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 |
|---|---|---|---|---|
108,409 | <p>Given a commutative ring <span class="math-container">$A$</span> with unity, Grothendieck used universal polynomials to define a <em>special</em> <span class="math-container">$\lambda$</span>-ring structure on <span class="math-container">$\Lambda(A):=1+t\:A[[t]]$</span>. Suppose <span class="math-container">$A$</sp... | Maarten Bergvelt | 26,886 | <p>Hazewinkel in <a href="https://arxiv.org/abs/0804.3888" rel="nofollow noreferrer">Witt vectors. Part 1</a> warns about an error on page 15, second paragraph of this book. In fact he advises to "steer clear" of the book!</p>
|
1,176,615 | <p>I am invited to calculate the minimum of the following set:</p>
<p>$\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$.</p>
<p>Is there any idea?</p>
<p>(The question changed because there is no maximum for the set (as proved in the following answers) and I assume ... | Ryan Vitale | 95,806 | <p>Solve for one variable using the equation $(x+1)(y+1)=2$. Solving for $x$ we get $x=\frac{1-y}{1+y}$. Then rewrite the floor function as $\lfloor{xy+\frac{1}{xy}}\rfloor=\lfloor{\frac{y^2(1-y)^2+(1+y)^2}{y(1-y^2)}}\rfloor$, so there is no max, taking $y \rightarrow 1$</p>
|
1,146,759 | <p>A covering of a group $G$ a family $\{S_i\}_{i \in I}$ of subsets of $G$ such that $G = \displaystyle \bigcup _{i \in I} S_i$.</p>
<p>Why is true that: A group covered by finitely many cyclic subgroups is either cyclic or finite?</p>
<p>Remark:
Is true that by Baer (see D. Robinson, Finiteness Conditions and Gener... | Derek Holt | 2,820 | <p>Here are some hints. I denote a cyclic group of order $n$ by $C_n$.</p>
<p>Step 1. First show that $C_{\infty} \times C_n$ is not covered by finitely many cyclic groups for any $n$, including $n=\infty$.</p>
<p>Note that the condition of being covered by finitely many cyclic ubgroups is inherited by subgroups. By ... |
879,640 | <p>Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that</p>
<p>$A,B \text{ have the same inverse matrix} \iff A=B$?</p>
| Michael Albanese | 39,599 | <p>Note that $GL(n, \mathbb{F})$, the set of invertible $n\times n$ matrices over the field $\mathbb{F}$, is a group. In any group, inverses are unique, so if $a^{-1} = b^{-1}$, by taking inverses it follows that $a = b$. In particular, this applies to the group $GL(n, \mathbb{F})$.</p>
|
3,845,475 | <p>Here's what I'm tasked with showing:</p>
<p>Let <span class="math-container">$(a_n)$</span> be a convergent sequence with <span class="math-container">$a_n\rightarrow a$</span> as <span class="math-container">$n\rightarrow\infty$</span>. By the Algebraic Limit Theorem, we know that <span class="math-container">$(a_n... | fleablood | 280,126 | <blockquote>
<p>I don't see how this (sentence in italics) can be true. For example if we have S={1,2,3} the number of orderings that can be obtained are 3!=6. Following the solution's reasoning we could calculate the orderings for S by ordering the cards different form 3 and then inserting in into that ordering, that ... |
1,859,810 | <p>Consider the function
$$
f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .
$$
This is a bizarre and fascinating function. A few properties of this function that SEEM to be true:</p>
<p>1) $f(x)$ is $2\pi$-periodic and odd around $\pi$.</p>
<p>2) $\lim_{x\rightarrow \pi_-} f(x) = \ln 2$. (Can... | John Barber | 73,626 | <p>Although I've already selected one of the excellent answers above, I thought I'd post an answer that I figured out to one of the questions I posed in the original post, namely: "I would guess that it diverges as $x\rightarrow 0_+$. How does it diverge there?"</p>
<p>So, as noted above, the term $\mathrm{sign}(\sin(... |
535,757 | <p>The exercise is to give an example for two sets $M$ and $N$, and functions $f$ and $g$, for which $f \circ g = id_M$, but $g \circ f \ne id_N$.</p>
<p>My idea is a bit based on my computer programming background, where <code>(x/2)*2</code> is <code>0</code> for integers. Here it is:</p>
<p>$$M=N=\mathbb{N_0}.$$
$$... | Clive Newstead | 19,542 | <p>A simpler example might be $M=[0,\infty)$ and $N = \mathbb{R}$ with $f(x) = \sqrt{x}$ and $g(x)=x^2$. You can check the details.</p>
|
2,721,836 | <p>I recently found a different method to compute prime number in $\mathcal O(\log(\log n))$ complexity. At present, that logic working fine for $300$ digits prime number, which I found on websites.I need to validate whether that logic will be working fine for a higher number of digits. At present, I have computed a pr... | Klangen | 186,296 | <p><a href="http://www.ellipsa.eu/public/primo/primo.html" rel="nofollow noreferrer">Primo</a> does this. It only runs on Linux, if you don't have one at home you can just create a virtual machine and run it as guest within your current OS. There are various tests that Primo can perform, please read the documentation o... |
1,137,565 | <blockquote>
<p>Let <span class="math-container">$A\Delta C\subseteq A\Delta B$</span>. (<span class="math-container">$\Delta$</span> denotes symmetric difference.)</p>
<p>Prove <span class="math-container">$A\cap B \subseteq C$</span>.</p>
</blockquote>
<p>I am getting ready for a test and I could really use proof ver... | Lucian | 93,448 | <blockquote>
<p><em>Where is the $x=+1$ point ?</em></p>
</blockquote>
<p>At infinity. Just let $t\to\infty$, and evaluate the two limits.</p>
|
4,341,172 | <p><span class="math-container">$(x+1)^2y''+(x+1)y'+y=x^2+2\sin(\ln(x+1)), y(0)=\frac{1}{5},y'(0)=2$</span></p>
<p>My solution:</p>
<p><span class="math-container">$y''+\frac{y'}{(x+1)}+\frac{y}{(x+1)^2}=\frac{x^2+2\sin(\ln(x+1))}{(x+1)^2}$</span></p>
