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 |
|---|---|---|---|---|
7,064 | <p>The <a href="http://en.wikipedia.org/wiki/Long_division" rel="nofollow">Wikipedia article</a> on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy s... | Yuval Filmus | 1,277 | <p>In Israel I was taught (in the 90's) the US notation, with the divisor on the right (might have something to do with the fact that we write Hebrew right to left).</p>
|
99,617 | <p>How can I animate a point on a polar curve? I have used <code>Animate</code> and <code>Show</code> together before in order to get the curve and the moving point together on the same plot, but combining the polar plot and the point doesn't seem to be working because point only works with Cartesian coordinates.</p>
... | kglr | 125 | <pre><code>Animate[PolarPlot[2 Sin[4 θ], {θ, 0, 2 Pi}, Axes -> False,
MeshFunctions -> {#3 &}, Mesh -> {{{θ, Directive[Red, PointSize[Large]]}}}],
{θ, 0, 2 Pi}, AnimationRunning -> False]
</code></pre>
<p><a href="https://i.stack.imgur.com/PI5cZ.gif" rel="nofollow noreferrer"><img ... |
1,404,960 | <p>Say you have a function $f(x)$ and a line $g(x)=ax+b$. How do you reflect $f$ about $g$?</p>
<p>I am apparently supposed to write more text, but the line above is all I am after, hence I wrote this sentence as well.</p>
| Ben Grossmann | 81,360 | <p>We can implement this as a translation of the line to the origin, a reflection about the line through the origin, and then a translation back in the same direction.</p>
<p>In particular: suppose we want to reflect a point $(x_0,y_0)$ across this line.</p>
<ul>
<li><p>First translate it to the point $(x_1,y_1) = (x... |
1,737,835 | <p>You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to offer Twix and 3 Musketeers. The cost of buying these two candies is<br>
$$C= 5T^2 + 2TM + 3M^2 + 800$$
Where T is... | marco trevi | 170,887 | <p>\begin{equation}
g(x,y):=\max\left(\frac{|x+y|}{\sqrt{2}},\frac{|x-y|}{\sqrt{2}}\right)+\max(|x|,|y|)=1
\end{equation}
gives an octagon in $\mathbb{R}^2$, but I'm not sure if it's a norm...</p>
|
206,636 | <p>What's the fastest way to find the local maxima of a 2D list? <em>E.g.</em></p>
<pre><code>nx = ny = 100;
dat = Table[Sin[2. \[Pi] x/nx] (0.1 + Cos[2. \[Pi] y/ny]), {y, 0, ny}, {x, 0, nx}];
ListPlot3D[dat]
</code></pre>
<p><img src="https://i.stack.imgur.com/MyGBl.png" alt="Mathematica graphics"></p>
<p>This (u... | Alexey Popkov | 280 | <blockquote>
<p>Why is <code>MaxDetect</code> so much slower when <code>Chop</code> is applied?</p>
</blockquote>
<p>Because <code>Chop</code> returns exact zeros what prevents packing of the matrix attempted by <code>MaxDetect</code>. You can detect an attempt to pack the array with <code>Trace</code>:</p>
<pre><c... |
206,636 | <p>What's the fastest way to find the local maxima of a 2D list? <em>E.g.</em></p>
<pre><code>nx = ny = 100;
dat = Table[Sin[2. \[Pi] x/nx] (0.1 + Cos[2. \[Pi] y/ny]), {y, 0, ny}, {x, 0, nx}];
ListPlot3D[dat]
</code></pre>
<p><img src="https://i.stack.imgur.com/MyGBl.png" alt="Mathematica graphics"></p>
<p>This (u... | Henrik Schumacher | 38,178 | <p>A severe problem is that <code>Table</code> generates an unpacked array, a thing that is often very annoying. When converting to an image, it is packed automatically (one can check that, e.g., with <code>Developer`PackedArrayQ[ImageData[Image[dat]]]</code>). And because many functions work faster on packed arrays th... |
2,946,408 | <p>Here is the question:
Find the volume of the region bounded by the <span class="math-container">$x$</span>-axis, <span class="math-container">$x=4$</span>, and <span class="math-container">$y=sqrt(x)$</span>. (rotated about x-axis)</p>
<p>I understand that we can find the cross sectional area at each point with res... | G Cab | 317,234 | <p>Since the rotation axis is the x, then<br>
- in the first (disks) you have to integrate <span class="math-container">$\pi y(x)^2 dx$</span>,<br>
- in the second (shells) you have to integrate <span class="math-container">$2 \pi (4-x(y)) dy$</span></p>
<p>Have a look at <a href="https://math.stackexchange.com/ques... |
2,245,010 | <p>Does there exist a topological group which can be covered by (nontrivial and proper) open subgroups of itself? If so, what are groups of these types called and is this a nice property for a topological group to have? Or is this just impossible?</p>
| DanielWainfleet | 254,665 | <p>(1). An ordered group is a group $G$ with a linear order $<$ such that $a<b\implies ((ac<bc)\land (ca<cb))$ for all $a,b,c \in G$. </p>
<p>(2). Consider the free Abelian group $G$ on $\{a_n:n\in \mathbb N\}$ where $m\ne n\implies a_m\ne a_n,$ with identity element $1.$ For $1\ne x\in G$ there is a uni... |
38,480 | <p>(Before reading, I apologize for my poor English ability.)</p>
<p>I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral below
$$ \int_0^{\frac{\pi}{2}} \arctan (1 - \sin^2 x \; \cos^2 x... | Giuseppe Negro | 8,157 | <p>I don't think your point of view is the right one. To compute an integral analytically and to compute an integral numerically are different things. A numerical analysis professor of mine once said that, in applications (engineering, physics...) it is often more convenient to directly evaluate integrals by numerical ... |
3,279,965 | <p>Let <span class="math-container">$A$</span> be a non-symmetric <span class="math-container">$n\times n$</span> real matrix. Assume that all eigenvalues of <span class="math-container">$A$</span> have positive real parts. Could you please show some simple conditions such that <span class="math-container">$A+A^T$</spa... | tortue | 140,475 | <p>Let <span class="math-container">$\lambda$</span> be an eigenvector for <span class="math-container">$A$</span> with eigenvector <span class="math-container">$v$</span>, then <span class="math-container">$A v = \lambda v$</span> and <span class="math-container">$v^{\dagger}A^{\dagger} = \overline{\lambda} v^{\dagger... |
3,701,175 | <p>If a function <span class="math-container">$h(x)$</span> satisfies:</p>
<p>exists partition <span class="math-container">$ P=\left\{ a_{0},a_{1},...,a_{n}\right\} $</span></p>
<p>of the interval [a,b], such that <span class="math-container">$ h $</span> is constant in the segment <span class="math-container">$(a_... | Divide1918 | 706,588 | <p>The upper sum or lower sum in the definition of Riemann integral is essentially the (definite) integral of a stepwise function. Consider a partition of <span class="math-container">$[a,b]: P=\{x_0=a,x_1,...,x_n=b\}$</span>. Now, define <span class="math-container">$ h(x)$</span> by <span class="math-container">$h(x)... |
3,809,026 | <p>I'm trying to solve this problem in the implicit differentiation section of the book I'm going through:</p>
<blockquote>
<p>The equation that implicitly defines <span class="math-container">$f$</span> can be written as</p>
<p><span class="math-container">$y = \dfrac{2 \sin x + \cos y}{3}$</span></p>
<p>In this probl... | Anderson Brasil | 388,949 | <p>An induction is meant to prove that a certain statement <span class="math-container">$A(n)$</span> is true for all integers <span class="math-container">$n$</span> equal or greater than a certain <span class="math-container">$n_0$</span>. If <span class="math-container">$A$</span> fails fails at <span class="math-co... |
449,617 | <p>I would like to know how do you solve summation expressions in an easy way (from my understanding).
