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 |
|---|---|---|---|---|
1,540,082 | <p>I need to prove that the following sequence converges as $n\to\infty$:
$\sum\limits_{k=n+1}^{2n} 1/k$</p>
<p>The problem is that I've only ever seen sums from i to n for example. I'm confused because not only is 2n alien to me, but also k and n are related.</p>
<p>So far I've tried substituting k for n+1 to simpli... | frog | 84,997 | <p>Observe that you sum $n$ terms in the sum, hence you can estimate the sum by $n$ times the biggest term, hence you get
$$
\sum_{k=n+1}^{2n}\frac{1}{k}\leqslant \frac{n}{n+1}\stackrel{n\to\infty}{\longrightarrow}1
$$
Hence the sequence is bounded and since it is monotonically increasing the convergence follows.</p>
|
3,779,674 | <p>I've been working on this question for some time now, and have not had any significant progress towards a solution:</p>
<blockquote>
<p>Let <span class="math-container">$\Omega = \{z = x +iy \in \mathbb{C} : |y| \leq 1\}.$</span> If <span class="math-container">$f(z) = z^2 +2$</span>, then draw a sketch of
<span cla... | Dietrich Burde | 83,966 | <p><em>Hint:</em> The result is a matrix with integer coeffcients, namely
<span class="math-container">$$
\begin{pmatrix} 2070 & 3722 & 4631 \cr
3722 & 6701 & 8353 \cr
4631 & 8353 & 10423 \end{pmatrix}
$$</span>
So you need to evaluate at a certain point.</p>
|
3,779,674 | <p>I've been working on this question for some time now, and have not had any significant progress towards a solution:</p>
<blockquote>
<p>Let <span class="math-container">$\Omega = \{z = x +iy \in \mathbb{C} : |y| \leq 1\}.$</span> If <span class="math-container">$f(z) = z^2 +2$</span>, then draw a sketch of
<span cla... | mechanodroid | 144,766 | <p>You can calculate the last expression as
<span class="math-container">$$(A+I)^9A+A = (((A+I)^3)^2+I)A = \begin{bmatrix} 2070 & 3722 & 4631 \cr
3722 & 6701 & 8353 \cr
4631 & 8353 & 10423 \end{bmatrix}$$</span>
with only <span class="math-container">$4$</span> matrix multiplicati... |
3,779,674 | <p>I've been working on this question for some time now, and have not had any significant progress towards a solution:</p>
<blockquote>
<p>Let <span class="math-container">$\Omega = \{z = x +iy \in \mathbb{C} : |y| \leq 1\}.$</span> If <span class="math-container">$f(z) = z^2 +2$</span>, then draw a sketch of
<span cla... | Disintegrating By Parts | 112,478 | <p><span class="math-container">\begin{align}
\det(\lambda I-A)&=\left|
\begin{array}{ccc}
\lambda & -1 & 0 \\
-1 & \lambda & -1 \\
0 & -1 & \lambda-1
\... |
2,232,677 | <p>A parallellogram $ABCD$ is given, where $E$ is the midpoint of $BC$ and $F$ is a point on $AD$ so that $|FD| = 3|AF|$. $G$ is the point where $BF$ and $AE$ intersects.</p>
<p>Express the vector $AG$ in terms of vectors $AB$ and $AD$.</p>
<p>My solution to the problem is the following: Impose an <em>affine transfor... | DHMO | 413,023 | <p>Forgive my lack of vector symbols below.</p>
<hr>
<p>$AE = AB+\dfrac12AD$</p>
<p>$BF = -AB+\dfrac14AD$</p>
<hr>
<p>Let $AG = hAB+kAD$.</p>
<p>Since $AG // AE$, we have $h=2k$.</p>
<p>Since $BG // BF$, we have $-(h-1)=4k$.</p>
<hr>
<p>Solving for $h$ and $k$ gives $h=\dfrac13$ and $k = \dfrac16$, so:</p>
<p... |
3,933,135 | <p>Let <span class="math-container">$m$</span> and <span class="math-container">$n$</span> are positive integer</p>
<blockquote>
<p>Can it be shown that, For every <span class="math-container">$m\ge 5$</span></p>
<p><span class="math-container">$$\sum_{i=1}^ni^m-m^i>0\iff n=2,3,...,m$$</span></p>
</blockquote>
<p>Ex... | John Omielan | 602,049 | <p>You're asking about proving for every <span class="math-container">$m \ge 5$</span> that</p>
<p><span class="math-container">$$\sum_{i=1}^{n}(i^m - m^i) \gt 0 \iff n = 2, 3, \, \ldots \, , m \tag{1}\label{eq1A}$$</span></p>
<p>First, consider the <span class="math-container">$\impliedby$</span> direction, starting a... |
2,911,849 | <p>Let $S_i$ be a countable family of sets and $f$ a continuous map (say everything is in the complex plane).
Is it true that
$$
f( \overline{ \cup_i S_i } ) =
\overline{ \cup_i f( S_i ) }
$$
or perhaps the inclusion $\subseteq$?</p>
| José Carlos Santos | 446,262 | <p>The inclusion holds. You have$$f\left(\overline{\bigcup_iU_i}\right)\subset\overline{f\left(\bigcup_iU_i\right)}$$and this has nothing to do with unions. It's just that, if $f$ is continuous and $A\subset D_f$, $f\left(\overline A\right)\subset\overline{f(A)}$.</p>
<p>However, the equality doesn't hold in general. ... |
63,601 | <p>I know that if a ring has a multiplicative identity, then the multiplicative identity must be unique. Are there simple-to-describe examples of rings with two (or more) multiplicative right-identities?</p>
| Qiaochu Yuan | 290 | <p>Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup
$$S = \langle a, b | ab = aa = a, ba = bb = b \rangle$$
works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every... |
3,907,519 | <p>Prove that for all <span class="math-container">$a,b \in\mathbb Z$</span>, if <span class="math-container">$a+b$</span> is even then <span class="math-container">$a-b$</span> is even.</p>
<p>I started off by assuming that a and <span class="math-container">$b$</span> are both odd integers, <span class="math-containe... | Rubenscube | 848,029 | <p>I think this is definitely a good start, but it is always a good idea to cover all possible cases, which means mentioning even and even, but also even and odd. I hope this helps!</p>
<p>Another way to proof it, is without using cases:
If <span class="math-container">$a+b$</span> is even, than there exists a whole nu... |
3,907,519 | <p>Prove that for all <span class="math-container">$a,b \in\mathbb Z$</span>, if <span class="math-container">$a+b$</span> is even then <span class="math-container">$a-b$</span> is even.</p>
<p>I started off by assuming that a and <span class="math-container">$b$</span> are both odd integers, <span class="math-containe... | Peter Szilas | 408,605 | <p><span class="math-container">$a+b$</span> is even.</p>
<p>Assume <span class="math-container">$a-b$</span> is odd.</p>
<p>Add: <span class="math-container">$(a+b)+(a-b)=2a;$</span></p>
<p>Adding an odd number <span class="math-container">$(a-b)$</span> to an even number <span class="math-container">$(a+b) $</span> g... |
1,736,360 | <p>Using the binomial expansion, find the taylor series of the function $$f(z) = \frac{1}{(1+z)^3}$$ and find its radius of convergence.</p>
<hr>
<p>The solution is</p>
<p>$$f(z) = \sum_{n=0}^\infty (-1)^n\frac{(n+1)(n+2)}2z^n$$</p>
<p>How do we find this? I thought the binomial expansion was</p>
<p>$$(1+z)^n = \s... | David Quinn | 187,299 | <p>Using the Binomial Theorem, for $|x|<1$, the coefficient of $x^r$ is $$\frac{(-3)(-4)...(-(r+2))}{r!}=(-1)^r\frac{1.2.3...(r+2)}{1.2.r!}$$
$$=(-1)^r\frac{(r+1)(r+2)}{2!}$$</p>
<p>Hence the result given</p>
|
918,693 | <p>I've been at this for a few hours now, and it's frighteningly similar to the problem stated here: </p>
<p><a href="https://math.stackexchange.com/questions/388356/how-to-prove-int-0-infty-e-x2cos2bx-dx-frac-sqrt-pi2-e-b2">How to prove $\int_0^\infty e^{-x^2}cos(2bx) dx = \frac{\sqrt{\pi}}{2} e^{-b^2}$</a></p>
<p>b... | André Nicolas | 6,312 | <p>In your integral, $a$ is presumably positive, else we would not have convergence.</p>
<p>Make the change of variable $t=x\sqrt{a}$ and you will be at an integral very close to the one linked to. </p>
|
138,173 | <p>$f(x, y) = 0$ and $g(x, y) = 0$,
both $f$ and $g$ are cubic polynomial equation (at most 10 coefficients for each).</p>
<p>Is there any fixed method to solve this degenerate equation system?
