and so there may not be a... | Maggie Myers | 545,995 | <p>You are correct. You found a counterexample. Every nonempty subset S has an m such that m≤k for all k in S (a minimum but not necessarily a maximum.)</p>
|
856,556 | <p>Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$</p>
<p>I have troubl... | StrangerLoop | 161,646 | <p>Yes, if we are working only on the space $[0,1]$, for example, then $f_n$ restricted to this interval is dominated by $1$, which is integrable on $[0,1]$. So the DCT applies, and tells us that the limit of the integral is the integral of the limit, which is $0$. This is easily verified, since $f_n$ integrates to $... |
973,101 | <p>I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid.</p>
<p>If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,x_2,x_3\}$, calculate the distance from the origin</p>
<p>$$d=\sqrt{x_1^2+x_2^2+x_3^2}$$</p>
<p>and calculat... | Max | 654,677 | <p>I have been working on this problem this week and I have found the following approach effective.</p>
<p>The ellipsoid is parameterized by</p>
<p><span class="math-container">$$\vec f(t, u) = \begin{bmatrix}a \cos u \cos t \\ b \cos u \sin t \\ c \sin u\end{bmatrix}, \quad
\begin{align}t &\in [0, 2 \pi] \\ u &... |
3,930,373 | <p>Hi so im working on a question about finding the <span class="math-container">$rank(A)$</span> and the <span class="math-container">$dim(Ker(A))$</span> of a 7x5 Matrix. Without being given an actual matrix to work from.</p>
<p>I have been told that that the homogeneous equation <span class="math-container">$A\vec x... | Adam Rubinson | 29,156 | <p>The function under consideration is <span class="math-container">$f:x \to \sqrt{-x^2} $</span> with domain {<span class="math-container">$0$</span>}. In other words, your function is, <span class="math-container">$f:$</span> {<span class="math-container">$0$</span>} <span class="math-container">$\to $</span>{<span ... |
2,972,611 | <p><strong>Q</strong>: Prove the given inequalities for positive a,b,c:<br><span class="math-container">$(i) \left[\frac{bc+ca+ab}{a+b+c}\right]^{a+b+c}>\sqrt{(bc)^a.(ca)^b.(ab)^c}$</span><br><span class="math-container">$(ii) \left(\frac{a+b+c}{3} \right)^{a+b+c}<a^ab^bc^c<\left(\frac{a^2+b^2+c^2}{a+b+c}\righ... | Batominovski | 72,152 | <p>First of all, you should have <span class="math-container">$\geq $</span> or <span class="math-container">$\leq$</span> instead of <span class="math-container">$>$</span> and <span class="math-container">$<$</span> in your inequalities. In each of them, the equality occurs if and only if <span class="math-con... |
195,065 | <p>Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing function.</p>
<p>Let $a<f(a)$ and $f(b)<b$. Prove that there is a $a<c<b$ such that $f(c)=c$. </p>
<p>My attempt at a proof is as follows. Let $c:=\sup\{x:a\leq x\leq b\text{, }x\leq f(x)\}$.</p>
<p>This is where I'm stuck. Since I can't u... | Brian M. Scott | 12,042 | <p>HINT: Let $L=\{x\in[a,b]:x<f(x)\}$. Is $L$ non-empty? Is $L$ bounded above?</p>
<p><strong>Added:</strong> In other words, your idea is reasonable, though I used a slightly different set from yours. $L$ <strong>is</strong> non-empty and bounded (why?), so we can let $c=\sup L$. What do you know about $f(x)$ for ... |
4,361,063 | <blockquote>
<p>Suppose that <span class="math-container">$(X, \tau_1)$</span> is compact and <span class="math-container">$\tau_1 \subset \tau_2$</span>. Is the space <span class="math-container">$(X, \tau_2)$</span> compact? Does the converse hold i.e if <span class="math-container">$(X, \tau_1)$</span> is compact an... | Sourav Ghosh | 977,780 | <p>Suppose, <span class="math-container">$card(X) \ge \aleph_{0}$</span></p>
<p>Consider two topology on <span class="math-container">$X$</span></p>
<p><span class="math-container">$\tau_{indiscrete}=\{\emptyset, X\}$</span> and</p>
<p><span class="math-container">$\tau_{discrete}={\scr{P}}(X) $</span></p>
<p>Then, <sp... |
258,893 | <p>I have a set of three moderately simple simultaneous equations that I'd like to simplify and eliminate a set of variables for (this is a simple example of a more general class of problem - but I'd like to get the simple example working before starting on the bigger ones). Asking Mathematica to <code>Eliminate</code>... | Akku14 | 34,287 | <p>The quite fastest way is to use Solve, which is not sensitive if some variables are not independent of each other, add identities for Sin, Cos and eliminate all Sin und Cos at the same time.</p>
<pre><code>Solve[{xt1 ==
Cos[t3] (x1 Cos[t2] + (x3 Cos[t1] + x2 Sin[t1]) Sin[t2]) - (x2 Cos[
t1] - x3 Sin[t1]) Sin[t... |
258,893 | <p>I have a set of three moderately simple simultaneous equations that I'd like to simplify and eliminate a set of variables for (this is a simple example of a more general class of problem - but I'd like to get the simple example working before starting on the bigger ones). Asking Mathematica to <code>Eliminate</code>... | J. M.'s persistent exhaustion | 50 | <p>If I use the <a href="https://en.wikipedia.org/wiki/Weierstrass_substitution" rel="nofollow noreferrer">Weierstrass substitution</a> along with <code>TrigExpand[]</code>, <code>GroebnerBasis[]</code> accomplishes the elimination pretty quickly:</p>
<pre><code>GroebnerBasis[TrigExpand[Thread[{xt1, xt2, xt3} == {x1, x... |
3,276,958 | <p>Given two points <span class="math-container">$P_1(x_1,y_1)$</span> and <span class="math-container">$P_2(x_2,y_2)$</span> with <span class="math-container">$y_1>0$</span> and <span class="math-container">$y_2>0$</span> I need to find the parameters <span class="math-container">$a$</span> and <span class="math... | Martin R | 42,969 | <p>Taking the logarithms of the desired identities <span class="math-container">$y_j = a b^{x_j}$</span> gives a <em>linear equation system</em>
<span class="math-container">$$
\begin{align}
1 \cdot \log a + x_1 \cdot \log b = \log y_1 \\