I am computer science student analyzing for loops and finding it's time complexity.</p>
<p>e.g</p>
<p><strong>Code</strong>:</p>
<pre><code> for i=1 to n
x++
end for
</code></pre>
<p><strong>Summation</stro... | Brian M. Scott | 12,042 | <p>For the second problem, suppose that $G$ is a universal countable locally finite graph, and let $V=\{v_n:n\in\omega\}$ be the vertex set of $G$. For each $n\in\omega$ define a function</p>
<p>$$f_n:V\to\omega:v\mapsto|\{w\in V:d_G(v,w)\le n\}|\;,$$</p>
<p>where $d_G(v,w)$ is the length of the shortest path from $v... |
1,977,345 | <p>Queston;</p>
<p>Given that 2 is a generator of cyclic group U(25), find all generators.</p>
<p>I am only conversant with the finding the mod which is very long with this question. Pls can someone enlighten me on how to get it done faster.</p>
<p>I am new to the cyclic group.</p>
<p>solution
U(25) = {1,2,3,4,6,7,... | lhf | 589 | <p>In a cyclic group of order $n$ generated by $g$, the order of $g^k$ is $\dfrac{n}{\gcd(n,k)}$.</p>
<p>In particular, the generators are $g^k$ with $\gcd(n,k)=1$.</p>
<p>In your case, $g=2$ and $n=\phi(25)=20$.</p>
<p>Therefore, the generators of $U(25)$ are $2^k$ for $k$ coprime with $20$, that is, $k$ odd not a ... |
18,564 | <p>I want to get the eight points of intersection from the equations <code>2 Abs[x] + Abs[y] == 1</code> and <code>Abs[x] + 2 Abs[y] == 1</code>. To solve these equations, I tried</p>
<pre><code>Solve[{2 Abs[x] + Abs[y] == 1, Abs[x] + 2 Abs[y] == 1}, {x, y}]
</code></pre>
<p>but could only get the four points. I then... | Pinguin Dirk | 5,274 | <p>Actually, the way I interpret your <code>ContourPlot</code>, there are only 4 points of intersection (red/blue curves). So I do interpret your question such that you want to include intersection with the coord axes as well. I hope this helps:</p>
<pre><code>eq1 = 2 Abs[x] + Abs[y] == 1;
eq2 = Abs[x] + 2 Abs[y] == 1... |
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>
| Noah Stein | 5,963 | <p>I'd suggest reading about <a href="http://en.wikipedia.org/wiki/Resultant" rel="nofollow">resultants</a>. The applications section of that article gives a method for solving your problem.</p>
<p>P.S. The level of this question seems maybe borderline for this forum (non-research-level questions can always be asked ... |
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>
| BallBoy | 512,865 | <p>You can use induction on $n$ and the Pascal identity $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$.</p>
<p>The only tricky case is when $j = \frac{n}{2}$, since then $j \not\leq \frac{n-1}{2}$, but you can use the fact that $\binom{n-1}{j} = \binom{n-1}{n-1-j} = \binom{n-1}{j-1}$.</p>
|
1,399,008 | <p>I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:</p>
<p>\begin{align*}
\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\... | Harish Chandra Rajpoot | 210,295 | <p>Notice, we have $$\lim_{n\to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\dots +\frac{n}{n^2+n^2}\right)=\lim_{n\to \infty}\sum_{r=1}^{n}\frac{n}{n^2+r^2}$$</p>
<p>$$\lim_{n\to \infty}\sum_{r=1}^{n}\frac{n}{n^2+r^2}=\lim_{n\to \infty}\sum_{r=1}^{n}\frac{\frac{1}{n}}{1+\left(\frac{r}{n}\right)^2}$$
... |
47,655 | <p>Let $X_n$ be the set of all word of the length $3 n$ over the alphabet $\{A,B,C\}$ which contain each of the three letters <em>n</em> times.</p>
<p>The amount of elements of $X_n$ is $\frac{(3n)!}{(n!)^3}$, but why?</p>
<p>I tried to split the problem into three smaller ones by only looking at the distribution of ... | kuch nahi | 8,365 | <p>The number of words out of length $n$ out of which $n_1$ are of one letter and $n_2$ are of another letter etc and $n_k$ are of another letter is same as the number of arrangements of $n$ objects of which $n_1$ are of one type, $n_2$ are of another type, etc is $$\frac{n!}{n_1!n_2!\cdots n_k!}$$ Can you prove this?... |
2,839,823 | <p>I don't know where to ask, but I'm trying. I just thing we cannot do hocus pocus methods.</p>
<hr>
<p>Solve $3^x+4^x=5^x$</p>
<p>Okay, so my friend gave me this equation, and his solution. But I don't belive it holds. It comes here:</p>
<p>"Solution: </p>
<p>$3^x+4^x=5^x\Leftrightarrow \frac{3^x}{3^x}+\frac{4^x... | hmakholm left over Monica | 14,366 | <p>It is obvious that $x=2$ is a solution.</p>
<p>To see that there cannot be other (real) solutions: We're looking for solutions of
$$ 5^x - 4^x - 3^x = 0 $$
Divide this by $4^x$ (which is always positive) on both sides, and we get
$$ (\tfrac54)^x - 1 - (\tfrac34)^x = 0 $$
Here, both the terms $(\tfrac54)^x$ and $-(\... |
142,007 | <p>Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$</p>
<p>Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.</p>
<p>I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion... | jojobo | 864,803 | <p>It is equivalent to <span class="math-container">$\nexists n: \frac{n^2+1}{p}\in \mathbb{N}$</span>, which is a case of the sum of squares theorem.</p>
|
185,900 | <p>What is the <strong>fastest way</strong> to find the smallest positive root of the following transcendental equation:</p>
<p><span class="math-container">$$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... | rmw | 57,128 | <pre><code>eq = 1 + 5 E^(-0.045 t) - (0.03 Sin[t] - 1.2 Cos[t]);
</code></pre>
<p>To find the first root, plot the equation and marker the roots.</p>
<pre><code>Plot[eq, {t, 50, 150}, Mesh -> {{0}}, MeshFunctions -> {#2 &}, MeshStyle -> Directive[Red, PointSize@Medium]]
</code></pre>
<p><a href="https:... |
1,911,601 | <p>Alright so I had to find the area of the common region determined by $y>=x^{0.5}$ and $x^2+y^2<2$. And I proceeded like this --
<a href="https://i.stack.imgur.com/R19kg.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R19kg.jpg" alt="enter image description here"></a></p>
<p>But I'm not get... | rogerl | 27,542 | <p>The integral you wrote is the integral of $\sqrt{x}$ between the vertical lines $x=0$ and $x=\sqrt{2}$. This is <em>not</em> the same as the picture, which should integrate only between $x=0$ and the boundary of the circle. The easiest approach here is to convert to polar coordinates.</p>
<p>If you aren't familiar ... |
1,798,710 | <blockquote>
<p>Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism</p>
</blockquote>
<p>Requirements for a homeomorphism $f:X \rightarrow Y$:</p>
<ol>
<li>$f$ is continuous</li>
<li>$f$ is bijective</li>
<li>$f^{-1}$ is continuous</li>
</ol>... | J.E.M.S | 73,275 | <p><strong>Hint:</strong> That map is open that means that image of an open set is open. </p>
|
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>
<... | Community | -1 | <p>Children are only taught the factorial of naturals because the definition is elementary (though <span class="math-container">$0!=1$</span> deserves special comments).</p>
<p>When you want to extend it to reals, a natural way is with the integral</p>
<p><span class="math-container">$$I_n=\int_0^\infty x^ne^{-x}dx$$</... |
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... | Anatoly Kochubei | 12,205 | <p>Some basic notions and results about Lie algebras are given in the generality you want in Chapter 1 of Bourbaki's "Lie Groups and Lie Algebras", and also in the book by J.-P. Serre "Lie Algebras and Lie Groups".</p>
