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 |
|---|---|---|---|---|
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | fleablood | 280,126 | <p>Basic pigeon hole. If all integers fall into either type <span class="math-container">$A$</span> or type <span class="math-container">$B$</span> and you have <span class="math-container">$n$</span> integers. Then at least <span class="math-container">$\lceil \frac n2 \rceil$</span> will be of the same type.</p>
<... |
3,347,391 | <blockquote>
<p>Find the maximum and minimum values of <span class="math-container">$x^2 + y^2 + z^2$</span> subject to the equality constraints <span class="math-container">$x + y + z = 1$</span> and <span class="math-container">$x y z + 1 = 0$</span></p>
</blockquote>
<p>My try:</p>
<p>Let <span class="math-conta... | Jack D'Aurizio | 44,121 | <p>Let us assume that <span class="math-container">$x,y,z$</span> are the roots of a monic, cubic polynomial in the <span class="math-container">$t$</span>-variable,
<span class="math-container">$$ q(t) = t^3-t^2+ct+1. $$</span>
This polynomial has three real roots iff its discriminant is non-negative, i.e. iff
<span ... |
226,346 | <p>I have the three dimensional Laplacian <span class="math-container">$\nabla^2 T(x,y,z)=0$</span> representing temperature distribution in a cuboid shaped wall which is exposed to two fluids flowing perpendicular to each other on either of the <span class="math-container">$z$</span> faces i.e. at <span class="math-co... | Alex Trounev | 58,388 | <p>We can solve this problem with using method explained in my answer <a href="https://mathematica.stackexchange.com/questions/262458/unbounded-solution-at-boundaries-with-few-combination-of-values-in-this-bvp-solu/262563?noredirect=1#comment654595_262563">here</a> and in my paper attached to this <a href="https://comm... |
15,033 | <p>I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .</p>
<p>The angle must start from the incoming edge ($e_1$) and ends at another ed... | picakhu | 4,728 | <p>Assuming that finding the coordinates(of your vectors) is not too difficult. </p>
<p>You can use the definition of dot and or cross products, and then solve for the angles. </p>
<p>So you can for example use $| a \times b | = |a||b| \sin\, \theta$</p>
|
2,390,670 | <p>For an array with range $n$ filled with random numbers ranging from 0 (inclusive) to $n$ (exclusive), what percent of the array contains unique numbers?</p>
<p>I was able to make a program that tries to calculate this with repeated trials and ended up with ~63.212%.</p>
<p>My Question is what equation could calcul... | kimchi lover | 457,779 | <p>Your number is suspiciously close to $1-1/e$. The fraction of values represented exactly $k$ times in your array should be close to $\exp(-1)/k!$, so it looks like your program counted the number of distinct values in the array, rather than the number of values represented only once.</p>
|
242,203 | <p>What's the derivative of the integral $$\int_1^x\sin(t) dt$$</p>
<p>Any ideas? I'm getting a little confused.</p>
| amWhy | 9,003 | <p>You can use the fundamental theorem of calculus, but if you have not yet covered that theorem, in short, you'll be taking the derivative - with respect to x - of the integral of $\sin(t)dt$ when the integral is evaluated from $1$ to $x$:</p>
<p>$$\frac{d}{dx}\left(\int_1^x \sin(t) \text{d}t\right) = \frac{d}{dx} [-... |
1,595,658 | <p>$$ \text { Given the function: }f:\mathcal{N^+} \to \mathcal{N^+} where f \left(k\right) = \sum_{i=0}^k \,4^i. $$ </p>
<p>Examining the prime factorizations of f(k) for k= 1...48, many factors appear in a regular pattern. </p>
<p>QUESTION: </p>
<ol>
<li>Is there a proof that these patterns continue for larger ... | Gottfried Helms | 1,714 | <p>I've once "invented" a small notation-scheme for the handy expression of the primefactorization of that function. </p>
<ol>
<li>Let's define $[a:p] = 1 $ if $p$ divides $a$ and $=0$ if not.
($p$ need not a simple primefactor here) </li>
<li>Let's define $\{a,p\} = m $ giving the (high... |
2,563,402 | <p>For which values of $a$ does the system
$$x_1 + x_2 + x_3 = 1$$
$$x_1 + 2x_2 + ax_3 = 2$$
$$2x_1 + ax_2 + 4x_3 = a^2$$
have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions? Where the system has infinitely many solutions, write the solutions in parametric form.</p>
<p>So I tried to row reduc... | Karn Watcharasupat | 501,685 | <p>You are on the right track :)</p>
<p>So you have
\begin{align}
&\left[\begin{array}{ccc|c}
1&1&1&1\\
0&1&a-1&1\\
0&a-2&2&a^2-2
\end{array}\right]\\
\xrightarrow{R_3-(a-2)R_2\to R_3}\
&\left[\begin{array}{ccc|c}
1&1&1&1\\
0&1&a-1&1\\
0&0&... |
2,563,402 | <p>For which values of $a$ does the system
$$x_1 + x_2 + x_3 = 1$$
$$x_1 + 2x_2 + ax_3 = 2$$
$$2x_1 + ax_2 + 4x_3 = a^2$$
have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions? Where the system has infinitely many solutions, write the solutions in parametric form.</p>
<p>So I tried to row reduc... | Raffaele | 83,382 | <p>$\det \left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & a \\
2 & a & 4 \\
\end{array}
\right)=3 a-a^2$</p>
<p>$3a-a^2\ne 0\to a\ne 0\lor a\ne 3$ the sistem has one and only one solution</p>
<p>If $a=0$ the system becomes </p>
<p>$\begin{cases}
x_1 + x_2 + x_3=1\\
x_1 + 2 x_2=2\\
2 x_1 + ... |
3,135,440 | <p>This is throwing me off a bit I believe mainly because the way the question is worded? Would this simply be <span class="math-container">$4$</span> out of <span class="math-container">$36$</span>?</p>
| Robert Shore | 640,080 | <p>By the way, the simplest way to solve this problem is probably to observe</p>
<p><span class="math-container">$$y= \frac{x+1}{x-1} = 1+\frac{2}{x-1} \Rightarrow \frac{dy}{dx} = -\frac{2}{(x-1)^2}.$$</span></p>
<p>I'm guessing the point of the problem was to convince yourself that the various techniques result in t... |
3,909,191 | <p>I am not too sure as to what the relation is but I think <span class="math-container">$R = \{(1, 2), (2, 3), (3, 4), ..., (n - 1, n)\} $</span>.</p>
<p>Any guidance would be appreciated.</p>
| Claude Leibovici | 82,404 | <p>For this equation, I should strongly recommend to use the trigonometric method for three real roots since
<span class="math-container">$$\Delta=6914880 \left(175 X^4 Y^2+230 X^2 Y^4+147 Y^6\right) >0 \quad \forall X,Y$$</span>
<span class="math-container">$$p=-\frac{49}{240} \left(5 X^2+9 Y^2\right)\qquad\qquad ... |
2,924,165 | <p>Assume that $\mathbb R$ is an ordered field (i.e. $\mathbb R$ is a model of real numbers). We define the set of natural numbers $\mathbb N$ as the smallest inductive set containing $1_\mathbb R$ (multiplicative identity of the field $\mathbb R$), where by definition a set $X\subset \mathbb R$ is inductive if $x\in X... | Liandarin | 595,286 | <p>It may be useful to consider the change of coordinates by rotating the $xy$-plane by $45^\circ$ as:
$$u = x+y,\; v=x-y.$$
Then the top surface of the cylinder is described by the simple equation $z=2-u$ and you can calculate the elliptical surface area in terms of principle axes.</p>
