qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,128,825 | <p>$$\int^{x}_{0}tf(t)dt = x\sin(x)+\cos(x)-1$$
find $f(\pi), f'(x)$</p>
<p>This question confuses me because usually, the way I have seen questions like these, they have been in the form:</p>
<p>$$H(x) = \int^{x}_{a} f(t)dt$$</p>
<p>This form is kind of different so I am not sure how I would solve it.</p>
<p>My t... | celtschk | 34,930 | <p>Well, strictly speaking the integral equation doesn't fix $f(\pi)$, nor does it guarantee even the existence of $f'(x)$. For example, for arbitrary $\alpha$ be
$$f_\alpha(x) = \cases{
\alpha & for $x\in\pi\mathbb Q$\\
\cos x & otherwise}$$
Then $f_\alpha$ fulfills the integral equality, $f_\alpha(\pi)=\alpha... |
1,360,560 | <p>Definition: If $ f $ is holomorpic in $G$ and gamma $\gamma$ is $G$-homotophic to a point then gamma is G-contratible and if gamma is G-contractible then $ \int_\gamma f = 0 $. </p>
<p>By splitting the integral using partial fraction we get
$$ \int_\gamma \frac {dz}{z^2 − 2z} = \frac12\int_\gamma \frac {dz}{z − 2... | Ivo Terek | 118,056 | <p>If both $0$ and $2$ are in the interior of $\gamma$, by the residue theorem we have: </p>
<p>$$\begin{align} \int_\gamma \frac{{\rm d}z}{z^2 - 2z} &= 2\pi i\left({\rm Res}(f,0) + {\rm Res}(f,2)\right) \\ &= 2\pi i\left(\frac{1}{2\cdot 0 - 2} + \frac{1}{2\cdot 2 - 2}\right) \\ &= 2\pi i\left(-\frac{1}{2}... |
1,360,560 | <p>Definition: If $ f $ is holomorpic in $G$ and gamma $\gamma$ is $G$-homotophic to a point then gamma is G-contratible and if gamma is G-contractible then $ \int_\gamma f = 0 $. </p>
<p>By splitting the integral using partial fraction we get
$$ \int_\gamma \frac {dz}{z^2 − 2z} = \frac12\int_\gamma \frac {dz}{z − 2... | A.Γ. | 253,273 | <p>A possible source of confusion can come from the following <strong>wrong</strong> reasoning: we want to calculate the integral of $f(z)=1/z$ along the unit circle $\gamma$. If we do the inversion of the complex plane $w=1/z$ then $\gamma\mapsto\gamma$ and our function becomes $f(1/z)=w$ which is analytic in the unit... |
241,683 | <p>EDIT: I adjusted the vertices to have labels that are integers (like the weights). Can the answer be adapted to this case?</p>
<p>I use simultaneous display of vertices and weights (this topic is related to another question posted <a href="https://mathematica.stackexchange.com/questions/154513/vertex-labels-of-graph... | Ulrich Neumann | 53,677 | <p>Add Assumptions :</p>
<pre><code>zw=FullSimplify[
Integrate[t1^2*E^((1 - h)*s0*t1), {t1, 0, T}],
Assumptions -> 0 < h < 1
]
</code></pre>
<p><strong>Addendum</strong></p>
<pre><code>zw /. h -> 1 - m /. m -> 1 - h
(*-((2 + E^((1 - h) s0 T) (-2 + (1 - h) s0 T (2 - (1 - h) s0 T)))/((1 -
h)^3 s0^3))... |
3,020,674 | <p>I need to use Fermat's Theorem to prove that 10001 is not prime. I understand that I just need to find a counterexample where <span class="math-container">$a^{10000}$</span> mod 10001 = 1 mod 10001 does not hold true, but this seems kind of difficult with such large numbers. Any ideas?</p>
| Patrick Stevens | 259,262 | <p>Fermat pseudoprimes to any given base are really very rare, so you might as well just launch in with <span class="math-container">$2$</span> and hope for the best. This is a bit tedious but perfectly doable by repeated squaring:
<span class="math-container">$$10000 = 2^{13}+2^{10}+2^9+2^8+2^4$$</span>
so you just ne... |
1,681,134 | <p>This is a problem from Harvard Stat 110 Probability Homework set 2, and Blitzstein's <em>Introduction to Probability</em> (2019 2 ed) Ch 1, Exercise 54, p 51.</p>
<blockquote>
<p>Alice attends a small college in which each class meets only once a week. She is deciding between 30 non-overlapping classes. There are 6 ... | Alex | 38,873 | <p>Apologies, I miscalculated the number of classes.</p>
<p>1) 3 classes in 1 day, 1 in the remaining: $5 \times\binom{6}{3} \times 6^4$</p>
<p>2) 2 classes in 2 days, 1 in the remaining: $\binom{5}{2}(\binom{6}{2})^2 \times 6^3$ </p>
|
632,029 | <blockquote>
<p>Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous such that
$$\lim_{x\to\infty} f(x) = \lim_{x\to -\infty}f(x) = 0$$
Show that there $\exists x_0\in\mathbb{R}: \, \mid f(x)\mid \, ≤ \, \mid f(x_0)\mid$ for $\forall x \in \mathbb{R}$.</p>
</blockquote>
<p>Basically, this means I have to sho... | DanZimm | 37,872 | <p>Let's look at a sub problem, </p>
<blockquote>
<p>Let $\, f : \mathbb{R} \to \mathbb{R}$ be continuous s.t.
$$
f(x) > 0 \; \forall x \in \mathbb{R}, \lim_{x \to \infty} f(x) = \lim_{x \to -\infty} f(x) = 0
$$
then $\exists \, x_0 \in \mathbb{R}$ s.t. $f(x) \le f(x_0) \; \forall x \in \mathbb{R}$</p>
</bloc... |
1,877,632 | <p>My integral calculus is rusty.
How do I calculate the interior area (blue region) of four bounding circles?<br><br>
<a href="https://i.stack.imgur.com/VtQIy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VtQIy.png" alt="enter image description here"></a></p>
| 5xum | 112,884 | <p>You don't need integrals for that. You can quickly show that the blue area in your picture is the same as the blue area in the picture below: <a href="https://i.stack.imgur.com/qSBaL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qSBaL.png" alt="enter image description here"></a></p>
|
2,487,297 | <p>I am reading Tu's book "Introduction to Manifolds". In Example 11.8, it says that $f(t) = (t^2 - 1,t^3-t)$ is an immersion however it says that the equation $f'(t) = (2t,3t^2 -1) = (0,0)$ has no solution in $t$ and this means the map $f$ is an immersion. This confuses me since I thought the definition of an immersio... | Kelvin Lois | 322,139 | <p>You have $f : \mathbb{R} \rightarrow \mathbb{R}^2 $ defined as $f(t) =(f^1(t), f^2(t)) =(t^2 - 1,t^3-t)$. The differential at $t_0$ is a map $d_{t}f : T_{t} \mathbb{R} \rightarrow T_{f(t)}\mathbb{R}^2$. At any point $t \in \mathbb{R}$ , basis of the tangent space at that point $T_t \mathbb{R}$ is $\frac{d}{dt}$. So... |
3,193,823 | <p>I am asked to evaluate: <span class="math-container">$\frac{4+i}{i}+\frac{3-4i}{1-i}$</span></p>
<p>The provided solution is: <span class="math-container">$\frac{9}{2}-\frac{9}{2}i$</span></p>
<p>I arrived at a divide by zero error which must be incorrect. My working:</p>
<p><span class="math-container">$\frac{4+... | Andrei | 331,661 | <p>Here is your error <span class="math-container">$1-i^2=1-(-1)=2\ne 0$</span></p>
|
1,333,605 | <p>Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$.</p>
<p>$BS$ is just a linear combination of $B$. Thus $BS$ should be a new basis. Am I right?</p>
| Leox | 97,339 | <p>To get a new basis the matrix $S$ should have nonzero determinant.</p>
|
209,757 | <p>Find the value of $c$ which makes it possible to solve:</p>
<p>$$u+v+2w=2,$$
$$2u+3v-w=5,$$
$$3u+4v+w=c$$</p>
| Ragib Zaman | 14,657 | <p><strong>HINT</strong>: Add the two first equations.</p>
|
2,428,160 | <p>In the video game Player Unknown's Battlegrounds, you start with a circular safezone of diameter $n$. After some time, the safezone instantly shrinks to a circle with a diameter $\frac{n}{2}$. </p>
<p>The catch is that the smaller safezone will be entirely contained within the bounds of the previous, larger circl... | fleablood | 280,126 | <p>Well, a bit handwavy but:</p>
<p>Not for any $n > 1$, $(x^n - y^n)= (x-y)(x^{n-1} + x^{n-2}y + .... + xy^{n-2}+ y^{n-1})$</p>
<p>So $h = (x+h) - x = [(x+h)^{\frac 14} - x^{\frac 14}][(x+h)^{\frac 34} + (x+h)^{\frac 12}x^{\frac 14} + (x+h)^{\frac 14}x^{\frac 12} + x^{\frac 34}]$</p>
<p>So for $h> 0$ then $\f... |
2,428,160 | <p>In the video game Player Unknown's Battlegrounds, you start with a circular safezone of diameter $n$. After some time, the safezone instantly shrinks to a circle with a diameter $\frac{n}{2}$. </p>