|
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... | Peter May | 14,447 | <p>There is a joke definition of a Lie algebra, due to my adviser John Moore,
that is relevant. His definition of a Lie algebra over a commutative ring
$R$ is that it is a module $L$ with a bracket operation such that there
exists an associative $R$-algebra $A$ and a monomorphism $L \to A$ of $R$-modules
that takes ... |
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... | Pasha Zusmanovich | 1,223 | <ol>
<li><p>J.F. Hurley in a series of papers studied Lie algebras obtained by taking the multiplication table (with integer coefficients, due to Chevalley) of simple Lie algebras of classical or exceptional type and considering them over a commutative ring. The results describe center, ideal structure, etc. of such al... |
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$$</span></p>
<p>where <span class="math-container">$\mathrm P$</span> is a positive definite matrix with appropriate dimension. How a... | Rodrigo de Azevedo | 339,790 | <p>Since $\mathrm P$ is positive definite, $\mathrm P^{\frac 12}$ exists and is symmetric. Hence,</p>
<p>$$\mbox{tr} (\mathrm P \mathrm M^T \mathrm M) = \mbox{tr} (\mathrm P^{\frac 12} \mathrm P^{\frac 12} \mathrm M^T \mathrm M) = \mbox{tr} (\mathrm M \mathrm P^{\frac 12} \mathrm P^{\frac 12} \mathrm M^T) = \mbox{tr} ... |
2,615,115 | <blockquote>
<p>For $S$ in the domain of a function $f$, let $f(S)= \{f(x): x\in S\}$. Let $C$ and $D$ be subsets of the domain of $f$. Give an example where equality doesn't hold in $f(C\cup D)\subseteq f(C)\cup f(D)$.</p>
</blockquote>
<p>I have proved $f(C\cup D)\subseteq f(C)\cup f(D)$ by definition, but I can't... | Nick A. | 412,202 | <p>What you are trying to prove is wrong. It is always true that $f$ and $\cup$ commute meaning that $f(\cup A_i)=\cup f(A_i)$. Take a look <a href="https://math.stackexchange.com/questions/1131956/image-of-the-union-and-intersection-of-sets">here</a>
!</p>
|
2,615,115 | <blockquote>
<p>For $S$ in the domain of a function $f$, let $f(S)= \{f(x): x\in S\}$. Let $C$ and $D$ be subsets of the domain of $f$. Give an example where equality doesn't hold in $f(C\cup D)\subseteq f(C)\cup f(D)$.</p>
</blockquote>
<p>I have proved $f(C\cup D)\subseteq f(C)\cup f(D)$ by definition, but I can't... | Akira | 368,425 | <p>I think you meant $f(C\cap D)\subseteq f(C)\cap f(D)$. Let me give you an example where inequality does not hold.</p>
<p>$\text{Let }X = \{1, 2\} \text{ and } f(1) = f(2) = 1$.</p>
<p>$\implies f[\{1\}\cap\{2\}]=\varnothing \text{ and }f(\{1\})\cap f[\{2\}]=\{1\}$</p>
<p>$\implies f[\{1\}\cap\{2\}] \neq f[\{1\}]\... |
735,338 | <p>In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral),</p>
<p>$$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int dx_1e^{\frac{im\tau}{2} \bigg((x_{2} - x_1)^2+(x_{1} - x_0)^2\bigg)}$$</p>
<p>After some manipulation, I end up with an integra... | Jeff Faraci | 115,030 | <p>I will calculate your integral
$$
I=\int_{-\infty}^\infty e^{i\alpha x^2}=2\int_0^\infty e^{i\alpha x^2}dx
$$
by using some residue methods. This will be a general proof for you, you can put in the constant factors at the end where you need them.
\begin{equation}
F(\alpha)\equiv C(\alpha)+iS(\alpha)=\int_{-\infty}^... |
735,338 | <p>In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral),</p>
<p>$$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int dx_1e^{\frac{im\tau}{2} \bigg((x_{2} - x_1)^2+(x_{1} - x_0)^2\bigg)}$$</p>
<p>After some manipulation, I end up with an integra... | robjohn | 13,854 | <p>To compute the integral
$$
\int_0^\infty e^{i\alpha x^2}\,\mathrm{d}x
$$
we will consider the contour integral
$$
\int_\gamma e^{i\alpha z^2}\,\mathrm{d}z=0
$$
where $\gamma=[0,R]\cup Re^{i[0,\pi/4]}\cup e^{i\pi/4}[R,0]$ as $R\to\infty$. </p>
<p>$\hspace{3cm}$<img src="https://i.stack.imgur.com/p6Mjx.png" alt="ent... |
2,943,637 | <p>I want to prove that if <span class="math-container">$|z|=1 $</span> then <span class="math-container">$z^8-3z^2+1 \neq 0$</span>. I tried to prove the reciprocal by taking norms in <span class="math-container">$z^8-3z^2+1= 0$</span> and then solving for <span class="math-container">$ |z|$</span> but it does not wo... | Martin R | 42,969 | <p>If <span class="math-container">$z^8-3z^2+1 = 0$</span> then
<span class="math-container">$$
3 |z|^2 = |3 z^2| = |z^8 + 1| \le |z|^8 + 1
$$</span>
and that is not possible if <span class="math-container">$|z| = 1$</span>.</p>
|
2,828,023 | <p><strong>Four cards are dealt off the top of a well-shuffled deck.
Find the chance that:</strong></p>
<p><strong>You get a queen or a king.</strong> </p>
<p>The solution shows $ 1- \frac { \binom {44}{4}}{ \binom {52}{4}}$ </p>
<p>I get that in this case it's easier to use compliment rule, however I'm trying anot... | Graham Kemp | 135,106 | <p>In addition to @InterstellarProbe 's answer, your method of Inclusion and Exclusion should be</p>
<p>$$\begin{align}\mathsf P(Q\geq 1\cup K\geq 1) &= \mathsf P(Q\geq 1)+\mathsf P(K\geq 1)-\mathsf P(Q\geq 1\cap K\geq 1)
\\&=\left(\tfrac{\tbinom 41\tbinom {48}3+\tbinom 42\tbinom {48}2+\tbinom 43\tbinom {48}1... |
119,492 | <p>First I define a function, just a sum of a few sin waves at different angular frequencies:</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40}
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, .001}];
pts = Length@fn
ListPlot[fn, Joined -> True, PlotRange -> {{0, 1000}, All}]
{20, 50, 100, 200, 400}
</code>... | Hugh | 12,558 | <p>As MarcoB remembered I have looked at this <a href="https://mathematica.stackexchange.com/a/91725/12558">before</a> and concluded that LowpassFilter is for image processing not for signal processing. Lets go through your example with random noise and see what we get. I start by making some random noise with zeros be... |
2,966,959 | <blockquote>
<p>Consider the simultaneous system of differential equations:
<span class="math-container">$$ \begin{equation}
x'(t)=y(t) -x(t)/2\\
y'(t)=x(t)/4-y(t)/2
\end{equation} $$</span>
If <span class="math-container">$ x(0)=2 $</span> and <span class="math-container">$ y(0)=3 $</span>, then what is <span cl... | Vasili | 469,083 | <p>Everything was fine until you've made a silly mistake. It should be <br> <span class="math-container">$$\large{v+x\frac{dv}{dx}=-\frac{v}{2+3v^2}}$$</span> <span class="math-container">$$\large{x\frac{dv}{dx}=-\frac{3v+3v^3}{2+3v^2}}$$</span>
<span class="math-container">$$\large{\frac{2+3v^2}{3v+3v^3}}dv=-xdx$$</sp... |
2,493,802 | <p>If $A \subset \mathbb R$, we have the distance between a point in $\mathbb R$ and $A$ as</p>
<p>$$d(x,A)=\inf\{|x-a| \mid a \in A\}.$$</p>
<p><strong>For $\varepsilon>0$, if we define $A(\varepsilon) = \{x \in \mathbb R \mid d(x,A) < \varepsilon \}$, how do we show this set is open?</strong></p>
<hr>