<p>I hope, this is helpful.</p>... |
4,006,571 | <p>Prove this statement using a proof by contradiction: <br />
Let <span class="math-container">$n$</span> be a natural number. If <span class="math-container">$x_1,\ldots,x_n \in \mathbb{N} \cup \{0\}$</span> and <span class="math-container">$\sum_{i=1}^{n}{x_i} = n+1$</span> then there is an <span class="math-contain... | Wuestenfux | 417,848 | <p>If <span class="math-container">$x_i\leq 1$</span> for all <span class="math-container">$i$</span>, then <span class="math-container">$x_1+\ldots+x_n\leq 1+\ldots +1 = n\cdot 1 = n$</span>.</p>
<p>Contraposition:
If <span class="math-container">$x_1+\ldots+x_n> n$</span>, there is an <span class="math-container">... |
2,762,323 | <p>I need to find the asymptotic behavior of $$\sum_{j=1}^N \frac{1}{1 - \cos\frac{\pi j}{N}}$$
as $N\to\infty$.</p>
<p>I found (using a computer) that this asymptotically will be equivalent to $\frac{1}{3}N^2$, but don't know how to prove it mathematically.</p>
| Mir Aaliya | 698,391 | <p>I came across a map like: (X, T1) and (X, T2) is a topological space with topologies T1 and T2 where T1 is stronger than T2. Can we say that a map from (X, T1) to (X, T2) is an inclusion map? Anybody please.</p>
|
2,543,215 | <p>I’ve read a post asking whether a subring of a PID is always a PID. The answer is no, but the post itself gave me more questions.</p>
<ol>
<li><p>Is that possible for a PID that is a subring of a non-PID?</p></li>
<li><p>Is that possible for a subring of a PID that is not a UFD? </p></li>
</ol>
<p>Some hints or ex... | Edward Evans | 312,721 | <p>The answer to your first question is yes, as in the comments, $\Bbb Z$ is a subring of $\Bbb Z[X]$, which is a PID, but $\Bbb Z[X]$ is not a PID (look at the ideal $(2, X)$ for the canonical example).</p>
<p>For your second question, pick your favourite ring which is not a unique factorisation domain and then exten... |
1,809,017 | <p>Let $U$ be an open set containing $0$ and $f:U \rightarrow C$ a holomorphic function such that $f(0)=0$ and $f^{'}(0)=2$.Prove that there exists an open neighbourhood $0 \in V \subset U $ and a holomorphic injective function $h:V \rightarrow V$ such that $h(f(z))=2h(z)$. Since I don't have any idea where to start, I... | Hrhm | 332,390 | <p>Let's integrate this in terms of $z$. Let $A(z)$ be the area of the horizontal cross section of $x^2+4y^2-(2-z)^2\leq 0$ at height $z$. Instead of integrating from $-\sqrt{2}\leq z\leq\sqrt{2}$, we will integrate from $4-\sqrt{2}\leq z \leq 4+\sqrt{2}$. (This is only so that we do not have to deal with ugly negative... |
1,331,063 | <p>Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that
$$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$</p>
| Martin Argerami | 22,857 | <p>No. Let
$$
S=\begin{bmatrix}0&1\\0&0\end{bmatrix},\ \ T=\begin{bmatrix}0&0\\1&0\end{bmatrix}.
$$
Then
$$
\text{Tr}(TS)=1,\ \ \text{ and } \|T\|\,|\text{Tr}(S)|=0.
$$</p>
|
264,594 | <p>I need to make a proof but I can't come to the solution:
<p>For every vertex of oriented graph with vertices $U_{1},U_{2},\ldots,U_{n}$ we've got $s_{+}(U)$ the number of edges, which come to the vertex $U$, and $s_{-}(U)$ the number of edges which leave from the vertex.
<p>Prove that: $\sum_{i=1}^{n} |(s_{+}(U_{i})... | Douglas S. Stones | 139 | <p>For any integer $a$, we can check that $|a| \equiv \pm a \equiv a \pmod 2$. Thus:
\begin{align*}
\sum_{i=1}^{n} |s_{+}(U_{i})-s_{-}(U_{i})| & \equiv \sum_{i=1}^{n} \big( s_{+}(U_{i})-s_{-}(U_{i}) \big) & \text{since } |a| \equiv a \pmod 2 \\
& \equiv \sum_{i=1}^{n} s_{+}(U_{i})- \sum_{i=1}^{n} s_{-}(U_... |
1,111,854 | <p>For example:</p>
<p><img src="https://i.stack.imgur.com/xEFpG.jpg" alt="enter image description here"></p>
<p>The last three lines have a |t=ti, what does that mean?</p>
| AlexR | 86,940 | <p>The notation is defined by
$$\begin{align*}
f(x) \Big|_{x=a}^b & := f(b) - f(a)\\
f(x) \Big|_{x=a} & := f(a)
\end{align*}$$
(note the "inconsistency" in the sign of $f(a)$)<br>
The notation allows for notational improvements of statements like
$$\left. \frac{\partial}{\partial x} \phi(x,s) \right|_{x=x_0}$$
... |
624,002 | <p>Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.</p>
<p>pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \righ... | zcn | 115,654 | <p>Your proof is not quite correct - an (abstract) homomorphism $\mathbb{Z}^2 \to \mathbb{Z}^3$ need not send $\mathbb{Z}^2$ to $\mathbb{Z}^2 \times \{0\}$. Here's my preferred way of showing they are not isomorphic (and the argument generalizes):</p>
<p>For any abelian group $A$, the set of group homomorphisms $\text... |
3,808,077 | <p>I'm trying to show that <span class="math-container">$P(A\triangle B)=P(A)+P(B)–2P(A\cap B)$</span>. Knowing that <span class="math-container">$A\triangle B=(A\cap B^{c})\cup(A^{c} \cap B)$</span>.</p>
<p>So, what I did was this:</p>
<p><span class="math-container">\begin{equation*}
\begin{aligned}
P(A\triangle B)&a... | Michael Rozenberg | 190,319 | <p>Because <span class="math-container">$$P(A\Delta B)=P\left((A\setminus B)\cup(B\setminus A)\right)=$$</span>
<span class="math-container">$$=P\left((A\setminus(A\cap B))\cup(B\setminus(A\cap B))\right)=$$</span>
<span class="math-container">$$=P\left((A\setminus(A\cap B)))+P(B\setminus(A\cap B))\right)=$$</span>
<sp... |
3,293,955 | <p>Here is the curve i wish to plot with a function:</p>
<p><a href="https://i.stack.imgur.com/4Smib.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4Smib.png" alt="eliptical curve"></a></p>
<p>I expect the curve to be 1/4 of an elipse but I only have the coordinates to work with (minx,miny and max... | rubikscube09 | 294,517 | <p>For the hint, first notice that by dominated convergence:</p>
<p><span class="math-container">$$
\|f\|_{L^1} = \int_\mathbb{R^k} |f(y)|\mathrm{d}y = \lim_{r \to \infty} \int_{B_r} |f(y)| \mathrm{d}y
$$</span>
for any <span class="math-container">$f \in L^1 (\mathbb{R}^k)$</span>. Note also that <span class="math-co... |
954,376 | <p>Beltrami made (out of thin paper or stiff or starched cloth not mentioned) a model of a surface of constant negative Gauss curvature <span class="math-container">$ K=-1/c^2$</span>. The original might have resembled a <em>large saddle shaped</em> Pringles chip, and frills might have developed by sagging with time, i... | Willemien | 88,985 | <p>Lets start at the beginning</p>
<p>There is no 3 dimensional Euclidean surface that represents the complete hyperbolic plane (Hilberts theorem )</p>
<p>There are Surfaces of revolution that have (outside some boundaries ) a constant negative curvature $K$ .</p>
<p>All surfacesof revolution revolving around the $x... |
2,508,499 | <p>How is this hold that $\mathbb R \subseteq B(0,2)$ where
$\big<\mathbb R,d\big>$ and d is a discrete metric?</p>
<p>By doing so we showed that $\mathbb R $ is bounded</p>
| John Griffin | 466,397 | <p>Mimicking my answer to your previous question:</p>
<p>Recall that
$$
B(0,2) = \{x\in\mathbb{R} \mid d(0,x)<2\}.