<p>First of all , I found the equation solution of <span class="math... | user577215664 | 475,762 | <p><span class="math-container">$$(x+1)^2y''+(x+1)y'+y=x^2+2\sin(\ln(x+1))$$</span> <span class="math-container">$$y(0)=\frac{1}{5},y'(0)=2$$</span>
Substitute <span class="math-container">$u=x+1$</span>:
<span class="math-container">$$u^2y''+uy'+y=(u-1)^2+2\sin(\ln(u))$$</span> <span class="math-container">$$y(1)=\fra... |
4,580,470 | <p>Suppose <span class="math-container">$X$</span> is a Geometric random variable (with parameter <span class="math-container">$p$</span> and range <span class="math-container">$\{k\geq 1\}$</span>).</p>
<p>Let <span class="math-container">$M$</span> be a positive integer.</p>
<p>Let <span class="math-container">$Z:=\m... | whoisit | 1,094,230 | <p>They are actually equal:
<span class="math-container">$\sum_{k=1}^\infty k\mathbb P[Z= k] = \sum_{k=1}^\infty \mathbb P[Z\geq k] $</span></p>
<p>Consider this sum:</p>
<p><span class="math-container">$
+ \; P(1) \\
+P(2) + P(2) \\
+P(3) + P(3) + P(3) \\
\vdots
$</span></p>
<p>The rowwise grouping of this sum is t... |
2,757,562 | <p>I'm work through some questions relating to connection coefficents. My question is more about the summation notation being used. Why is there no starting point (or end point) for the summation here? For the i, j, k, I take these to range from 1 to 2 for a 2 dimensional surface. Does this just imply that l does likew... | Lee Mosher | 26,501 | <p>This formula comes from differential geometry, it takes place in the Cartesian coordinate space $\mathbb{R}^n$ for some value of the dimension $n$. Once $n$ has been specified, the indices $i,j,k,l$, by convention, take values in the index set $\{1,...,n\}$. </p>
<p>In this formula, the values $i,j,k$ are fixed and... |
1,552,055 | <p>Why is it true that all irrational numbers are non-terminating/non-repeating decimals?</p>
<p>By definition, an irrational number is one that can't be expressed as a ratio of integers.</p>
| Hagen von Eitzen | 39,174 | <ul>
<li>If the decimal expansion of a number $x$ is terminating, with $n$ digits after the decimal point, say, then $10^nx$ is an integer $m$ (the decimal expansion is shifted by $n$ places to the left, hence has nothing after the decimal point) so that $x =\frac{m}{10^n}$ is a fraction of integers, aka. rational numb... |
813,716 | <p>I am supposed to calculate the following as simple as possible.</p>
<p>Calcute:
$$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$
Tip: $$ 101^8 = 10828567056280801$$</p>
<p>I have absolutely no idea how this tip is supposed to help me.<br>
Do I still have to calculate each potency?<br>
Can I somehow solv... | Michael Albanese | 39,599 | <p><strong>Hint:</strong> $$1 + x + x^2 + \dots + x^n = \frac{x^{n+1} - 1}{x - 1}.$$</p>
|
3,488,405 | <blockquote>
<p>Let <span class="math-container">$L(n)$</span> denote the number of positive divisors of a number <span class="math-container">$n$</span>. Prove that <span class="math-container">$\sum_{n=1}^N L(n)=\lfloor{\sqrt N}\rfloor\pmod 2$</span>.</p>
</blockquote>
<p>I wanted to prove that by induction.
For <... | Will Jagy | 10,400 | <p>Your <span class="math-container">$L(n)$</span> can be odd only when <span class="math-container">$n$</span> itself is a perfect square. So, mod 2, you are just counting the squares up to the upper bound, which you called <span class="math-container">$N$</span>.</p>
<p>First, <span class="math-container">$L(1) = 1.... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | Abhra Abir Kundu | 48,639 | <p>Suppose you have to select n balls from a collection of $R$ black balls and $M$ white balls.</p>
<p>Then we must select $k$ black balls and $n-k$ white balls in whatever way we do.(for $0\le k\le n$)</p>
<p>For a fixed $k\in N,0\le k\le n$ we can do this in $\binom{R}{k}\binom{M}{(n-k)}$ ways.</p>
<p>so to get th... |
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| Betty Mock | 89,003 | <p>Let $f(x) = x^2\sin(1/x)$ on $(0,1)$ and $f(0) = 0$. Then $f$ is differentiable throughout $[0,1]$. All derivatives satisfy the intermediate value property (Darboux's Theorem); but $f'(x)$ is discontinuous at $0$.</p>
|
3,785,982 | <p>Given the following ODE,</p>
<p><span class="math-container">$$\frac{{dy}}{{dx}}=\cos ({x})-\sin ({y})+{x}^{2}; \quad {y}\left({x}_{0}=-1\right)=y_0=3$$</span></p>
<p>I have to use the Taylor Series Method to compute the value of <span class="math-container">$y(x)$</span> at <span class="math-container">$x=-0.8$</sp... | PrincessEev | 597,568 | <p>Note that, if <span class="math-container">$$x_n = \left( \frac{n+2}{n} \right)^{n^2} = \frac{(n+2)^{n^2}}{n^{n^2}}$$</span></p>
<p>then</p>
<p><span class="math-container">$$\frac{x_{n+1}}{x_n} = \underbrace{\frac{(n+3)^{(n+1)^2}}{(n+1)^{(n+1)^2}}}_{\displaystyle x_{n+1}} \cdot \underbrace{\frac{n^{n^2}}{(n+2)^{n^2... |
1,262,305 | <p>$$f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$$
I need to find the $x$ and $y$ intercepts, and the inflection points of the function $f(x)$ (with both $x$ and $y$ coordinates). I need to find it through the calculator and explain my answer. How do I find the antiderivative?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint.</strong> You may observe that, by partial fraction decomposition, you have
$$
\frac{t^3-2t^2-4}{t^2+1}=t-2-\frac{t+2}{t^2+1}
$$ giving, for the antiderivative that vanishes at $x=0$:
$$
\begin{align}
f(x)=\int_0^x\frac{t^3-2t^2-4}{t^2+1}dt&=\int_0^x\left(t-2-\frac{t+2}{t^2+1}\right)dt\\\\
&=\in... |
279,043 | <p>I would like to plot a complex graph of the Riemann zeta function on the Argand diagram <span class="math-container">$ς(s)$</span>, where <span class="math-container">$s = \frac{1}{2} + i t $</span>, and the value of <span class="math-container">$t$</span> is varied to get a graph in the polar form.</p>