thanks.</p>
| Peter Mueller | 18,739 | <p>Resultants are good, but for practical computations the Bezout Lemma (it's the same as the extended Euclidean algorithm) can be less expensive: Assume that $f(x,y)$ and $g(x,y)$ are relatively prime. Consider $f$ and $g$ as polynomials in $y$ over the field $k(x)$, where $k$ is your (unspecified) base field. Then th... |
2,581,361 | <p>For even $n \in \mathbb{N}$, prove $\binom{n}{i}< \binom{n}{j} $ if $0\leq i<j\leq \frac{n}{2}$.</p>
<p>So far all I have been able to come up with are a bunch of seemingly useless inequalities. </p>
<p>Any hints would be greatly appreciated.</p>
| P. Koymans | 385,053 | <p>Try proving
$$
\binom{n}{i} < \binom{n}{i + 1}
$$
by using the formula
$$
\binom{n}{i} = \frac{n!}{i! \cdot (n - i)!}.
$$</p>
|
962,301 | <p>$$(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)…$$</p>
<p>I have found partial sums equal to natural numbers. The first 3 addends sum to 1. The first 8 sum to 2. The first 15 sum to 3. When the minuend in an addend is the square root of a perfect square, the partial sum is a natural number... | Ron Gordon | 53,268 | <p>The $n$ term may be rewritten as </p>
<p>$$\frac1{\sqrt{n}+\sqrt{n+1}}$$</p>
<p>which behaves as $1/(2 \sqrt{n})$ as $n \to\infty$, so by comparison with the harmonic series, this series diverges.</p>
|
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>
| Alicia Garcia-Raboso | 1,797 | <p>This is not a complete answer by any means, but here are the two most basic arguments. First of all, you have that every projective scheme that can be embedded in $\mathbb{P}^n$ can be covered by $n+1$ open affines, namely the closed subschemes of the affines $U_i = \lbrace z_i \neq 0 \rbrace \cong \mathbb{A}^n$.</p... |
2,839,823 | <p>I don't know where to ask, but I'm trying. I just thing we cannot do hocus pocus methods.</p>
<hr>
<p>Solve $3^x+4^x=5^x$</p>
<p>Okay, so my friend gave me this equation, and his solution. But I don't belive it holds. It comes here:</p>
<p>"Solution: </p>
<p>$3^x+4^x=5^x\Leftrightarrow \frac{3^x}{3^x}+\frac{4^x... | Phil H | 554,494 | <p>Another method..........</p>
<p>$f(x) = 3^x + 4^x - 5^x$</p>
<p>$f'(x) = \ln(3)3^x + \ln(4)4^x - \ln(5)5^x$</p>
<p>$x_0 = 3$</p>
<p>$x_1 = 3 - \frac{27 + 64 -125}{-82.79} = 2.59$</p>
<p>$x_2 = 2.59 - \frac{-11.15}{-40.73} = 2.32$</p>
<p>$x_3 = 2.32 - \frac{-4.12}{-18.72} = 2.10$</p>
<p>$x_4 = 2.10 - \frac{-.9... |
1,028,301 | <p>What I've first done is show that $Aut(\mathbb Z / 24\mathbb Z)$ is isomorphic to $\mathbb Z_2\oplus\mathbb Z_2\oplus\mathbb Z_2$ due to it having 8 members and the greatest rank out of all of them is 2.</p>
<p>Now $M_2(\mathbb Z/3\mathbb Z)$ is isomophic to $\mathbb Z_3\oplus\mathbb Z_3\oplus\mathbb Z_3\oplus\math... | yatima2975 | 360 | <p>You're looking for a non-trivial homomorphism $\phi$ of a group (let's call it $G$) of size 81 to one of size 8 (your isomorphisms are indeed correct!). </p>
<p>What can you say about $|Ker(\phi)|$, $|G/Ker(\phi)|$ and $|Im(\phi)|$ in terms of their factorisations? </p>
|
142,007 | <p>Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$</p>
<p>Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.</p>
<p>I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion... | Arturo Magidin | 742 | <p>Suppose <span class="math-container">$x^2\equiv -1\pmod{p}$</span>. Then <span class="math-container">$x^4\equiv 1\pmod{p}$</span>. Since <span class="math-container">$p = 4k+3$</span>, we have
<span class="math-container">$$x^{p-1} = x^{4k+2} = x^2x^{4k} \equiv -1(x^4)^k\equiv -1\pmod{p},$$</span>
which contradicts... |
185,900 | <p>What is the <strong>fastest way</strong> to find the smallest positive root of the following transcendental equation:</p>
<p><span class="math-container">$$a + b\cdot e^{-0.045 t} = n \sin(t) - m \cos(t)$$</span></p>
<pre><code>eq = a + b E^(-0.045 t) == n Sin[t] - m Cos[t];
</code></pre>
<p>where
<span class="m... | Bob Hanlon | 9,362 | <p>Amplifying on comment by @bills; for your specific example there are no roots</p>
<pre><code>eq = 11 + 5 E^(-0.045 t) == 0.03 Sin[t] - 1.2 Cos[t] // Rationalize;
</code></pre>
<p>The <a href="https://reference.wolfram.com/language/ref/FunctionRange.html" rel="noreferrer"><code>FunctionRange</code></a> of the LHS a... |
618,166 | <p>Does there exists a real valued function which is everywhere continuous and differentiable at exactly one point on the real line?</p>
| Martin Argerami | 22,857 | <p>Let <span class="math-container">$f(t) $</span> be the <a href="https://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow noreferrer">Weierstrass function</a>, continuous everywhere and nowhere differentiable. Define
<span class="math-container">$$
g(t)=t^2f(t).
$$</span>
Then <span class="math-container">$g... |
1,195,166 | <p>for a>0 can someone give me some clues to prove this inequality. </p>
<p><img src="https://i.stack.imgur.com/WFQA2.png" alt="enter image description here"></p>
<p>I found its derivative but I can't express the f(0) to use the Mean Value Theorem. Can you give me at least a point where to start at</p>
| ASB | 111,607 | <p>$ \dfrac{f(x+\delta x)-f(x)}{(x+\delta x)-x}=\dfrac{\sqrt{x+\delta x}-\sqrt{x}}{\delta x}=\dfrac{(\sqrt{x+\delta x}-\sqrt{x})(\sqrt{x+\delta x}+\sqrt{x})}{\delta x (\sqrt{x+\delta x}+\sqrt{x})}=\dfrac{1}{\sqrt{x+\delta x}+\sqrt{x}} $</p>
|
1,463,887 | <p>$f(x,y)=x^2-y^2$ constraint to $x^2+y^2=1$</p>
<p>$f_x=2x$ and $f_y=-2y$ $\implies$ the critical point is at $(0,0)$</p>
<p>However, $(0,0)$ does not occur in the constraint. does that mean i don't have to consider it?</p>
<p>So if we work of the boundary curves $y=\pm\sqrt{1-x^2}$</p>
<p>we get a min @ $(0,1)$ ... | Jack D'Aurizio | 44,121 | <p>A bit of an overkill, don't you think? The problem is equivalent to finding the extreme values of $z-(1-z)$, i.e. of a line, subject to the constraint $z\in[0,1]$.</p>
|
2,901,914 | <blockquote>
<p>Given the function $f:[0,2\pi)\to S^1$, $\varphi\mapsto (\cos(\varphi), \sin(\varphi))^t$. Show that $f$ is continuous and a bijection, but not a homeomorphism.</p>
</blockquote>
<p>That $f$ is continuous is clear, since every component is continuous.