$$
If $d$ is the discrete metric, then
$$
d(0,x)=\begin{cases}
0 & \text{if $x=0$}\\
1 & \text{otherwise}
\end{cases}
$$ so that $d(0,x)<2$ for every $x\in\mathbb{R}$. Therefore $B(0,2)=\mat... |
3,061,575 | <p>It is a principle and proof from Introduction to Set Theory, Hrbacek and Jech. </p>
<p>In the proof, line 1 and 2, I couldn't understand why <span class="math-container">$Q(0)$</span> is true. </p>
<p><span class="math-container">$Q(0)$</span> means that "<span class="math-container">$P(k)$</span> holds for all <s... | Andreas | 317,854 | <p>Apply the integration <span class="math-container">$\int_0^1 \cos(\pi x) f(x) dx$</span>, then
<span class="math-container">$$\int_0^1 \cos(\pi x) \sin(\pi x) dx=a_0 \int_0^1 \cos(\pi x) dx +\sum_{n=1}^{\infty}a_n \int_0^1 \cos(\pi x) \cos(n\pi x) dx$$</span></p>
<p>Since <span class="math-container">$\int_0^1 \co... |
939,110 | <p>This is a different but related question to one I asked earlier. I link to it here:</p>
<p><a href="https://math.stackexchange.com/questions/938953/to-show-that-f-is-injective-i-dont-get-this-statement">"To show that f is injective" - I don't get this statement</a></p>
<p>I am pretty new to "function... | Nishant | 100,692 | <p>Hmm, how about $\mathbb Q/\mathbb Z$?</p>
|
1,711,087 | <p>Find the number of elements in the set if the average of these numbers is seven less than the number of elements in the set.</p>
| Stella Biderman | 123,230 | <p>We know that $\sum a_i=30$, $\frac{1}{N}\sum a_i= N-7$. Can you figure out how to solve these two equations? Try substitution.</p>
|
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| Miguel Mars | 226,969 | <p>I think the other answers fit best to solve what you were thinking, but I will write this in the case you want to know more.</p>
<p>If we are working in a polynomial ring like $\mathbb Z _2 \left[ x \right]$, then this equality is always true, as we get that $\left(x-a \right)^2 = \left(x+a \right)^2$ for all $a\in... |
523,932 | <p>I've got a system of equations which is:<br></p>
<p>$\begin{cases} x=2y+1\\xy=10\end{cases}$</p>
<p>I have gone into this: $x=\dfrac {10}y$.
<br>
How can I find the $x$ and $y$?</p>
| Felix Marin | 85,343 | <p>$$
1 = \left(x - 2y\right)^{2} = x^{2} - 4xy + 4y^{2}
$$</p>
<p>$$
1 + 80 = x^{2} + 4xy + 4y^{2} = \left(x + 2y\right)^{2}
$$</p>
<p>$$
x + 2y = \pm 9\,,
\quad
x = {1 \pm 9 \over 2}\,,
\quad
y = {\pm 9 - 1 \over 4}
$$</p>
<p>$$
\color{#ff0000}{\large\left(x, y\right)
=
\quad
\left(5,2\right)\,,\quad
\left(-4, -\,... |
1,618,699 | <p>Let A be a non-empty subset of $\mathbb{R}$ that is bounded above and put $s=\sup A$<br>
Show that if $s\notin A$ the the set $A\cap (s-ε,s)$ is infinite for any $ε>0$ </p>
<p>This has to be solved using contradiction, by supposing $A\cap (s-ε,s)$ is an finite set. But I am not sure how to proceed after this.</... | JMP | 210,189 | <p>The construction comes from analytic continuation.</p>
<p>A function can only be considered valid where it is finite. Often if we extend its domain to complex numbers it remains finite for longer, indeed almost entirely.</p>
<p>A classic example is Riemann's extension of the zeta function.</p>
<p>$\zeta(-1)=-\fra... |
3,050,295 | <p>I'm having a problem with following equation:</p>
<p>I'm applying <span class="math-container">$(a+b)^2$</span> and <span class="math-container">$(a-b)^2$</span>, but am unable to get the correct answer.</p>
<blockquote>
<p><span class="math-container">$$(\sqrt{3}+i)^{2017} + (\sqrt{3} - i)^{2017}$$</span></p>
<... | Coupeau | 626,733 | <p>I would use the Euler's formula. Say <span class="math-container">$z = \sqrt{3} + i$</span>.<br>
So <span class="math-container">$tan(\phi) = 1/\sqrt{3}$</span><br>
and <span class="math-container">$\phi = \pi/6$</span><br>
and <span class="math-container">$r = |z|= \sqrt{1^2 + \sqrt{3}^2} = \sqrt{4} = 2$</span><br>... |
345,844 | <p>Should be simple enough, yet I can't show that there are no monomorphisms $\mathbb{Z}^3\rightarrow \mathbb{Z}^2$. (It is true, right?)</p>
| N. S. | 9,176 | <p><strong>Hint</strong> If $f(x): \mathbb Z^3 \to \mathbb Z^2$ is any morphism then </p>
<p>$$f(1,0,0), f(0,1,0), f(0,0,1) \in \mathbb Z^2 \subset \mathbb Q^2$$</p>
<p>Then, they must be linearly dependent over $\mathbb Q$ since $ \mathbb Q^2$ is a 2 dimensional $\mathbb Q$-vector space. It is trivial to prove that... |
345,844 | <p>Should be simple enough, yet I can't show that there are no monomorphisms $\mathbb{Z}^3\rightarrow \mathbb{Z}^2$. (It is true, right?)</p>
| Mikasa | 8,581 | <p>If we have such that monomorphism $$\phi:\mathbb Z^3\to\mathbb Z^2$$ then according to first theorem of homomorphism we get $\mathbb Z^3\leq\mathbb Z^2$.</p>
|
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | Davislor | 422,808 | <p>As others have said, the formula works for a logarithm of any base, because of the change-of-base formula. However, the accepted answer says, “The practical reason for using base 10 was a little old fashioned: it allowed the use of tables of logarithms instead of a calculator and reducing these calculations to addit... |
77,379 | <p>It is to show for an $a\in \mathbb{C}^{\ast}$ that $aB_{1}(1)= B_{|a|}(a)$ </p>
<p>where B denotes a disc </p>
<p>Okay, maybe this is correct: </p>
<p>$aB_{1}(1) = a(e^{i\phi}) = ae^{i\phi} = |a|e^{i\phi} = B_{|a|}(a)$</p>
<p>But this seems very wrong! </p>
<p>V</p>
| savick01 | 18,493 | <p>Well, your idea is OK. But you should improve some things:</p>
<p>First: If B is a disk and you write $e^{i\phi}$, then it is a parametrization of a circle not the whole disk. But actually a disk is a sum of circles plus the middle point, so the strategy is OK.</p>
<p>Second: As @Adam wrote, you can't write $ae^{i... |
727,752 | <blockquote>
<p>If S is a compact subset of R and T is a closed subset of S,then T is compact.</p>
<p>(a) Prove this using definition of compactness.</p>
<p>(b) Prove this using the Heine-Borel theorem.</p>
</blockquote>
<p>My solution: firstly I should suppose a open cover of T, and I still need to think of the
set S-... | user137301 | 137,301 | <p>$T$ is a closed subset of $S$ if and only if $T=C\cap S$ for some $C$ closed in $\mathbb{R}$. But $S$ is closed too, being compact, so $T$ is closed in $\mathbb{R}$ because it is the intersection of two closed sets. This takes care of the remaining part of $(b)$. For $(a)$, $\mathbb{R}\setminus T$ is an open set con... |
405,772 | <p>I encountered a conformal mapping on the complex plane:$$z\rightarrow e^{i\pi z}$$
and I am not sure about where it does send the point at infinity. If I could say something along the lines: $$\text{Im}(\infty) = \infty$$
Then it would map it to the origin but there is still a voice in my head saying that this equal... | gt6989b | 16,192 | <p>You are seeking</p>
<p>$$
\lim_{z \to \infty} e^{i\pi z}.
$$</p>
<p>Note that if you consider $z = x + iy$, then by De Moivre's Theorem, we have
$$
e^{i\pi z} = e^{i\pi (x+iy)} = \frac{\cos(\pi x) + i \sin(\pi x)}{e^y}.