<p>Can anyon... | Nasser | 70 | <p><a href="https://i.stack.imgur.com/Pogqu.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pogqu.gif" alt="enter image description here" /></a></p>
<pre><code>ClearAll["Global`*"]
s[t_] := 1/2 + I*t
Manipulate[
tick;
Module[{x, y},
x = Re[Zeta[s[t]]]; y = Im[Zeta[s[t]]];
AppendTo[coo... |
84,183 | <p>I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, <a href="http://arxiv.org/abs/gr-qc/9706069" rel="nofollow">http://arxiv.org/abs/gr-qc/9706069</a>). So one identifies all the rays $\mathbb{C} \cdot \phi$ with the vector $\phi$ itself.</p>
... | Alain Valette | 14,497 | <p>Look at section 5 of this paper by Helmick and Helminck:
<a href="http://eprints.eemcs.utwente.nl/3487/01/1667.pdf" rel="nofollow">http://eprints.eemcs.utwente.nl/3487/01/1667.pdf</a></p>
|
84,183 | <p>I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, <a href="http://arxiv.org/abs/gr-qc/9706069" rel="nofollow">http://arxiv.org/abs/gr-qc/9706069</a>). So one identifies all the rays $\mathbb{C} \cdot \phi$ with the vector $\phi$ itself.</p>
... | Richard Montgomery | 2,906 | <p>Feynman, in his lecture notes, argues convincingly that you will understand the guts of Quantum Mechanics [QM] as best you can by looking at the two-slit experiment whose Hilbert space is two-dimensional. Or you could look at the Stern-Gerlach experiment, where it is three-dimensional. I recommend you stick to the... |
4,315,449 | <p>Is it correct to say that <span class="math-container">$\mathbb{R}^n \subset \mathbb{R}^n$</span> ?</p>
<p><strong>EDIT</strong>: The context of may question is that I am having a function that is defined as <span class="math-container">$f \colon D \subset \mathbb{R}^n \to \mathbb{R}^n $</span> and I am wondering if... | Masacroso | 173,262 | <p>The most used convention is that the symbol <span class="math-container">$\subset $</span> applies to any subset of a set, so <span class="math-container">$A\subset A$</span> is correct for every set <span class="math-container">$A$</span>.</p>
|
1,600,063 | <p>I am trying to prove the following statement:</p>
<p>Given any two real numbers $x,y$ with $x<y$, there exists a rational number $q$ that satisfies $x<q<y$.</p>
<p>I got stuck at one point of the proof, so this is what I thought of:</p>
<p>I want to find a rational number $q$, which can be expressed as $... | Dave Neary | 122,612 | <p>Once you notice that <span class="math-container">$\frac{1}{y-x} < n$</span> for some positive integer <span class="math-container">$n$</span> you are home and dry. Inverting this inequality, you get <span class="math-container">$y-x>\frac{1}{n}$</span>, so you can find an integer <span class="math-container">... |
858,576 | <p>Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two.</p>
<p>I can do this problem when I am working in only two subspaces of $V$ but I don't know how to do it with three. </p>
<p>What I tried is:
If one of the subspaces contains the other two, Then their union... | Gina | 102,040 | <p>The statement is false. Consider the following counterexample:</p>
<p>Consider the vector space $V=(\mathbb{Z}/2\mathbb{Z})^{2}$ where $F=\mathbb{Z}/2\mathbb{Z}$. Let $V_{1}$ be spanned by
$(1,0)$. Let $V_{2}$ be spanned by $(0,1)$. Let $V_{3}$ be spanned by $(1,1)$. Then we have $V=V_{1}\cup V_{2}\cup V_{3}$, but... |
2,611,676 | <p>Or consider the general problem-
Find the value of n for which x^n is just greater than x!</p>
<p>I dont know even if it is possible to find the solution or not...</p>
| Salech Alhasov | 25,654 | <pre><code>import math
n=1
while 100**n < math.factorial(100):
n+=1
print(n)
</code></pre>
<p>For completeness, this python code prints the number $79$.</p>
|
446,499 | <p>I have just learned the definition of connectedness and wikipedia gives an example of a disconnected set: <span class="math-container">$(0,1)\cup \left\{ 3 \right\}$</span> (<a href="https://en.wikipedia.org/wiki/Connected_space#Examples" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Connected_space#Exampl... | palio | 10,463 | <p>A disconnected space $X$ is a space written as a disjoint union of open or closed sets, that is $X=Y\sqcup Z$ where $Y$ and $Z$ are both open or both closed in $X$ relatively to the topology you put on $X$. For example the disjoint union $\{0\}\sqcup (0,1]$ gives the interval $[0,1]$ which is connected, this is beca... |
2,694,740 | <p>$$\frac{2.10^{-7} - 0,4.10^{-6}}{10^{-8}} = ? $$</p>
<p>These questions are making me confused because we're dealing with the terms like $10^x$. What are your professional tips? </p>
<p><strong>My attempt:</strong></p>
<p>$$\frac{2.10^{-7} - 4.10^{-7}}{10^{-8}} \tag{1} $$
$$\frac{ -8.10^{-7}}{10^{-8}} \tag{2} $$<... | Mohammad Riazi-Kermani | 514,496 | <p>$$\frac{2.10^{-7} - 0,4.10^{-6}}{10^{-8}} = 10^8 (2.10^{-7} - 0,4.10^{-6})=20-40=-20$$</p>
|
1,656,145 | <p>Let the real function of two real variables$$u(x,y) =
\begin{cases}
x, & \quad \text{if } |y|>|x| \\
-x, & \quad \text{if } otherwise
\\ \end{cases} $$</p>
<p>Is there a sequence $\{(x_n,y_n)\}_{n \geq 0}$ which converge to $(0,0)$ such that $\lim_{n \to \infty} u(x_n,y_n) \not= u(0,0)$?</p... | Eric Wofsey | 86,856 | <p>If $\{(x_n,y_n)\}$ converges to $(0,0)$, then in particular $\{x_n\}$ converges to $0$. Thus for any $\epsilon>0$, there is an $N$ such that $|x_n|<\epsilon$ for all $n>N$. But $|u(x,y)|=|x|$ for all $(x,y)$, so this means $|u(x_n,y_n)|<\epsilon$ for all $n>N$. Thus $\{u(x_n,y_n)\}$ converges to $0... |
2,012,532 | <p>The following is all confirmed to be true:</p>
<p>Matrix A =
$
\begin{bmatrix}
0 & 1 & -2 \\
-1 & 2 & -1 \\
2 & -4 & 3 \\
1 & -3 & 2 \\
\end{bmatrix}
$</p>
<p>U =
$