<p>The catch is that the smaller safezone will be entirely contained within the bounds of the previous, larger circl... | Simply Beautiful Art | 272,831 | <p>You have the special case of $a=1$ and $b=4$ of the derivative of $x^{a/b}$, which may be found using the geometric series:</p>
<p>$$\frac{x^n-h^n}{x-h}=x^{n-1}+x^{n-2}h+\dots+xh^{n-2}+h^{n-1}=\sum_{k=0}^{n-1}x^{n-k-1}h^k$$</p>
<p>We then proceed as follows:</p>
<p>\begin{align}\frac d{dx}x^{a/b}&=\lim_{h\to ... |
4,274,526 | <p>When proving a limit at <span class="math-container">$a$</span> with value <span class="math-container">$L$</span> with the definition, we must show that for all <span class="math-container">$\epsilon >0$</span>, there is <span class="math-container">$\delta >0$</span> such that:</p>
<p><span class="math-conta... | SV-97 | 960,107 | <p>To add to the other answers:
the inequality <span class="math-container">$|x-a| \leq |f(x) - L|$</span> is a local property - this may hold for any <span class="math-container">$x$</span> but note that the right side still depends on <span class="math-container">$x$</span>, so at best you can use this to show <span ... |
666,103 | <p>I am studying elementary number theory, and just started learning about divisors. I always, try to read several other sources mostly because it helps me understand ideas better, also the textbook I am using- is not always clear for me. </p>
<p>Some sources state that 0|0 is not possible, while others allow 0|0. <... | Cameron Buie | 28,900 | <p>Some books may avoid allowing $a=0$ because it is in some ways natural to translate the statement "$a\mid b$" to the statement "$\frac{b}{a}$ is an integer," which of course makes no sense when $a=0.$ Some books may avoid the $0\mid 0$ case because we do not have uniqueness of factorization of $0$ up to units, as we... |
333,645 | <p>$$f(z)=\frac 1{\cos(z^4)-1}$$</p>
<p>$z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does Wolfram Alpha claim that the pole is of order $2$?</p>
| Julien | 38,053 | <p>You are right. And <a href="http://www.wolframalpha.com/input/?i=1/%28cos%20%28z%5E4%29-1%29" rel="nofollow">Wolfram seems to agree</a>.</p>
<p>By Taylor expansion $\cos z=1-z^2/2+O(z^4)$ so
$$
\cos(z^4)-1=-\frac{z^8}{2}+O(z^{16})=-\frac{z^8}{2}(1+o(1)).
$$
Hence
$$
f(z)=\frac{1}{\cos(z^4)-1}=\frac{1}{-\frac{z^8}{2... |
419,205 | <p>My question is: is it possible to find pentagonal numbers which are also tetrahedral?
A pentagonal number is obtained by the formula:
$$P_k=\frac{1}{2}k(3k-1)$$
The equivalent formula for the tetrahedral number $T_n$
is:
$$T_n=\frac{1}{6}n(n+1)(n+2)$$
So the problem is to find a $T_n=P_k$ that means to solve:
$$n(n+... | Ogen | 70,527 | <p>$\det \left(\begin{bmatrix} 5 -\lambda & -2\\ 1 & 2-\lambda\end{bmatrix}\right)=0$
therefore,</p>
<p>$(5 - \lambda)(2 - \lambda) - (-2 \times 1) = 0$</p>
<p>$(5 - \lambda)(2 - \lambda) = -2$</p>
<p>Solving that gives $\lambda\in \{4,3\}$.</p>
|
1,525,470 | <p>Let $K$ be a finite separable extension of a field $k$ of prime degree $p$. Let $\theta$ in K be such that $K = k(\theta)$ and $\theta_1 ..., \theta_p$ be the conjugates of $\theta$ over $k$ in some algebraic closure $k^a$. Let $\theta=\theta_1$ and if $\theta_2 \in k(\theta)$ Then show that $K/k$ Galois.</p>
<p>As... | MooS | 211,913 | <p>Here is another approach: The galois group is a subgroup of $S_p$ and contains an element of order $p$ by Cauchy's Theorem, let us call it $\sigma$. This is a $p$-cycle. After possibly replacing $\sigma$ by a suitable power, we may assume $\sigma(\theta)=\theta_2$, in particular the restriction of $\sigma$ to $K$ is... |
354,365 | <p>Let $X_i$ be pairwise-uncorrelated random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$. Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum_{i\in \mathbf{n}} X_i$.</p>
<p>Then, as s... | Ittay Weiss | 30,953 | <p>Here is a geometric way of looking at it (it's not a rigorous proof, but gives a nice way to understanding, non-algebraically, why it's true): The geometric meaning of the determinant of a matrix is that it is the algebraic volume of the parallelepiped spanned by the columns (or rows) of the matrix (algebraic volume... |
354,365 | <p>Let $X_i$ be pairwise-uncorrelated random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$. Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum_{i\in \mathbf{n}} X_i$.</p>
<p>Then, as s... | James Georgas | 88,439 | <p>Take your <em>m</em> x <em>n</em> matrix <strong>A</strong>, row reduce it to <strong>A'</strong>, and determine that its rank is <em>r</em>. Form a new <em>m</em> x <em>r</em> matrix <strong>B'</strong> from the linearly independent pivot columns of <strong>A'</strong>. Note that the rank of <strong>B'</strong> is ... |
169,998 | <p>If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within the angle created by those points?</p>
<p>The process I'm currently using is to get the angle of all three lines tha... | Robert Mastragostino | 28,869 | <p>Shift your coordinates so that the point of the angle is the origin. Normalize the other points (so we're replacing all other points by ones that are a unit distance from the "origin". Say the cone is made by $O,A,B$ (in the new coordinates) and the point to find is $C$. If $A\cdot C$ and $B\cdot C$ are greater than... |
169,998 | <p>If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within the angle created by those points?</p>
<p>The process I'm currently using is to get the angle of all three lines tha... | Community | -1 | <p>No need to solve systems of linear equations, call trigonometric functions, or even normalize the vectors. Let $a$, $b$, and $c$ be the vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$, and denote $$u \wedge v = \begin{vmatrix}u_x & v_x \\ u_y & v_y\end{vmatrix} = u_x v_y - u_y... |
1,973,500 | <p>What is the number of abelian groups of order 40? I thought the number is just $3$. More specifically, they are</p>
<p>$$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_5$$
$$\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_5$$
$$\mathbb{Z}_8\times\mathbb{Z}_5$$</p>
<p>However, my answer says:<a... | Alexis Olson | 11,246 | <p>The ones listed in the book are not distinct.</p>
<p>$$\Bbb Z_{40} \cong \Bbb Z_{5} \times \Bbb Z_8 $$</p>
<p>$$\Bbb Z_{20} \times \Bbb Z_2 \cong \Bbb Z_{10} \times \Bbb Z_4 \cong \Bbb Z_{5} \times \Bbb Z_4 \times \Bbb Z_2 $$</p>
<p>$$\Bbb Z_{10} \times \Bbb Z_2 \times \Bbb Z_2 \cong \Bbb Z_{5} \times \Bbb Z_2 \... |
124,955 | <p>Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by decomposing $x^3 + y^3$ into $(x + y)(x + y\omega)(x + y\omega^2)$ where $\omega$ is the primitive cube root of unity or ... | daniel | 18,124 | <p>I recommend Harold Edwards' <em>Fermat's Last Theorem.</em> Euler's method of infinite descent for the case n = 3 (with a careful explanation of the gap in Euler's proof) is given and corrected in sections 2.2, 2.5 of this book. This also takes about 5 pages. </p>
<p>This may be the version in Hardy and Wright but... |
247,335 | <p>Prove, formally that:
$\log_2 n! \ge n$ for all integers $n>3$. </p>
<p>Hint: first prove that $n! ≥2^n$, for all integers $n >3$.</p>
<p>So far what I have: </p>
<p>Base case, $n = 4$,</p>
<p>$4! = 24$</p>
<p>$2^4 = 16$.</p>
<p>Therefore, it is true when $n = 4$.</p>
<p>So how do I proceed from here?</... | martini | 15,379 | <p>For the induction step just note that you can assume $2^{n-1} \le (n-1)!$ as induction hypotheses, giving you (we have $2 \le 3 < n$):
$$ 2^n = 2\cdot 2^{n-1} \le 2 \cdot (n-1)! < n \cdot (n-1)! = n! $$
By induction, now $2^n \le n!$ for any $n > 3$. Applying $\log_2$ to both sides (note that $\log_2$ is mo... |
267,618 | <p>So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$?