$$</p>
<p>Now everything depends on along which path $z \to \infty$: if $x$ is fixed and $y \t... |
4,188,020 | <p>I know that there exists a connection on a principal bundle and via parallel transport it is possible to define a a covariant derivative on the associated bundle.</p>
<p>However, can we also define a covariant derivative on the principal bundle. I.e. something that can differentiate a section along a vector field? O... | jw_ | 671,015 | <p>It is defind in <a href="https://en.wikipedia.org/wiki/Exterior_covariant_derivative" rel="nofollow noreferrer">Wikipedia</a> as <span class="math-container">$D\phi(v_0,...,v_k)=d\phi(hv_0,...,hv_k)$</span> where <span class="math-container">$h$</span> is the projection to horizontal subspace according to the given ... |
4,613,214 | <p>I have to do a large modulo but my answer is incorrect.<br />
I am given:<br />
<span class="math-container">$$ 111^{4733} \mod 9467 $$</span></p>
<ul>
<li>9467 prime</li>
<li>111 and 9467 are coprime</li>
<li>Also note that 4733*2=9466<br />
So we can Apply Euler's theorem</li>
</ul>
<p><span class="math-container"... | Ted | 15,012 | <p>Your calculation is wrong because of your 1/2 exponent. The operation <span class="math-container">$x^{1/2}$</span> doesn't make sense when you are working modulo <span class="math-container">$p$</span>.</p>
<p>A number may have 2 square roots mod <span class="math-container">$p$</span>. This is of course also true ... |
4,242,561 | <p>Let <span class="math-container">$T: R^3 \rightarrow R^3$</span> be a linear transformation such that <span class="math-container">$T(x,y,z) = (x,0,0)$</span>. Which implies that the matrix that represents the transformation is <span class="math-container">\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\e... | Mark S. | 26,369 | <p>It depends on your book/the context.</p>
<p>If the context is such that matrices assume the standard basis and stand for the corresponding linear transformation, then the first (writing that <span class="math-container">$T$</span> is the matrix) is fine.</p>
<p>If we distinguish matrices from linear transformations ... |
2,086,006 | <p>You have $7$ boxes in front of you and $140$ kittens are sitting side-by-side inside the
boxes, $20$ in each box. You want to take some kittens as your pets. However the
kittens are very cowardly. Each time you chose a kitten from a box, the kittens that
are in that box to the left of it go to the box in the left, t... | DonAntonio | 31,254 | <p>$$n\left(\sqrt[n]{ea}-\sqrt[n]a\right)=n\sqrt[n]a\left(\sqrt[n]e-1\right)=\sqrt[n]a\,\frac{\sqrt[n]e-1}{\frac1n}\xrightarrow[n\to\infty]{}1\cdot\left[\left(e^x\right)'\right]_{x=0}=1\cdot e^0=1$$</p>
|
2,080,716 | <p>I have the quadratic form
$$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$</p>
<p>I want to diagonalize the matrix of Q. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.</p>
| Kanwaljit Singh | 401,635 | <p>Its simple find the sum of terms divisible by 21.
And add this sum.</p>
<p><strong>Sum of first 100 terms excluding terms divisible by 3 and 7 = Sum of first 100 terms - Sum of terms divisible by 3 - Sum of terms divisible by 21 + Sum of terms divisible by 21.</strong></p>
<p>Because when you are subtracting the s... |
3,355,542 | <p>Let <span class="math-container">$f \in L^{1} [0,1]$</span> such that for all smooth function <span class="math-container">$h: [0,1] \to \mathbb R$</span> with <span class="math-container">$h(0) = h(1) = 0$</span> one has <span class="math-container">$\int_{0}^{1} f(t) h'(t) = 0$</span>. Prove that <span class="mat... | Aphelli | 556,825 | <p>Consider the linear form <span class="math-container">$L_f:\phi \longmapsto \int_0^1{f\phi}$</span>, defined on the vector space <span class="math-container">$V$</span> of functions <span class="math-container">$\phi$</span> that are smooth and compactly supported in <span class="math-container">$(0,1)$</span>. </p>... |
1,986,172 | <p>I am asked to simplify $(\sqrt{t^3}) \times (\sqrt{t^5})$.</p>
<p>I get up to $\sqrt[3]{t^3}\times \sqrt{t^5}$ but I am not sure how to simplify this further as now roots are involved and not just powers.</p>
<p>When I checked the solutions the final answer should be $t^4$ but I'm not sure how this is achieved.</p... | Emilio Novati | 187,568 | <p>If my edit is correct you have:
$$
\sqrt{t^3}\times \sqrt{t^5}=\sqrt{t^3\times t^5 }=\sqrt{t^8}=t^4
$$</p>
<p>or, with fractional exponents:
$$
\sqrt{t^3}\times \sqrt{t^5}=t^{\frac{3}{2}}t^{\frac{5}{2}}=t^{\frac{3}{2}+\frac{5}{2}}=t^{\frac{8}{2}}=t^4
$$</p>
|
211,290 | <p>Is it possible to import graphs generated by <code>geng</code> (a tool from <a href="http://pallini.di.uniroma1.it/" rel="noreferrer">the nauty suite</a>) one by one, rather than all at once. If one could also specify not only the order but also the number of edges that would be great, but the main thing is to be a... | Bill | 18,890 | <p>I guessed where to insert a missing <code>)</code> in Eq2. If that was right then</p>
<pre><code>Plot[ReIm[Eq1[d]-0.08*Eq2[d]],{d,-1/2,1/2}]
</code></pre>
<p>shows you approximately where the two roots are so you can give <code>FindRoot</code> good starting estimates.</p>
|
4,041,140 | <p>this is a problem which is for homework in my math course. The problem states that you must find two distinct, non-zero matrices, (Size 2x2) such that A * B + A + B = 0.</p>
<p>I'm not really looking for an answer, but rather the methodology I should be using to come to this answer. It seems like the easiest way to ... | José Carlos Santos | 446,262 | <p>Take any number <span class="math-container">$a\ne-1$</span>. Then take <span class="math-container">$b=-\frac a{a+1}$</span>. With this choice, <span class="math-container">$ab+a+b=0$</span>. Now, take <span class="math-container">$A=\left[\begin{smallmatrix}a&0\\0&a\end{smallmatrix}\right]$</span> and <spa... |
1,048,526 | <p>I'm trying to bound the quantity
<span class="math-container">$\langle \nabla \Psi(x),\bar{x}-x \rangle$</span> above, with the bound depending on <span class="math-container">$\|x-\bar{x}\|$</span> and perhaps also of <span class="math-container">$\|x-y\|$</span> for fixed (but not varying) points <span class="math... | Dirk | 3,148 | <p>The answer is no. On the real line consider $\Phi(x)=|x| $ (and add some smooth convex function with minimum in zero if you like). Then the minimum is in zero but the subgradient at any positive point is about 1.</p>
|
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | Michael Rozenberg | 190,319 | <p>It's <span class="math-container">$$x^2(x-4)-4(x-4)=0$$</span> or
<span class="math-container">$$(x-4)(x^2-4)=0.$$</span>
Can you end it now?</p>
|
3,008,162 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be well-ordered sets, and suppose <span class="math-container">$f:A\to B$</span> is an
order-reversing function. Prove that the image of <span class="math-container">$f$</span> is finite.</p>
<p>I started by supposing not. Then... | Hagen von Eitzen | 39,174 | <p>By the givens, each non-empty subset of <span class="math-container">$f(A)$</span> has both a minimal and a maximal element. Conclude that <span class="math-container">$f(A)$</span> cannot contain a subset isomorphic to <span class="math-container">$\omega$</span></p>
|
2,199,303 | <p>Consider the DE $$y''+\lambda y=0$$ where $\lambda$ is a constant </p>
<p>subject to the boundary conditions $$y(0)=0$$ and $$y(a)=0$$ where $a$ is a positive constant</p>
<p>I want to find the eigenvalues and eigenfunctions related to this problem</p>
<p>My attempt:</p>
<p>The auxiliary equation is $$m^2-\lambd... | Disintegrating By Parts | 112,478 | <p>I'll suggest a general method that separates the equations at the endpoints. This does not directly answer your question, but the method is worth knowing.</p>
<p>For a solution $y$ of the second order equation $y''+\lambda y$, there is no non-trivial solution with $y(0)=0=y'(0)$. So, by scaling $y$ if necessary, it... |
1,325,432 | <p>$f(x) = x$ , $f(x+2\pi) = f(x) $ on $ [-\pi , \pi] $ </p>
<p>How do I know that this function is even or odd? My book says odd, but I don't understand how to work this out? </p>
<p>also why does $a_0 = 0$ and $a_n = 0$? </p>
<p>since its an odd function I thought we use the even extension? </p>
<p>i.e $$ a... | albo | 22,610 | <p>If $f(-x)=-f(x)$, then we say $f$ is odd. On the other hand if $f(-x)=f(x)$ we say $f$ is even.</p>
<p>The general Fourier representation of $f$ is </p>
<p>$$
f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty \biggl[ a_{n}\cos\biggl(\frac{n\pi x}{L}\biggl)+b_{n} \sin\biggl(\frac{n\pi x}{L}\biggl) \biggl]\qquad for~-L\leq x\leq... |
4,160 | <p>I am a guest here, having responded to a general invitation extended to the <a href="https://stats.stackexchange.com/questions">Cross Validated</a> community, to possibly contribute answers whenever some question related to Statistics comes up in this site.