\begin{bmatrix}
-1 & 2 & -1 \\
0 &am... | dantopa | 206,581 | <h2>Problem</h2>
<p>$$
\mathbf{A} =
\left[
\begin{array}{rrr}
0 & 1 & -2 \\
-1 & 2 & -1 \\
2 & -4 & 3 \\
1 & -3 & 2 \\
\end{array}
\right]
$$</p>
<h2>Associated Permutation Matrix</h2>
<p>Don't start with a $0$ pivot element. Move the first row down. The permutation matrix inte... |
3,995,728 | <p>For the integral
<span class="math-container">$$
\int_1^2 \frac {3^x + 2}{3^{2x} + 3^x} dx
$$</span>
choose the interval of its result: <span class="math-container">$(-\infty, 0], (0, \frac 12], (\frac 12, 1], (1, 3], (3, \infty)$</span>. According to the author of this task you do not have to compute the integral i... | Steven Alexis Gregory | 75,410 | <p><a href="https://i.stack.imgur.com/IfSqx.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IfSqx.jpg" alt="enter image description here" /></a></p>
<p>Is this what they were asking for?</p>
<p><span class="math-container">$\alpha = m \angle BAC = m \operatorname{arc} DE$</span></p>
<p><span class="m... |
3,512,521 | <p>Let <span class="math-container">$x_1, x_2,\dots, x_n$</span> be positive real numbers. Let <span class="math-container">$A$</span> be the <span class="math-container">$n\times n$</span> matrix whose <span class="math-container">$i,j^\text{th}$</span> entry is <span class="math-container">$$a_{ij}=\frac{1}{x_i+x_j}.... | user1551 | 1,551 | <p>There is also a proof that is similar in spirit to the hint you've mentioned. See exercise 1.6.3 (pp.24-25) of Bhatia, <em>Positive Definite Matrices</em>. The idea is that, instead of writing the Cauchy matrix as an integral of Gramians, we write it as an infinite sum of Gramians. More specifically, let <span class... |
3,512,521 | <p>Let <span class="math-container">$x_1, x_2,\dots, x_n$</span> be positive real numbers. Let <span class="math-container">$A$</span> be the <span class="math-container">$n\times n$</span> matrix whose <span class="math-container">$i,j^\text{th}$</span> entry is <span class="math-container">$$a_{ij}=\frac{1}{x_i+x_j}.... | Rodrigo de Azevedo | 339,790 | <p>Complementing A.Γ.'s answer, and rephrasing a bit:</p>
<blockquote>
<p>Given <span class="math-container">$a_1, a_2, \dots, a_n > 0$</span>, we build the <span class="math-container">$n \times n$</span> symmetric Cauchy matrix <span class="math-container">$\rm C$</span> whose entries are <span class="math-contain... |
3,151,452 | <p>The context of the question is that a bakery bakes cakes and the mass of cake is demoted by <span class="math-container">$X$</span> such that <span class="math-container">$X \sim N(300, 40^2)$</span>. A sample of 12 cakes is taken and the mean of the sample is 292g. The question wants me to find the <span class="mat... | Minus One-Twelfth | 643,882 | <p><span class="math-container">$\newcommand{\P}{\mathbb{P}}$</span>It appears that you have <span class="math-container">$X\sim N(\mu, 40^2)$</span> (known variance), and your null hypothesis is that <span class="math-container">$\mu = 300$</span>, with alternative hypothesis <span class="math-container">$\mu\ne 300$<... |
3,748,739 | <p>Let <span class="math-container">$X\perp Y$</span> with <span class="math-container">$X,Y\sim N(0,1)$</span>. Let <span class="math-container">$U=\frac{(X+Y)}{\sqrt{2}}$</span> and <span class="math-container">$V=\frac{(X-Y)}{\sqrt{2}}$</span>.</p>
<ol>
<li>Find the law of <span class="math-container">$(U,V)$</span>... | Ethan Bolker | 72,858 | <p>To rule out (D) start with a nonnegative continuous function that's not differentiable - perhaps <span class="math-container">$g(x) = |x|$</span>. Then construct <span class="math-container">$f$</span> by integrating twice, so that <span class="math-container">$g$</span> is its second derivative. Then the integral o... |
3,748,739 | <p>Let <span class="math-container">$X\perp Y$</span> with <span class="math-container">$X,Y\sim N(0,1)$</span>. Let <span class="math-container">$U=\frac{(X+Y)}{\sqrt{2}}$</span> and <span class="math-container">$V=\frac{(X-Y)}{\sqrt{2}}$</span>.</p>
<ol>
<li>Find the law of <span class="math-container">$(U,V)$</span>... | Robert Israel | 8,508 | <p>Hint for (C): if <span class="math-container">$f'' \ge 0$</span> and <span class="math-container">$f'(x) > 0$</span>, then <span class="math-container">$f'(t) \ge f'(x)$</span> for all <span class="math-container">$t \ge x$</span>, so <span class="math-container">$f(t) \ge f(x) + (t-x) f'(x)$</span> for <span cla... |
1,222,909 | <p>I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$
I was wondering if there was an easy way as it is hard to get values from this.</p>
<p>Edit: The equation ends up simplifying ... | Poppy | 41,695 | <p>The curve is $\alpha(\theta)=(2sin{\theta}\cos{\theta},2\sin{\theta}\sin{\theta})$. The velocity vector is $\alpha'(\theta)=(2\cos{2\theta},2\sin{2\theta})$. Now we have to find all points so that this vector is proportional to $(1,1)$.</p>
<p>$\theta=\dfrac{\pi}{8}$ or $\theta=\dfrac{5\pi}{8}$</p>
|
272,057 | <p>Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$. </p>
<p>My question is the following: Let $X$ be a random variable defined on some probability space (rich enough) with l... | Benoît Kloeckner | 4,961 | <p>Yes, you can even ensure $\mathbb{E}(|X-Y|)=d$ and all you need for $G$ is that it has an atomless law, $\rho$ say.</p>
<p>Take the disintegration $(\lambda_x)$ of the optimal coupling $\Pi$ with respect to $\mu$ (note your formula has a $dx$ where one should read a $dy$). What you want is that $f(x,\cdot)$ sends t... |
4,363,409 | <blockquote>
<p>Define <span class="math-container">$X_0=\alpha\in(0,1)$</span> the initial capital and <span class="math-container">$X_n$</span> as the remaining capital after each game.