<p>I ... | Cameron Buie | 28,900 | <p>Well, I'd start by taking an arbitrary $x\in A(\varepsilon).$ Since $d(x,A)<\varepsilon,$ then there is some $a\in A$ such that $|x-a|<\varepsilon.$ (Can you see why?) Thus, for any $x\in A(\varepsilon),$ we have that $$x\in\bigcup_{a\in A}(a-\varepsilon,a+\varepsilon).\tag{$\heartsuit$}$$ On the other hand, g... |
2,493,802 | <p>If $A \subset \mathbb R$, we have the distance between a point in $\mathbb R$ and $A$ as</p>
<p>$$d(x,A)=\inf\{|x-a| \mid a \in A\}.$$</p>
<p><strong>For $\varepsilon>0$, if we define $A(\varepsilon) = \{x \in \mathbb R \mid d(x,A) < \varepsilon \}$, how do we show this set is open?</strong></p>
<hr>
<p>I ... | 7697 | 480,457 | <p>A set S in metric space is open if for any $x \in S$, there is $\delta>0$(this $\delta$ depends of $x$) such that $B_{\delta}(x) \subset S$.</p>
<p>Take $x \in A(\epsilon)$. In this case we choose $\delta = \epsilon - d(x,A)$.</p>
<p>Take $y \in B_{\delta}(x)$. Then we have $d(y,A) \leq d(y,x) + d(x,A)< \eps... |
973,101 | <p>I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid.</p>
<p>If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,x_2,x_3\}$, calculate the distance from the origin</p>
<p>$$d=\sqrt{x_1^2+x_2^2+x_3^2}$$</p>
<p>and calculat... | mercio | 17,445 | <p>One way to proceed is to generate a point uniformly on the sphere, apply the mapping $f : (x,y,z) \mapsto (x'=ax,y'=by,z'=cz)$ and then correct the distortion created by the map by discarding the point randomly with some probability $p(x,y,z)$ (after discarding you restart the whole thing).</p>
<p>When we apply $f$... |
913,239 | <p>Given a circle with known center $c$, known radius $r$ and perimeter point $x$:
$$
(x - c_x)^2 + (y - c_y)^2 = r^2
$$
with a tangent line that also goes through a point $p$ lying outside the circle. How do I find the point $x$ at which the line touches the circle?</p>
<p>Given that the tangent line is orthogonal to... | mathlove | 78,967 | <p>What you got is correct. You can solve them in the following way.</p>
<p>We have to solve the following two :
$$(x-a)^2+(y-b)^2=r^2\tag1$$
$$(x-a)(x-s)+(y-b)(y-t)=0\tag2$$
where $x=p_x,y=p_y,a=c_x,b=c_y,s=pp_x,t=pp_y$.</p>
<p>Note that
$$(1)\iff \color{red}{x^2}-2ax+a^2+\color{blue}{y^2}-2by+b^2=r^2$$
$$(2)\iff \c... |
913,239 | <p>Given a circle with known center $c$, known radius $r$ and perimeter point $x$:
$$
(x - c_x)^2 + (y - c_y)^2 = r^2
$$
with a tangent line that also goes through a point $p$ lying outside the circle. How do I find the point $x$ at which the line touches the circle?</p>
<p>Given that the tangent line is orthogonal to... | georg | 144,937 | <p>Equation of circle center (m,n) radius r: $(x-m)^2+(y-n)^2=r^2$, the point $P(x_p,y_p)$ lying outside the circle. Solve using polar.</p>
<p>The coordinates of the point of contact of tangents from P - solve equations:</p>
<p>$(x-m)^2+(y-n)^2=r^2$ </p>
<p>and </p>
<p>$(x-m)(x_p-m)+(y-n)(y_p-n)=r^2$ </p>
<p>Edi... |
3,259,910 | <p>My question is: if G:= Group of polynomials under addition with elements from <span class="math-container">$\{0,1,2,3,4,5,6,7,8,9\}$</span>
<span class="math-container">$f(x)=7x^2+5^x+4$</span> and <span class="math-container">$g(x)=4x^2+8^x+6$</span> ,<span class="math-container">$f(x)+g(x)=x^2+3x$</span>
what ... | egreg | 62,967 | <p>Note that
<span class="math-container">$$
2\sin\alpha\sec^2\alpha=\frac{2\sin\alpha}{\cos^2\alpha}=\frac{2\sin\alpha}{1-\sin^2\alpha}
$$</span>
If
<span class="math-container">$$
f(x)=\arctan\frac{2x}{1-x^2}
$$</span>
then
<span class="math-container">$$
f'(x)=\frac{2}{1+x^2}
$$</span>
so <span class="math-containe... |
3,259,910 | <p>My question is: if G:= Group of polynomials under addition with elements from <span class="math-container">$\{0,1,2,3,4,5,6,7,8,9\}$</span>
<span class="math-container">$f(x)=7x^2+5^x+4$</span> and <span class="math-container">$g(x)=4x^2+8^x+6$</span> ,<span class="math-container">$f(x)+g(x)=x^2+3x$</span>
what ... | lab bhattacharjee | 33,337 | <p>Using <a href="https://math.stackexchange.com/questions/1837410/inverse-trigonometric-function-identity-doubt-tan-1x-tan-1y-pi-tan/1837799#1837799">Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$</a>,</p>
<p>... |
4,361,063 | <blockquote>
<p>Suppose that <span class="math-container">$(X, \tau_1)$</span> is compact and <span class="math-container">$\tau_1 \subset \tau_2$</span>. Is the space <span class="math-container">$(X, \tau_2)$</span> compact? Does the converse hold i.e if <span class="math-container">$(X, \tau_1)$</span> is compact an... | Arctic Char | 629,362 | <p>The first one holds if <span class="math-container">$X$</span> is finite, since every topology on finite sets are compact. If <span class="math-container">$X$</span> is infinite, for any topology <span class="math-container">$\tau_1$</span> on <span class="math-container">$X$</span> one can take <span class="math-co... |
4,494,489 | <p>These are questions I’m stuck on:</p>
<p>(i) Prove for <span class="math-container">$k\in \mathbb{R}^+:\frac{2}{\sqrt{k+1}+\sqrt{k}}\lt\frac{1}{\sqrt{k}}$</span></p>
<p>(ii) Prove <span class="math-container">$16\lt\sum_{k=1}^{80}\frac{1}{\sqrt{k}}\lt17$</span></p>
<p>I did the first one but I just equated them unti... | Li Kwok Keung | 1,072,805 | <p>Sorry that I still stick to counting <span class="math-container">$i$</span> from <span class="math-container">$1$</span> to <span class="math-container">$n$</span>. This sounds more natural to me.</p>
<p>I think that the calculation is incorrect because</p>
<p>First, in your formula, <span class="math-container">$... |
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... | Cameron Williams | 22,551 | <p>Think of it like this: </p>
<p>$$\int_{-\infty} ^{\infty} \delta(x-a) f(x) dx = f(a). $$</p>
<p>(Assuming $f$ is sufficiently nice.) Here your $f$ is given by $(e^{-s})^t\cos(t)H(t)$, where $H$ is the Heaviside step function. Does this help? </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>
| CopyPasteIt | 432,081 | <p>Even before defining the rational numbers, one usually learns the 'prime factor cancellation' game, or how to divide without really trying.</p>
<p>For example, if you want apply Euclid's algorithm to $28$ and $16$, you can put $28$ 'on top' and $16$ on bottom, writing<br>
$\frac{28}{16} = \frac{2^2\,7}{2^4} = \frac... |
2,922,494 | <p>I came across the following problem today.</p>
<blockquote>
<p>Flip four coins. For every head, you get <span class="math-container">$\$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... | joriki | 6,622 | <p>Your temptation is right and your gut is wrong. You do get an extra $\frac12$ if you got tails at least once. The probability that you don't have a tail to reflip is $\frac1{16}$, so you get an extra $\frac12\left(1-\frac1{16}\right)=\frac{15}{32}$. This added to the base expectation of $2 = \frac{64}{32}$ gives $\f... |
2,922,494 | <p>I came across the following problem today.</p>
<blockquote>
<p>Flip four coins. For every head, you get <span class="math-container">$\$1$</span>. You may reflip one coin after the four flips. Calculate the expected returns.</p>
</blockquote>
<p>I know that the expected value without the extra flip is <span class="m... | AlanDouglas | 595,237 | <p>Your gut is wrong, as pointed out already.