Using the Chernoff bound we know that $\frac{1}{\epsilon^2}\log(1/\delta)$ is enough. And I know that this is als... | Henry.L | 25,437 | <p><strong>This argument based on modification of [2].</strong></p>
<p><strong>(1)</strong> Let A be the event that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\geq\frac{1}{2}$ , let $X_{i}=0$ mean heads of coin and $X_{i}=1$ mean tails.</p>
<p>$Pr(A)\leq Pr\left\{ \left|\frac{1}{n}\sum_{i=1}^{n}X_{i}-\left(\frac{1}{2}-\epsilon\r... |
3,784,663 | <p>Pretty self explanatory.</p>
<blockquote>
<p>If I had for example something like <span class="math-container">$\sin^{-1}(\pi/12)$</span> in an expression, is it ever possible to express that expression without inverse trig functions?</p>
</blockquote>
| vonbrand | 43,946 | <p>If you have <em>fixed</em> arguments, the inverse trigonometric function (or any other, for that matter) is just a fixed value. Finding out what the value is (giving it a "name" that isn't just the expression to be replaced) might or might not be possible or simple to do. Or even desirable, if e.g. you wan... |
4,561,921 | <p>I'm studying newtonian dynamical systems and they can be described by the differential equation
<span class="math-container">$$1)\space m\ddot{x} = F(x)$$</span>
supposing <span class="math-container">$F$</span> sufficiently regular we could define the potential <span class="math-container">$V$</span> as its primit... | Dominik Kern | 1,106,909 | <p>I would like to emphasize the fundamental relation
<span class="math-container">\begin{equation}
\text{d}x = v\,\text{d}t.
\end{equation}</span>
The crucial point is
<span class="math-container">\begin{equation}
\dot{v} = \frac{\text{d}v}{\text{d}t} \ne \frac{\text{d}v}{\text{d}x}.
\end{equation}</span>
And so you o... |
117,558 | <p>Is there anything known about the existence of <a href="http://mathworld.wolfram.com/HeronianTriangle.html" rel="nofollow noreferrer">Heronian triangles</a> ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? Equivalently, is there a degenerate ... | Noam D. Elkies | 14,830 | <p>Yes, for example the 13-14-15 triangle can be scaled by 11
to find a point $D$ at distance $80$, $91$, $102$ from the
vertex opposite the side of length
$11 \cdot 13$, $11 \cdot 14$, $11 \cdot 15$ respectively:</p>
<p><img src="https://i.stack.imgur.com/LIjItm.png">
<a href="http://math.harvard.edu/~elkies/mo11755... |
117,558 | <p>Is there anything known about the existence of <a href="http://mathworld.wolfram.com/HeronianTriangle.html" rel="nofollow noreferrer">Heronian triangles</a> ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? Equivalently, is there a degenerate ... | Rosie F | 94,242 | <p>If it is required that <span class="math-container">$D$</span> be interior to <span class="math-container">$\triangle ABC$</span>, then there are smaller examples than the one shown by Noam Elkies. Each of the following two has the bonus that it is an orthic system and can be embedded on the integer lattice with one... |
75,421 | <p>Can someone help me finish my solution?</p>
<p><strong>Question:</strong> Show that there are sets $A_{ij}$ for $i,j$ ∈ $\mathbb N$ such that for no <em>countable</em> $\space$H$\subseteq\mathbb N^{\mathbb N}$</p>
<p>$\bigcup_{i=0}^\infty\Bigg(\bigcap_{j=0}^\infty{A_{ij}}\Bigg)=\bigcap\Bigg\lbrace\Bigg(\bigcup... | Brian M. Scott | 12,042 | <p>I don’t at the moment see a way to make your approach work, so I’m going to suggest a different one. Each of the sets $A_{ij}$ will be a subset of $\mathbb{N}^\mathbb{N}$. Specifically, try letting $$A_{ij} = \{f\in\mathbb{N}^\mathbb{N}:f(i)=j\}$$ for each $\langle i,j \rangle \in \mathbb{N}^2$. Note that with this ... |
3,541,869 | <p>I was reading from <a href="https://books.google.com.gh/books/about/Ordinary_Differential_Equations.html?id=iU4zDAAAQBAJ&source=kp_book_description&redir_esc=y" rel="nofollow noreferrer"><em>Ordinary Differential Equations</em></a> <strong>(Lesson 13 Example 13.3 page 110)</strong> and came across this quest... | Ted Shifrin | 71,348 | <p>Note that <span class="math-container">$(f+g)-(b+c) = (f-b)+(g-c)$</span>, so that's how the triangle inequality was applied.</p>
<p>The <span class="math-container">$\epsilon/2$</span> arrived so that you would have <span class="math-container">$$\|(f+g)-(b+c)\|\le \|f-b\| + \|g-c\|<\epsilon/2 + \epsilon/2 = \e... |
3,541,869 | <p>I was reading from <a href="https://books.google.com.gh/books/about/Ordinary_Differential_Equations.html?id=iU4zDAAAQBAJ&source=kp_book_description&redir_esc=y" rel="nofollow noreferrer"><em>Ordinary Differential Equations</em></a> <strong>(Lesson 13 Example 13.3 page 110)</strong> and came across this quest... | Izaak van Dongen | 473,276 | <ol>
<li>This comes from first rewriting <span class="math-container">$(f + g) - (b + c)$</span> as <span class="math-container">$(f - b) + (g - c)$</span> and applying the triangle inequality.</li>
<li><p>This is using the fact that <span class="math-container">$f(x) \to b$</span>. This precisely means that for <em>an... |
84,036 | <p>I was doing an optimization but facing a problem getting what exactly Minimize function do. I run the following code:</p>
<pre><code> Log1[x_] := If[x == 0, 0, Log2[Abs[x]]];
VEntropy[x_] := -(x Log1[x] + (1 - x) Log1[1 - x]);
Prob[a_, b_, x_, y_] := 1 - 1/((a/x)^2 + (b/y)^2);
Cost[a_, b_, x_, y_] :=... | MarcoB | 27,951 | <p>With the constraint <code>x^2 + y^2 == 1</code>, you are minimizing the <code>Cost</code> function <em>on the boundary of the unit disk</em> centered on the origin, i.e. only along the rim of that circle. </p>
<p>What you seem to want is optimizing anywhere <em>within the unit disk</em>. You want an inequality ther... |
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| Martin Brandenburg | 1,650 | <p>Since it was also asked for examples, let me mention how to compute the product of two ideals (beyond the already mentioned principal ideals).</p>