A player bets <span class="math-container">$1-X_n$</span> if <span class="math-container">$X_n>1/2$</span> and <span class="math-... | David Quinn | 187,299 | <p>Imagine that the surface of the water is part of a large flat mirror that extends through the earth and up to the horizon.</p>
<p>For each object you want to reflect, such as the large tree in the foreground, decide how high the base of it is above the level of the water, and imagine where the level of the mirror wo... |
4,363,409 | <blockquote>
<p>Define <span class="math-container">$X_0=\alpha\in(0,1)$</span> the initial capital and <span class="math-container">$X_n$</span> as the remaining capital after each game.
A player bets <span class="math-container">$1-X_n$</span> if <span class="math-container">$X_n>1/2$</span> and <span class="math-... | Michael Thwaites | 627,654 | <p>All previous comments taken into account, I should add that reflections of objects often appear taller than the actual image of the object seen directly. When I worked north of Lake Merritt in Oakland, the reflections of the buildings across the lake were noticeably taller than the buildings. I concluded this is do ... |
126,549 | <p>For a quadratic form $q(\mathbf{v})$, when you change the basis do you <em>always</em> change the quadratic form? Can you have the same quadratic form with respect to different basis? Or is the quadratic form unique to the basis. </p>
<p>Also, if you're given a quadratic form say $q(\mathbf{v}) = 3x^2 + y^2 - 2z^2 ... | Sunni | 10,800 | <p>The basis for your quadratic form is the natural basis, i.e., $(1, 0,\cdots, 0), \cdots, (0, \cdots, 0, 1)$. The matrix of quadratic form depends on the basis. Canonical form of a symmetric matrix is a diagonal matrix. You may compare these two forms...</p>
|
126,549 | <p>For a quadratic form $q(\mathbf{v})$, when you change the basis do you <em>always</em> change the quadratic form? Can you have the same quadratic form with respect to different basis? Or is the quadratic form unique to the basis. </p>
<p>Also, if you're given a quadratic form say $q(\mathbf{v}) = 3x^2 + y^2 - 2z^2 ... | Sonu Lamba | 573,334 | <p>This <a href="https://drive.google.com/file/d/1cNxG-Mf6Uq4W48Wi186p9e6mOJ7fvwKt/view?usp=drivesdk" rel="nofollow noreferrer">link</a> may help you. See <strong><em>proposition 2.3</em></strong>. I think that implies that matrix of quad form changes on changing basis. That is there are many other basis too (other tha... |
2,208,755 | <p>I got stuck on this question: find all solutions $x$ for $a\in R$:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{(a^2-a+1)^3}{a^2(a-1)^2}$$</p>
<p>I see that if we simplify we get:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{[(x-{\frac 12})^2+{\frac 34}]^3}{[(x-{\frac 12})^2-{\frac 14}]^2}$$</p>
<p>From the ex... | Jean Marie | 305,862 | <p>Let <span class="math-container">$$f(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}\tag{1}$$</span></p>
<p>(representative curve in Fig. 1).</p>
<p>Your question can be reformulated in the following way : </p>
<p><span class="math-container">$$\text{for a given} \ a, \ \ \text{find all} \ x \ \text{such that} \ \ f(x)=f(a) \... |
2,947,953 | <p>Given the two functions <span class="math-container">$$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$</span> and <span class="math-container">$$g(x) = \ln(x+1)-\ln(x-1)$$</span>
I can justify independently why <span class="math-container">$\text{dom}(f) = (-\infty, -1) \cup (1,\infty)$</span>, and <span class="math-contai... | Anik Bhowmick | 354,636 | <p>For <span class="math-container">$f(x)$</span>, you are considering about the domain of <span class="math-container">$ln$</span> and also of <span class="math-container">$\frac{x+1}{x-1}$</span>. But for <span class="math-container">$g(x)$</span>, you are considering about the domain of <span class="math-container">... |
2,947,953 | <p>Given the two functions <span class="math-container">$$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$</span> and <span class="math-container">$$g(x) = \ln(x+1)-\ln(x-1)$$</span>
I can justify independently why <span class="math-container">$\text{dom}(f) = (-\infty, -1) \cup (1,\infty)$</span>, and <span class="math-contai... | Jean-Luc Bouchot | 24,153 | <p>Going back to the definition, a mapping <span class="math-container">$f: \begin{array}{ccc} D & \to & E \\ x &\mapsto &f(x)\end{array}$</span>is characterized by three elements: </p>
<ol>
<li>A domain <span class="math-container">$D$</span> where the mapping is defined (i.e. an <em>input</em> space)... |
99,799 | <p>I have a <code>Solve</code> similar to the following:</p>
<pre><code>Solve[e^2 - c^2 == -15, {e, c}, Integers]
(* {{e -> -7, c -> -8}, {e -> -7, c -> 8}, {e -> -1, c -> -4},
{e -> -1, c -> 4}, {e -> 1, c -> -4}, {e -> 1, c -> 4},
{e -> 7, c -> -8}, {e -> 7, c -... | Adam Strzebonski | 6,258 | <p>Interval is a 1D region, so <code>Element[e, Interval[...]]</code> makes e a 1D vector
not a scalar. If you want <code>e</code> to be a scalar use <code>Element[{e}, Interval[...]]</code>.</p>
<pre><code>In[1]:= Solve[e^2 - c^2 == -15 && Element[{e}|{c}, Interval[{0, 4}]], {e, c}, Integers]
Out[1]= {{e -&g... |
78,641 | <p>I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.</p>
| Stefan Geschke | 7,743 | <p>Nathan's answer seems to indicate that there are some strange spaces that are ccc but not separable. But in fact, such spaces are rather common:</p>
<p>All products of separable Hausdorff spaces are ccc, but if the spaces have at least two different points, then products with more than $2^{\aleph_0}$ factors are n... |
834,228 | <p>$$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?</p>
<p>In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.</p>
| AnnieOK | 130,682 | <p>Assuming the claim is true for $n$: $u_n < u_{n-1}$ </p>
<p>We'll show the claim is also right for $n+1$. Indeed:<br>
$$u_{n+1} = \frac{1}{3-u_n} < \frac{1}{3-u_{n-1}} = u_n$$</p>
<p>The inequality is of course based on the assumption.