I do not teach Mathematics, but I do occasionally teach Sta... | DavidButlerUofA | 1,853 | <p><strong>On the (false) distinction between descriptive and inferential statistics</strong></p>
<p>In my view, it is rare that your only purpose is simply to describe the data you have. Even a simple graph is usually used to make a tentative hypothesis about the relationship between variables or the distribution of ... |
1,968,541 | <p>$$144x^5 − 121x^4 + 100x^3 − 81x^2 − 64x + 49 = 0 $$</p>
<p>I re-wrote it as </p>
<p>$$ 12^2x^5 - 11^2x^4 + 10^2x^3 - 9^2x^2 - 8^2x + 7^2 = 0 $$</p>
<p>And then as $$ \sum_{k=0}^{5} (k+7)^2(-1)^kr^k = 0 $$</p>
<p>But I don't know what to do with that. Thanks for any help!</p>
| Len West | 377,227 | <p>Assume. n is an integer solution.</p>
<p>Then (x-n) would be a factor of the polynomial.</p>
<p>Then n would have to be a divisor of 49.</p>
<p>The only possibilities for n are positive or negative 1, 7, & 49. </p>
<p>Substitution of these 6 possibilities shows that none are solutions.</p>
<p>Therefore... |
2,106,003 | <p>I was just reading about the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">Banach–Tarski paradox</a>, and after trying to wrap my head around it for a while, it occurred to me that it is basically saying that for any set A of infinite size, it is possible to divide it into two sets B and C su... | hexomino | 314,970 | <p>In addition to the other answer, you might be interested to learn that extending the statement of the Banach-Tarski paradox to $\mathbb{R}$ or $\mathbb{R}^2$ doesn't work.</p>
<p>See <a href="https://terrytao.wordpress.com/2009/01/08/245b-notes-2-amenability-the-ping-pong-lemma-and-the-banach-tarski-paradox-optiona... |
1,255,311 | <p><img src="https://i.stack.imgur.com/5V9e0.png" alt="enter image description here"></p>
<p>I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something with notation here. Any help/... | Brandon Suarez | 235,266 | <p>\begin{align}Z&=Ae^it+Be^{-i}t\\
&=A(\cos(t)+i\sin(t))+B(\cos(t)-i\sin(t))\\
&=(A+B)\cos(t)+i(A-B)\sin(t)\\&=x+iy\\
\implies
x&=(A+B)\cos(t)\\
y&=(A-B)\sin(t)\end{align}</p>
|
2,506,279 | <blockquote>
<p>If $\lim_{x\to \infty}xf(x^2+1)=2$ then find
$$\lim_{x\to 0}\dfrac{2f'(1/x)}{x\sqrt{x}}=?$$</p>
</blockquote>
<p>My Try :
$$g(x):=xf(x^2+1)\\g'(x)=f(x^2+1)+2xf'(x^2+1)$$
Now what?</p>
| algoHolic | 409,509 | <p>What a difference a day makes :¬)</p>
<p>It turns out that <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic#Floating-point_numbers" rel="nofollow noreferrer"><em>the Wikipedia page's sigma notation</em></a> was correct after all. It's just that their worked-out calculation is in fact wrong. Misleadi... |
618,986 | <p>I'm having trouble with this question, I'd like someone to point me in the right direction.</p>
<p>let $A$ be a n by n matrix with real values.
show that there is another n by n real matrix $B$ such that $B^3=A$, and that $B$ is symmetric. Are there more matrices like this $B$ or is it the only one?</p>
<p>What I... | Thomas Russell | 32,374 | <p>Note that $\forall \mathbf{M}\in\mathbb{R}^{n\times n}$ such that $\mathbf{M}$ is symmetric, we have $\mathbf{M}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{-1}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{T}$, for some orthogonal matrix $\mathbf{P}$.</p>
<p>Therefore we have $\mathbf{B}^{3}=\left(\mathbf{P}\mathbf{\Lambda}_{\m... |
293,245 | <p>Most true statements independent of PA that I know of is equivalent to some consistency statement. For example</p>
<ul>
<li>Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$</li>
<li>Goodstein's theorem is equivalent to Con(PA)</li>
<li>Any conjunction or disjunction of the above.</li>
</ul... | user103227 | 103,227 | <p>The theory $PA + Con(PA)$ has the property you are asking for, this is the so called Friedman-Goldfarb-Harrington principle (see, e.g., <a href="https://projecteuclid.org/euclid.ndjfl/1093883515" rel="nofollow noreferrer" title="Fifty years of self-reference in arithmetic">Fifty years of self-reference in arithmetic... |
2,553,284 | <p>I know that
$$\ln e^2=2$$
But what about this?
$$(\ln e)^2$$
A calculator gave 1. I'm really confused.</p>
| Reader Manifold | 376,599 | <p>$\ln e$ is 1. So $(\ln e ) ^ 2 $ is one.</p>
|
157,497 | <p>Let's suppose I have created a 3d image of gray scale Images with:</p>
<pre><code>image3d = Image3D[Table[readImage[i], {i, numberOfImages}]];
</code></pre>
<p>and </p>
<pre><code>image3dSlices = Image3DSlices[image3d]
</code></pre>
<p>To show the 3d image I can use:</p>
<pre><code>image3d
</code></pre>
<p>or... | Akku14 | 34,287 | <p>Here another simple, quite mechanical way. </p>
<p>Straightforword apply operators that are suited to simplify the expression, simplify and later apply the inverse operator.</p>
<pre><code>t1 = Sum[Sin[n x]/n, {n, \[Infinity]}]
(* 1/2 I (Log[1 - E^(I x)] - Log[E^(-I x) (-1 + E^(I x))]) *)
t2 = t1 // Exp... |
3,266,930 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a positive random variable on the <span class="math-container">$(\Omega,\mathscr{A},P)$</span>. Show that if <span class="math-container">$X\in L_p$</span> for <span class="math-container">$1<p<\infty$</span>.
Prove <span class="math-container">$\... | jgon | 90,543 | <p>Let the sum be <span class="math-container">$S$</span>. Then multiply by <span class="math-container">$a$</span>.