Furthermore it is continuous differentiable.</p>
... | Sri-Amirthan Theivendran | 302,692 | <p>To see that the inverse is not continuous note that there exists a sequence of points $(y_n)$ s.t $y_n\to (1,0)$ but $f^{-1}(y_n)\to 2\pi\ne f^{-1}(1,0)=0.$ Consider the sequence of points given by
$$
(\cos (2\pi-n^{-1}), \sin (2\pi-n^{-1}) )
$$
for example.</p>
|
2,901,914 | <blockquote>
<p>Given the function $f:[0,2\pi)\to S^1$, $\varphi\mapsto (\cos(\varphi), \sin(\varphi))^t$. Show that $f$ is continuous and a bijection, but not a homeomorphism.</p>
</blockquote>
<p>That $f$ is continuous is clear, since every component is continuous.
Furthermore it is continuous differentiable.</p>
... | José Carlos Santos | 446,262 | <p>The function $f$ is surjective becasue if $(x,y)\in S^1$, there is a $\theta\in[0,2\pi)$ such that $(x,y)=(\cos\theta,\sin\theta)$; just take $\theta=\arccos x$ if $y\geqslant0$ and $\theta=2\pi-\arccos x$ otherwise.</p>
<p>And $f^{-1}$ is discontinuous because $\lim_{n\to\infty}\left(\cos\left(2\pi-\frac1n\right),... |
1,324,824 | <p>Compute the flux integral with vector field,
$\vec F(x,y,z)= <2x,y,3z>$
and a sphere of radius 36 centered at the point $(1,2,-1)$</p>
<p>so what I did was I used divergence theorem which is</p>
<p>$\int\int \vec F \cdot \vec n dS = \int \int \int (div \vec F) dV$</p>
<p>I used the right hand side and so</p... | user1337 | 62,839 | <p>Since the divergence of the field $\vec{F}$ is constant, the given integral doesn't depend on the origin of the sphere. Moreover, the surface mustn't even be sphere shaped, as long as it encloses the same volume.</p>
|
1,836,998 | <p>$$
\lim_{x\to 0^{+}}\left[\left(1+\frac{1}{x}\right)^x+\left(\frac{1}{x}\right)^x+\left(\tan(x)\right)^{\frac{1}{x}}\right]$$</p>
<p>Attempt: I used $\tan(x)\approx x$ also $(1+n)^{1/n}=e$ so I let $x=0+h$ and lim changes to lim h tending to $0$. But that gives me $e+\infty+h^{1/h}$. While the answer is an integer ... | Mark Viola | 218,419 | <p>Note that we can write </p>
<p>$$\begin{align}
\left(1+\frac1x\right)^x&=e^{x\log\left(1+\frac1x\right)}\\\\
&=e^{x\left(\log(1+x)-\log(x)\right)}\\\\
&=e^{-x\log(x)}e^{x\log(1+x)}
\end{align}$$</p>
<p>Since $\lim_{x\to 0^+}x\log(x)=0$, we find using the continuity of the exponential function that</p>
... |
204,780 | <p>Let $X$ and $Y$ be independent discrete random variables, each taking values $1$ and $2$ with probability $1/2$ each. How do I calculate the covariance between $max(X,Y)$ and $min(X,Y)$?</p>
| Brian M. Scott | 12,042 | <p>Let $W=\min\{X,Y\}$ and $Z=\max\{X,Y\}$; then the desired covariance is $\mathrm{E}[WZ]-\mathrm{E}[W]\mathrm{E}[Z]$. $W=1$ unless $X=Y=2$, so $\mathrm{Pr}(W=1)=\frac34$ and $\mathrm{Pr}(W=2)=\frac14$, and therefore</p>
<p>$$\mathrm{E}[W]=\frac34\cdot1+\frac14\cdot2=\frac54\;.$$ </p>
<p>The calculation of $\mathrm{... |
459,124 | <p>Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental don't seem to apply. This is such a special case, does any one know if the series</p>
<p>$$\sum_{i=0}^{\infty} \fra... | Jared | 65,034 | <p>(What I learned reading the Wikipedia article on <a href="http://en.wikipedia.org/wiki/Transcendental_number" rel="nofollow">transcendental numbers</a>)</p>
<p>If $a$ is an algebraic number in $(-1,1)$, it has been shown that any number of the form:</p>
<p>$$\sum_{i=0}^{\infty}a^{2^i}$$</p>
<p>is transcendental. ... |
3,820,947 | <p>Now I have always been rather intrigued with factorial, at first, in high school, teachers told me that factorials are only defined for whole numbers. As I studied, I found factorials for positive reals and negative fractions. But the integral with which we define factorial falls flat on the negative integers.</p>
<... | Jordan Mitchell Barrett | 649,843 | <p>The recursive definition of the factorial <span class="math-container">$(n+1)! = n! \cdot (n+1)$</span> can be rearranged to deduce <span class="math-container">$$n! = \dfrac{(n+1)!}{n+1}$$</span></p>
<p>This is one way to work out that <span class="math-container">$0!=1$</span>, taking <span class="math-container">... |
84,255 | <p>It is just like the linear algebra over commutative ring (maybe advanced linear algebra), that is a nature extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic.</p>
<p>I just know Weibel’s book “Homological Algebra” discuss something, but that is just a fo... | Primoz | 6,339 | <p>Lie algebras over rings (Lie rings) are important in group theory. For instance, to every group $G$ one can associate a Lie ring</p>
<p>$$L(G)=\bigoplus _{i=1}^\infty \gamma _i(G)/\gamma _{i+1}(G),$$</p>
<p>where $\gamma _i(G)$ is the $i$-th term of the lower central series of $G$. The addition is defined by the a... |
84,255 | <p>It is just like the linear algebra over commutative ring (maybe advanced linear algebra), that is a nature extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic.</p>
<p>I just know Weibel’s book “Homological Algebra” discuss something, but that is just a fo... | Fernando Muro | 12,166 | <p>Integral Lie algebras are also important in homotopy theory, if $X$ is a simply connected space and $\pi_*(X)=\bigoplus_{n\geq 1}\pi_n(X)$ are its homotopy groups, $\pi_*(X)[-1]$ is a graded Lie algebra. The Lie bracket is the Whitehead product. This Lie algebras satisfy the feature that $[x,x]\neq 0$ in general sin... |
1,898,839 | <p>Is the following inequality true?</p>
<p><span class="math-container">$$\mbox{Tr} \left( \mathrm P \, \mathrm M^T \mathrm M \right) \leq \lambda_{\max}(\mathrm P) \, \|\mathrm M\|_F^2$$</span></p>
<p>where <span class="math-container">$\mathrm P$</span> is a positive definite matrix with appropriate dimension. How a... | Rodrigo de Azevedo | 339,790 | <p><span class="math-container">$$\mbox{tr}(\mathrm A \mathrm B) = \left\langle \mathrm A^\top, \mathrm B \right\rangle = \langle \mbox{vec} (\mathrm A^\top), \mbox{vec} (\mathrm B) \rangle \leq \|\mbox{vec} (\mathrm A^\top)\|_2 \|\mbox{vec} (\mathrm B)\|_2 = \|\mathrm A\|_F \|\mathrm B\|_F$$</span></p>
|
4,802 | <p>Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups a... | Benjamin Antieau | 100 | <p>This may not be particularly helpful, but when $X=\mathrm{spec} k$, for a field $k$, then $CH^{2n}(X,n)=K_n^M(k)$, the Milnor $K$-theory of $k$. I do not know if there are any other useful characterizations. Very little is known, except for the formal properties like $A^1$-invariance and such.</p>
<p>I believe that... |
4,802 | <p>Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups a... | Jinhyun Park | 3,168 | <p>Benjamin Antineau's answer needs a minor correction. For $X= spec (k)$, we have $CH^n (X, n) = K^M _n (k)$, not $CH^{2n} (X, n)$. Indeed not too many is known, but it is a very interesting subject (at least for me) to pursue. A good start would be Burt Totaro's paper 'Milnor K-theory is the simplest part of K-theory... |
352,501 | <p>There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.</p>
<blockquote>
<p><strong>Lemma</strong> Let <span class="math-container">$G$</span> be a group of order <span class="math-container">$30$</span>. Then the <span class="math-container">$5$</span>-Syl... | pavan | 423,856 | <p><span class="math-container">$o(G) = 30 = 2*3*5$</span></p>
<p>Since the number of <span class="math-container">$5$</span> - Sylow subgroups is of form, <span class="math-container">$N_G(5) = 5k +1 $</span> and <span class="math-container">$N_G(5) \Big | o(G)$</span>,</p>
<p><span class="math-container">$\implies N... |