For a full proof you should add the base case as well.</p>
|
781,776 | <blockquote>
<p>A red die, a blue die, and a yellow die (all six sided) are rolled. Given that no two of the dice land on the same number, what is the conditional probability that blue is less than yellow which is less than red?</p>
</blockquote>
<p>The Answer is a sixth. I have absolutely no idea how to do this tho... | Caleb Stanford | 68,107 | <p>Your answer seems to be mixing computations where the computers are distinguishable, and computations where the computers are indistinguishable. In fact the problem specifies every computer is the same, so refer to David's answer for the correct solution.</p>
<p>If the computers <em>were</em> distinguishable, the ... |
448 | <p>Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?</p>
| mau | 89 | <p>The case of two colors is simple: if you have m yellow balls and n blue ones you only need to choose m positions among (m+n) possibilities, that is (m+n)!/(m!·n!). The other balls' positions are automatically set up.</p>
|
3,419,276 | <p>I'm reading about the directional derivative:</p>
<blockquote>
<p>Let <span class="math-container">$(E,\|\cdot\|)$</span> and <span class="math-container">$(F,\|\cdot\|)$</span> be Banach spaces over the field <span class="math-container">$\mathbb{K}$</span>, and <span class="math-container">$X$</span> an open su... | fleablood | 280,126 | <p>Assuming <span class="math-container">$b >0; b\ne 1$</span> then <span class="math-container">$\log_b u$</span> is, by definition, <span class="math-container">$\text{"whatever power we must raise b to in order to get u"}$</span>.</p>
<p>So <span class="math-container">$b^{\log_b u} = b^{\text{"whatever power we... |
2,990,580 | <p>I am doing my maths A-level*. Often when I am at home I get questions about why we solve certain problem types in a certain way. One example is "why does completing the square work?"</p>
<p>Is there a website which collects explanations like these together for me to read? <strong><em>Preferably one that is aimed at... | M. Damon | 573,313 | <p>Would this work instead, </p>
<p><span class="math-container">$$x \in A \rightarrow x \in A\cup C \rightarrow x\in B\cup C\\\text{so we either have that:}\\(1)\;x\in B , x\notin C\\(2)\;x\in C, x\notin B \\(3)\;x\in B, x\in C\\ \text{For (1) we have that $x \in B$. For (3) we have that $x \in B$. So we would just n... |
2,007,173 | <p>I've managed to severely confuse myself in my attempts to simplify this seemingly straightforward expression:
$$
\arctan(\cot(\alpha)),\quad\text{with $0<\alpha\leq\pi$.}
$$
It seems like maybe there are some issues with domain, as $\cot$ and $\tan$ are defined on $(0,\pi)$ and $(-\frac{\pi}{2},\frac{\pi}{2})$, r... | Stefano | 387,021 | <p>One way to see this is to calculate the derivative of $\arctan(\cot(\alpha))$, which is identically equal to $-1$. Therefore, since you are restricting yourself to the interval $(0, \pi)$, you can conclude that your expression is equal, on that interval, to $-\alpha + c$ for some $c \in \mathbb R$. Finally, setting ... |
1,200,919 | <p>Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?</p>
| Bart Michels | 43,288 | <p>Suppose $x$ is rational. Clearly $x<2$ and $x>1$, so $x$ is not an integer. Now from this question we obtain that $x^x=2$ is irrational, a contradiction: <a href="https://math.stackexchange.com/questions/978174">If $x$ is a positive rational but not an integer, is $x^x$ irrational?</a></p>
|
85,309 | <p>Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. </p>
<p>My question is: Is there a constant c (independent of A and B and the dimension) such that</p>
<p>$$(A-B)^2 \leq c (A+B)^2?$$</p>
<p>Th... | Noam D. Elkies | 14,830 | <p>There is no such $c$ even if we use only $2 \times 2$ matrices.
For any $c \geq 1$ let $A,B$ be the positive-semidefinite matrices
$$
A = \left( \begin{array}{lc} c^2 & c \cr c & 1 \end{array} \right),
\phantom\infty
B = \left( \begin{array}{cc} 1 & 0 \cr 0 & 0 \end{array} \right).
$$
of rank $1$. T... |
356,306 | <p>If $f:X_1 \rightarrow X_2$ and $g:X_2 \rightarrow X_3$ are homomorphisms.