<p>If $I$ is generated by elements $\{a_i\}$ and $J$ is generated by elements $\{b_j\}$, then $I \cdot J$ is generated by the elements $\{a_i \cdot b_j\}$. You can verif... |
627,749 | <blockquote>
<p>Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial.</p>
</blockquote>
<p>OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})</p>
| user66733 | 66,733 | <p>If we name the homomorphism $\varphi: G \to A$ then the kernel of $\varphi$ must be a normal subgroup of $G$. So, since $G$ is simple and $\ker{\varphi}$ is normal in $G$ two cases can happen: $\ker{\varphi}=\{e\}$ or $\ker{\varphi}=G$.</p>
<p>If $\ker{\varphi}=\{e\}$ then $\varphi$ is one-to-one and the image of $... |
2,744,496 | <p>Consider the vector space $\Bbb{R}^3$ with coordinates $(x_1, x_2, x_3)$ equipped with the inner product $$\langle(a_1, a_2, a_3),(b_1, b_2, b_3)\rangle= 2(a_1b_1 + a_2b_2 + a_3b_3) − (a_1b_2 + a_2b_1 + a_2b_3 + a_3b_2).$$</p>
<p>Write down all vectors in $\Bbb{R}^3$ which are orthogonal to the plane $x_1 − 2x_2 + ... | Mohammad Riazi-Kermani | 514,496 | <p>Note that $$\begin{bmatrix}1\\-2\\2\end{bmatrix} $$ is not your normal vector to the plane any more .</p>
<p>You need to find the normal by first finding two linearly independent vectors in the plane, such as $(0,1,1)$ and $(-2,0,1)$ and find the normal vector perpendicular to these vectors with the new inner produ... |
328,811 | <p>I already asked a question (<a href="https://math.stackexchange.com/questions/328680/order-of-operations-in-rotation-matrix-notation">Order of operations in rotation matrix notation.</a>) about the order in which a particular equation is "processed" and now I need to generalise that and learn the rules of math notat... | AJMansfield | 50,951 | <p>The basic answer is that trig functions are special. Other than those, math symbols usually behave pretty much the same way. I personally almost always mean $f(x)^2$ when I write $f^2(x)$, and I have a bunch of other conventions I use to ensure that I understand my notes, but as long as you get to the right answer... |
3,824,267 | <p>I have to calculate real, imaginary, modulus and Arg of the following</p>
<blockquote>
<p><span class="math-container">$(\frac{(\sqrt3 + i)}{1-i})^{25}$</span></p>
</blockquote>
<p>I stucked on the following part</p>
<blockquote>
<p><span class="math-container">$(\frac{(\sqrt3 + i)(1+i)}{2})^{25} = (\frac{(\sqrt 3-1... | TonyK | 1,508 | <p>You should not have expanded <span class="math-container">$(\sqrt 3+i)(1+i)$</span> as you did. It's much easier if you calculate <span class="math-container">$(\sqrt 3+i)^{25}$</span> and <span class="math-container">$(1+i)^{25}$</span> separately, because both can be expressed as <span class="math-container">$re^{... |
3,824,267 | <p>I have to calculate real, imaginary, modulus and Arg of the following</p>
<blockquote>
<p><span class="math-container">$(\frac{(\sqrt3 + i)}{1-i})^{25}$</span></p>
</blockquote>
<p>I stucked on the following part</p>
<blockquote>
<p><span class="math-container">$(\frac{(\sqrt3 + i)(1+i)}{2})^{25} = (\frac{(\sqrt 3-1... | Community | -1 | <p><span class="math-container">$$\sqrt3 + i = 2(\frac{\sqrt3}{2}+\frac{1}{2}i) = 2(cos(\frac{\pi}{6}) + sin(\frac{\pi}{6})i)$$</span><br />
applying euler identity,<br />
<span class="math-container">$$\sqrt3 + i = 2e^{\frac{\pi}{6}i}$$</span><br />
Similarly
<span class="math-container">$$1 - i = \sqrt2(\frac{1}{\sqr... |
3,064,458 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow\mathbb{R}$</span> be a continuous function. </p>
<ol>
<li>Show that for each <span class="math-container">$\epsilon\in(0,1)$</span>, <span class="math-container">$\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon... | h3fr43nd | 485,385 | <p>Another approach would be to use some measure theory:</p>
<p>Clearly for all <span class="math-container">$x \in [0,1-\epsilon)$</span> we have that <span class="math-container">$x^n \to 0$</span> as <span class="math-container">$n \to \infty$</span>, and hence <span class="math-container">$f(x^n) \to f(0)$</span> ... |
2,584,969 | <blockquote>
<p>Given the second-order ordinary differential equation:
$$
{y}''+y=f(x)
$$
prove that:
$$
y_p(x)=\int_{0}^{x}f(u)\sin(x-u)du
$$
is the particular solution of the equation.</p>
</blockquote>
<p>I know this is homework but I've been trying to solve it for the past few days and I can't. I even a... | Lutz Lehmann | 115,115 | <p>Variation of constants tells you to find $y_p$ in the form
$$
y_p(x)=c_1(x)\cos(x)+c_2(x)\sin(x)
$$
where the coefficient functions satisfy the differential equation system
\begin{align}
c_1'(x)\cos x+c_2'(x)\sin x &= 0\\
-c_1'(x)\sin x+c_2'(x)\cos x &= f(x)
\end{align}
which leads to
$$
c_1'(x)=-f(x)\sin x,... |
3,426,704 | <p>How to interpret the results
<span class="math-container">$$
1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}
\\
1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}
$$</span></p>
<p>I want to find a clear argument (combinatorial example,etc.) to prove this, other than induction or merely use the form... | antkam | 546,005 | <p>The sum of cubes formula can also be derived by slightly generalizing <a href="https://math.stackexchange.com/a/3113228/546005">this answer by Mike Earnest</a> for the sum of squares case.</p>
<p>Consider counting ordered quadruplets of integers <span class="math-container">$(w,x,y,z)$</span> s.t.</p>
<ul>
<li><p>... |
4,550,913 | <p>I want to show that <span class="math-container">$f:[0,2\pi]\longrightarrow\mathbf{S}^1$</span> defined as <span class="math-container">$f(x)=(\cos x,\sin x)$</span> is closed, surjective, and continuous but not open. I already prooved that it is surjective but I still can't show that it is closed and not open.</p>
| Damian Pavlyshyn | 154,826 | <p>The topology on <span class="math-container">$[0, 2\pi]$</span> is very nearly the same as the one on <span class="math-container">$\mathbf{S}^1$</span>; only the endpoints <span class="math-container">$0, 2\pi$</span> are cause problems.