1 \cdot\log a + x_2 \cdot\log b = \log y_2
\end{align}
$$</span>
which can be... |
1,401,116 | <p>An aeroplane is observed by two persons travelling at $60 \frac{km}{hr} $ in two vehicles moving in opposite directions on a straight road. To an observer in one vehicle the plane appears to cross the road track at right angles while to the observer in the other vehicle the angle appears to be $45°$. At what angle ... | triple_sec | 87,778 | <p>Found it. It is the “discrete version” of Liouville’s theorem. I was wrong about Alaoglu: the functional-analytic approach uses the Krein–Milman theorem (although proofs based on martingale theory are available, too). See more <a href="https://math.stackexchange.com/questions/849284/looking-for-different-proofs-of-d... |
1,124,417 | <p>I was working my way through some Propositional Logic and had the following doubt :</p>
<blockquote>
<p><strong>Why is this true :</strong></p>
<p>((p <span class="math-container">$\Rightarrow$</span> r) <span class="math-container">$\land$</span> (q <span class="math-container">$\Rightarrow$</span> r)) <span class=... | mercio | 17,445 | <p>$2$ and $3$ are invertible, so we can look at
$6F(x,y,z) = 3(2x+y-az)^2 + (3y-az)^2 + 2(a^2-3b^2)z^2$.<br>
After an invertible change of basis, the equation $F(x,y,z) = 0$ has the same number of solutions as $3X^2+Y^2 + 2(a^2-3b^2)Z^2 = 0$.</p>
<p>According to wether $3$ and $2(a^2-3b^2)$ (which are nonzero) are... |
2,369,133 | <p>I just found out that
if you want to get 1 with the fraction: $$\frac{5}{2}$$
Then you multiply it with: $$ \frac{2}{5} $$
Does anyone have a good way to think about this? </p>
| Bill Dubuque | 242 | <p>$x = a/b\iff bx = a.\,$ If there is a number $x'$ with $\,\color{#c00}{xx' = 1}$ then scaling this by $x'$ yields the</p>
<p>equation $\ b= b\color{#c00}{xx'} = ax',\,$ i.e $\,x' = b/a$</p>
<hr>
<p>More generally if $\,r\,$ is a root of a polynomial $f(x)$ then its reciprocal $\,1/r\,$ is a root of the <a href=... |
2,922,494 | <p>I came across the following problem today.</p>
<blockquote>
<p>Flip four coins. For every head, you get <span class="math-container">$\$1$</span>. You may reflip one coin after the four flips. Calculate the expected returns.</p>
</blockquote>
<p>I know that the expected value without the extra flip is <span class="m... | William Wong | 589,804 | <p>Expectation of first 4 flips is $2.</p>
<p>Expectation of the fifth flip, condition on there is a tail in the first 4 flips, is 0.5.</p>
<p>Expectation of the fifth flip, condition on there is no tail, is 0.</p>
<p>Probability there is a tail in the first 4 flips is 15/16.</p>
<p>Total expectation = 2 + 0.5*(15/... |
1,203,116 | <p>I'm trying to prove that in such triangle, AB>AC, but I can't suceed. I tried to use some properties of isósceles triangle since the sides are equal, but couldn't find something nice. Also, whats the criteria needed to prove that a side in a triangle is greater than the other? Thank you so much!</p>
| Joffan | 206,402 | <p>Since $|AC|=|AD|, \triangle ACD$ is isosceles and $\angle ADB$ is obtuse.</p>
<p><img src="https://i.stack.imgur.com/kdtD9.png" alt="enter image description here"></p>
<p>Therefore $|AB| > |BD| = |AC|$</p>
|
4,541,392 | <p>I have an integral that looks like the following:
<span class="math-container">$$
\int_a^b \frac{1}{\left(1 + cx^2\right)^{3/2}} \mathrm{d}x
$$</span>
I have seen a method of solving it being to substitute <span class="math-container">$x = \frac{\mathrm{tan}(u)}{\sqrt{c}}$</span>; however, this seems somewhat sloppy... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
&\text { Let } y=\frac{1}{x^2} \text {, then } d x=-\frac{1}{2y^{\frac{3}{2}}} d y\\
I&=\int_{\frac{1}{a^2}}^{\frac{1}{b^2}} \frac{1}{\left(1+\frac{c}{y}\right)^{\frac{3}{2}}}\left(-\frac{d y}{2y \frac{3}{2}}\right)\\
&=\frac{1}{2} \int_{\frac{1}{b^2}}^{\fr... |
163,654 | <p>Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.</p>
| jacob | 4,181 | <p>To expand on my answer above, consider the sequence $\dots\subset Y_2\subset Y_1$
of spaces where $Y_i=X_i$ for $i\leq n$ and $Y_i=X_n$ for $i>n$.</p>
<p>Milnors proof proceeds by taking $M_i$ to be the mapping cylinder of $X_{i+1}\rightarrow X_i$ and $M_0$ the cone over $X_0$ with vertex $t$. Then one takes t... |
163,654 | <p>Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.</p>
| Włodzimierz Holsztyński | 8,385 | <p>There is no reason for a homological surjection. And it's false all the time, so-to-speak. Thus let's assume that we are asking either about a homological injection or about the cohomological surjection which goes in the other direction (naturally--as a categorist would say).</p>
<p>Now the answer is YES under some... |
100,957 | <p>I have this picture of small particles in a polymer film. I want to count how many particles in the figure, so that I can have a rough estimation of the particle density. But the image quality is poor, I had a hard time to do it.</p>
<p><a href="https://i.stack.imgur.com/ez50R.jpg" rel="noreferrer"><img src="https:... | Graumagier | 25,421 | <p>You could try something like this</p>
<pre><code>HighlightImage[
ImageTake[img, {100, 400}, {100, 400}],
ImageCorners[
ImageTake[img, {100, 400}, {100, 400}],
1.0,
0.0002,
10
]
]
</code></pre>
<p><a href="https://i.stack.imgur.com/N6qB7.png" rel="noreferrer"><img src="https://i.stack.imgur.co... |
3,954,010 | <p>The case <span class="math-container">$n=3$</span> is from <a href="https://usamts.org/Tests/Problems_31_1.pdf" rel="nofollow noreferrer">here</a>. It's straightforward to prove it's true:</p>
<p>First we notice that if any two of <span class="math-container">$x_1, x_2, x_3$</span> are equal then all must be equal.<... | Andreas | 317,854 | <p>Consider the question as a sequence. It starts with two values and has the recursive definition <span class="math-container">$x(n) = x(n-1)\frac{\log x(n-2)}{\log x(n-1)}$</span>. For <span class="math-container">$x_1^{x_2} = x_2^{x_3} = \cdots = x_{n-1}^{x_n} = x_n^{x_1} $</span> to hold, this sequence has to be p... |