If $g \circ f =0$ does it imply that $Im f \subseteq ker g$? and how to show that? do you have an example?
thanks :)</p>
| Federica Maggioni | 49,358 | <p>Take an element $y\in Im f$. Then $y$ is of the form $y=f(x)$, for some $x\in X_1$. Now apply $g$ to $y$. You get $g(y)=g(f(x))=(g\circ f)(x)$, which is zero by assumption. As an example, take $f$ arbitrary and $g(y)=0$ for every $y\in X_2$.</p>
|
1,234,471 | <p>Given two sequences $(a_k),(b_k)$ with $a_k\geq0,b_k>0$ such that the power series $\sum_{k=0}^\infty a_k b_kr^{k}$ and $\sum_{k=0}^\infty a_kr^k$ converge for each $r>0$. My question now is: Does there exist a constant $c$ (depending only on $(b_k))$ such that
\begin{align*}
\sum_{k=0}^\infty a_kb_kr^{k}\geq ... | String | 94,971 | <p>Let $1_n$ be the set of such strings of length $n$ ending in $1$. Similarly let $0_n$ be the set of such strings ending in $0$. Then we have
$$
\begin{align}
1_{n+1}&=\{x\parallel 1\mid x\in 1_n\cup 0_n\}\\
0_{n+1}&=\{x\parallel 0\mid x\in 1_n\}
\end{align}
$$</p>
<hr>
<p>Define $A_n=|1_n|$ and $B_n=|0_n|$... |
1,384,053 | <p>Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$?</p>
<p>Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those kind of problem</p>
| Ben Grossmann | 81,360 | <p><strong>Hint for the first:</strong> With $x = 0.01$ and $n = 101$, note that
$$
(1 + x)^n = 1 + n\,x + \binom n2 x^2 + \cdots \geq 1 + n\,x
$$</p>
<p><strong>Hint for the second:</strong> It is equivalent to show that $e^{1/e} > \pi^{1/\pi}$. In order to do so, consider the function $f(x) = x^{1/x}$ and use ... |
1,384,053 | <p>Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$?</p>
<p>Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those kind of problem</p>
| Claude Leibovici | 82,404 | <p>Consider $$A=1.01^{101}$$ So $$\log(A)=101\log(1.01)$$ Now use the very fast convergent expansion $$\log\Big(\frac{1+x}{1-x}\Big)=2\,\Big(\frac {x}{1}+\frac {x^3}{3}+\frac {x^5}{5}+\cdots\Big)$$ and make $\frac{1+x}{1-x}=\frac{101}{100}$ that is to say $x=\frac{1}{201}$. The first term (to which only positive terms ... |
3,450,581 | <p>I know convergence-preserving functions have been discussed a fair amount in the past; however, I was a looking at <a href="https://math.stackexchange.com/questions/1337042/sum-a-n-converges-iff-sum-fa-n-converges/1337057#1337057">another post</a>, and I saw the following result: if <span class="math-container">$f$<... | Cye Waldman | 424,641 | <p>I think I understand your dilemma now. Say that <span class="math-container">$m=a+b$</span> is fixed and we let <span class="math-container">$b=1/n$</span>, where <span class="math-container">$n$</span> is an integer, so that we satisfy the condition that <span class="math-container">$m/b$</span> is indeed an intege... |
632,891 | <p>I'm trying to solve this limit, for which I already know the solution thanks to Wolfram|Alpha to be $\sqrt[3]{abc}$:</p>
<p>$$\lim_{n\rightarrow\infty}\left(\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}\right)^n:\forall a,b,c\in\mathbb{R}^+$$</p>
<p>As this limit is an indeterminate form of the type $1^\inft... | Yiorgos S. Smyrlis | 57,021 | <p>Fact I.
$$
\lim_{n\to\infty}\left(1+\frac{a}{n}+\frac{b}{n^2}\right)^{\!n}=\mathrm{e}^a.
$$</p>
<p>Fact II. For every $a>0$, there exists a $b>0$, such that
$$
1+\frac{\ln a}{n}\le a^{1/n} \le 1+\frac{\ln a}{n}+\frac{b}{n^2}.
$$</p>
<p>Using the two facts:
$$
1+\frac{\ln a+\ln b+\ln c}{3n}\le\frac{a^{1/n}+b... |
51,390 | <p>Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(\log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and $h_d$ is another matrix. </p>
<ul>
<li>For such a set of arbitrary matrices, how can one power-series expand $\sqrt... | Carl Woll | 45,431 | <p>You can use my function <a href="https://mathematica.stackexchange.com/a/162063/45431">MatrixD</a> from the question <a href="https://mathematica.stackexchange.com/q/161918/45431">det simplification</a>:</p>
<pre><code>MatrixD[expr_, x__] := With[
{old = OptionValue[SystemOptions[], "DifferentiationOptions"->... |
3,082,080 | <p><a href="https://i.stack.imgur.com/Dcf40.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Dcf40.png" alt="enter image description here"></a></p>
<blockquote>
<p>The sides <span class="math-container">$AB, BC, CD$</span> of trapezoid <span class="math-container">$ABCD$</span> touches the circle w... | dfnu | 480,425 | <p><strong>Hint</strong>. </p>
<p>Once you showed that <span class="math-container">$ABO$</span>, <span class="math-container">$OBC$</span> and <span class="math-container">$COD$</span> are equilater triangles, note that the altitude of such triangles is equal to <span class="math-container">$r=1$</span>. By Pythagore... |
4,347,308 | <p><em><strong>Definition:</strong></em></p>
<p>Let <span class="math-container">$(X,\mathscr{A},\mu)$</span> be a measurable space, an atom of the measure <span class="math-container">$\mu$</span> is a set <span class="math-container">$A \in\mathscr{A}$</span> with the property that
<span class="math-container">$\mu(A... | Paul | 202,111 | <p><span class="math-container">$$\frac{1+x^3}{1+x^2} =x+\frac{1-x}{1+x^2}=x+\frac{1}{1+x^2}-\frac{x}{1+x^2}$$</span>
So you can use the Maclaurin series for <span class="math-container">$\frac{1}{1+x^2}$</span> to get the answer.</p>
|
3,009,477 | <p>I want to determine <span class="math-container">$f(x) = x+\sin x$</span> is homeomorphic or not on <span class="math-container">$\mathbb{R}$</span>?</p>
<p>A bijective continuous function is homeomorphic if its inverse is also continuous.