In particular, sets of the form <span class="math-container">$[0, x)$</span> o... |
328,197 | <p>Let $R$ be a commutative ring with unity, and let $S\subset R$ be any finite set. Then
$$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x,$$
which is easy enough to show by induction.</p>
<p>Does this follow from any sort of general principle? Perhaps inclusion-exclusion, in some form?</p>
| sdcvvc | 12,523 | <p>Let $S'$ be $\{x : x-1 \in S\}$, in other words we shift $S$ by one.</p>
<p>The equality
$$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x$$</p>
<p>reduces to</p>
<p>$$ \sum_{L \subset S'} \prod_{x \in L} x = \prod_{x \in S'} (x+1)$$</p>
<p>but this is obviously distributivity: LHS is the expanded ... |
255,652 | <p>I came across a problem that says:</p>
<p>Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?</p>
<blockquote>
<p>(a) The function $f$ is bounded.<br>
(b) The limit $\lim_{x\to\infty}f(x)$ exists.<br>
(c) The function $f$ is unifor... | Asaf Karagila | 622 | <p><strong>HINT:</strong> Think about $\ln x$ and $\sin x$ as examples. </p>
|
1,584,653 | <p>Let linear transformation is defined as </p>
<p>$\mathcal{A}(1,1,1)=(1,0,0)$</p>
<p>$\mathcal{A}(1,-1,0)=(1,1,1)$</p>
<p>$\mathcal{A}(1,0,1)=(1,1,1)$</p>
<p>Find matrix of $\mathcal{A}$ and inverse (not in matrix representation, if exists).</p>
<p>Attempt:</p>
<p>Transformation $\mathcal{A}$ can be represented... | user84413 | 84,413 | <p>Let S be the matrix for $\mathcal{A}$.</p>
<p>Then $S\begin{bmatrix}1&1&1\\1&-1&0\\1&0&1\end{bmatrix}=\begin{bmatrix}1&1&1\\0&1&1\\0&1&1\end{bmatrix}$, </p>
<p>so $S=\begin{bmatrix}1&1&1\\0&1&1\\0&1&1\end{bmatrix}\begin{bmatrix}1&1&1\\... |
1,639,390 | <p>I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know <em>why</em> it's that exactly.</p>
<p>Also, I am having troubles justifying htat $\frac{n!}{k!} = n(n-1)(n-2)....k(k+1)$</p>
| Eli Rose | 123,848 | <p>$\require{cancel}$</p>
<p>$$\frac{n!}{k!} = \frac{n(n-1)(n-2)\dots}{k(k-1)(k-2)\dots} = \frac{n(n-1)(n-2)\dots(k+1)k(k-1)(k-2)\dots}{k(k-1)(k-2)\dots} = \frac{n(n-1)(n-2)\dots(k+1)\cancel{k(k-1)(k-2)\dots}}{\cancel{k(k-1)(k-2)\dots}} = n(n-1)(n-2)\dots(k+1)$$</p>
|
3,979,912 | <p>Let <span class="math-container">$f:[0,\infty)\rightarrow\mathbb{R}$</span> be a function with a "blow up" in finite time i.e. <span class="math-container">$$\limsup\limits_{t\uparrow T_{max}}|f(t)|=\infty.$$</span>
I don't unterstand the difference between lim and lim sup in this case. How would the defin... | Hume2 | 673,267 | <p>The sequence may have more limit points. In that case, the limit doesn't exist and limes superior is the supremum of those limit points. For example take sequence:<span class="math-container">$$
0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,\ldots
$$</span>
This sequence has two limit points: <span class="math-container">$0$</spa... |
1,203,799 | <p>Let $m,n,k$ be nonnegative integers. How might I go about evaluating the following integral?</p>
<p>$$ \int_{-\infty}^\infty \left( \frac{\mathrm{d}^m}{\mathrm{d}x^m} e^{-x^2} \right) \left( \frac{\mathrm{d}^n}{\mathrm{d}x^n} e^{-x^2} \right) x^k e^{x^2} \mathrm{d}x $$</p>
| giorgiomugnaini | 215,560 | <p><strong>HINT</strong>, not yet a complete answer.</p>
<p>Hermite polynomials:
$$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$</p>
<p>so integral becomes:</p>
<p>$$ I_{m,n,k}=(-1)^{m+n}\int_{-\infty}^\infty H_m(x) H_n(x) x^k e^{-x^2} \mathrm{d}x $$</p>
<p>Particular case $k=0$:</p>
<p>$$ I_{m,n,0}=(-1)^{m+n}\i... |
2,626,919 | <p>I'm attempting to calculate the first-order perturbation energy shift for the quantum harmonic oscillator with a perturbing potential of $V(x)=A\cos(kx)$. Omitting the relevant physical factors, I've gotten to the point where I need to calculate:
\begin{equation}
\int_{-\infty}^{\infty} e^{-x^2}(H_n (x))^2 e^{ikx}dx... | Jack D'Aurizio | 44,121 | <p>Starting with the generating function
$$ e^{2xt-t^2}=\sum_{n\geq 0}H_n(x)\frac{t^n}{n!}\tag{1} $$
then replacing $t$ with $t e^{i\theta}$ we have
$$ \exp\left[2xt e^{i\theta}-t^2 e^{2i\theta}\right] = \sum_{n\geq 0}H_n(x) e^{ni\theta}\frac{t^n}{n!}\tag{2} $$
and by Parseval's identity
$$ \int_{-\pi}^{\pi}\exp\left[4... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Dániel G. | 81,032 | <p>There are a lot of good answers already to this question, but I thought I could contribute with some of my favourite "elementary" problems (ones which barely require any prerequisites). In my opinion all of these problems have beautiful solutions which are a joy to find, and altough they don't necessarily lead to de... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | P i | 3,096 | <p><em>(Note: Originally posted as comment, writing as an answer upon request.)</em></p>
<p><strong>Seek to understand!</strong> </p>
<p>Every single result you have learned to apply, now seek to understand that result from the ground up so that you see it with the clarity of 1+1=2.</p>
<p>Differential and integral ... |
697,506 | <p>Assume the following summation,</p>
<p>$$
\sum_{i=0}^{1000}\left(-1\right)^{i}{1000 \choose i}\left(100 - i\right)^{500}.
$$</p>
<p>I know that this summation is zero, $0$ ( I've checked it with Maple, though ). But I cannot find any proof for that!. Can you provide any help ?.</p>
<p>P.S. This is not a homework ... | Lutz Lehmann | 115,115 | <p>If $(Δ p)(x)=p(x+1)-p(x)$ denotes the step-one difference operator, your expression is equal to
$$0=(Δ^{1000} p)(x)\quad\text{where}\quad p(x)=(100-x)^{500}$$</p>
<p>Since each application of $Δ$ lowers the degree of the polynomial by one, already $(Δ^{501} p)(x)=0$, and moreso the higher order differences.</p>
<h... |
1,487,992 | <blockquote>
<p>Determine: $(0.064)^{\frac{1}{3}}$ is transcendental or algebraic </p>
</blockquote>
<p>To show a number is transcendental/algebraic do I need to show there is a monic polynomial with integer coefficients such that the number is/is not its root ?</p>
<p>in this case:</p>
<p>$x=(0.064)^{\frac{1}{3}}... | 5xum | 112,884 | <p><a href="https://en.wikipedia.org/wiki/Algebraic_number" rel="nofollow">https://en.wikipedia.org/wiki/Algebraic_number</a></p>
<p>A number is algebraic if it is a root of a non-zero polynomial with rational coefficients. There is no demand that the polynomial be monic.</p>
|
1,425,042 | <p>Let B be the set of all irrational numbers together with the numbers 0, 1, and -1. Let addition and multiplication be defined on B in the same way they are defined for real numbers. Determine the field properties that are satisfied by B. Is B a field?</p>
| molarmass | 119,376 | <p>Since $\sqrt{2}$ is irrational, $\sqrt{2} \in B$.</p>
<p>However, $\sqrt{2} \cdot \sqrt{2} = 2 \notin B$. Therefore $B$ is not closed under multiplication and hence $B$ is not a field.</p>
|
946,902 | <p>Is it possible in some cases that using the ILATE rule does not yield an explicit antiderivative but making another choice does yields one? If so, please give examples.</p>
| user84413 | 84,413 | <p>Another example is $\int\frac{xe^x}{(x+1)^2} dx$.</p>
<p>According to this rule, you would let $u=\frac{x}{(x+1)^2}$ and $dv=e^x dx$,</p>
<p>which would give $\frac{xe^x}{(x+1)^2}-\int\frac{(1-x)e^x}{(x+1)^3}dx$.</p>
<p>Instead, you want to let $u=xe^x$ and $dv=\frac{1}{(x+1)^2}dx$,</p>
<p>which gives $-\frac{xe... |
87,636 | <p>I'm following the book <em>Measure and Integral</em> of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.</p>
<p>Consider <span class="math-container">$E\subseteq \mathbb{R}^n$</span> a measurable set. In the following all the integrals are taken over <span class="math-container">$E$</span>, <sp... | t.b. | 5,363 | <p>On leo's request I'm posting my comment as an answer.</p>
<p>Your treatment of the equality cases of Hölder's and Minkowski's inequalities are perfectly fine and clean. There's a small typo when you write that $\int|fg| = \|f\|_p\|g\|_q$ if and only if $|f|^p$ is a constant times of $|g|^q$ almost everywhere (... |