The possible outcomes from the initial flip are:</p>
<p>"4 heads" x 1</p>
<p>"3 heads" x 4</p>
<p>"2 heads" x 6</p>
<p>"1 heads" x 4</p>
<p>"0 heads" x 1</p>
<p>This gives an expected return of (4 + 12 + 12 + 4 + 0)/16 = 2</p>
<p>If you add 0.5 to each case except 4 ... |
3,879,009 | <p>How to prove this lemma?</p>
<blockquote>
<p><span class="math-container">$\min(x,y,z) \leq ax+by+cz \leq \max(x,y,z)$</span>, with <span class="math-container">$a+b+c = 1$</span> for any real numbers <span class="math-container">$x,y,z$</span> and <span class="math-container">$a,b,c$</span> positive.</p>
</blockqu... | Farouk Deutsch | 455,437 | <p><span class="math-container">$a \min(x,y,z)+b \min(x,y,z)+c \min(x,y,z) \leq ax+by+cz \leq a \max(x,y,z)+b \max(x,y,z)+c \max(x,y,z)$</span></p>
|
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 $P(x) = a_{0}x^{n} + \cdots + a_{n - 1}x + a_{n... | W-t-P | 181,098 | <p>Here is yet another solution; it is a little long, but is in fact very simple and non-technical.</p>
<p><strong>Outline of the proof</strong></p>
<p>The crucial observation is that the polynomial <span class="math-container">$Q(x):=((1-x)P(1-x))'$</span> also has the property in question: integrating by parts, cha... |
261,605 | <p>Applying the <a href="https://en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem" rel="nofollow noreferrer">Müntz–Szász theorem</a> on $[0,1]$ repeatedly, we can represent
$$
x= \sum_{n\geq 2} c_n x^n
$$
as a uniformly convergent series (<strong>edit:</strong> only over some subsequence, see edits below) o... | Robert Israel | 13,650 | <p>It can't be in $\ell^p$ or bounded, in fact you can't have $|c_n| = O(t^{-n})$ for any $t > a$ where the subsequence converges on $[a,1]$. This is because if $|c_n| = O(t^{-n})$, $\sum_n c_n z^n$ is analytic in $|z|<t$, and by uniqueness...</p>
|
2,058,560 | <p>Can anyone please simplify this boolean expression ? My answer always reduces to a single variable i.e $x$ but my instructor reduced to three literals.
$$x'y'z+x'yz'+xy'z'+xyz$$ </p>
| DanielV | 97,045 | <p>$$x'y'z+x'yz'+xy'z'+xyz = x \text{ xor } y \text{ xor } z$$</p>
<p>There's an even number of negations in each term.</p>
|
100,957 | <p>I have this picture of small particles in a polymer film. I want to count how many particles in the figure, so that I can have a rough estimation of the particle density. But the image quality is poor, I had a hard time to do it.</p>
<p><a href="https://i.stack.imgur.com/ez50R.jpg" rel="noreferrer"><img src="https:... | bill s | 1,783 | <p>The local binarization can help with the uneven lighting. <code>Dilation</code> helps disconnect some particles that remain connected, and <code>DeleteSmallComponents</code> removes small portions caused by noise.</p>
<pre><code>img = Import["http://i.stack.imgur.com/0OJd7.jpg"];
comps = MorphologicalComponents[Del... |
3,954,010 | <p>The case <span class="math-container">$n=3$</span> is from <a href="https://usamts.org/Tests/Problems_31_1.pdf" rel="nofollow noreferrer">here</a>. It's straightforward to prove it's true:</p>
<p>First we notice that if any two of <span class="math-container">$x_1, x_2, x_3$</span> are equal then all must be equal.<... | void_117 | 843,560 | <p>It can be proved to be false for n = 2.</p>
<p>Let x1^x2 = x2^x1. Then we raise both sides with power 1/(x1*x2).</p>
<p>This means x1^((x2/x2)*x1) = x2^((x1/x1)*x2). This means x1^(1/x1) = x2^(1/x2).</p>
<p>This means we have to check if the function f(x) = x^(1/x) is one-one or not. It turns out it is not.</p>
<p>A... |
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>$$\lim \limits_{n \to \infty} \frac{-\alpha_n}{\beta_n -\alpha_n}, \lim \limits_{n \to \infty} \frac{\beta_n}{\beta_n - \... | Chappers | 221,811 | <p>Your issue is in evaluating $f(0)$ as $(\sin{0})/0$: you haven't used the definition you have given, which has $f(0)=1$.</p>
<p>Then you have
$$ \lim_{h \to 0} \frac{\frac{\sin{h}}{h}-1}{h} = \lim_{h \to 0} \frac{\sin{h}-h}{h^2} = \lim_{h\to 0} \frac{\cos{h}-1}{2h} = \lim_{h \to 0} \frac{-\sin{h}}{2} = 0, $$
using ... |
990,467 | <p>I am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end.</p>
<p>6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2} $</p>
<p>6 a) Find the equation of the vertical asymptotes. Explain how.</p>
<p>For the above question how ... | Antony | 180,827 | <p>Graphs of $f(x)=|\sin x|$ and $f(x)=2^{\cos(x)}$. For the point $A\ $ $\cos x=0.56424...$
<img src="https://i.stack.imgur.com/OuwBt.jpg" alt="enter image description here"></p>
|
3,251,928 | <p><span class="math-container">$$\frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-a)(b-c)} + \frac{1}{c(c-a)(c-b)} $$</span></p>
<p>I tried to get everything to the same denominator, and then simplify numerators first but it is very complicated and long if I just use brute force, to multiply all the expressions given from the pr... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$\sum_{cyc}\frac{1}{a(a-b)(a-c)}=\sum_{cyc}\frac{bc(c-b)}{abc(a-b)(b-c)(c-a)}=\frac{(a-b)(b-c)(c-a)}{abc(a-b)(b-c)(c-a)}=\frac{1}{abc}.$$</span>
I used the following property of the Schur's polynomial:
<span class="math-container">$$\sum_{cyc}(a^2c-a^2b)=(a-b)(b-c)(c-a).$$</span></p>
|
2,385,949 | <p>While playing around with random binomial coefficients , I observed that the following <em>identity</em> seems to hold for all positive integers $n$:</p>
<p>$$ \sum_{k=0}^{2n} (-1)^k \binom{4n}{2k}\cdot\binom{2n}{k}^{-1}=\frac{1}{1-2n}.$$</p>
<p>However, I am unable to furnish a proof for it ( though this result i... | Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*} S(n) &=& (2n+1)\sum_{k=0}^{2n}(-1)^k\binom{4n}{2k}\int_{0}^{1}(1-x)^k x^{2n-k}\,dx\tag{Euler Beta}\\&=& (2n+1)\sum_{k=0}^{2n}(-1)^k\binom{4n}{2k}\int_{0}^{1}2z(1-z^2)^k z^{4n-2k}\,dz\tag{$x\mapsto z^2$}\\&=&(2n+1)\int_{0}^{\pi/2}\sin(\theta)\cos(\theta)\left[e^{4ni\theta}+... |
14,893 | <p>What is the best way to draw a path (red color, dashed) along the surface of the smoothHistogram3D. Assume the paths desired to be the X=0 and Y=0 Axes.</p>
<pre><code>data = {{0.798333, 1.21167}, {-0.415, 0.915}, {-0.675, 0.715}, {0.785,
0.675}, {-0.55, 0.645}, {-0.125, 0.57}, {0.15, 0.27}, {-0.3,
0.115}, {0.925... | Rojo | 109 | <p>Perhaps</p>
<pre><code>typingEffect[str_String, secPerLetter_: 0.3] :=
Dynamic[StringTake[str,
Clock[{0, StringLength@str, 1}, StringLength@str secPerLetter]]]
</code></pre>
<p>(thanks @SjoerdC.deVries)</p>
<p>So this gives an animation</p>
<pre><code>typingEffect["hello my dear"]
</code></pre>
<p>Other e... |
14,893 | <p>What is the best way to draw a path (red color, dashed) along the surface of the smoothHistogram3D. Assume the paths desired to be the X=0 and Y=0 Axes.</p>
<pre><code>data = {{0.798333, 1.21167}, {-0.415, 0.915}, {-0.675, 0.715}, {0.785,
0.675}, {-0.55, 0.645}, {-0.125, 0.57}, {0.15, 0.27}, {-0.3,
0.115}, {0.925... | István Zachar | 89 | <p>Implementation with <code>ScheduledTask</code>. When the <code>Type</code> button is pushed, a scheduled task is started that increases the display length of the temporary string <code>temp</code>. It can be paused (<code>Stop</code>) and continued (<code>Type</code> again).</p>
<pre><code>text = StringTake[Example... |
26,986 | <p>Here are the differential equations that set's up the 11 coupled oscillators.</p>