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... | Mike Parks | 496,458 | <p>Statistically your odds of winning indeed are increased by picking 10 times out of 100 assuming the 10 picks are distinct and without replacement.</p>
<p>1 pick of 10 = 10% Chance of selecting the winning ticket</p>
<p>1 of 100= 1%
1 of 99= 1.01%
1 of 98 = 1.02%
1 of 97 = 1.03%
1 of 96 = 1.04%
1 of 95 = 1.05%
1 o... |
2,007,123 | <p>Suppose $R=\mathbb{Q}[x]$ is a ring and define $T$ to be $T=\{f(x) \in R : f(0) \neq 0\}$ </p>
<p>Prove that $\frac{x}{1}$ is the only irreducible element in $T^{-1}R$ (disregarding associates).</p>
<hr>
<p>My approach: assume $\frac{x}{1}$ is reducible. So there exist non-unit elements $\frac{a}{b}, \frac{c}{d}$... | Nominal Animal | 318,422 | <p>Rewritten on 2016-11-12. The OP raised very good questions in the comments. Note that this is not intended as an exhaustive answer (as one might expect from, say, a mathematician?), but more like observations from someone who routinely uses bilinear interpolation for numerical data.</p>
<blockquote>
<p>How can a ... |
3,401 | <p>Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where pe... | Igor Rivin | 11,142 | <p>I think the answers are not to the question asked (at least <em>as</em> it is asked). The ergodicity of the geodesic flow (which, by the way holds for all negatively curved surfaces -- a fact surprising not mentioned in any of the above answers) does not mean that a fixed geodesic will hit <em>every</em> point on th... |
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... | Alan Abraham | 823,763 | <p>No, the probability distribution of the sum of the faces of rolling <span class="math-container">$10$</span> <span class="math-container">$6$</span>-sided dice is different than that of rolling a single <span class="math-container">$60$</span>-sided die.</p>
<p>For obvious reasons, this is because you can roll any n... |
119,492 | <p>First I define a function, just a sum of a few sin waves at different angular frequencies:</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40}
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, .001}];
pts = Length@fn
ListPlot[fn, Joined -> True, PlotRange -> {{0, 1000}, All}]
{20, 50, 100, 200, 400}
</code>... | Hugh | 12,558 | <p>I could not make LowpassFilter work for one-dimensional time histories either. I suggest you use a Butterworth filter which is more standard. Also I like to use random for testing because you can look at the transfer function. For the test below I add a few zeros at the end of the random to allow for filter ringing.... |
2,966,959 | <blockquote>
<p>Consider the simultaneous system of differential equations:
<span class="math-container">$$ \begin{equation}
x'(t)=y(t) -x(t)/2\\
y'(t)=x(t)/4-y(t)/2
\end{equation} $$</span>
If <span class="math-container">$ x(0)=2 $</span> and <span class="math-container">$ y(0)=3 $</span>, then what is <span cl... | Aleksas Domarkas | 562,074 | <p>Integrating factor is <span class="math-container">$\mu=y^3$</span>. Then solution of equation <span class="math-container">$$xydx+(2x^2+3y^2)dy=0$$</span> is
<span class="math-container">$$y^6+x^2y^4=C$$</span></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 ... | Pedro | 23,350 | <p>I have what I think is an equivalent situation:</p>
<blockquote>
<p>Suppose $f$ is such that $f(x)-f(y)\leq (x-y)^2$ for all $x,y$. Then $f$ is constant. </p>
</blockquote>
<p><strong>PROOF</strong> Choose any $x,y$. The hypothesis means that</p>
<p>$f(y)-f(x)=-(f(x)-f(y))\leq (y-x)^2$ whence $|f(x)-f(y)|\leq (... |
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 ... | wj32 | 35,914 | <p><strong>Step 1.</strong> Let $x,y\in\mathbb{R}$ with $x \ne y$. For all $r>1$ we have $|x-y|<r|x-y|$, so $|f(x)-f(y)|<r^2|x-y|^2$. This is true for all $r>1$, so we must have $|f(x)-f(y)| \le |x-y|^2$.</p>
<p><strong>Step 2.</strong> This is just Exercise 5.1 in Rudin; rewrite the above as
$$\left\vert\... |
2,709,532 | <p>(True/False) If $S$ is a non-empty subset of $\mathbb{N}$, then there exists an element $m \in S$ such that $m\ge k$ for all $k \in S$.</p>
<p>So my reasoning is that the above statement is false. Since $S$ is a non-empty subset of $\mathbb{N}$, it may also be the case that $S=\mathbb{N}$, and so there may not be a... | Mohammad Riazi-Kermani | 514,496 | <p>You are on the right track but your counter-example is an overkill.</p>
<p>Every infinite subset of $\mathbb N$ can serve as a counter-example. </p>
<p>For example, multiples of 5 or the set of prime numbers or the set of perfect squares are such counter-examples. </p>
|
4,626,019 | <p>I recently encountered a quant interview question, which I think is not super hard, but still I am not very sure about my results.</p>
<p>Could you guys give me some hints on it?</p>
<p>The question goes as follows:</p>
<blockquote>
<p>An ant is put on a vertex of a simplex, and at each vertex, it has equal probabil... | Sangchul Lee | 9,340 | <p><em>Sketch of Proof.</em> Suppose the ant is moving on an <span class="math-container">$n$</span>-simplex. Then the crucial observation is that we can come up with a Markov chain which moves in a single direction:</p>
<p><span class="math-container">$$ 0
\overset{\frac{n}{n}}{\longrightarrow} 1
\overset{\frac{n-1}{n... |
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... | Martín-Blas Pérez Pinilla | 98,199 | <p>Idea for an approximate solution: divide the ellipsoid in small enough, flat enough quasi-rectangles $P_i$, choose an almost-area-preserving parametrization of each piece:
$$f_i:R_i\longrightarrow P_i$$
with $R_i\subset\Bbb R^2$ a rectangle. Choose randomly an index $j$ with probability $\text{area}(R_j)/\sum_i\text... |
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... | Artashes | 1,054,066 | <p>Here's the code, This works in ANY dimension :</p>
<pre><code>import numpy as np
dim = 2
r=1
a = 10
b = 4
A = np.array([[1/a**2, 0],
[0, 1/b**2]])
L = np.linalg.cholesky(A).T
x = np.random.normal(0,1,(200,dim))
z = np.linalg.norm(x, axis=1)
z = z.reshape(-1,1).repeat(x.shape[1], axis=1)
y = x/z... |
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... | robjohn | 13,854 | <p>From the definition of a limit in a metric space:
<span class="math-container">$$
\forall\epsilon\gt0,\exists\delta\gt0:0\lt|x-0|\le\delta\implies|f(x)-0|\le\epsilon\tag1
$$</span>
Since there are no <span class="math-container">$x$</span> in the domain of <span class="math-container">$f(x)=\sqrt{-x^2}$</span> that ... |
1,123,513 | <p>The question is$$\text{Is }\mathbb{R}^2\text{ a subspace of }\mathbb{C}^2?$$My first thing to think about it now is $$\text{Is }\mathbb{R}^2\text{ a subset of }\mathbb{C}^2?$$ I think no because what is $\mathbb{R}^2$ it's like all those pairs $(0,1)$ right? and what is $\mathbb{C}^2$ it's like euh $((a,b),(c,d))$ (... | TonyK | 1,508 | <p>According to most of the usual definitions, $\mathbb R$ is not a subset of $\mathbb C$. You are absolutely correct about this. In the same way, $\mathbb N$ is not a subset of $\mathbb Z$, which is not a subset of $\mathbb Q$, which is not a subset of $\mathbb R$.</p>
<p>However, $\mathbb N$ has a <em>natural embedd... |
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 ... | Robert Israel | 8,508 | <p>You do not remember correctly. Try $f(x,y) = x$. Perhaps there was an assumption that $f$ is bounded?</p>
|
728,503 | <p>\begin{align*}
f(x) = \left\{\begin{array}{ll}
0 & \text{ if } x=0\\
x^\alpha \sin(x^{-\beta}) & \text{ otherwise }
\end{array}\right.