555,921 | <p>Apparently, I'm not understanding this simple concept. What are the differences between the two? Can a person have multiple pure strategies that change throughout the game?</p>
| Gregory Fenn | 389,331 | <p>I want to add an observation that might make mixed strategies more intuitive. This observation is based on [1].</p>
<p>If you are player $1$ and the other players are $\{2,3,4, \dots, n\}$; and the other players are playing strategies $p_2, p_3, \dots$ . </p>
<p>Suppose that you have a choice between strategies $p... |
363,910 | <p>Given two positive rational number $a,b$. How to show that almost surely Brownian motion
attains a local maximum at some time in $(a,b)$?</p>
| Brian Rushton | 51,970 | <p>They may have wanted you to think of the 3-sphere as $\mathbb{R}^3$ plus a point. Then if $U$ is the sphere minus any other point, $U\cap\mathbb{R}^3$ is homeomorphic to $\mathbb{S}^2\times \mathbb{R}$.</p>
|
1,852,240 | <p>Making r the subject </p>
<p>$ S = 2xr^2 + xrl $ </p>
<p>On the right side of the equation , I factorise $r$-</p>
<p>$ r ( 2xr + xl ) $ </p>
<p>I realised that I still have a $r$ inside $2xr$ . How do I remove that $r$ ? Thanks for the help ! </p>
| Ahmed S. Attaalla | 229,023 | <p>$$2xr^2 + xlr-S=0$$</p>
<p>Now solve using the quadratic formula with $a=2x$ ,$b=xl$, $c=-S$. Or by completing the square.</p>
|
1,852,240 | <p>Making r the subject </p>
<p>$ S = 2xr^2 + xrl $ </p>
<p>On the right side of the equation , I factorise $r$-</p>
<p>$ r ( 2xr + xl ) $ </p>
<p>I realised that I still have a $r$ inside $2xr$ . How do I remove that $r$ ? Thanks for the help ! </p>
| Ross Millikan | 1,827 | <p>You have a <a href="https://en.wikipedia.org/wiki/Quadratic_equation" rel="nofollow">quadratic equation</a> $2xr^2+xrl-S=0$ and can use the quadratic formula to get $r=\frac 1{4x}(-xl\pm\sqrt{r^2l^2-4\cdot 2x \cdot S})$</p>
|
429,844 | <p>If I have a $4\times 4$ matrix $A$ with real entries that has all $1$'s on the main diagonal, $A$ is singular and we know one eigenvalue $k_{1}=3+2i$. What about the others three eigenvalues?</p>
<p>I think one should be $k_{2}=3-2i$ because they always come in pairs, right?</p>
<p>Then, since $A$ is singular I th... | DonAntonio | 31,254 | <p>Hints:</p>
<p>Yes, the the roots of <strong>real</strong> polynomials appear in pair of conjugate complex ones, so you already have three roots of the char. polynomial:</p>
<p>$$3\pm 2i\;,\;0$$</p>
<p>Now, we also have that</p>
<p>$$\text{tr.}(A)=4$$</p>
<p>and since</p>
<p>$$+3+2i+3-2i+\alpha=\;-\text{cubic c... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | Peter Smith | 35,151 | <p>The title suggests puzzlement about the very idea of a proof by induction (maybe I am misconstruing the question). Anyway, let's take the simplest case, where we want to show that <em>all</em> natural numbers have some property $P.$ We obviously can't give separate proofs, one for each $n$, that $n$ has $P$, becaus... |
3,501,763 | <p>My question is:</p>
<blockquote>
<p>Is there a way to stack marbles by using only a single one-marble stacking operation such that an infinite 3-dimensional stack is constructed?</p>
</blockquote>
<p>For example:</p>
<p>In 1-dimension one can start with a stack such as</p>
<pre><code>-o-o-o-o
</code></pre>
<p... | Ross Millikan | 1,827 | <p>A more usual way to stack the marbles in 2D is
<span class="math-container">$$\begin{align} &6\ \ 9\\&3\ \ 5\ \ 8\\&1\ \ 2\ \ 4\ \ 7\end {align}$$</span>, putting a new one on the end and going up diagonally to the left. The same idea works in any number of dimensions. In 3D you would place <span clas... |
2,633,975 | <p>Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega\subset\mathbb{R}^n$ open and bounded and $\lambda>0$ sufficiently small so that $2\lambda u<1$. Define $w$ by $w\leq\frac{1}{\lambda}$ and
$$
u(x)=w(x)-\frac{\lambda}{2}w(x)^2.
$$
I have to prove that $w\rightarrow u$ uniformly as $\lambda\rightarrow0$... | mathreadler | 213,607 | <p>You can rewrite $\bf Bx=v$ to $${\bf B}=\min_{\bf B}\{\|{\bf Bx-v}\|_F^2\}$$</p>
<p>with equality when (and only when) that norm equals 0. So you can add it to your cost function</p>
<p>$${\bf B}=\min_{\bf B}\{\|{\bf A-B}\|_F^2+\lambda\|{\bf Bx-v}\|_F^2\}$$</p>
<p>The larger $\lambda$ the more important to fulfil... |
2,694,875 | <p>Let A= {x is reals:x>0} and define a relation on A by x relation y
If xy=0 for x,y in A .</p>
<p>I was wondering if this is reflexive relation. So far I thought
If x=1 and y= 0, then 1*0=0 and
0*1 is also =0. It can be reflexive not sure if I am doing it right by using this counter example . How do I prove wha... | José Carlos Santos | 446,262 | <p>It is not reflexive. If $x>0$, then $x\times x>0$. In particular, $x\times x\neq0$.</p>
|
2,659,001 | <p>I would like to compute the subdifferential of the function</p>
<p>$$ f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx} $$
where $\alpha>0$ and $B$ is symmetric positive definite. </p>
<p><strong>Attempted Solution</strong> (I am brand new to subdifferentiability)</p>
<p>Since subderivatives, like normal derivatives,... | arriopolis | 346,605 | <p><a href="http://www.wolframalpha.com/input/?i=u%5E(2%2F3)+-+5u%5E(1%2F3)+%2B+7+%3D+0" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=u%5E(2%2F3)+-+5u%5E(1%2F3)+%2B+7+%3D+0</a>
<a href="http://www.wolframalpha.com/input/?i=u%C2%B3+-+20u+%2B+343+%3D+0" rel="nofollow noreferrer">http://www.wolframalpha.... |
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Christopher A. Wong | 7,378 | <p>There are way too many approaches to ODEs to have any one book cover them all. I occasionally use a book called <em>Differential Equations and Dynamical Systems</em>, by Lawrence Perko. The focus of this book is on <em>qualitative</em> behavior - existence of fixed points, limit cycles, blow-up solutions, etc.</p>
... |
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Julien Puydt | 12,664 | <p>Henri Cartan's course in differential calculus does cover quite a few useful things for differential equations, from a high-level point of view : you'll find the notion of differentiation in a generic form, the big theorems are proven (local inversion, Cauchy-Lipschitz, ...).</p>
<p>For the low-level and the explic... |
1,874,555 | <p>If $4x/3y = 7/2$, what is the value of $y/x$?</p>
<p>This is a multiple choice question, and the choices are as follows:</p>
<p>A. $3/14$ </p>
<p>B. $8/21$</p>
<p>C. $21/8$</p>
<p>D. $14/3$</p>
<p>I started off answering this by cross multiplying it down to $8x=21y$</p>
<p>From there, $x=21y/8$ and $y=8x/21$ ... | Community | -1 | <p>$$\frac{4x}{3y}=\frac72$$ </p>
<p>Then Multiply by $\frac{y}{x}$:</p>
<p>$$\frac{4x}{3y}\frac{y}{x}=\frac72\frac{y}{x}$$</p>
<p>Then:</p>
<p>$$\frac43=\frac72\frac{y}{x}$$</p>
<p>Then multiply by $\frac27$ to get...</p>
|
1,874,555 | <p>If $4x/3y = 7/2$, what is the value of $y/x$?</p>
<p>This is a multiple choice question, and the choices are as follows:</p>
<p>A. $3/14$ </p>
<p>B. $8/21$</p>
<p>C. $21/8$</p>
<p>D. $14/3$</p>
<p>I started off answering this by cross multiplying it down to $8x=21y$</p>
<p>From there, $x=21y/8$ and $y=8x/21$ ... | Community | -1 | <p>You were correct in reducing the original equation down to $8x=21y$. From here, similar to what you did, divide both sides by 21 and solve for $y$; namely, $$y=\frac{8x}{21}$$
Now divide both sides by $x$ to obtain your final result of $$\frac{y}{x}=\frac{8}{21}$$</p>
<p>Note: you could have just as easily obtained... |
1,298,970 | <p>I am trying to wrap my head around some integration applications.</p>
<p>I went through the exercise of integrating the circumference of a circle, $2*\pi*r$, to get the area of a circle. I simply used the power rule and get $\pi*r^2$.</p>