<pre><code>new = Join[
Table[x[i]''[t] == - x[i][t] +
0.1*(x[i + 1][t] - 2*x[i][t] + x[i - 1][t]), {i, 1,
9}], {x[0]''[t] == -x[0][t], x[10]''[t] == x[9][t], x[0][0] == 1,
x[0]'[0] == 1, x[1]'[0] == 0, x[1][0] == 0},
Table[x[i][0... | Kuba | 5,478 | <p><strong>After edit</strong></p>
<p>I think oscillation directions should be parallel. </p>
<pre><code>g[t_] = Table[{Cos[i*2 Pi/11], Sin[i*2 Pi/11], x[i][t]} /. Loin[[1]], {i, 0, 10}];
Animate[
Show[
ListPointPlot3D[g[t], PlotRange -> 1.5, BoxRatios -> 1, Filling -> Axis,
PlotStyle -> ... |
2,657,112 | <p>I am currently reading the paper "PRIMES is in p" and have come across some notation that I don't quite understand in this following sentence</p>
<blockquote>
<p>Consider a prime $q$ that is a factor of $n$ and let $q^k || n$. Then...</p>
</blockquote>
<p>What does the notation $q^k || n$ mean here?</p>
<p>The ... | user | 505,767 | <p>It means "<strong>divides exactly</strong>" in the sense that $q^{k+1}$ does not divide n.</p>
|
1,755,107 | <p>Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $.</p>
<p>What will be the value of $d(x, \emptyset )$? I am confused between $+ \infty $ and $- \infty$. Also, is it possible to find $ \min \{ d(x,a) : a \in \emptyse... | Wes | 320,529 | <p>Fix $x \in X$. If $\{d(x,a) \mid a \in \emptyset\} \neq \emptyset$, then there exists $a \in \emptyset$.</p>
<p>It is vacuously true that for each $M \in \mathbb{R}$ we have $a \geq M$, for all $a \in \emptyset$, so it's reasonable to define $inf\{\emptyset\}=\infty$.</p>
<p>The $min$ function returns a minimal el... |
1,262,444 | <p>I study hard throughout the year and I am able to solve most problems in the text assigned to us and I am frequently the only one who can solve the hardest problems in the assignments or the problem sets. However, I can rarely do exceptionally well at math tests in my University and I frequently run out of time whil... | Kushashwa Ravi Shrimali | 232,558 | <p>Sometimes, you do questions, you practice a lot but you don't get the same output. Don't care much about what you're getting as the output unless and until your input is in doubtful situation. Give it your best and the results will be shown, no matter the conditions be! </p>
<p>Mathematics is a subject, which requi... |
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... | Daniel | 150,142 | <p>The notation $\sqrt[3]{-1}$ is a little bit ambiguous, since there are exactly three third roots of $-1$ over the complex numbers (in general, there are exactly $n$ $n-$roots of any complex number $z$ so the notation $\sqrt[n]{z}$ is ambiguous too).</p>
<p>Since</p>
<p>$$(-1)^3 = -1$$</p>
<p>$-1$ is one of those ... |
3,929,893 | <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$H$</span>, <span class="math-container">$K$</span> normal subgroups of <span class="math-container">$G$</span>. Prove that <span class="math-container">$G=HK$</span> if and only if <span class="math-container">$G/(H\cap K... | Berci | 41,488 | <p>Working with <span class="math-container">$\phi$</span> and the first isomorphism theorem is indeed a proper way.</p>
<p><strong>Hints:</strong><br>
If <span class="math-container">$x,y\in KH=HK$</span> with <span class="math-container">$x=kh,\,y=h'k'$</span>, then <span class="math-container">$\phi(kh')=(xH,yK)$</s... |
578,968 | <p>I studied when H is a separable Hilbert space, every seminorm is norm;but, if H is not separable, it isnot correct. Is it true? pelese help me </p>
| Rasmus | 367 | <p>What you say is false. For instance, on every Hilbert space we have the constant zero seminorm, which is not a norm unless the Hilbert space is zero-dimensional.</p>
|
578,968 | <p>I studied when H is a separable Hilbert space, every seminorm is norm;but, if H is not separable, it isnot correct. Is it true? pelese help me </p>
| Martin Argerami | 22,857 | <p>It is not true that every seminorm on a Hilbert space is a norm. To see a very explicit example, let $H=\mathbb C^2$, and consider the seminorm
$$
\|(a,b)\|=|a|.
$$</p>
|
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>
| user | 505,767 | <p>Applying <strong>sum to product rule</strong> on LHS and <strong>sine of the sum of 2 angles</strong> on RHS</p>
<p>$$2\sin(\alpha)+2\sin(\beta)=3\sin(\alpha+\beta)$$</p>
<p>$$4\sin \left(\frac{\alpha+\beta}{2}\right)\cos \left(\frac{\alpha-\beta}{2}\right)=6\sin \left(\frac{\alpha+\beta}{2}\right)\cos \left(\frac... |
4,139,003 | <p>For this I used the method of making the determinant nonzero. First I create the array</p>
<p><span class="math-container">\begin{equation}
\begin{pmatrix}
a^2 & 0 & 1\\
0 & a & 2\\
1 & 0 & 1
\end{pmatrix}
\end{equation}</span></p>
<p>Then <span class="math-container">$\det(A)=a^3-a=a(a^2-1)$... | Ritam_Dasgupta | 925,091 | <p>There could be a better approach. You have been given the lines <span class="math-container">$y=x$</span> and <span class="math-container">$y=2x$</span>. You should just consider coordinates of centre of circle to be <span class="math-container">$(h,k)$</span>, and then simply write the equation of perpendicular dis... |
4,139,003 | <p>For this I used the method of making the determinant nonzero. First I create the array</p>
<p><span class="math-container">\begin{equation}
\begin{pmatrix}
a^2 & 0 & 1\\
0 & a & 2\\
1 & 0 & 1
\end{pmatrix}
\end{equation}</span></p>
<p>Then <span class="math-container">$\det(A)=a^3-a=a(a^2-1)$... | Math Lover | 801,574 | <p>For two given lines that are not parallel to each other, the angle bisector is given by</p>
<p><span class="math-container">$ \displaystyle \frac{Ax+By+C}{\sqrt{A^2+B^2}} = \pm \frac{ax+by+c}{\sqrt{a^2+b^2}}$</span></p>
<p>where the given lines are <span class="math-container">$Ax + By + C =0$</span> and <span class... |
872,275 | <p>here's the question, how can I solve this:</p>
<p>$$\lim_{x \rightarrow \infty} x\sin (1/x) $$</p>
<p>Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is reformed in the following way </p>
<p>$$\frac{\sin t}{t}$$</p>
<p>then, and this is what I really can´t ... | Joe | 107,639 | <p>You can simply observe that
$$
\lim_{x\rightarrow+\infty}x\sin(1/x)=
\lim_{x\rightarrow+\infty}x\frac{1}{x}\frac{\sin(1/x)}{\frac{1}{x}}=
\lim_{x\rightarrow+\infty}\frac{\sin(1/x)}{1/x}
$$</p>
<p>Now it should be clear that $1/x$ tends to $0$ as $x$ approaches to $+\infty$. Hence it will be equivalent to write $1/x... |
2,675,954 | <p>Fractals, when viewed as functions, are everywhere continuous and nowhere differentiable. Can this also be used as a definition for fractals?</p>
<p>i.e. Are all fractals everywhere continuous and nowhere differentiable? And also: Are all functions that are everywhere continuous and nowhere differentiable fractals?... | SAHEB PAL | 309,736 | <p><strong>By taking elementary operation:</strong>$$\Delta=\begin{vmatrix}
yz-x^2&zx-y^2&xy-z^2\\
zx-y^2&xy-z^2&yz-x^2\\
xy-z^2&yz-x^2&zx-y^2
\end{vmatrix}=\begin{vmatrix}
xy+yz+zx-x^2-y^2-z^2&zx-y^2&xy-z^2\\
xy+yz+zx-x^2-y^2-z^2&xy-z^2&yz-x^2\\
xy+yz+zx-x^2-y^2-z^2&yz-x^2&a... |
3,988,595 | <p>Suppose <span class="math-container">$A \subset \mathbb{R}^n$</span>. Define
<span class="math-container">$$D^m=\{x \in \mathbb{R}^{m+1} : ||x|| \leq 1 \}$$</span></p>