We get
<span class="math-container">$$aS=\sum_{i=1}^{\phi(n)}a^i=a^{\phi(n)}+\sum_{i=1}^{\phi(n)-1}a^i =S,$$</span>
since even if <span class="math-container">$a$</span> doesn't have order <span class=... |
1,724,812 | <p>I'm struggling with the following limit:</p>
<p>$$\lim_{x \to 0} \frac{1-(\cos x)^{\sin x}}{x^2}$$</p>
<p>Don't know where to start with this. Hints/solutions very appreciated.</p>
| Paramanand Singh | 72,031 | <p>We have
\begin{align}
L &= \lim_{x \to 0}\frac{1 - (\cos x)^{\sin x}}{x^{2}}\notag\\
&= \lim_{x \to 0}\frac{1 - \exp(\sin x\log \cos x)}{x^{2}}\notag\\
&= -\lim_{x \to 0}\frac{\exp(\sin x\log \cos x) - 1}{\sin x\log \cos x}\cdot\frac{\sin x\log \cos x}{x^{2}}\notag\\
&= -\lim_{x \to 0}\frac{\log\cos ... |
3,854,286 | <p>This was an exercise in my class, please help:</p>
<blockquote>
<p>Put <span class="math-container">$A = {\mathbb Q}[x,y]$</span> and <span class="math-container">$B = {\mathbb Q}[x,z]$</span>. Consider the morphism <span class="math-container">$f \colon A \to B$</span> of <span class="math-container">${\mathbb Q}$<... | Aphelli | 556,825 | <p>Given what the element is, it is enough to show (by the property of the tensor product) that there is an <span class="math-container">$A$</span>-bilinear <span class="math-container">$\beta: (x,y) \times B \rightarrow C$</span> with <span class="math-container">$\beta(x,z) \neq \beta(y,1)$</span>.</p>
<p>Take <span ... |
529,260 | <p>Let $V$ be a complex vector space of dimension $n$ with a scalar product, and let $u$ be an unitary vector in $V$. Let $H_u: V \to V$ be defined as</p>
<p>$$H_u(v) = v - 2 \langle v,u \rangle u$$</p>
<p>for all $v \in V$. I need to find the minimal polynomial and the characteristic polynomial of this linear operat... | James | 68,019 | <p>$H_{u}=I-2P_{u}$, where $P_{u}:v\mapsto\left\langle v,u\right\rangle u$
is the projection onto the one-dimensional subspace spanned by $u$.
Decompose the space as $V=\mathbb{C}u\oplus\left(V-\mathbb{C}u\right)$,
then </p>
<p>$$
P_{u}=\left[\begin{array}{cc}
1 & 0\\
0 & 0_{n-1}
\end{array}\right]
$$
and
$$
... |
970,062 | <p>To show to quadratic forms are not equivalent, we can find rank, or discriminant or some element which is represented by either one only etc. But Is there a general criterion to show that two binary(right now I am only concerned for binary) quadratic forms are equivalent.
Like here is an example which uses variable ... | Nero | 88,078 | <p>Revuz and Yor: Continuous Martingales and Brownian Motion.
Karatzas and Shreve: Brownian Motion and Stochastic Calculus.</p>
<p>Both are indispensable.</p>
|
3,492,435 | <p>I am reading <em><strong>Foundations of Constructive Analysis</strong></em> by Errett Bishop. In the first chapter he describes a particular construction of the real numbers. There is a intermediate definition before his primary introduction of the Real numbers:</p>
<blockquote>
<p>A sequence <span class="math-con... | Michael Hardy | 11,667 | <p>Replacing <span class="math-container">$m$</span> by <span class="math-container">$m+1$</span> in <span class="math-container">$x^m$</span> is antidifferentiating (modulo multiplication by a constant).</p>
<p>Replacing <span class="math-container">$n$</span> by <span class="math-container">$n-1$</span> in <span cla... |
3,492,435 | <p>I am reading <em><strong>Foundations of Constructive Analysis</strong></em> by Errett Bishop. In the first chapter he describes a particular construction of the real numbers. There is a intermediate definition before his primary introduction of the Real numbers:</p>
<blockquote>
<p>A sequence <span class="math-con... | marwalix | 441 | <p>Just integrate <span class="math-container">$I_{m,n}$</span> by parts. Taking <span class="math-container">$u=(1-x)^n$</span> and <span class="math-container">$dv=x^mdx$</span> the integration by parts formula</p>
<p><span class="math-container">$$\int_0^1u\cdot dv =\left[u\cdot v\right]_0^1-\int_0^1v\cdot du$$</sp... |
423,159 | <p>What do you call a linear map of the form <span class="math-container">$\alpha X$</span>, where <span class="math-container">$\alpha\in\Bbb R$</span> and <span class="math-container">$X\in\mathrm O(V)$</span> is an orthogonal map (<span class="math-container">$V$</span> being some linear space with inner product)? A... | paul garrett | 15,629 | <p>"Orthogonal similitude" would be consistent with a very-common use of "symplectic similitude" (in automorphic forms and repn theory) for <span class="math-container">$g\in GL_n$</span> such that <span class="math-container">$g^\top J g=\nu(g)\cdot J$</span> for skew-symmetric matrix <span class="... |
277,594 | <p><a href="https://i.stack.imgur.com/yX9my.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/yX9my.gif" alt="enter image description here" /></a></p>
<pre><code>Manipulate[
ParametricPlot[{Sec[t], Tan[t]}, {t, 0, u}, PlotStyle -> Dashed,
PerformanceGoal -> "Quality", Exclusions -> All,
... | bmf | 85,558 | <p>I think you want something like the following (?)</p>
<pre><code>Sum[A[i, j] x[i] x[j], {i, 1, 2}, {j, 1, 2}] /. {x[i_] x[j_] ->
c[i, j], x[i_] x[i_] -> c[i, i]}
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/RXsBV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RXsBV.j... |
1,710,304 | <p>I have a boolean algebra equation that i'm not able to simplify fully.</p>
<p>\begin{align}
&(c+ab)(d+b(a+c))\\
&(c+ab)(d+ba+bc)\\
&cd+ abc + bc^2+abd+a^2 b^2 + ab^2 c\\
&\text{using boolean laws $x^2=x$ and $x+x=x$}\\
&cd + bc + abd + ab + (abc + abc)\\
&cd + bc + abd + ab + abc
\end{align... | parsiad | 64,601 | <p>You might have typed this into Mathematica incorrectly. Here's the solution:</p>
<p>\begin{align*}
(c+ab)(d+b(a+c)) & =(c+ab)(d+ab+bc))\\
& =cd+abc+bc+abd+ab+abc\\
& =cd+abc+bc+abd+ab\\
& =cd+bc+ab(c+d+1)\\
& =cd+bc+ab
\end{align*}</p>
<p>Here's <a href="http://www.wolframalpha.com/input/?i... |
2,172,975 | <p>I am reading <a href="http://www.deeplearningbook.org/contents/linear_algebra.html" rel="nofollow noreferrer">http://www.deeplearningbook.org/contents/linear_algebra.html</a> Chapter $2$, page $44$ ($3$rd paragraph) of this book and got confused. Can any body help me to understand this paragraph? Thanks in advance.<... | DonAntonio | 31,254 | <p>I think it simply says: if $\;u,v\;$ two <em>linearly independent</em> eigenvectors corresponding to one same eigenvalue $\;\lambda\;$ , then <strong>any</strong> linear combination $\;\alpha u+\beta v\;$ is also an eigenvector for the same eigenvalue, and we can thus choose on of these lin. comb.'s instead of $\;u\... |
1,981,928 | <p>While I was studying properties of limit and sequences, I found a theorm that says 'if {$s_n$}, {$t_n$} are convergent sequences, then $s_n \le t_n$ for all $n \in \mathbb{N}$ implies that $$\lim_{n\rightarrow \infty} s_n \le \lim_{n\rightarrow \infty} t_n$$