<p>However when I extend this to a square, I calculate the length around a... | André Nicolas | 6,312 | <p>Consider a square such that the distance from the centre to any side is $r$. Then the area of the square is $4r^2$, and the perimeter of the square is $8r$, which is the derivative of $4r^2$. </p>
<p>So your circle rule works for the square, if we use the right parameter to describe its size.</p>
<p><strong>Explor... |
206,026 | <p>Let $k$ be a field.</p>
<p>$G/k$ be a simply connected semisimple algebraic group. </p>
<p>Let $X/k$ be a smooth affine $k$-scheme. </p>
<p><strong>Question</strong>: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back from $X$?</p>
| Matthias Wendt | 50,846 | <p>In full generality, the answer is no. There are examples of Parimala of non-extended torsors for special orthogonal groups over $\mathbb{A}^2_{\mathbb{R}}$, see e.g. Amer J. Math. 100 (1978), 913-924, (admittedly the group is not simply-connected in this case). These examples are also discussed in Lam's book "Serre'... |
3,795,174 | <p>Let <span class="math-container">$a_n$</span> be a positive sequence.</p>
<p>We define <span class="math-container">$b_n$</span> as following:</p>
<p><span class="math-container">$$b_n = \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$</span></p>
<p><strong>Question:</strong> Prov... | user | 505,767 | <p>In your counterexample something doesn't work, indeed you are assuming</p>
<p><span class="math-container">$$\large {a_n=2^{\frac{(n-1)(n-2)}{2}}}\to \infty$$</span></p>
<p>and therefore</p>
<p><span class="math-container">$$b_n= \frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{2^{n-1}} + a_n\ge a_n \to \infty$$</span>... |
3,196,706 | <p>Let <span class="math-container">$K = \mathbb{Q}(\theta)$</span> be a numberfield and <span class="math-container">$[K:\mathbb{Q}]=n$</span>. When <span class="math-container">$\mathbb{Q}_p$</span> is the field of <span class="math-container">$p$</span>-adic numbers and <span class="math-container">$K_p=\mathbb{Q}_... | Xabu | 284,503 | <p>I think an illustrative example is to look at cyclotomic extensions. Hensel's Lemma states that if a primitive polynomial <span class="math-container">$f$</span> in <span class="math-container">$\mathbb{Z}_p[x]$</span> admits a modulo <span class="math-container">$p$</span> factorization</p>
<p><span class="math-co... |
3,674,924 | <p>The angular momentum components in Cartesians are
<span class="math-container">$$\hat L_x=\hat y\hat p_z-\hat z\hat p_y$$</span>
<span class="math-container">$$\hat L_y=\hat z\hat p_x-\hat x\hat p_z$$</span>
<span class="math-container">$$\hat L_z=\hat x\hat p_y-\hat y\hat p_x$$</span></p>
<p>Starting from
<span cl... | DanielWainfleet | 254,665 | <p>Hint.</p>
<p><span class="math-container">$\sum_{n=0}^{\infty}(\pi x)^{2n-1}(-1)^n/(2n-1)!=\sin \pi x=\pi x\prod_{n=1}^{\infty}(1-x^2/n^2).$</span></p>
<p>Compare the co-efficients of <span class="math-container">$x^3$</span> and <span class="math-container">$x^5$</span> on the far LHS to those on the far RHS.</p>... |
62,967 | <p><code>CoefficientRules</code> acts like the following.</p>
<pre><code>In[1]:= CoefficientRules[2 x^3 + 3 x^2 y + 4 x y^2 - 5 x + 1]
Out[1]= {{3, 0} -> 2, {2, 1} -> 3, {1, 2} -> 4, {1, 0} -> -5, {0, 0} -> 1}
</code></pre>
<p>My question is how one can "extend" this function so that it may allow the n... | Junho Lee | 16,245 | <p>This is my try that is something trick.</p>
<pre><code>function[eq_] := CoefficientRules[eq /.
Power[a_, b_?(# < 0 &)] -> Power[a, -10^10 b]] /.
a_?(# > 10^9 &) -> -a/10^10
function[2 x^3 + 3 x^(-2) y + 4 x y^2 - 5 y^(-3) + 1]
</code></pre>
<blockquote>
<p><code>{{-2, 1} -> 3, {3, 0... |
2,821,413 | <p>$$f'(x) = 2\frac{x^{1/3}-1}{x^{1/3}}$$</p>
<p>critical numbers: x = 1,0</p>
<p>What does it mean by this function is continuous at zero, but not differentiable at zero.</p>
| SystematicDisintegration | 453,502 | <p>$f'(x)$ is not defined at $x=0$, so we say that $f(x)$ is not differentiable at $x=0$ (these are NOT the same things though - an example will follow later). However, by integrating $f'(x)$ we get $f(x) = 2x-3x^{2/3}$ up to a constant, which is defined everywhere. Now $x^{1/n}$ is continuous for $n\in \mathbb{N}$, an... |
827,467 | <p>How do you prove that $n^5 \equiv n\pmod {10}$ Hint given was- Fermat little theorem. Kindly help me out. This is applicable to all positive integers $n$</p>
| Michael Albanese | 39,599 | <p><strong>Hint:</strong> In order to use Fermat's Little Theorem, you need the modulus to be prime which $10$ is not. However, $10 = 2\times 5$. Use Fermat's Little Theorem for each prime and see if you can combine these results to arrive at your claim.</p>
|
827,467 | <p>How do you prove that $n^5 \equiv n\pmod {10}$ Hint given was- Fermat little theorem. Kindly help me out. This is applicable to all positive integers $n$</p>
| Clive Newstead | 19,542 | <p>Here's a direct proof: note that
$$n^5 - n = n(n^4-1) = n(n-1)(n+1)(n^2+1)$$</p>
<p>You need to prove that this number is divisible by $2$ and $5$.</p>
<ul>
<li><p>For divisibility by $2$, note that either $n$ or $n+1$ must be even.</p></li>
<li><p>For divisibility by $5$, you should prove that if neither $n$ nor ... |
4,021,746 | <p>Euclidean distance is not linear in high dimensions. However, in multiple regression the idea is to minimize square distances from data points to a hyperplane.</p>
<p>Other data analysis techniques have been considered problematic for their reliance on Euclidean distances (nearest neighbors), and dimensionality redu... | Math Lover | 801,574 | <p>We have one flag each of <span class="math-container">$4$</span> colors - <span class="math-container">$R, B, G, O$</span>. We need to make a signal with two flags and order matters.</p>
<p>So if we pick any of the color as first, we have <span class="math-container">$3$</span> choices for the second color. If <span... |
3,484,791 | <p>.I am trying to prove the following statement (Linear Algebra Done Right, Section 2.A, #17):</p>
<p>Suppose <span class="math-container">$p_0,p_1,\ldots,p_m$</span> are polynomials in <span class="math-container">$P_m(F)$</span> such that <span class="math-container">$p_j(2)=0$</span> for each <span class="math-con... | Michael Hardy | 11,667 | <p>This doesn't work. Consider the set <span class="math-container">$\{x-2,(x-2)^2\}.$</span> That is a linearly independent set, despite the fact that this same argument would appear to prove it is linearly dependent.</p>
<p>Linear independence means this:</p>
<ul>
<li>For <b>every</b> sequence of coefficients <span... |
3,707,182 | <p>I just got done with an exam and one question was to determine the possible minimal polynomials of <span class="math-container">$A$</span>, if <span class="math-container">$A^3$</span> is the identity matrix. Note that <span class="math-container">$A$</span> is just some square matrix over <span class="math-containe... | Blacks | 671,435 | <p>I think you're right, since the mononic polynomial <span class="math-container">$x^2+x+1$</span> doesn't splits over <span class="math-container">$\mathbb{Q}$</span> and <span class="math-container">$x-1$</span> splits over that field. Something that would be more efficient is having the characteristic polynomial so... |
3,135,085 | <p><span class="math-container">$\lim_{x\to 2} {x^2 - 4\over x^3 - 4x^2 +4x}$</span></p>
<p>I used L'Hospital's rule twice on this, and got a solution, but my textbook says it's an indeterminate form. Is using L'Hospital's rule twice wrong, and if yes, why so?</p>
| fleablood | 280,126 | <p>You can use L'Hoptital rules as many times as you like so long as the numerator and denominator make an inderterminate form.</p>
<p><span class="math-container">$\frac {x^2 -4}{x^3-4x^2 +4x}_{x=2}\to \frac {2^2-4}{2^3 - 4*2^2 + 4*2} \to \frac 00$</span>.</p>
<p>So we can use it.</p>
<p>First time:</p>
<p><span c... |
3,349,312 | <p>I have some confusion in this question