<p>Suppose <span class="math-container">$A$</span> and <span class="math-container">$D^m$</span> are homeomorphic , then is it necessary that <span c... | TravorLZH | 748,964 | <p>Let <span class="math-container">$(a,b)$</span> denote the gcd of <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, then</p>
<p><span class="math-container">$$
\varphi(n)\triangleq\sum_{k\le n,(k,n)=1}1=\sum_{k\le n}\left\lfloor1\over(k,n)\right\rfloor
$$</span></p>
<p>where <span ... |
3,988,595 | <p>Suppose <span class="math-container">$A \subset \mathbb{R}^n$</span>. Define
<span class="math-container">$$D^m=\{x \in \mathbb{R}^{m+1} : ||x|| \leq 1 \}$$</span></p>
<p>Suppose <span class="math-container">$A$</span> and <span class="math-container">$D^m$</span> are homeomorphic , then is it necessary that <span c... | Quade | 790,417 | <p>Why (3) is true intuitively:</p>
<p>For each <span class="math-container">$m_{i}$</span>, <span class="math-container">$n_{j}$</span> respectively coprime and less than <span class="math-container">$m$</span>, <span class="math-container">$n$</span>, we can build <strong>one</strong> and <strong>only one</strong> <s... |
2,391,812 | <p>The question is to find the largest integer that divides all $p^4-1$, where p is a prime greater than 5. Being asked this question, I just assume this number exists. Set $p = 7$, then $p^4-1=2400$. I don't have any background in number theory and not sure what to do next. Thank you for your help! </p>
| Aaron | 9,863 | <p>Since $p^4-1$ is relatively prime to $p$, we know that the common divisor will not have any large prime factors. Indeed, from your calculation, it will have to be a divisor of $2400=2^5*3*5^2$. However, you could make an even better guess by trying out a few more primes and then taking the greatest common divisor.... |
2,391,812 | <p>The question is to find the largest integer that divides all $p^4-1$, where p is a prime greater than 5. Being asked this question, I just assume this number exists. Set $p = 7$, then $p^4-1=2400$. I don't have any background in number theory and not sure what to do next. Thank you for your help! </p>
| DanielWainfleet | 254,665 | <p>Let $n$ be the largest integer that divides $p^4-1$ for all prime $p\geq 7.$ </p>
<p>We have $11^4-1=14640$ and $7^4-1=2400.$ The $gcd$ of $14640$ and $2400$ is $240.$ So $$n\leq 240.$$ If $p$ is odd then modulo $16$ we have $p^4\in \{(\pm 1)^4, (\pm 3)^4,(\pm 5)^4,(\pm 7)^4\}=\{1^2,9^2, 25^2, 49^2\}=\{1^2,9^2,9^2... |
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... | Claude Leibovici | 82,404 | <p>As Robert Israel answered, it does not seem that the antiderivative could be computed even using special functions.</p>
<p>However, considering $$I=\int_0^a\frac{\text{erf}\left(\frac{x}{c}\right)}{\sqrt{1-x^2}}\,dx$$ hoping that $a$ is not too large, you could expand the integrand as a truncated Taylor series and ... |
2,062,638 | <blockquote>
<p>Consider a curve $\gamma(t): [a,b] \to \mathbb{R}^n$. The curve
$$-\gamma(t)=\gamma(a+b-t) \,\,\,\,\,\,\,\,\,\,\,\,\,\, t \in [a,b]$$</p>
<p>is called the "reverse" curve (or path) of $\gamma(t)$.</p>
</blockquote>
<p>This definition is clear, but how is the "reverse" path defined in the follo... | πr8 | 302,863 | <p>A picture would really help here - the intuitive idea behind this is a bit like how, when looking at compositions of functions (or even group elements), $(fg)^{-1}=g^{-1}f^{-1}$. </p>
<p>Here, you're dealing with a <em>concatenation</em> of paths, which is an analog of composition (well, almost) in this setting. Yo... |
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... | nombre | 246,859 | <p>If you define $\Bbb{R}$ using Dedekind cuts over $\mathbb{Q}$, then $(**)$ can be proven without using $(*)$. (the Dedekind completion of a dense linear order without endpoints always has the least upper bound property). </p>
<p>It is tricky to say that $(*)$ follows from $(**)$ because $(*)$ is a defining caracte... |
183,536 | <p>What is the best way to take a part of an expression and put it somewhere else in the same expression? Essentially I need to combine a <code>Extract</code>, <code>Delete</code> and <code>Insert</code> atomically, that is, handle corner cases where the deletion might cause a shift of the position where I want to inse... | kglr | 125 | <pre><code>ClearAll[changePos]
changePos = Insert[Delete[#, #2], #[[#2]], {#3}] &;
</code></pre>
<p><strong>Examples:</strong></p>
<pre><code>changePos[CharacterRange["a", "k"], 7, 3]
</code></pre>
<blockquote>
<p>{"a", "b", "g", "c", "d", "e", "f", "h", "i", "j", "k"}</p>
</blockquote>
<pre><code>changePos[C... |
183,536 | <p>What is the best way to take a part of an expression and put it somewhere else in the same expression? Essentially I need to combine a <code>Extract</code>, <code>Delete</code> and <code>Insert</code> atomically, that is, handle corner cases where the deletion might cause a shift of the position where I want to inse... | Carl Woll | 45,431 | <p>If the parts you want to move are at the same level, you can use <a href="http://reference.wolfram.com/language/ref/Part" rel="nofollow noreferrer"><code>Part</code></a>:</p>
<pre><code>list = Range[10];
list[[{3, 7}]] = list[[{7, 3}]];
list
</code></pre>
<blockquote>
<p>{1, 2, 7, 4, 5, 6, 3, 8, 9, 10}</p>
</bl... |
3,676,003 | <p>I would like to find the parametric equation for a curve starting at radius of curvature 10 at angle 0 degrees and ending at radius of curvature 100 at 90 degrees. The equation for change in radius of curvature along the path will be specified. </p>
<p>One can imagine this as a 90 degree arc of a circle, except the... | Jean Marie | 305,862 | <p>There is a simple curve, the ellipse </p>
<p><span class="math-container">$$\begin{cases}x&=&a \cos(t)\\ y&=&b\sin(t)\end{cases} \ \ \ \iff \ \ \ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \ \ \ \ $$</span></p>
<p>providing an answer to your issue. Indeed, its curvature is</p>
<p><span class="math-contai... |
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>
| Math Gems | 75,092 | <p>If we mimic amWhy's proof for a general summation we obtain a powerful result.</p>
<p>$$ \sum_{i=1}^n f(i) = g(n)$$</p>
<p><strong>Base case:</strong> </p>
<ul>
<li><p>Let $n=1$ and test: $$\sum_{i=1}^1 f(i) = f(1) \color{#c00}{=?}\ g(1)$$</p></li>
<li><p>The $ $ Base Case $\ \,n = 1\ $ holds true $\iff \color{#... |
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... | pseudoeuclidean | 325,914 | <p>Look at the second image that you provided. Let $A$ represent the region under the line; let $B$ represent the region under the parabola; and let $C$ represent the region in between the line and the parabola. Note that $B$ and $C$ do not overlap. We want to find the area of region $C$.</p>
<p>Notice that the combin... |
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... | Han de Bruijn | 96,057 | <p>Come on! Some of us use integrals, or even the
<A HREF="http://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola/2076079#2076079"><I>fundamental idiom of integral calculus</I></A>
to explain the idea that areas can be subtracted. However,
the concept of subtracting areas is much more... |
2,444,376 | <p>Let be $u:\mathbb{R}\rightarrow(-\infty,+\infty]$ a convex function and I suppose that $u$ admits a point of minimum.