this proof is quite easy to construct, as you can say </p>... | Eff | 112,061 | <p><strong>It is not true</strong>. However, if <span class="math-container">$a_n < b_n$</span> (or <span class="math-container">$a_n \leq s_n$</span> as seen in your theorem) for all <span class="math-container">$n\in\mathbb{N}$</span> and their limits exist, then you <strong>can</strong> instead say that
<span cla... |
9,335 | <p>How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!</p>
| Gabriel Ebner | 474 | <p>Another nice way is to use characteristic functions:</p>
<p>The map $\chi : \mathcal{P}(\Omega) \to \{0,1\}^\Omega$ assigns to every subset of $\Omega$ its characteristic function.</p>
<ul>
<li>$\chi$ is bijective.</li>
<li>$\chi$ is continuous, i.e. $\chi_{\lim\sup_{n\to\infty} A_n} = \lim\sup_{n\to\infty}\, \chi... |
323,559 | <p>Developable surfaces in <span class="math-container">$\mathbb{R}^{3}$</span> have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).</p>
<p>I am a curious about potential or actual applications to other fields of mathematics and science of flat submanifolds of <span class="math-... | Francesco Polizzi | 7,460 | <p>The torus <span class="math-container">$T$</span> can be embedded as a flat submanifold of <span class="math-container">$\mathbb{R}^4$</span>, the so-called <a href="https://en.wikipedia.org/wiki/Clifford_torus" rel="nofollow noreferrer">Clifford torus</a>. It is possible to put infinitely many different complex str... |
323,559 | <p>Developable surfaces in <span class="math-container">$\mathbb{R}^{3}$</span> have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).</p>
<p>I am a curious about potential or actual applications to other fields of mathematics and science of flat submanifolds of <span class="math-... | Piotr Hajlasz | 121,665 | <ol>
<li><p>Crystallographic groups define flat compact manifolds and they are used to describe symmetries of crystals. </p></li>
<li><p>Flat tori are used in computational physics and chemistry: if you want to investigate dynamics, say of a gas and and for computational reasons you can only consider 1000 particles, yo... |
118,275 | <p>In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with <em>split</em> Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objec... | Angelo | 4,790 | <p>The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Science in 1981. It is used to define and study finite vector bundles. Nori does not give any references, so as far I know... |
3,964,862 | <p>The arithmetic mean has the nice property of minimising the sum of squares, or in other words, minimising the sum of quadratic-euclidean distances. Formally, given a set of points <span class="math-container">$x_0, \dots, x_n \in \mathbb{R}^d$</span>, the arithmetic mean, <span class="math-container">$\mu = \frac{1}... | Matthew H. | 801,306 | <p>Consider the metric <span class="math-container">$d:(0,\infty)^2 \rightarrow [0,\infty)$</span> defined by <span class="math-container">$$d(x,y)=\Bigg|\ln\Big(\frac{y}{x}\Big)\Bigg|$$</span> This is the metric you're looking for. Given a dataset <span class="math-container">$\{x_1,\ldots ,x_n\}\subseteq (0,\infty)$<... |
202,247 | <p>I'm working on some problem in algebraic geometry. I need a reference to the following result:</p>
<p>Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non zero polynomial. The complement manifold $\mathbb{C}^{h}\setminus\left\lbrace F=0\right\rbrace$ is a
nonempt... | Georges Elencwajg | 450 | <p><strong>Claim:</strong> The complement $U=\mathbb C^h\setminus \{F=0\}$ is path-connected and thus connected.<br>
<strong>Proof:</strong><br>
Given $a,b\in U$ consider the affine complex line $L_{a,b}=L$ joining $a$ to $b$.<br>
The polynomial $F\mid L$ is not zero since it is not zero at $a$ nor at $b$.<br>
Thus i... |
3,276,572 | <p>Let be <span class="math-container">$\lVert \cdot \rVert$</span> a matrix norm (submultiplicative).</p>
<p>Do we have for all matrices of determinant 1, the following lower bound:</p>
<p><span class="math-container">$$\lVert M \rVert \geq 1$$</span></p>
<p>I'm very confused and could not find any counterexample a... | Raito | 112,314 | <p>Well, in fact, I just got it.</p>
<p><span class="math-container">$SL_n(\Bbb K)$</span> is closed.</p>
<p>If I suppose that <span class="math-container">$\lVert M \rVert < 1$</span>, then: <span class="math-container">$M^n \in SL_n(\Bbb K)$</span> for all <span class="math-container">$n$</span>, and then: <span... |
1,728,920 | <p>I am a software engineer trying to wrap his head around <strong>Fast Fourier Transform (FFT)</strong>. Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of the algorithm/operations, and in fact will likely just use an open source math library to do most o... | mathreadler | 213,607 | <p>Sound is moving waves in matter. Sin and Cos are waves too.</p>
<p>Heat transfer is often modelled with differential equations. Complex exponentials (which are the basis functions in the Fourier Transform) are eigenfunctions to the operation of differentiation.</p>
|
1,946,881 | <p>Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn... | Hagen von Eitzen | 39,174 | <p>Consider any $G$ with at least two elements and define $a*b=a$.
Then for all $x\in G$, there exists $e\in G$ (namely, $e=x$) such that $x*e=e*x=x$. However, there is no $e\in G$ such that for all $x\in G$, we have $e*x=x$ (because $e*x=e$)</p>
|
1,946,881 | <p>Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn... | celtschk | 34,930 | <p>For example, consider the set $G=\{a,b,c\}$ with $x*y=y$ for all $x,y\in G$.</p>
<p>Let's first test the group axiom:</p>
<blockquote>
<p>There exists an $e\in G$ such that for all $x\in G$, $x∗e=e∗x=x$</p>
</blockquote>
<p>So let's check:</p>
<ul>
<li>Could it be that $e=a$? No, because $a*b=b\ne a$.</li>
<li... |
2,655,518 | <p>$2ac=bc$
find the ratio ( $K$ )
what is the ratio of their area?
<a href="https://i.stack.imgur.com/9NPRi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NPRi.png" alt="enter image description here"></a>I found out it is $2$ or $1/2$
is it true? </p>
<p>if the question isn't clear, make sure to... | TheSimpliFire | 471,884 | <p>If you have an operator that yields more than one solution, then obviously the solutions are going to be different (otherwise there would only be one). But this does <em>not</em> mean they are equal.</p>
<p>Your example involves the function $f(x)=\sqrt x$, with $f(25)=\pm5$. Although both $-5$ and $5$ satisfy $\sq... |
1,231,095 | <p>How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$?</p>
<p>I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of any way to actually take the inverse transform of $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}$.</p>
| Fatemeh Shiravand | 230,494 | <p>You can write
$$
\begin{align}
F^{\prime}(s)&=[\ln(s^{2}+a^{2})-\ln(s^{2}+b^{2})]^\prime\\
&=\frac{2s+2a}{s^{2}+a^{2}}-\frac{2s+2b}{s^{2}+b^{2}}\\
&=\frac{2s}{s^{2}+a^{2}}+\frac{2a}{s^{2}+a^{2}}-\frac{2s}{s^{2}+b^{2}}-\frac{2b}{s^{2}+b^{2}}\\
&\to 2(\cos at)+2(\sin at)-2(\cos bt)-2(\sin bt)
\end{ali... |
2,751,819 | <p>I need some help solving this.