<a href="https://math.stackexchange.com/questions/624611/a-problem-on-comparison-of-dimension-between-two-subspace-of-polynoamial-vector">A problem on comparison of dimension between two subspace of polynomial vector space.</a></p>
<blockquote>
<p>Let <span class="math-contain... | Community | -1 | <p>The polynomials of <span class="math-container">$V$</span> have form <span class="math-container">$ a_0 + a_1 x+ a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}$</span> and therefore <span class="math-container">$\dim (V)=2n+1.$</span></p>
<p>The relation <span class="math-container">$P(1) + P(-1) = 0$</span> reduces the number... |
489,848 | <p>So, I have the following points: $\left( \begin{matrix} 5 \\ 0 \\ 0 \end{matrix} \right), \left( \begin{matrix} 0 \\ 4 \\ -1 \end{matrix} \right), \left( \begin{matrix} -4 \\ 4 \\ 3 \end{matrix} \right)$ and I need to find the equation of the circle passing through them.</p>
<p>Here is how I solved it, but it wa... | Servaes | 30,382 | <p>Another way to solve the problem is by determining the center of the circle. First, determine the plane containing the three given points, hopefully you will find
$$2X+3Y+2Z=10.$$
Next, find parametrisations for the perpendicular bisectors of two pairs of points on the circle. The perpendicular bisector of $(0,4,-1)... |
611,230 | <p>I'm not a mathematician at all, but I'm looking for a formula that will help with the following problem:</p>
<p>It's coming up to Christmas and our employees want to take leave, but some employees will have accrued less leave than than the amount they wish to take in which case they are entitled to take all of thei... | Jim | 56,747 | <p>Let $x$ be the amount of leave they end up taking and $a$ the amount they have already accrued. Their total leave $x$ should be the leave they already have plus the leave they will accrue while on leave, so
$$x = a + 0.038462x$$
You can solve this for $x$ to get
$$x = \frac{a}{1 - 0.038462}$$</p>
|
1,445,561 | <p>Let us say we have the indicator function $\chi_{\{|x|\leq 1\}}$ in $\mathbb{R}^2$.</p>
<p>How can I write out the weak derivative of this indicator function?</p>
<p>Is it $\delta_{|x|=1}$? Or it should be vector valued measure like $\bigg(\frac{\partial}{\partial x_1} \chi_{{|x|\leq 1}},\frac{\partial}{\partial x... | Jean-Marie Aubry | 695,535 | <p>My edit of the other answer (which is wrong) was rejected, so I'll copy it with correction and due credit to its original author.</p>
<p>The weak gradient of the characteristic function of a domain <span class="math-container">$\Omega$</span> with a smooth boundary is the vector-valued measure <span class="math-con... |
1,337,451 | <p>I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$2^x(1+2^{x^3-x})=2^{x^2+1}\iff 1+2^{x^3-x}=2^{x^2-x+1}$$</p>
<p>If $x^3-x>0,1+2^{x^3-x}>1$ is odd unlike $2^{x^2-x+1}$ as $x^2-x+1>0$ for real $x$</p>
<p>So, $x^3-x$ must be $0$</p>
<p>and consequently, $2=2^{x^2-x+1}\implies 1=x^2-x+1$</p>
|
1,337,451 | <p>I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.</p>
| Community | -1 | <p>For $x>0\land x\ne1$,
$$0<x(x-1)^2\implies x^2<\frac{x+x^3}2.$$</p>
<p>Then by monotonicity and convexity of the exponential</p>
<p>$$2^{x^2}<2^{\frac{x+x^3}2}<\frac12\left(2^x+2^{x^3}\right).$$</p>
<p>For $x<0$,
$$\dfrac12\left(2^x+2^{x^3}\right)<1<2^{x^2}.$$</p>
<p>Two cases remain, $x=... |
1,836,538 | <p>In the figure, the ratio of AD to DC is 3 to 2. If area of $\Delta ABC$ is 40 $cm ^ {2}$ , what is the area of $\Delta BDC $ </p>
<p><a href="https://i.stack.imgur.com/wa1bW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wa1bW.png" alt="Note - Image NOT drawn to scale"></a></p>
| Vineet Mangal | 346,869 | <p>Assume D as origin and A and C on x axis. So cordinates of A $(-3h,0)$ and C $(2h,0)$ where $h$ can be any real number. Now BD line is a line passing through origin. SO write it's equation $y=kx$ and assume cordinates of B as $(p,kp)$. Now area of triangle ABC is $A_1=\frac{5kph}{2}$ and area of triangle BDC is $A_2... |
3,248,730 | <p>Consider the problem below:</p>
<blockquote>
<p>Let <span class="math-container">$S(A)$</span> represent the sum of elements in set <span class="math-container">$A$</span> of size <span class="math-container">$n$</span>. We shall
call it a special sum set if for any two non-empty disjoint subsets, <span class="... | Hagen von Eitzen | 39,174 | <p>Each element of or <span class="math-container">$n$</span>-element set <span class="math-container">$A$</span> has three possibilities: It can be element of <span class="math-container">$B$</span>, or of <span class="math-container">$C$</span>, or of neither. Therefore, we count <span class="math-container">$3^n$</s... |
343,894 | <p>I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) reason to ignore the negative square root.</p>
<p>I always thought that the square root always gave two values, a p... | ryang | 21,813 | <blockquote>
<p>If <span class="math-container">$(\pm x)^2 = X,$</span> then <span class="math-container">$\sqrt{X} = x\,?$</span></p>
</blockquote>
<p>The counterexample <span class="math-container">$(x,X)=(-3,9)$</span> shows that the answer to the title question is No.</p>
|
2,223,267 | <p>For<br>
$$e^{-j\pi n}$$</p>
<p>How does this become
$$(-1)^n$$</p>
<p>or is it actually
$$(-1)^{-n}$$
I have checked on calculator and values are all the same when the same n value is used</p>
| mins | 874,001 | <p>Expression <span class="math-container">$\large -j \pi n$</span> is equivalent to:</p>
<ul>
<li><span class="math-container">$\large \quad j \pi (-n)$</span> and</li>
<li><span class="math-container">$\large \quad j (-\pi) n$</span>.</li>
</ul>
<p>This leads to two possible uses of <a href="https://en.wikipedia.org/... |
264,025 | <p>Suppose we start with a $n\times n$ matrix with entries sampled independently and uniformly at random from $[0,1]$. The weight of a set of entries will simply be the sum of those entries. A permutation refers to a set of $n$ entries, no two on the same row or column.</p>
<p>Pick a permutation whose corresponding en... | verret | 22,377 | <p>A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. </p>
<p>Semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. </p>
<p>This answers your question, as it is known that there are infinitely m... |
1,496,438 | <p>Let $A=[a_{ij}]$ be a square matrix of order $2$ where $a_{ij}\in\left\{0,1,2,3,4,6\right\}$.Find the number of matrices $A$ with distinct elements such that $AA^{-1}=I$,where $I$ is unit matrix of order $2$.<br></p>
<hr>
<p>My Attempt:<br>
Let $A=\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$<br>... | Empty | 174,970 | <p>$r^3+1=(r+1)(r^2-r+1)=0 \implies r=-1,\frac{1\pm \sqrt 3i}{2}$ </p>
|
19,590 | <p><a href="http://www.xamuel.com/graphs-of-implicit-equations/" rel="noreferrer">Here</a> are several equations, it seems that Mathematica couldn't plot them well, although I set PlotPoints>100</p>
<pre><code> ContourPlot[Csc[1. - x^2] Cot[2. - y^2] - x*y == 0,
{x, -10, 10}, {y, -10, 10}, PlotPoints -> 120]
</... | whuber | 91 | <p><strong>The heart of the problem lies in the removable singularities where the cosecant or cotangent become undefined.</strong></p>
<p>Recalling that $\csc(x) = 1/\sin(x)$ and $\cot(x) = \cos(x)/\sin(x)$, notice that</p>
<p>$$\csc(1-x^2)\cot(2-y^2) = xy$$</p>
<p>implies (<em>via</em> clearing the denominators) th... |
16,848 | <p>Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained from one another by a sequence of isotopies and Kirby moves.</p>
<p>The original proof by R. Kirby (Inv Math 45, 35... | Daniel Moskovich | 2,051 | <p>There is Bob Craggs' 1974 proof, which was never published. It relies on Wall's result, that any two 2-handle cobordisms between S^3 and itself are stably homeomorphic if the associated bilinear forms have the same signature and type. It's very much what you are after. I have a hard-copy in my office. I do not full... |
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