1,227,330 | <p>I have the following hypotheses: $\alpha_n \to 0, \beta_n \to 0, \alpha_n < 0 < \beta_n$. I need to show that the following two limits exist <strong>so that I may add them</strong>:</p>
<p>$$\lim \limits_{n \to \infty} \frac{-\alpha_n}{\beta_n -\alpha_n}, \lim \limits_{n \to \infty} \frac{\beta_n}{\beta_n - \... | Rolf Hoyer | 228,612 | <p>Note that in the usual proof of the limit $\lim_{h\to 0} \frac{\sin(h)}{h} = 1$, it is shown that the inequality $$\cos(h) \le \frac{\sin(h)}{h} \le 1$$ holds for all $h$ in an interval around $0$. Assuming this result, we have $$\cos(h) - 1 \le \frac{\sin{h}}{h} - 1 \le 0$$ in this interval. Taking absolute valu... |
228,105 | <p>I want the last number after the "/". How can I do that?</p>
<pre><code>"http://arxiv.org/abs/math/0208009v1"
</code></pre>
<p>I just want <strong>0208009v1</strong></p>
<p>I want to import multiple links from the following:</p>
<pre><code>{{"http://arxiv.org/abs/math/0208009v1"}, \
{&q... | user1066 | 106 | <p>In addition, using a modification of this <a href="https://stackoverflow.com/a/8798197/499167">regex</a>:</p>
<pre><code>(StringCases[#, RegularExpression["[^/]+$"]]&/@list)//Flatten
</code></pre>
<blockquote>
<p>{0208009v1, 0905.0227v1, 0907.5143v2, 0509348v1, 0608711v2, 0002018v2}</p>
</blockquote>
<... |
1,235,973 | <p>Im reaching a point in programing where I need to create basic shapes which I simply cant since my math skills are very bad. After finding out that the skills required are trigonometry I read a few books. The result was that I couldnt understand any word at all.</p>
<p>What is required in order to understand basic ... | Michael Hardy | 11,667 | <p>You certainly do not need calculus. You need some basic algebra. There are a few things in basic <b>geometry</b> that you show be thoroughly aware of:</p>
<ul>
<li>The number $\pi$ is the ratio of circumference to diameter of a circle. For example, a circle whose diamater is $1$ foot has a circumference of $\pi$... |
3,148,623 | <p>I want to find all prime ideals in <span class="math-container">$\mathbb{Z}/n\mathbb{Z}$</span> where <span class="math-container">$n>1$</span>.</p>
<p>I think I have to use the following theorem (because they asked me to prove it right before this exercise, which wasn't too complicated):</p>
<blockquote>
<p>... | Bernard | 202,857 | <p>Another theorem says the prime ideals of <span class="math-container">$R/I$</span> correspond bijectively to the prime ideals of <span class="math-container">$R$</span> containing <span class="math-container">$I$</span>.</p>
<p>In the case of <span class="math-container">$\mathbf Z/n\mathbf Z$</span>, this means i... |
3,148,623 | <p>I want to find all prime ideals in <span class="math-container">$\mathbb{Z}/n\mathbb{Z}$</span> where <span class="math-container">$n>1$</span>.</p>
<p>I think I have to use the following theorem (because they asked me to prove it right before this exercise, which wasn't too complicated):</p>
<blockquote>
<p>... | tomasz | 30,222 | <p><strong>Hint</strong>: to solve this exercise, it is enough to use the following facts:</p>
<ul>
<li>an ideal <span class="math-container">$I\unlhd R$</span> is prime if and only if <span class="math-container">$R/I$</span> is a domain,</li>
<li>the characteristic of a domain is prime.</li>
</ul>
<p>Using these, i... |
26,986 | <p>Here are the differential equations that set's up the 11 coupled oscillators.</p>
<pre><code>new = Join[
Table[x[i]''[t] == - x[i][t] +
0.1*(x[i + 1][t] - 2*x[i][t] + x[i - 1][t]), {i, 1,
9}], {x[0]''[t] == -x[0][t], x[10]''[t] == x[9][t], x[0][0] == 1,
x[0]'[0] == 1, x[1]'[0] == 0, x[1][0] == 0},
Table[x[i][0... | mmal | 1,210 | <p>Solution by @Kuba can be easily extended to put the oscillators on a circle.</p>
<pre><code>Loin = NDSolve[Stew, Table[x[i], {i, 0, 10}], {t, 50}];
fr[t_] = Transpose@{Most@Range[0, 2 Pi, 2 Pi/11], x[#][t]&/@Range[0, 10]} /. First@Loin;
r0 = 3;
Animate[
ListPlot[
Function[{th, r}, {(r0 + r) Cos[th], ... |
3,547,768 | <p>Given a function <span class="math-container">$f:\mathbb{R}\rightarrow\mathbb{R}$</span> which is differentiable and given that <span class="math-container">$f(0)=2$</span> and <span class="math-container">$(e^x + 1)f'(x) = e^x f(x)$</span> for every <span class="math-container">$x \in \mathbb{R}$</span>, prove that... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p><span class="math-container">$${d\over dx} \ln f(x)={f'(x)\over f(x)}$$</span></p>
|
3,525,356 | <p><a href="https://i.stack.imgur.com/8OitL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8OitL.png" alt="enter image description here"></a></p>
<p>Can someone help me on this? Any help will be greatly appreciated. </p>
| Community | -1 | <p><span class="math-container">$\begin{pmatrix}1&k\\0&1\end{pmatrix}^n=\begin{pmatrix}1&nk\\0&1\end{pmatrix}$</span>, so you can replace the super diagonal <span class="math-container">$1$</span> by <span class="math-container">$k\ne0\in\Bbb Z_p$</span>.</p>
<p>Plus you can do a subdiagonal version.<... |
2,137,269 | <p>I just sat in on a lecture on exponential generating functions in combinatorics (I have no formal education in combinatorics myself). It was quite interesting, but I'm afraid I don't actually understand what the generating function is/does. I've tried doing some minimal research online, but everything I've seen seem... | User8976 | 98,414 | <p>I recommend you to see the book <a href="https://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow noreferrer">generating functionology</a> by Wilf. It is available online. </p>
|
3,929,893 | <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$H$</span>, <span class="math-container">$K$</span> normal subgroups of <span class="math-container">$G$</span>. Prove that <span class="math-container">$G=HK$</span> if and only if <span class="math-container">$G/(H\cap K... | ArsenBerk | 505,611 | <p><strong>HINT:</strong> For the first part, we know that