I have tried:</p>
<p>$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
=\frac{1}{\operatorname{det}A}\cdot \begin{bmatrix}
d & -b \\
-c & a \\
\end{bmatrix}$$
I ended up with $$a=\frac{d}{\operatorname{det}A},$$
and
$$d=\frac{a}{\operat... | lhf | 589 | <p>By the Cayley–Hamilton theorem or <a href="https://math.stackexchange.com/questions/1494369/show-that-a-matrix-a-pmatrixab-cd-satisfies-a2-adaad-bci-o">direct verification</a>, we have $A^{2}-\operatorname {tr}(A)A+\det(A)I=0$.</p>
<p>From $A^2=I$, we get $\operatorname {tr}(A)A=(\det(A)+1)I$.</p>
<p>Taking traces... |
3,715,475 | <p>About a year ago I asked <a href="https://math.stackexchange.com/questions/3258617/alaoglu-theorem-over-the-p-adics">here</a> whether the Banach-Alaoglu Theorem works over the <span class="math-container">$p$</span>-adics. The satisfactory answer I got is that the "usual" proof only uses local compactness, and so th... | Chilote | 113,061 | <p>In the general non-Archimedean case, instead of using the concept of compactness, it is more suitable to use the concept of compactoidness. I think the theorem you are looking for is
<a href="https://i.stack.imgur.com/1GihY.png" rel="noreferrer"><img src="https://i.stack.imgur.com/1GihY.png" alt="enter image descrip... |
11,073 | <p>I have three simple graphs in one Plot. Now I am trying to make a button for each graph so you can hide or show it in the plot. Until now I was just able to make a checkbox with the Manipulate function, but I don't now how to tell the checkbox that it should hide my graph when unchecked an display it when checked. <... | Vitaliy Kaurov | 13 | <p>One way to do this is to use <code>Opacity</code> to hide a graph and empty label "" to hide a label:</p>
<pre><code>Manipulate[
Plot[{0.5 x + 1, x, 2 x - 2}, {x, -1, 5},
PlotRange -> {-1, 5}, AspectRatio -> 1,
PlotStyle -> {Opacity[a], Opacity[b], Opacity[c]},
Epilog -> {
Text[If[a... |
11,073 | <p>I have three simple graphs in one Plot. Now I am trying to make a button for each graph so you can hide or show it in the plot. Until now I was just able to make a checkbox with the Manipulate function, but I don't now how to tell the checkbox that it should hide my graph when unchecked an display it when checked. <... | Mike Honeychurch | 77 | <p><code>Manipulate</code> is probably the easiest for this specific case but here is an alternative:</p>
<pre><code> DynamicModule[{select = {1, 2, 3}},
Column[{
CheckboxBar[
Dynamic[select], {1 -> "g(x)", 2 -> "y=x",
3 -> "f(x)"}],
Dynamic@Plot[Evaluate@{0.5 x + 1, 2 x - 2, x}[[select]], {... |
3,989,921 | <p>Answered (I think!):</p>
<p>The triple product's purpose is to find a direction to the origin, perpendicular to the baseline, which is super trivial in 2D as there is only two perpendicular orientations, but the "cylinder" distinction is made in 3D because there are infinite perpendicular orientations - he... | Community | -1 | <p>I must be missing something, but taking the question in isolation, namely "Let X be the vector between points A and B. The direction to the origin, apparently, is:"</p>
<p>... Welp, If we know all about the endpoints of the line, namely <span class="math-container">$\vec{A}$</span> and <span class="math-co... |
3,989,921 | <p>Answered (I think!):</p>
<p>The triple product's purpose is to find a direction to the origin, perpendicular to the baseline, which is super trivial in 2D as there is only two perpendicular orientations, but the "cylinder" distinction is made in 3D because there are infinite perpendicular orientations - he... | Aleksejs Fomins | 250,854 | <p>Double cross product is a very common technique to project a vector onto the surface. Consider the triple product</p>
<p><span class="math-container">$$-\vec{n} \times \vec{n} \times \vec{v}$$</span></p>
<p>where <span class="math-container">$\vec{n}$</span> is a normal vector to some surface. <span class="math-cont... |
1,488,501 | <p>Let $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ a periodic function so that forall $x\in \mathbb{R}$ $$\varphi(x+4)=\frac{\varphi(x)-5}{\varphi(x)-3}$$ Find the period the $\varphi$.</p>
| Paul Sinclair | 258,282 | <p>Defining $f(x) = {x - 5\over x-3}$, we have $\varphi(x + 4) = f(\varphi(x))$. More generally, $\varphi(x + 4k) = f^{(k)}(\varphi(x))$. Assuming that the period is a multiple of $4$, for some $k$, $\varphi(x) = y = f^{(k)}(y)$.</p>
<p>I suspect the trick is to find a value of $k$ such that $y = f^{(k)}(y)$ has real ... |
736,036 | <p><strong>Problem</strong> - The least number which leaves remainders 2, 3, 4, 5 and 6 on dividing by 3, 4, 5, 6 and 7 is?</p>
<p><strong>Solution</strong> - Here 3-2 = 1, 4-3 = 1, 5-4 = 1 and so on.</p>
<p>So required number is (LCM of 3, 4, 5, 6, 7) - 1 = 419</p>
<p><strong>My confusion</strong> - </p>
<p>I didn... | Charles | 1,778 | <p>You can see by inspection that -1 works. Then any other solution must be congruent to -1 mod lcm(3,4,5,6,7) = 420 and hence the least positive solution is 419.</p>
|
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | Greg Markowsky | 387,394 | <p>Another fun way is to use Parseval's identity (one should never pass up an opportunity to do so). Suppose your region contains <span class="math-container">$0$</span> (if not, just translate it so that it does). Then <span class="math-container">$f$</span> has a power series representation <span class="math-containe... |
1,649,320 | <p>I am having a hard time proving this simple and natural identity of sets. what I do is go round and round in circles:</p>
<p>$$A\cup( A\cap B) = (A\cup A) \cap (A\cup B)$$
$$= A \cap(A\cup B)$$</p>
<p>Now what? I apply the distributive property again and reach the first expression. How can I show this using set pr... | Brian M. Scott | 12,042 | <p>Assume that $A$ and $B$ are subsets of some universal set $X$. Then</p>
<p>$$\begin{align*}
A\cup(A\cap B)&=(\color{red}A\cap X)\cup(\color{red}A\cap B)\\
&=\color{red}A\cap(X\cup B)\\
&=A\cap X\\
&=A\;.
\end{align*}$$</p>
|
1,649,320 | <p>I am having a hard time proving this simple and natural identity of sets. what I do is go round and round in circles:</p>
<p>$$A\cup( A\cap B) = (A\cup A) \cap (A\cup B)$$
$$= A \cap(A\cup B)$$</p>
<p>Now what? I apply the distributive property again and reach the first expression. How can I show this using set pr... | user 1 | 133,030 | <blockquote>
<p><strong>Answer 1</strong>. Clearly $A \subseteq A\cup (A\cap B).$<br>
For the converse, Note that $\color{maroon}A \subseteq A$ and $\color{lime}{A\cap B} \subseteq A$. So $\color{maroon}A\cup \color{lime}{(A\cap B)} \subseteq A.$ </p>
<hr>
</blockquote>
<p><strong>Answer 2</strong>. If $x\in... |
873,439 | <p>This is a follow-up <a href="https://math.stackexchange.com/questions/872921/prove-that-if-the-square-of-a-number-m-is-a-multiple-of-3-then-the-number-m/872927">question</a>.</p>
<p>The problem is:</p>
<blockquote>
<p>Given two natural numbers, $m$ and $n$, and $n \vert m^2$.</p>
<p>Find necessary and suffi... | gammatester | 61,216 | <p>Your statement is <strong>wrong</strong>. For every $n$ set $m=n$ and you have $n|m^2$ and $n|m.\;$ But $n$ is not necessarily square-free! Ex. $4|16$ and $4|4$ but $4$ is not square-free.</p>
|
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Davislor | 422,808 | <p>To flesh out Mark Bennet’s answer with a little more detail, there are three questions here:</p>
<ol>
<li><p>If a group of six people all try something they’ll succeed at one time in five, how often will at least one of them succeed?</p></li>
<li><p>If a group of six people all try something they’ll succeed at one ... |
2,771,034 | <p>$\frac{a_n}{b_n} \rightarrow 1$ and $\sum_{n=1}^\infty b_n$ converges, can it be concluded that $\sum_{n=1}^\infty a_n$ converges?<br>
My attempt at an answer to this question: since $\sum_{n=1}^\infty b_n$ converges, $b_n \rightarrow 0$. Because of this, $a_n \rightarrow 0$ equally fast. However, I'm well aware tha... | G Tony Jacobs | 92,129 | <p>This is the limit comparison test. As long as $\sum b_n$ converges, the limit of $\frac{a_n}{b_n}$ being any real number is enough to guarantee that $\sum a_n$ converges.</p>
<p>Indeed, since $\sum b_n$ is convergent, then so is $\sum kb_n$ for any real $k$. Whatever limit you obtain for $\frac{a_n}{b_n}$, choose s... |
71,031 | <p>In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare conjecture.</p>
<p>In a series of papers, Akbulut and Gompf have shown most of these Cappell-Shaneson knots actually are kn... | Scott Carter | 36,108 | <p>There is a paper by Iain Aitcheson (possible mis-spelling of the last name) and Hyam Rubenstein published in a Contemporary Mathematics Series of the AMS (Conference Proceedings) that is the most explicit description of which I know. I wanted to to try and draw the corresponding knot diagrams or Yoshikawa diagrams a... |
95,819 | <p>I have a set of parametric equations in spherical coordinates that supposedly form circle trajectories. See below:</p>
<pre><code>r=C1
theta=C2*Sin[beta]*Sin[phi[t]]
phi=(C2*Sin[beta]*(Cos[theta[t]]/Sin[theta[t]])*Cos[phi[t]])+(C2*Cos[beta])
</code></pre>
<p>C1 and C2 are constants and beta is some angle, say 15... | gpap | 1,079 | <p>I didn't have time to look at this properly as your functions don't work straight out of the box because they require additional definitions. Also, I think there is a derivative missing in your original funciton definitions. </p>
<p>Anyway, I will use the parameters from Jack LaVigne's answer (but I have modified t... |
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