\end{align*}</p>
<p>Determine the values of $\alpha$ and $\beta$ for which this function is differentiable at $x=0$.</p>
<p>I found the derivative, but I do... | Community | -1 | <p>Remember that the delta function has the special property that</p>
<p>$$\int_{-\infty}^{\infty} \delta(t - a) f(t) dt = f(a)$$</p>
<p>(or more particularly, $\int_0^{\infty} \delta(t - a) f(t) dt = f(a)$ whenever $a > 0$) together with the definition of the Laplace transform. Can you take it from here?</p>
|
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>
| Shuri2060 | 243,059 | <p>$$\frac{a}{b}\times\frac{b}{a}=\frac{a\times b}{b\times a}=\frac{ab}{ab}=1\quad\quad\quad a\neq 0\,,\, b\neq0$$</p>
|
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... | abc... | 470,090 | <p>Your answer is $\frac {80}{32}$. The $\frac1 {32}$ is from where all four flips are heads.</p>
|
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... | Siong Thye Goh | 306,553 | <p>Let $X_i=1$ if $i$-th toss is head and $0$ otherwise.</p>
<p>The reward is $$\sum_{i=1}^4X_i + X_5\left( 1-\prod_{i=1}^4X_i\right)=\sum_{i=1}^5X_i-\prod_{i=1}^5X_i$$ </p>
<p>Hence </p>
<p>$$\mathbb{E}\left(\sum_{i=1}^5X_i-\prod_{i=1}^5X_i\right)=\left(\sum_{i=1}^5\mathbb{E}[X_i]-\prod_{i=1}^5\mathbb{E}[X_i]\right... |
392,441 | <p>Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.</p>
<p>I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation $f\delta′=f(0)\delta′−f′(0)\delta$.</p>
| mrf | 19,440 | <p>You're missing an $f$ in the left hand side.</p>
<p>Compute $(f\delta)'$ in two different ways.</p>
<p>On one hand $(f\delta)' = f'\delta + f\delta' = f'(0)\delta + f\delta'$.</p>
<p>On the other hand $(f\delta)' = (f(0)\delta)' = f(0)\delta'$.</p>
<p>Hence $f\delta' = f(0)\delta' - f'(0)\delta$.</p>
|
392,441 | <p>Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.</p>
<p>I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation $f\delta′=f(0)\delta′−f′(0)\delta$.</p>
| Mårten W | 58,780 | <p>(I suppose that with $\delta$ you mean the Dirac function.)
Just evaluate $(f(t)\delta(t))'$ in two ways.</p>
<p>First, use Leibniz' rule:
$$
(f(t)\delta(t))'=f(t)\delta'(t)+f'(t)\delta(t)=f(t)\delta'(t)+f'(0)\delta(t).
$$
We also have
$$
(f(t)\delta(t))'=(f(0)\delta(t))'=f(0)\delta'(t).
$$</p>
<p>Since the two ex... |
43,646 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/43531/havel-hakimi-theorem">Havel-Hakimi Theorem</a> </p>
</blockquote>
<p>Hi. I'm a beginner at graph theory, and I recently came across the Havel-Hakimi Theorem which is used to determine whether a seque... | Dai | 1,864 | <p>The proof in the book "Pearls in Graph Theory" <a href="http://books.google.ca/books?id=R6pq0fbQG0QC&lpg=PP1&dq=pearls%20of%20graph%20theory&pg=PA10#v=onepage&q=pearls%20of%20graph%20theory&f=false" rel="nofollow">here</a> is quite clear. </p>
|
4,541,392 | <p>I have an integral that looks like the following:
<span class="math-container">$$
\int_a^b \frac{1}{\left(1 + cx^2\right)^{3/2}} \mathrm{d}x
$$</span>
I have seen a method of solving it being to substitute <span class="math-container">$x = \frac{\mathrm{tan}(u)}{\sqrt{c}}$</span>; however, this seems somewhat sloppy... | MathDona | 1,084,986 | <p>Let
<span class="math-container">$$
I=\int \frac{1}{\left(1 + cx^2\right)^\frac{3}{2}} \mathrm{d}x= \int \frac{1}{x^3(1/x^2+c)^{3/2}}$$</span>.
Let <span class="math-container">$1/x=u \implies -dx/x^2=du$</span> and then <span class="math-container">$u^2=v$</span>
<span class="math-container">$$I=-\int \frac{udu}{(u... |
732,132 | <blockquote>
<p>Suppose $P(x)$ is a polynomial of degree $n \geq 1$ such that $\int_{0}^{1}x^{k}P(x)\,dx = 0$ for $k = 1, 2, \ldots, n$. Show that $$\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$$</p>
</blockquote>
<p>If we assume that $P(x) = a_{0}x^{n} + \cdots + a_{n - 1}x + a_{n... | orangeskid | 168,051 | <p>Note that <span class="math-container">$P(x)$</span> is a linear combination of <span class="math-container">$1$</span> and <span class="math-container">$x, x^2$</span>, <span class="math-container">$\ldots$</span>, <span class="math-container">$x^n$</span> and is perpendicular to the subspace generated by <span cla... |
119,527 | <p>I'm trying to prove the following:</p>
<blockquote>
<p>Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences such that $(a_n)_{n\in\mathbb{N}}$ converges and $(b_n)_{n\in\mathbb{N}}$ is bounded. If $a=\lim_{n\to\infty} a_n$, prove that $$\limsup_{n\to\infty} (a_n+b_n) = a +\limsup_{n\to\infty... | Aryabhata | 1,102 | <p>Let $\displaystyle b_{n_k}$ be a subsequence of $\displaystyle b_n$ which converges to $\displaystyle \limsup b_n$.</p>
<p>Show that $\displaystyle a_{n_k} + b_{n_k}$ converges. </p>
<p>How does this limit relate to $\displaystyle \limsup (a_n + b_n)$?</p>
|
1,227,330 | <p>I have the following hypotheses: $\alpha_n \to 0, \beta_n \to 0, \alpha_n < 0 < \beta_n$. I need to show that the following two limits exist <strong>so that I may add them</strong>:</p>
<p>$$\lim \limits_{n \to \infty} \frac{-\alpha_n}{\beta_n -\alpha_n}, \lim \limits_{n \to \infty} \frac{\beta_n}{\beta_n - \... | robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/75151">this answer</a>, it is shown that for $0\le x\le\frac\pi2$,
$$
\cos(x)\le\frac{\sin(x)}x\le1\tag{1}
$$
So, for $0\lt x\le\frac\pi2$, and symmetrically for $-\frac\pi2\le x\lt0$, subtract $1$ from $(1)$ and divide by $x$:
$$
0=\lim_{x\to0}\frac{\cos(x)-1}{x}\le\lim_... |
467,318 | <blockquote>
<p><span class="math-container">$ABC$</span> is a triangle with integral side lengths. Given that <span class="math-container">$\angle A=3\angle B$</span>, find the minimum possible perimeter of <span class="math-container">$ABC$</span>.</p>
</blockquote>
<p>I got this problem from an old book (which did n... | wendy.krieger | 78,024 | <p>Such triangles exist. I found one 3:8:10, where the angle opposite 8 is 1/3 of that opposite 10. The trick here is to pick really small primes in the form of $x+y\sqrt{-n}$, and cube the result. Note here that we're using $n=7$, and the prime is $1\frac 12 + \frac 12\sqrt{-7}$. This is a pretty tiny cube, one ge... |
467,318 | <blockquote>
<p><span class="math-container">$ABC$</span> is a triangle with integral side lengths. Given that <span class="math-container">$\angle A=3\angle B$</span>, find the minimum possible perimeter of <span class="math-container">$ABC$</span>.</p>
</blockquote>
<p>I got this problem from an old book (which did n... | cosmo5 | 818,799 | <p>Giving it a try using elementary techniques.</p>
<p>In <span class="math-container">$\triangle ABC$</span>, let <span class="math-container">$B=\theta$</span>, <span class="math-container">$A=3\theta$</span>, <span class="math-container">$C=\pi-4\theta$</span>. By sine-rule