<span class="math-container">$$|HK| = \frac{|H|\cdot |K|}{|H \cap K|}$$</span>
So, what can we say about <span class="math-container">$|H \cap K|$</span> if <span class="math-container">$G = HK$</span> considering <span class="math-container">$HK \le G$</span>?... |
4,139,003 | <p>For this I used the method of making the determinant nonzero. First I create the array</p>
<p><span class="math-container">\begin{equation}
\begin{pmatrix}
a^2 & 0 & 1\\
0 & a & 2\\
1 & 0 & 1
\end{pmatrix}
\end{equation}</span></p>
<p>Then <span class="math-container">$\det(A)=a^3-a=a(a^2-1)$... | 2'5 9'2 | 11,123 | <p>My approach would be to first put the circle at the origin. Then find where tangent lines of the given slopes interesect, and shift from there.</p>
<p>The circle <span class="math-container">$x^2+y^2=9$</span> has tangent lines at <span class="math-container">$(x,y)$</span> with slope <span class="math-container">$-... |
3,714,021 | <p>I've been working with the power series <span class="math-container">$\sum_{n=0}^{\infty}n(1-2^{-n})z^n$</span> for the last few hours and I'm starting to get a little lost in trying to figure out a few things for an exercise.</p>
<p>First of all, I've managed to find the radius of convergence ofr the series with r... | TonyK | 1,508 | <blockquote>
<p>This must mean that the series diverges</p>
</blockquote>
<p>As you have written it, it only means that the series diverges at <span class="math-container">$z=1$</span>. You want to show that the series diverges whenever <span class="math-container">$|z|=1$</span>. But you can use the same argument.<... |
227,732 | <p>Are there any general relations between the eigenvalues of a matrix $M$ and those of $M^2$?</p>
| Emily | 31,475 | <p>If the eigenvalues are distinct, then the square matrix $A$ is diagonalizable, namely
$$A = Q^{-1}DQ.$$</p>
<p>Then, $$A^2 = (Q^{-1}DQ)^2 = Q^{-1}DQQ^{-1}DQ = Q^{-1}D^2Q.$$</p>
<p>The diagonal entries of $D^2$ are the diagonal entries of $D$, squared.</p>
|
4,477,354 | <p>Fourier series for function <span class="math-container">$f(x)=c^x$</span>, <span class="math-container">$c\in\mathbb Z$</span>, <span class="math-container">$c>1$</span> on interval <span class="math-container">$(a,b)$</span>, where <span class="math-container">$a,b\in\mathbb R$</span>, <span class="math-contain... | V.S.e.H. | 443,030 | <p>Note that using AM-GM:
<span class="math-container">$$
x_k = x_k \prod_{i=1}^{k-1}1 \leq \frac{k-1 + x_k^k}{k}\implies kx_k\leq k-1 + x_k^k
$$</span>
for <span class="math-container">$k \geq 1$</span>. Now sum each inequality along <span class="math-container">$k$</span> and you get the desired inequality.</p>
|
6,056 | <p>Are there any good places for me to sell off my mathematics books online especially Springer and Dover books?</p>
<p>(I thought perhaps this was off topic, but then I thought everybody in math probably has the problem of having bought pricey books that they would prefer to get some money back for, at one point or a... | Ryan Budney | 642 | <p>I find it's pretty easy to sell books on e-bay. Local used bookstores rarely go for good math books, unfortunately. </p>
<p>You might also want to try the big technical book resellers -- like Powell's Books, they're based in Portland Oregon. Most on-line searches for books (like Google Books) will search Powell's... |
2,447,549 | <p>If so, could someone give me an example?</p>
| Dirk | 379,594 | <p>$V = \mathbb{R}$ and $W = \{0\}$. Take any linear map $f : W \to V$. :)</p>
|
241,138 | <p>I think I didn't pay attention to uniform distributions because they're too easy.
So I have this problem</p>
<blockquote>
<ol>
<li>It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9.00 a.m. and he leaves home at 8.37 a.m., find the prob... | Aseem Dua | 48,261 | <p>In case of a double root, the method fails, we need to start over with a different assumption of the solution. This time though, we approach the problem in a more informed manner.</p>
<p>hint: when we solve a simple homogeneous system, when we have a double root, to obtain a second LI solution we simply multiply ro... |
3,401,055 | <blockquote>
<p>If the distance of any point <span class="math-container">$(x,y)$</span> from the origin is defined as <span class="math-container">$d(x,y)=max\{|x|,|y|\}$</span> where <span class="math-container">$|.|$</span> represents the absolute value function, and <span class="math-container">$d(x,y)=a$</span> wh... | Sangchul Lee | 9,340 | <blockquote>
<p><strong>Assumption.</strong> <span class="math-container">$V$</span> is a finite-dimensional inner product space and <span class="math-container">$A : V \to V$</span> is a linear operator such that the following condition holds:</p>
<p><span class="math-container">$$\langle v, Av\rangle \leq -\|v... |
3,401,055 | <blockquote>
<p>If the distance of any point <span class="math-container">$(x,y)$</span> from the origin is defined as <span class="math-container">$d(x,y)=max\{|x|,|y|\}$</span> where <span class="math-container">$|.|$</span> represents the absolute value function, and <span class="math-container">$d(x,y)=a$</span> wh... | The One | 619,244 | <p>Solution without using spectral theory.</p>
<p>Choose <span class="math-container">$f:R\rightarrow R$</span> for given <span class="math-container">$v\in V $</span> with <span class="math-container">$f(t) = e^{tA}(v) \cdot e^{tA}(v) $</span></p>
<p>Then <span class="math-container">$f^{'}(t)=Ae^{tA}(v)\cdot e^{tA}... |
2,722,170 | <p>I am a grade 11 student and I have to learn vectors for the IB exam. I know that to find the distance of a vector between two points in a 3 dimensional space for lets say point A and B, then you would use the formula
\begin{align*}
|AB|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.