I define:</p>
<p>$$(\varphi_\epsilon*u)(x)=\int_{\mathbb{R}^n}\varphi_\epsilon(y)u(x-y)dy, $$
where $\varphi_{\epsilon}$ is the standard mollifier.
Let's introduce the notation:
$$\tilde{u}_i=\varph... | Community | -1 | <p>Neither the convergence of derivative almost everywhere, nor uniform convergence on compact sets imply the convergence of minima on their own. Both modes of convergence leave the possibility of $u_i$ having some small values near infinity (if we don't know anything else about $u_i$). </p>
<p>Here we know that $u_i\... |
2,121,895 | <blockquote>
<p>How many three digit sequences using the numbers 0 ... 9 with conditions<br>
(a) Repetition of digits is allowed and the sequence cannot start with 0?<br>
(b) Repetition of digits is not allowed and the sequence cannot start with 0? </p>
</blockquote>
<p>For part (a) I did the following:<br>
(9)... | Kiran | 82,744 | <p>for part a, it is $9 \times 10 \times 10$ as you did.</p>
<p>for part b, it is $9 \times 9 \times 8$</p>
|
2,121,895 | <blockquote>
<p>How many three digit sequences using the numbers 0 ... 9 with conditions<br>
(a) Repetition of digits is allowed and the sequence cannot start with 0?<br>
(b) Repetition of digits is not allowed and the sequence cannot start with 0? </p>
</blockquote>
<p>For part (a) I did the following:<br>
(9)... | Joffan | 206,402 | <p>For part (b), using $\dfrac{n!}{(n-r)!}$ will get you $\dfrac{10!}{(10-3)!}=\dfrac{10!}{7!} = 10\cdot 9\cdot 8$ options, ignoring the leading zero restriction. Then $\frac{1}{10}$ of these options will have a leading zero, so we can multiply by $\frac {9}{10}$ to get back to the valid solutions, $ \frac {9}{10}\cdot... |
873,755 | <p>Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $</p>
<p>Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$.</p>
<p>I'm not sure how to do this, as the only way I know of dividing polynomials other than long division is synthetic div... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ {\rm mod\ }x^{\large 2}\!-1\!:\,\ x^{\large 2}\equiv 1\,\Rightarrow\,\color{#0a0}{x^{\large 2n}\equiv 1}\,\Rightarrow\,\color{#c00}{x^{\large 2n+1}\equiv x},\ $</span> hence</p>
<p><span class="math-container">$ f(x) =\, \overbrace{(c_0 + c_2\color{#0a0}{ x^2} ... |
2,174 | <p>As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of several he had seen.</p>
<p>The problem:</p>
<p>For an <span class="math-container">$n$</span> dimensional space, develo... | Isaac | 72 | <p>Though not discussed in the full generality you describe, the problem is discussed at length for n = 1, 2, and 3 in <em>Mathematics for High School Teachers: An Advanced Persepective</em> by Usiskin, Peressini, Machisotto, and Stanley (section 5.1.4; © 2003; published by Pearson Education). Specifically, the t... |
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}^... | vadim123 | 73,324 | <p>The eleven options listed are not equally likely. There are $5^2=25$ ways to roll $[1,1,x,x]$, but only one way to roll $[1,1,1,1]$. Hence, what you need is $$\frac{25\cdot 6+5\cdot4+1}{6^4}=\frac{171}{1296}\approx 0.132$$
To deal with three rolls, the simplest way to compute it is to compute the probability that ... |
83,648 | <p><a href="http://reference.wolfram.com/language/ref/FullGraphics.html" rel="nofollow noreferrer"><code>FullGraphics</code></a> hasn't worked entirely for a long time and the situation appears to be getting worse instead of better. In <em>Mathematica</em> 10.0, 10.1, 11.3, 12.3 up to 13.1 a simple usage throws numero... | akpc | 61,134 | <p>It seems that the problem appeared in the version 5-> 6 transition.</p>
<pre><code><< Version6`Graphics`
InputForm@ListPlot[{{0, 0}}]
(* Graphics[{...
Frame -> {{False, False}, {False, False}},...] *)
</code></pre>
<p>which has the same form in later version.
This subtle difference is causing the trouble.... |
710,146 | <p>Conversely, is it true that if every sequence of pointwise equicontinuous functions from $M$ to $\mathbb{R}$ is uniformly equicontinuous, them $M$ is compact?</p>
| Daniel Fischer | 83,702 | <p>Such a space need not be compact. Take any metric space with the discrete metric, $d(x,y) = 1$ for $x\neq y$. Then every family of functions $M \to \mathbb{R}$ is uniformly equicontinuous. But if the space contains infinitely many points, it is not compact.</p>
<p>A connected metric space on which every pointwise e... |
312,696 | <p>I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in 3-dimensional space)?</p>
<p>I understand that for original points holds $M\cdot pt_i = gd_i$, so combining all $pt_i$ into... | robjohn | 13,854 | <p>Form matrices whose rows are $pt_k$ and $gd_k$. Then
$$
\begin{bmatrix}pt_1\\pt_2\\pt_3\end{bmatrix}M=\begin{bmatrix}gd_1\\gd_2\\gd_3\end{bmatrix}
$$
Then we have
$$
M=\begin{bmatrix}pt_1\\pt_2\\pt_3\end{bmatrix}^{-1}\begin{bmatrix}gd_1\\gd_2\\gd_3\end{bmatrix}
$$
So your idea is correct.</p>
|
147,441 | <p>I've just been looking through my Linear Algebra notes recently, and while revising the topic of change of basis matrices I've been trying something:</p>
<p>"Suppose that our coordinates are $x$ in the standard basis and $y$ in a different basis, so that $x = Fy$, where $F$ is our change of basis matrix, then any m... | Keivan | 10,020 | <p>Without saying much, here is how I usually remember the statement and also the proof in one big picture:</p>
<p>\begin{array}{ccc}
x_{1},\dots,x_{n} & \underrightarrow{\;\;\; A\;\;\;} & Ax_{1},\dots,Ax_{n}\\
\\
\uparrow F & & \downarrow F^{-1}\\
\\
y_{1},\dots,y_{n} & \underrightarrow{\;\;\; B\... |
4,004,157 | <p>A firm wants to know how many of its employees have drug problems. Realizing the sensitivity of this issue, the personnel director decides to use a randomized response survey.</p>
<p>Each employee is asked to flip a fair coin,</p>
<p>If head (H), answer the question “Do you carpool to work?”</p>
<p>If tail (T), answ... | Community | -1 | <p>Consider <span class="math-container">$\Bbb F_5$</span> and consider <span class="math-container">$a=2$</span>, <span class="math-container">$b=3$</span> and <span class="math-container">$\sqrt b=2\sqrt a$</span>. Then, <span class="math-container">$x^2+2$</span> has roots <span class="math-container">$\pm(\sqrt a+\... |
4,179,720 | <p>I'm starting to study triple integrals. In general, I have been doing problems which require me to sketch the projection on the <span class="math-container">$xy$</span> plane so I can figure out the boundaries for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. For example, I had... | N. F. Taussig | 173,070 | <p>By convention, in a circular permutation, only the relative order of the people matters. Therefore, we can consider seating arrangements relative to one of the people at the table.</p>
<p>Suppose Julia is one of the eight people. Seat her. We will use her as our reference point. There are <span class="math-conta... |
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