<span class="math-container">$$\frac{a}{\s... |
1,296,669 | <p>A tapas bar serves 15 dishes, of which 7 are vegetarian, 4 are fish and 4 are meat. A table of customers decides to order 8 dishes, possibly including repetitions.</p>
<p>a) Calculate the number of possible dish combinations. </p>
<p>b) The customers decide to order 3 vegetarian dishes, 3 fish and 2 meat. Calculat... | eloiprime | 180,579 | <p><strong>a)</strong> First, we note that repetition is allowed, and the order in which we order the dishes is unimportant. Therefore, we use the formula ${k+n-1 \choose k}$. In this case, $n=15$ and $k=8$. So, the number of possible combinations of dishes is
$${8+15-1 \choose 8} = {22 \choose 8} = 319770.$$</p>
<p><... |
3,251,928 | <p><span class="math-container">$$\frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-a)(b-c)} + \frac{1}{c(c-a)(c-b)} $$</span></p>
<p>I tried to get everything to the same denominator, and then simplify numerators first but it is very complicated and long if I just use brute force, to multiply all the expressions given from the pr... | Thomas Andrews | 7,933 | <p>Via partial fractions, you have:
<span class="math-container">$$\begin{align}\frac{1}{a(a-b)(a-c) }&= \frac{1}{bc}\cdot\frac{1}{a}+\frac{1}{b(b-c)}\cdot\frac{1}{(a-b)} + \frac{1}{c(c-b)}\cdot\frac{1}{(a-c)}\\
&=\frac{1}{abc}-\frac{1}{b(b-a)(b-c)}-\frac{1}{c(c-a)(c-b)}
\end{align}$$</span></p>
<p>[Idea stole... |
2,385,949 | <p>While playing around with random binomial coefficients , I observed that the following <em>identity</em> seems to hold for all positive integers $n$:</p>
<p>$$ \sum_{k=0}^{2n} (-1)^k \binom{4n}{2k}\cdot\binom{2n}{k}^{-1}=\frac{1}{1-2n}.$$</p>
<p>However, I am unable to furnish a proof for it ( though this result i... | epi163sqrt | 132,007 | <p>When going through Jack's nice answer I did some intermediate steps to better see what's going on. Here is a somewhat more elaborated version, which might also be convenient for other readers.</p>
<blockquote>
<p>We obtain
\begin{align*}
\color{blue}{\sum_{k=0}^{2n}}&\color{blue}{(-1)^k\binom{4n}{2k}\binom... |
2,925,450 | <p>I am trying to solve the following problem: <span class="math-container">$|4-x| \leq |x|-2$</span>. I am trying to do it algebraically, but I'm getting a solution to the problem that makes no sense. I fail to see the error in my reasoning though. I hope to get an explanation where I went wrong.</p>
<p><span class="... | John Wayland Bales | 246,513 | <p>Subtract all terms to the left side of the inequality to obtain</p>
<p><span class="math-container">$$ |4-x|-|x|+2\le0 $$</span></p>
<p>Then define <span class="math-container">$f(x)=|4-x|-|x|+2$</span>. We want to find all values of <span class="math-container">$x$</span> for which <span class="math-container">$f... |
1,808,183 | <p>Consider the Cauchy problem:
$$\left\{\begin{array}{lll}
x^2\partial_x u+y^2\partial_yu=u^2\\
u(x,2x)=1
\end{array}\right.$$
It is easy to show that the characteristic equations are given by:
$$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{dz}{px^2+qy^2}=\frac{dp}{2pu-2xp}=\frac{dq}{2qu-2yq}=dt$$
By the Lagrange-Charpit's met... | Canardini | 341,007 | <p>In equation (3), Let $E_1$ denotes the conditional expectation$E(|X(1)=0)$ we use the iterated expectation , that is $E_{1}(X(s)X(t))=E_{1}(E(X(s)X(t)|X(t))$.</p>
<p>In equation (4), since we conditioned to X(t), that means that we know its value , hence X(t) is not random anymore under that assumption: we can rem... |
2,070,353 | <p>Let there be two arbitrary functions $f$ and $g$, and let $g \circ f$ be their composition.</p>
<ol>
<li><p>Suppose $g \circ f$ is one-one. Then prove that this implies $f$ is also one-one.</p></li>
<li><p>Suppose $g \circ f$ is onto. Then prove that this implies $g$ is also onto.</p></li>
</ol>
<p>I tried to do b... | user160738 | 160,738 | <p>Suppose that $m$ is then number of terms in the sequence. Denote the first term by $a_1$, and then you can see that </p>
<p>$$
a_m=5+2m
$$</p>
<p>then $a_m=2n+1$, $5+2m=2n+1$ so $2m=2n-4$ whence you get $m=n-2$</p>
|
1,410,185 | <p>I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction.</p>
<p>Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: $$\sqrt[3]{-1}=(-1)^{\frac{1}{3}}=(-1)^{\frac{2}{6}}=\sqrt[6]{(-1)^2}=\sqrt[6]{1}=1$$ which is a clear contradiction to what... | hunter | 108,129 | <p>You say</p>
<blockquote>
<p>since the properties of powers are preserved</p>
</blockquote>
<p>but this is not true, and you have given a proof that exactly the opposite is true:</p>
<p>There is no rule that, for $a$ negative, one has $a^{bc} = (a^b)^c$. Indeed, we can see that this is not a rule even without us... |
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... | awkward | 76,172 | <p>This is really a too-long comment, not a solution, but I can't resist sharing what the statistician Frederick Mosteller had to say about his first encounter with generating functions. He was talking about ordinary power series generating functions (I think), not the exponential generating functions asked about in t... |
4,645,582 | <h2><strong>Problem</strong></h2>
<p>Let <span class="math-container">$f$</span> be a continuous function, <span class="math-container">$f:[0,1]\to\mathbb{R}$</span> with <span class="math-container">$\int_{0}^1 (2x-1)f(x) dx = 0$</span>.</p>
<p>Prove that there exists a point c between <span class="math-container">$(0... | FShrike | 815,585 | <p>Meaning no disrespect to the existing answer, there is a much simpler solution.</p>
<p>Consider the continuously differentiable function <span class="math-container">$\Lambda:[0,1]\to\Bbb R$</span> given by: <span class="math-container">$$c\mapsto\int_0^c(2xc-c^2)f(x)\,\mathrm{d}x$$</span></p>
<p><span class="math-c... |
2,557,608 | <p>Let $X$ be integer valued with Characteristic Function $\phi$. How to show that </p>
<p>$\ P(|X| = k)= \frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} \phi(t) dt$</p>
<p>$P(S_n =k) =\frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} (\phi(t))^n dt $</p>
| Badam Baplan | 164,860 | <p>Recall the angle addition formula $$\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)$$ for arbitrary angles.</p>
<p>From what you are given,</p>
<p>$$\frac{2}{3}\sin(\alpha)+\frac{2}{3}\sin(\beta)=\sin(\alpha+\beta)$$</p>
<p>So let's say $\cos(\beta) = \frac{2}{3} = cos(\alpha)$</p>
<p>No... |
4,610,640 | <p>I have this equation :</p>
<p><span class="math-container">$$\frac{d\alpha}{dz} = - \frac{dr}{dz} * \frac{\tan(\alpha)}r $$</span></p>
<p>I searched for some similar examples but non of these equations was like this one.