\end{align*}
I am having difficulty... | Christian Blatter | 1,303 | <p>Assume $0<\theta<{\pi\over2}$.</p>
<p><em>Claim.</em> The resulting orbit is closed iff $\theta$ is a rational multiple of $\pi$.</p>
<p><em>Proof.</em> We draw our figure in the complex plane. Let $\ell_1$ be the real axis, and choose $\ell_2$ through the origin. We denote the points on $\ell_1$ by $p_k$ a... |
621,544 | <p>I have multiple data-sets from a Fourier series of a function $f(t,x)$ (the data-sets where obtained by varying $x$), so $A_n=\frac{2}{T}\int_0^T{f(t,x)\sin{\frac{2\pi nt}{T}}dt}$, which seems to follow an equation of the form
$$
A=an^b
$$
I know that $b$ will be negative, since $A$ approximately equal to $0$ for la... | Claude Leibovici | 82,404 | <p>As you know, the key problem in nonlinear regression is the starting point (initial values of the model parameters). When the problem can be linearized (such via a logarithmic transform), the problem is simpler since the solution of the linearized model wil give you good (or at least reasonable) starting values for ... |
2,542,259 | <p>The question says to "find geometrically the set of points $(x,y) \in \Bbb R^2$ such that $$x+2yi=|x+yi|$$</p>
<p>I'm kinda lost on this one, I've tried solving the equation directly but I don't even know what to do with that. Here's what I have right now:</p>
<p>$$x+2yi = |x+yi|$$
$$x+2yi = \sqrt{x^2+y^2}$$
$$x^2... | B. Mehta | 418,148 | <p>Since the right hand side is strictly real (and non-negative), we must have $y=0$, as $(x,y) \in \mathbb{R}^2$. So, the equation becomes $x = |x|$, and the solution set is $\{(x,0) \in \mathbb{R}^2 \mid x \geq 0\}$.</p>
<p>Geometrically, this is the non-negative real axis, or the halfline $[0, \infty) \times \{0\}$... |
1,376,627 | <p>I would like to know how to rewrite the following equations:</p>
<p>$$
\frac{d (f(x))}{d(x+c)} =0\\
\frac{d^2 (f(x))}{d(x+c)^2} =0\\
$$</p>
<p>Here $x$ is a variable, $c$ is a constant and $f(x)$ is a function of x.
I would also like to know the reasoning behind the answer.</p>
| mvw | 86,776 | <p>Introducing $y=x+c$ one gets
$$
0
= \frac{d(f(x))}{d(x+c)}
= \frac{d(f(y-c))}{dy}
= f'(y-c) \cdot 1
= f'(x)
= \frac{d(f(x))}{dx}
$$
For the second derivative we get:
$$
0
= \frac{d^2(f(x))}{d(x+c)^2}
= \frac{d}{d(x+c)} \frac{d(f(x))}{d(x+c)}
= \frac{d(f'(x))}{d(x+c)} = (f')'(x) = f''(x)
= \frac{d^2(f(x))}{dx^2}
$$... |
1,376,627 | <p>I would like to know how to rewrite the following equations:</p>
<p>$$
\frac{d (f(x))}{d(x+c)} =0\\
\frac{d^2 (f(x))}{d(x+c)^2} =0\\
$$</p>
<p>Here $x$ is a variable, $c$ is a constant and $f(x)$ is a function of x.
I would also like to know the reasoning behind the answer.</p>
| haqnatural | 247,767 | <p>$$\frac { d }{ dx } \left( f\left( x \right) \left( x+c \right) ^{ -1 } \right) =0\\ \frac { df\left( x \right) }{ dx } \left( x+c \right) ^{ -1 }+f\left( x \right) \frac { d\left( \left( x+c \right) ^{ -1 } \right) }{ dx } =0\\ \frac { df\left( x \right) }{ dx } \frac { 1 }{ \left( x+c \right) } -f\left( x \rig... |
286,312 | <p>I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically:</p>
<p>$$
\sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2}
$$</p>
<p>Any hints would be greatly appreciate</p>
| amWhy | 9,003 | <p><strong>Base case:</strong> </p>
<ul>
<li><p>Let $n=1$ and test: $$\sum_{i=1}^1(3i-2)=3-2=1=\frac{1(3\cdot 1-1)}{2}$$</p></li>
<li><p>True for $n = 1$</p></li>
</ul>
<p><strong>Induction Hypothesis</strong>: </p>
<ul>
<li>Assume that it is true for $k$: assume that $$\sum_{i=1}^k(3i-2)=\frac{k(3k-1)}{2}.$$ </li>
... |
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | Logan Luther | 347,317 | <p>In this context, to find the area shaded, think about the area under each curve separately, and then try to find the area NOT shared by them. What will you get?</p>
<p>Since the area is a plain number, to find a difference you simply subtract two areas:<img src="https://i.stack.imgur.com/6IfNB.png" alt="The area be... |
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | user39390 | 403,342 | <p>Suppose the topmost graph is 1 then 2, 3, 4 . We have to find the area between the Red-Curve and Blue line in the first quadrant ( Graph 1). When we Integrate the Red-Curve with the limit from 0 to some positive x , We get the area of red region ( Graph 2). In the same manner, When we Integrate blue line with limit ... |
2,646,749 | <p>I'm solving the following problem:</p>
<blockquote>
<p>Identify all maximal ideals in the ring $\mathbb R[x]/(x^2-3x+2)$.</p>
</blockquote>
<p>I want to solve it in two ways.</p>
<p><strong>First method.</strong> Consider the surjective quotient homomorphism $\mathbb R[x]\rightarrow \mathbb R[x]/(x^2-3x+2)$ wit... | David Hill | 145,687 | <p>For method 1, the correspondence theorem proves this immediately: There is a 1-1 order preserving correspondence between the set of ideals in $R/I$ and the set of ideals in $R$ that contain $I$.</p>
<p>For method 2, the Chinese Remainder theorem gives an isomorphism
$$\mathbb{R}[x]/(x^2-3x+2)\cong \mathbb{R}[x]/(x-... |
1,255,629 | <p>Show that </p>
<p>$$\sin\left(\frac\pi3(x-2)\right)$$ </p>
<p>is equal to </p>
<p>$$\cos\left(\frac\pi3(x-7/2)\right)$$</p>
<p>I know that $\cos(x + \frac\pi2) = −\sin(x)$ but i'm not sure how i can apply it to this question.</p>
| ZeroCool | 109,653 | <p>Maybe you can try
\begin{align*}\sin\left(\frac{\pi}{3}(x-2)\right) &= \sin\left(\frac{\pi}{3}x-\frac{2\pi}{3}\right)\\
&=\sin\left( \pi - \frac{\pi}{3}x+\frac{2\pi}{3}\right) \\
&=\sin\left(\frac{5\pi}{3}-\frac{\pi}{3}x\right) \\
&=\sin\left(\frac{\pi}{2}+\frac{7}{2}\frac{\pi}{3} -\frac{\pi}{3}x\rig... |
1,313,056 | <p>Got this matrix: </p>
<p>\begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix}</p>
<p>I should determine if the matrix is diagonalizable or not.
I found the eigenvalues ( only one) = 3.
My eigenvector is then \begin{bmatrix} 1 \\ 1 \end{bmatrix}
This matrix is not diagonizable (from my teachers notes) but i do... | marwalix | 441 | <p>If a $2\times 2$ matrix $A$ with one double eigenvalue say $\lambda$ is diagonalisable. It means there exists an invertible matrix $P$ such that</p>
<p>$$A=P\cdot\lambda I_2\cdot P^{-1}=\lambda P\cdot P^{-1}=\lambda I_2$$</p>
<p>This means the only $2\times 2$ with a double eigenvalue that is diagonalisable is dia... |
446,959 | <p>I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points:</p>
<ul>
<li>in the volume of a n-dimensional hypersphere</li>
<li>on the surface of a n-dimensional hypersphere</li>
</ul>
<p>knowing the dimension $n$, the center of the hypersphere $\vec{x}$ and its radiu... | nbubis | 28,743 | <p><a href="http://mathworld.wolfram.com/HyperspherePointPicking.html" rel="nofollow">Aperantly</a>, for picking points on the <em>surface</em> of a hypersphere is to generate $n$ Gaussian random variables $x_1, x_2, ...x_n$, and then use the vectors:</p>
<p>$$\frac{1}{\sqrt{x_1^2+x_2^2+...+x_n^2}} \left(
\begin{array... |
312,696 | <p>I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in 3-dimensional space)?</p>
<p>I understand that for original points holds $M\cdot pt_i = gd_i$, so combining all $pt_i$ into... | Wesner Moise | 168,911 | <p>I don't believe the accepted answer works for translations. If we have a three-dimensional point and a 3x3 transformation matrix, the zero point will always map to the zero point. </p>
<p>A general affine transformation requires a four-dimensional matrix.
It may be possible to use a four-dimensional point with the ... |
1,680,210 | <p>Three cards are drawn at random from a full deck. what is the probability of getting a three , a seven, and an ace?</p>
<p>Can someone help me with this question please, thanks!</p>
| Barry Cipra | 86,747 | <p>Inspired by Ojas's answer (which I think is incomplete because it seems to assume <span class="math-container">$a$</span> is positive):</p>
<p><span class="math-container">$x=0\implies a\mid90$</span></p>
<p><span class="math-container">$x=1\implies a\mid92$</span></p>
<p>Thus <span class="math-container">$a\mid2... |
4,285,189 | <p>I am aware of the following identity</p>
<p><span class="math-container">$\det\begin{bmatrix}A & B\\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B)$</span></p>
<p>When <span class="math-container">$A = D$</span> and <span class="math-container">$B = C$</span> and when <span class="math-container">$AB = BA$</... | Just a user | 977,740 | <p><span class="math-container">$\begin{pmatrix} A & B \\ B & A \end{pmatrix}\xrightarrow{\text{row1 -= row2}}\begin{pmatrix} A-B & B-A \\ B & A\end{pmatrix}\xrightarrow{\text{col2 += col1}}\begin{pmatrix} A-B & O \\ B & A+B\end{pmatrix}$</span></p>
|
2,680,297 | <p>I had been given in my complex analysis examination the following problem. </p>
<p><strong>Evaluate the following integral by using contour integration:</strong></p>
<p>$$
\int_0^{\pi}\dfrac{\sin(2\theta)}{5-3\cos(\theta)}d\theta
$$</p>
<p>The answer to which is:
$$
\dfrac{2}{9}(log_e(1024)-6).
$$ </p>
<p>I trie... | K B Dave | 534,616 | <p>Your question is basically whether
$$\int_0^{\pi}\frac{\sin 2\theta \,\mathrm{d}\theta}{5-3\cos\theta}=\int_{-1}^1\frac{2x\,\mathrm{d}x}{5-3x}= \tfrac{4}{9}(5\,\ln 2-3)$$
can be written as an integral about a <em>closed</em> curve of some algebraic function over the rationals. This statement is believed to be false... |
1,237,618 | <p>I really need Your help.</p>
<p>I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||x||^2-x^2}{||x||^3}$</p>
<p>Thank You!</p>
| science | 212,657 | <p>By the <a href="http://en.wikipedia.org/wiki/Convex_function" rel="nofollow">definition of convexity</a> and the triangle inequality we have</p>
<blockquote>
<p>$$ ||ax+(1-a)y|| \leq |a| ||x||+ |1-a|||y||,\quad 0\leq a\leq1. $$</p>
</blockquote>
|
4,179,720 | <p>I'm starting to study triple integrals. In general, I have been doing problems which require me to sketch the projection on the <span class="math-container">$xy$</span> plane so I can figure out the boundaries for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. For example, I had... | Ritam_Dasgupta | 925,091 | <p>For convenience, imagine the round table to be the unit circle. It is clear that we have <span class="math-container">$4$</span> seats, <span class="math-container">$2$</span> each on <span class="math-container">$x$</span>- and <span class="math-container">$y$</span>- axes, and <span class="math-container">$2$</spa... |
267,121 | <h2>UPDATE on FINAL RESULT</h2>
<p>Thanks to @SquareOne effort I generated higher-resolution videos with smoothing transitions that can be seen here:</p>
<ul>
<li><p><a href="https://www.linkedin.com/feed/update/urn:li:activity:6926902980323512320/" rel="noreferrer">https://www.linkedin.com/feed/update/urn:li:activity:... | Ben Izd | 77,079 | <blockquote>
<p>With my limited knowledge I could do <strong>Programmatic data access</strong>. I hope it would be useful for other answers.</p>