I'm confused about this one. As far as I know, I used RK4 for equations like this : <span class=... | B. Goddard | 362,009 | <p><span class="math-container">$$\frac{s+1}{s+2} =\frac{(s+2)-1}{s+2}=\frac{s+2}{s+2}-\frac{1}{s+2}.$$</span></p>
<p>You can also do long division of polynomials. The quotient is <span class="math-container">$1$</span> and the remainder is <span class="math-container">$1$</span>.</p>
|
2,675,954 | <p>Fractals, when viewed as functions, are everywhere continuous and nowhere differentiable. Can this also be used as a definition for fractals?</p>
<p>i.e. Are all fractals everywhere continuous and nowhere differentiable? And also: Are all functions that are everywhere continuous and nowhere differentiable fractals?... | flawr | 109,451 | <p>Hint: Note that your matrix has the form</p>
<p>$$\pmatrix{a&b&c\cr b&c&a\cr c&a&b\cr }$$</p>
<p>Which has the determinant</p>
<p>$$3abc-c^3-b^3-a^3$$</p>
<p>which can again be factored into</p>
<p>$$-\left(a+b+c\right)\,\left(a^2+b^2+c^2-ab-bc-ac\right)$$</p>
|
4,631,014 | <blockquote>
<p>Finding range of function</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\displaystyle f(x)=\cos(x)\sin(x)+\cos(x)\sqrt{\sin^2(x)+\sin^2(\alpha)}$</span></p>
</blockquote>
<p>I have use Algebric inequality</p>
<p><span class="math-container">$\displaystyle -(a^2+b^2)\leq 2ab\leq (a^2+b^... | mathlove | 78,967 | <p>I'm going to have a look on your try, and then add a claim and an example which might help you.</p>
<p>From
<span class="math-container">$$-(\cos^2x+\sin^2x)\leqslant 2\cos x\sin x\leqslant \cos^2x+\sin^2x\tag1$$</span>
and
<span class="math-container">$$\small -(\cos^2 x+\sin^2 x+\sin^2\alpha)\leqslant 2\cos x\sqrt... |
406,252 | <p>How to define the amount of additions. E.g. $1+2+3+4+5+6+7+8+9$</p>
<p>Are there $9$ additions, because of the nine numbers that are added together.
Or can you also say that there are $8$ additions, because there are only $8$ '$+$' signs.</p>
| response | 76,635 | <p>Addition is an operation and hence we are performing 8 additions as we repeat the operation of 'adding' 8 times. In contrast, I would say that we are adding 9 numbers.</p>
|
961,889 | <p>In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$:
\begin{equation}
\mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g)
\end{equation}
where $P$ is the point in the intersection we are interested in, $\mathcal{O}_P$ is its local ring in $\... | Slade | 33,433 | <p>We have $\operatorname{length}_{\mathcal{O}_P} \mathcal{O}_P/(f,g) = \operatorname{length}_{\mathcal{O}_P/(f,g)} \mathcal{O}_P/(f,g)$. So it is enough to show that the latter equals $\operatorname{length}_k \mathcal{O}_P/(f,g)$.</p>
<p>Let $A = \mathcal{O}_P / (f,g)$, $\mathfrak{m}$ its maximal ideal, and assume t... |
961,889 | <p>In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$:
\begin{equation}
\mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g)
\end{equation}
where $P$ is the point in the intersection we are interested in, $\mathcal{O}_P$ is its local ring in $\... | Georges Elencwajg | 3,217 | <p>Given a finitely generated module $M$ over a noetherian ring $A$, there exists a filtration of $M$ by submodules $M=M_0\supset M_1\cdots \supset M_n=0$ such that $M_i/M_{i+1}\cong A/\mathfrak p_i$ for some prime ideals $\mathfrak p_i\subset A$ (Bourbaki, <em>Commutative Algebra</em>, Chapter IV, §1, Theorem 1, page... |
2,179,012 | <p>Let $V_1$ be a Banach space and let $A_1 \subset V_1$ be a compact subset of $V_1$</p>
<p>Let $V_2$ be a Banach space and let $A_2 \subset V_2$ be a compact subset of $V_2$</p>
<p>Let $f: A_1 \to A_2 $ be a homeomorphism</p>
<p>Let $x_1,y_1 \in A_1$ such that $$\left\| x_1-y_1 \right\|=\sup_{v,u \in A_1} \left\| ... | Arpit Kansal | 175,006 | <p>The following are good books of complex Analysis,you can have a look at them and decide which one suits to your taste:</p>
<p><strong>1:</strong> Tristan Needham's Visual Complex Analysis.</p>
<p><strong>2:</strong> Complex Analysis by Stein and Shakarchi</p>
<p><strong>3:</strong> Complex Analysis by Ahlfors</p... |
627,718 | <p>I need help evaluating the integrals in Fourier Series.</p>
<p>For example, for the function <span class="math-container">$\cos^{2}x$</span>, I can evaluate <span class="math-container">$a_0$</span>, <span class="math-container">$a_n$</span>, and <span class="math-container">$b_n$</span>, where <span class="math-con... | user44197 | 117,158 | <p>This was answered just a few minutes back. I don't know how to link to that so I copied and pasted it here. Not sure how you can give someone else the credit!</p>
<p>Following up on lab Bhattacharjee's answer (Please give him the credit).</p>
<p>If you use complex Fourier series, then you just have to use Binomial... |
2,387,501 | <p>I'm struggling with the following integral:</p>
<p>$$
I = \int_{a}^{b}
{\frac{\mathrm{Erf}\left(\,{x/c}\,\right)}{\,\sqrt {\,{1 - {x^2}}\,}\,}\,\mathrm{d}x}
$$</p>
<p>Honestly, I do not know any approaches to solve it, except trying with Mathematica and searching for a possible solution in tables of integrals invo... | Robert Israel | 8,508 | <p>Since $a$ and $b$ are unspecified, you're looking for an antiderivative.
Integration by parts gives</p>
<p>$$ \arcsin(x) \text{erf}(x/c) - \frac{2}{c \sqrt{\pi}} \int \arcsin(x)\; e^{-x^2/c^2}\; dx $$</p>
<p>But this does not seem to have an elementary antiderivative.</p>
|
3,687,521 | <p>Recently I came accross this integral :
<span class="math-container">$$ \int \:\frac{1}{\sqrt[3]{\left(1+x\right)^2\left(1-x\right)^4}}dx $$</span>
I would evaluate it like this, first start by the substitution:
<span class="math-container">$$ x=\cos(2u) $$</span>
<span class="math-container">$$ dx=-2\sin(2u)du $$</... | Quanto | 686,284 | <p>Your result is correct, which can be obtained alternatively by substituting <span class="math-container">$t= \frac{1+x}{1-x}$</span> to arrive at</p>
<p><span class="math-container">$$ \int \:\frac{dx}{\sqrt[3]{\left(1+x\right)^2\left(1-x\right)^4}}
=\frac12\int t^{-2/3}dt = \frac32 t^{1/3}+C=\frac32 \sqrt[3]\frac{... |
567,428 | <p>Let $A \ne V$ be a subspace of $V$ and $B$ a linearly independent subset of $V$. Prove that $B$ can be completed to a basis of $V$ with vectors from $V \setminus A$.</p>
<p>OK, I started with: </p>
<p>$K=\operatorname{lin}(v_1,v_2,v_3,\dotsc)$</p>
<p>and if any of $v_1,v_2,\dotsc$ belongs to $A$ we can replace th... | egreg | 62,967 | <p>I assume that in the hypotheses you have that $V$ is finite dimensional.</p>
<p><strong>Lemma.</strong> <em>The only subspace of $V$ containing $V\setminus A$ is $V$.</em></p>
<p><em>Proof.</em> Let $v\in V$. If $v\notin A$ we have nothing to prove. If $v\in A$, then $v=(v-w)+w$ where $w\in V\setminus A$ (one such... |
23,980 | <p>I am calculating huge data files with an external program. I would like then to import the data into Mathematica for analysis. The files are 2 columns and up to many millions of rows. </p>
<p>So for small data files I have just been using:</p>
<pre><code>dataTable = Import["data.txt", "Table"];
</code></pre>
... | chuy | 237 | <p>Surely not the fastest or ideal solution, but this should work (it assumes that you have two numbers in each row):</p>
<pre><code>randomline[str_InputStream, num_] := (
SetStreamPosition[ str, 0];
Skip[str, Record, num-1];
Read[str, {Number, Number}])
str = OpenRead["data.txt"];
</code></pre>
<p>Now, if you... |
745,382 | <p>I'm interested in hypercomplexes, or number systems with many square roots of $-1$. Now, I know that <a href="http://en.wikipedia.org/wiki/Quaternion" rel="nofollow">quaternions</a> are non-commutative, but associative. I'm wondering if it's possible to have a number system with more "elements" (or square roots of... | Anixx | 2,513 | <p>Tessarines, also known as bicomplex numbers, are commutative and associative. You can build a tessarine-line system for any dimension <span class="math-container">$2^n$</span>.</p>
<p>You also can build any dimensional analogs of split-complex numbers.</p>
<p>You also can build any dimensional analogs of dual number... |
310,238 | <p>The <a href="https://en.wikipedia.org/wiki/Weyl_character_formula" rel="nofollow noreferrer">Weyl character formula</a> and <a href="https://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_denominator_formula" rel="nofollow noreferrer">the denominator identity</a> play important roles in the representation theory ... | Thorsten Heidersdorf | 129,060 | <p>Since you ask for formulas for the character, I will first assume that you are interested in finite dimensional representations.</p>
<p>If <span class="math-container">$\mathfrak{g}$</span> is a basic classical Lie superalgebra, such as <span class="math-container">$\mathfrak{sl}(m|n)$</span>, one distinguishes bet... |
1,545,258 | <p>In my abstract algebra book one of the first facts stated is the Well Ordering Principle:</p>
<p>(*) Every non-empty set of positive integers has a smallest member.</p>
<p>In real analysis on the other hand one of the first things introduced are the real numbers and their Completeness Axiom:</p>
<p>Every nonempty... | Emmbee1 | 446,283 | <p>I see the completeness axiom of $\mathbb{R}$ as a generalisation of the well ordering principle of $\mathbb{N}$. As every subset $S\subset\mathbb{Z}$ is bounded below, this condition need to be imposed on $S\subset\mathbb{R}$ to make the extension possible even though the bound doesn't have to be in $S$.</p>
|
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
$$
I know that $b$ will be negative, since $A$ approximately equal to $0$ for la... | Ross Millikan | 1,827 | <p>The traditional way is to take the log of each side. The equation becomes $\log A=\log a + b \log n$. Your data changes from $(A_i,n_i)$ to $(\log A_i, \log n_i)$ and you can do a linear fit in $a,b$. There are issues with the errors being transformed, which will change the best fit parameters. You are correct th... |
1,564,750 | <p>Imagine that your country's postal system only issues 2 cent and 5 cent stamps. Prove that it possible to pay for postage using only these stamps for any amount n cents, where n is at least 4.</p>
<hr>
<p>My attempt (using Strong induction, I know we can use induction but since they say you can can apply strong in... | Justpassingby | 293,332 | <p>You can avoid explicit induction by verifying a finite number of cases like 4,5,6,7,8 and then note that any higher natural number is separated from one of these by a multiple of 5.</p>
<p>Strictly logically that last step requires induction as well but the proof is much easier.</p>
|
1,564,750 | <p>Imagine that your country's postal system only issues 2 cent and 5 cent stamps. Prove that it possible to pay for postage using only these stamps for any amount n cents, where n is at least 4.</p>
<hr>
<p>My attempt (using Strong induction, I know we can use induction but since they say you can can apply strong in... | HEKTO | 92,112 | <p>Let's assume you know how to pay $n$ cents using only 2- and 5-cents stamps. You want to find a way to pay $n+1$ cents. There are two cases:</p>
<ul>
<li>If your sequence for the $n$ contains a 5-cents stamp, take it out and replace it by three 2-cent stamps.</li>
<li>If your sequence contains only 2-cents stamps, ... |
2,542,259 | <p>The question says to "find geometrically the set of points $(x,y) \in \Bbb R^2$ such that $$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... | avz2611 | 142,634 | <p>L.H.S is complex in nature where as R.H.S is Real. Thus $y$ has to be zero. So your solution is all Real for $x$ and $y$ is zero. </p>
<p>Now if $x$ and $y$ were complex you had to make sure the complex parts were canceled in L.H.S and solve for the rest</p>
|
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>
| wltrup | 232,040 | <p>Use the chain rule. Define $u = x + c$ then use the fact that $$\frac{d\cdot}{dx} = \frac{du}{dx} \frac{d\cdot}{du}$$ where the $\cdot$ represents any function, so</p>
<p>$$\frac{df}{dx} = \frac{du}{dx} \frac{df}{du}$$</p>
<p>It also follows that </p>
<p>$$
\begin{array}{rcll}
\frac{d^2f}{dx^2} &=& \frac{... |
286,312 | <p>I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically:</p>
<p>$$
\sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2}
$$</p>
<p>Any hints would be greatly appreciate</p>
| omid saba | 52,535 | <p>I will prove it in two way for you:</p>
<p>1- Mathematical Induction:</p>
<p>If $n=1$ then the left side is $1$ and also the right side is $1$ too.</p>
<p>Now think that we have $\sum_{i=1}^{n}(3i-2)=\frac{n(3n-1)}{2}$, we should show $\sum_{i=1}^{n+1}(3i-2)=\frac{(n+1)(3(n+1)-1)}{2}$ that is $\sum_{i=1}^{n+1}(3i... |
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | Community | -1 | <p>If you decompose the area in thin vertical rectangles, the height of the rectangles is the difference between the ordinates.</p>
<p><a href="https://i.stack.imgur.com/uGEQs.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uGEQs.gif" alt="enter image description here"></a></p>
|
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | Kanwaljit Singh | 401,635 | <p>The given curves are :</p>
<p>y = f(x). .........(1)</p>
<p>y = g(x). ...........(2)</p>
<p>between a and b. So (1) and (2) intersect at x = a and x = b.</p>
<p><a href="https://i.stack.imgur.com/rgdOq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rgdOq.png" alt="page 1"></a></p... |
2,652,662 | <blockquote>
<p>Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Define $A_{f,x} : (0, \infty) \to \mathbb{R}$ by $$A_{f,x} (r) = \text{diam}\left( f((x-r,x+r) \cap [a,b])\right).$$ Show that for all $\epsilon >0$ the set $\{x\in [a,b] : \lim_{r\to 0^+} A_{f,x} (r)\geq \epsilon \}$ is a closed set.</p>
</blockq... | trancelocation | 467,003 | <p>Here, the (conditional) sample spaces with equally likely outcomes are easily listed. </p>
<p>So let $\Omega$ denote the original sample space of all possible gender combinations of siblings.</p>
<p>Let $\Omega_{C}$ denote the restricted sample space induced by a given condition $C$. (Actually, $\Omega_{C} = C$).<... |
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Venus | 146,687 | <p>Alternatively, using Taylor series of exponential function and Euler's formula we have $$e^{\Large e^{i\theta}}=1+(\cos\theta+i\sin\theta)+\frac{(\cos2\theta+i\sin2\theta)}{2!}+\frac{(\cos3\theta+i\sin3\theta)}{3!}+\cdots$$
We also have
$$\int_0^{\pi/2}\cos(n\theta)\;d\theta=\frac{\sin\left(\frac{\pi n}{2}\right)}{... |
2,610,048 | <p>Ok, so I'm trying to prove statement in the header. I have read the following discussion on it, but I can't seem to follow it all the way through:</p>
<p><a href="https://math.stackexchange.com/questions/1198735/proving-sum-k-1n-k-k-n1-1/1198743#1198743?newreg=abbcf872d9904cbfa76cd161c4fecdd0">Proving $\sum_{k=1}^... | David | 119,775 | <p>Your $x$s can have any one of five values, so for example $[1,1,x,x]$ is really $25$ options, not $1$. The total number of successful rolls of four dice is
$$6\times25+4\times5+1=171$$
and the probability is
$$\frac{171}{1296}\ .$$
The chance of failing on one roll of four dice is
$$1-\frac{171}{1296}=\frac{1125}{1... |
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