</blockquote>
<blockquote>
<p>The code provided in this answer is introduced to show the procedure and the author does not hold any responsibility for its usage.</p>
</blockqu... |
4,070,996 | <p>I'm a bit lost in this integral: <span class="math-container">$$\int \frac{1}{1+\sin^4(x)} \, \mathrm dx$$</span>
I have tried solving with Wolfram, but I was getting a cosecant solution which doesn't seem as the correct method.</p>
<p>Do you have any ideas? :)</p>
<p>EDIT:
Do you please have step-by-step solution, ... | J.G. | 56,861 | <p>Double-check my arithmetic, but here's the strategy.</p>
<p>@Bernard's suggested substitution <span class="math-container">$t=\tan x$</span> gives
<span class="math-container">\begin{align}
& \int\left(1-\frac{t^4}{(t^2\sqrt{2}+1)^2-2(\sqrt{2}-1)t^2}\right) dt \\
= {} & t-\int\frac{t^4}{(t^2\sqrt{2}+ct+1)(t^... |
3,194,534 | <p>Integrate the following <span class="math-container">$\int\frac{x+1}{x(1+xe^x)^2}dx$</span></p>
<p>I tried to multiply the numerator and denominator by <span class="math-container">$e^{-2x}$</span></p>
<p>Getting the following simplification
<span class="math-container">$\int\frac{e^{-2x}(x+1)}{x(e^{-x}+x)^2}dx$</... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write your integrand in the form<span class="math-container">$$-\frac{e^x x+e^x}{e^x x+1}-\frac{e^x x+e^x}{\left(e^x x+1\right)^2}+\frac{1}{x}+1$$</span></p>
|
1,088,687 | <p>I have this example here:</p>
<p>$\lim_{n \to \infty} \frac{n^{2}+1}{n^{2}+n+1} = 1$</p>
<p>so I have to prove that, $\forall \epsilon > 0, \exists n_{0}\in N, \forall n(n \in N) \geq n_{0} \Rightarrow |\frac{n^{2}+1}{n^{2}+n+1} - 1| < \epsilon$</p>
<p>I have done in these steps <br/> <br/>
$|\frac{n^{2}+... | Khosrotash | 104,171 | <p>$$\left|\frac{n}{n^2+n+1}\right| <\epsilon\\\left|\frac{n}{n^2+n+1}\right| <\left|\frac{n}{n^2+n}\right| <\left|\frac{n}{n(n+1)}\right| <\left|\frac{1}{n+1}\right| <\epsilon\\n+1 >\frac{1}{\epsilon}\\n>\frac{1}{\epsilon}-1\\n\geqslant \left \lfloor \frac{1}{\epsilon}-1 \right \rfloor $$</p>
|
1,509,443 | <p>Prove that for $x\geq 1$ the following inequality is true: $$\dfrac{1}{x}\leq \ln(2x+1)-\ln(2x-1).$$</p>
<p>Can anyone help with this problem? I thought about some hours but any ideas</p>
| mathlove | 78,967 | <p>Let $f(x)=\ln(2x+1)-\ln(2x-1)-\frac 1x$. Then, we have
$$f'(x)=\frac{2}{2x+1}-\frac{2}{2x-1}+\frac 1{x^2}=\frac{-1}{x^2(2x+1)(2x-1)}\lt 0$$
for $x\ge 1$ with $\lim_{x\to +\infty}f(x)=0$.</p>
|
2,152,721 | <p>I have been messing around analyzing some self-made algorithms and then I stumbled upon this summation:</p>
<p>$$\sum_{j=0}^{n}\left\lfloor{2^{i-j}}\right\rfloor$$</p>
<p>I am thinking that my common algebraic manipulations involving series won't apply because of the floor function.</p>
<p>I also tried entering t... | Community | -1 | <p>Irrespective of the value of $i $, whether it is or not greater than $j $, we can write our sum as: $$S = \sum_{j=0}^{i} \lfloor 2^{i-j} \rfloor + \sum_{j=i+1}^{n} \lfloor 2^{i-j} \rfloor$$</p>
<p>Case $1$: If $i \geq n $, $$S = \sum_{j=0}^{n} \lfloor 2^{i-j} \rfloor = \sum_{j=0}^{n} 2^{i-j} (\text {why ?}) $$</p>... |
3,659,942 | <p>I have the following problem:</p>
<p>Find all <span class="math-container">$n$</span> that are natural numbers such that <span class="math-container">$\sqrt{n}+\sqrt{n+m}\in\mathbb{N}$</span>, where <span class="math-container">$m$</span> is a positive natural number that is the product of two primes that are not e... | DanielWainfleet | 254,665 | <p>Without recursion.</p>
<p>Let <span class="math-container">$<$</span> be a well-order on <span class="math-container">$A.$</span> A proper initial segment (<span class="math-container">$pis$</span>) of <span class="math-container">$A$</span> is <span class="math-container">$pred_<(a)=\{a'\in A:a'<a\}$</spa... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | V.G | 746,921 | <p>As you said, we have to fill <span class="math-container">$6$</span> gaps in <span class="math-container">$\text{_K_O_A_L_A_}$</span> with <span class="math-container">$7$</span> letters, so if <span class="math-container">$a$</span> letters go in the first gap, <span class="math-container">$b$</span> letters go in ... |
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | Adam Rubinson | 29,156 | <p>Suppose <span class="math-container">$a<b$</span>. We want to prove that <span class="math-container">$a^3 < b^3$</span>.</p>
<p>Well, <span class="math-container">$a<b, \ \therefore \exists \delta>0$</span> such that <span class="math-container">$b = a+\delta$</span>. So:</p>
<p><span class="math-contai... |
2,362,500 | <p>When you are solving an integral, and you have a square root of a square, for instance</p>
<p>$$ \int \sqrt{1+\cos x} = \int \sqrt{\frac{\sin^2x}{1-\cos x}}$$</p>
<p>Do I have to take the absolute value of $\sin x$, or can I just take the positive value? </p>
<p>$$= \int \frac{|\sin x|}{\sqrt{1-\cos x}}$$ OR
$$= ... | Tolure | 239,222 | <p>Again another way.
Since both conditions are independent of each other I did it in two steps.
The first I calculated the number of combinations for the first condition (Either A and B are on the team or they are not).</p>
<p><span class="math-container">$\;\;\displaystyle \binom{7}{3}+\binom{7}{5}= 56$</span></p>
... |
1,912,660 | <p>I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.</p>
<p>Assuming that the function is differentiable at the point in question,<br>
a) I had a look at a few resources online and also looked at this
<a href="https://math.stackexchange.com/questions/2... | Doug M | 317,162 | <p>I had first learned it as if $f(x,y,z) = k$ is a surface $\nabla f$ is a vector perpendicular to the surface. </p>
<p>i.e. the plane tangent to the surface at $\mathbf x = (x_1,y_1,z_1)$ is$\frac {\partial f}{\partial x}(\mathbf x) (x-x_1) + \frac {\partial f}{\partial y}(\mathbf x) (y-y_1) + \frac {\partial f}{\pa... |
1,912,660 | <p>I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.</p>
<p>Assuming that the function is differentiable at the point in question,<br>
a) I had a look at a few resources online and also looked at this
<a href="https://math.stackexchange.com/questions/2... | Matthew Leingang | 2,785 | <p>I think something missing in the accepted answer to the linked question is the connection between gradient and directional derivative. For a function $f(x,y)$ defined at $P=(x_0,y_0)$, and unit a vector $\mathbf{u}=\left<a,b\right>$, we define the <strong>directional derivative of $f$ in the direction $\mathb... |
1,506,805 | <p>$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi }{2}\right)}{x}\right)\:$</p>
| Jan Eerland | 226,665 | <p>$$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi}{2}\right)}{x}\right)=$$
$$\lim _{x\to 0}\left(-\frac{\sin(x)}{x}\right)=$$
$$-\left(\lim _{x\to 0}\frac{\sin(x)}{x}\right)=$$
$$-\left(\lim _{x\to 0}\frac{\frac{\text{d}}{\text{d}x}\sin(x)}{\frac{\text{d}}{\text{d}x}x}\right)=$$
$$-\left(\lim _{x\to 0}\frac{\cos(x)}... |
3,758 | <p>When implicitly finding the derivative of: </p>
<blockquote>
<p>$xy^3 - xy^3\sin(x) = 1$</p>
</blockquote>
<p>How do you find the implicit derivative of:</p>
<blockquote>
<p>$xy^3\sin(x)$</p>
</blockquote>
<p>Is it using a <em>triple</em> product rule of sorts?</p>
| Bill Dubuque | 242 | <p><strong>HINT</strong> $\;$ <a href="http://en.wikipedia.org/wiki/Logarithmic_differentiation" rel="nofollow"><strong>logarithmic differentiation</strong></a> makes the n-ary generalization obvious:</p>
<p>$$\rm (abc)'= abc \; log(abc)' = abc \;(log\; a + log\; b + log\; c)' = abc \; \bigg(\frac{a'}{a} + \frac{b'}{... |
300,067 | <p>They ask me to find all invertible matrices $A$ of the form: $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ and satisfying $A=A^{-1}$ and $A^t=A^{-1}$. I find that rather complex; does it have anything to do with orthogonality? Not sure. Any help please??</p>
| Ittay Weiss | 30,953 | <p>Hint: Given a 2x2 matrix, do you know how to compute its inverse? If not then figure out how to do that. Now that you do, take an arbitrary invertible 2x2 matrix (that is call its entries $a,b,c,d$), compute its inverse and solve the equations $A=A^{-1}$ and $A^t=A^{-1}$.</p>
|
763,073 | <p>I came across the symbol $|v_1 \wedge \dots \wedge v_m|^{-1}$ in a paper - this is the norm of the wedge product of vectors $v_k \in \mathbb{R}^n$ .
I thought it's meaning was self-evident until I tried to compute it. </p>
<p>First of all $|v_1 \wedge \dots \wedge v_m|$ is a <em>number</em> since we are taking th... | Umberto P. | 67,536 | <p>Write each $v_k$ as a column vector. The $m$-volume is the square root of the sum of the squares of the $m \times m$ minor determinants of the matrix $[v_1 | v_2 | \cdots | v_m]$. </p>
|
3,714,244 | <p>In a (finite horizon) sequential game of complete information, any subgame perfect Nash equilibrium (SPNE) necessarily must specify the equilibrium of the last stage game as a Nash equilibrium.</p>
<p>I am wondering if an analogous result holds for sequential games of incomplete information, that is, in a perfect B... | Community | -1 | <p>In PBE, the information sets are not subgames, and a subgame begins at a singleton information set, where the current state is common knowledge. For example, the standard signaling game with two types, two messages for the sender, and two actions for the receiver has only one subgame, the whole game. So the "last su... |
3,714,244 | <p>In a (finite horizon) sequential game of complete information, any subgame perfect Nash equilibrium (SPNE) necessarily must specify the equilibrium of the last stage game as a Nash equilibrium.</p>
<p>I am wondering if an analogous result holds for sequential games of incomplete information, that is, in a perfect B... | brunosalcedo | 161,776 | <blockquote>
<p>In a perfect Bayesian equilibrium (PBE) does the last stage game form
a Bayesian Nash equilibrium (BNE)?</p>
</blockquote>
<p>The short answer is yes, but it can be misleading. </p>
<p>First, any proper subgame of an incomplete information game will have complete information (among the players tha... |
3,557,398 | <p>Here are two second-order differential equations.</p>
<p><span class="math-container">$$ y''+9y=\sin(2t) \tag 1 $$</span></p>
<p><span class="math-container">$$ y'' +4y =\sin(2t) \tag 2 $$</span></p>
<p>I am told to use undetermine coefficients method to solve.</p>
<p>For 1), I use <span class="math-container">... | RobPratt | 683,666 | <p>Yes, it is true, and the property is called <a href="https://en.m.wikipedia.org/wiki/Subadditivity" rel="nofollow noreferrer"><em>subadditivity</em></a>.</p>
|
1,799,559 | <blockquote>
<p>Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$
for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall x\in[a,b]$</p>
</blockquote>
<p>I have difficulties proving this question. Consider for example $g(x)=0$ $\fora... | Patrick Stevens | 259,262 | <p>Your reasoning is wrong: the hypothesis asks that the integral be zero for <em>all</em> continuous $g: [a,b] \to \mathbb{R}$ with $g(a) = g(b) = 0$, not just for the particular choice $g = 0$.</p>
<p>So $f = 1$ is invalidated because of the function $g$ which is a nonlinear quadratic which is $0$ at both $a$ and $b... |
312,012 | <p>What is the exact difference between $\arg\max$ and $\max$ of a function?</p>
<p>Is it right to say the following?</p>
<blockquote>
<p>$\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is maximal. And $\max f(x)$ means value of $f(x)$ for which it is maximum.</p>
</blockquote>
... | Community | -1 | <p>Yes, you are right. $\max f(x)$ is the maximum value (if it exists) of $f(x)$ as $x$ varies through some domain, while $\arg\max f(x)$ is the value of $x$ at which this maximum is attained.</p>
<p>However, there could be more than one $x$ value which gives rise to the maximum $f(x)$, in which case $\arg\max f(x)$ ... |
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