I know that <span class="math-container">$f$</span> is bijective. Also <sp... | egreg | 62,967 | <p>Since <span class="math-container">$H$</span> is commutative, <span class="math-container">$x\circ y=xy+yx=2xy$</span>. This new multiplication is obviously commutative, but need not have a neutral element. For instance, if <span class="math-container">$H$</span> has characteristic <span class="math-container">$2$</... |
550,188 | <p>Okay so I have an equation in my book which is as follows..
$$
\frac {a}{s(s+a)}
$$
it says "using partial fractions this can be expanded to
$$
\frac {1}{s} + \frac {-1}{s+a}
$$</p>
<p>My usual method would be to cross multiply and do something like this
$$
\frac {a}{s(s+a)} = \frac {A(s+a)}{s(s+a)} + \frac {B(s)}... | Martin Argerami | 22,857 | <p>The equality you have is
$$
a=(A-B)s+aA.
$$
This suggests taking $A=B=1$. </p>
|
3,024,169 | <p>First, my apologies. This question may have been asked many times before but I do not know the correct terms to search on..... and my school trigonometry is many years ago. Pointing me to an appropriate already-answered question would be an ideal solution for me.</p>
<p>I am writing a program to do 3D view from a... | Andrei | 331,661 | <p>I think you don't define your axes correctly. You have rotation around <span class="math-container">$z$</span>, and in the next sentence you say it never happens. </p>
<p>I think you should use polar coordinates. If you call <span class="math-container">$\theta$</span> the angle from the vertical direction <span cl... |
3,845,570 | <p>Premises: <span class="math-container">$\neg(A \to B)\ ,\ \neg B \to C$</span> .</p>
<p>Conclusion: <span class="math-container">$C$</span></p>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could... | DanielV | 97,045 | <p>Start by assuming that B is true.</p>
<blockquote class="spoiler">
<p> You can prove a contradiction from that, to establish <span class="math-container">$\lnot B$</span>.</p>
</blockquote>
<blockquote class="spoiler">
<p> Within the assumption you can prove <span class="math-container">$A \to B$</span></p>
</blockq... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | Toby Mak | 285,313 | <p>This is a method similar to @GTonyJacobs and @adhg's answers, but in this answer I use a slightly different way of thinking: </p>
<p>The maximum number possible is a set of six $4$'s $(6, 6, 6, 6)$, which totals $6 * 4 = 24$. To reach $22$, we need to take off $2$ in total from the set. </p>
<p>There are $2$ ways ... |
271,824 | <p>I have a list= {4, 8, 10, 11, 12, 14, 16, 7, 9}</p>
<p>How can i partition the list by group of Arithmetic Progression with common difference 1 :</p>
<p>{{4}, {8}, { 10, 11, 12}, {14}, {16}, {7}, {9}}</p>
| Lukas Lang | 36,508 | <p>The reason <code>Except</code> is not working for you is because the entire list <code>li1</code> also matches that pattern, so everything is simply replaced by <code>x</code>. To fix it, you need to make sure <code>Except</code> matches only what you want. The easiest is to simply use <a href="https://reference.wol... |
4,041,842 | <p>I want to solve for <span class="math-container">$t \in \mathbb{R}, u'(t)=-u(t)\ln \lvert u(t) \rvert$</span>.</p>
<p>I defined two cases: <span class="math-container">$\mathbb{R^*_+}$</span> and <span class="math-container">$\mathbb{R^*_-}$</span>.</p>
<p>For <span class="math-container">$\mathbb{R^*_+}$</span>:</p... | JJacquelin | 108,514 | <p><span class="math-container">$$\frac{du}{u}=-\ln(u(t))dt$$</span>
You cannot integrate <span class="math-container">$\int-\ln(u(t))dt$</span> directly.</p>
<p>The mistake is : You integrate <span class="math-container">$\quad \int-\ln(u(t))du= -u\ln(u)+u\quad$</span> but this is not <span class="math-container">$\qu... |
578,487 | <p>Maybe this is a well know result, however, I could not find it. Before stating it, let me write here a well know result (at least for me)</p>
<blockquote>
<p>Assume that $\Omega\subset\mathbb{R}^N$ is a open domain and $f:\Omega\to\mathbb{R}$. If there is constants $L>0$ and $\alpha>1$ such that $$|f(x)-f(y... | Disintegrating By Parts | 112,478 | <p>Claim: For $f$ as stated, and $p > 1$, assume that the region $\Omega$ is path connected by continuous paths of finite total variation. Then $f$ is constant on $\Omega$.</p>
<p>To see this, let $x$, $y$ be given, and choose a path $v(t) : [0,1]\rightarrow \Omega$ of finite arc length $l(v)$ that connects $x$ to ... |
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | abel | 9,252 | <p>the convergence of $\int_0^\infty \sin(x^{2/3}$ at the lower limit $x = 0$ is not a problem. the trouble is at the upper limit $x = \infty$</p>
<p>to handle the upper limit, i will make a change of variable $x = t^{3/2}, dx = 3/2 t^{1/2} dt.$
then
$\int_0^\infty \sin x^{2/3} dx = \frac{3}{2} \int_0^\infty t^{1/2} ... |
4,547,918 | <p>Given the torus and given the point p <span class="math-container">$\in$</span> M corresponding to the parameters <span class="math-container">$s=\frac{\pi }{4}$</span> and <span class="math-container">$t=\frac{\pi }{3}$</span>.
Determine the cartesian equation of the tangent plane to M in p.</p>
<p><span class="mat... | Z Ahmed | 671,540 | <p>Use <span class="math-container">$\cos^2x=1-\sin^2x$</span>. Then <span class="math-container">$I=\int (1-\sin x) dx$</span></p>
|
3,154,244 | <p>I tried to ask this in a different way and did not correctly explain myself.</p>
<p>I am ok integrating the line <span class="math-container">$y = x$</span> , let us say from <span class="math-container">$0$</span> to <span class="math-container">$2$</span> using calculus.
If I want to get the square I can easily m... | Paras Khosla | 478,779 | <p>The dimensions do work correctly in case of a rectangle as well. Let us say the height of the rectangle is <span class="math-container">$k$</span> and width is <span class="math-container">$b-a$</span>.</p>
<p>Recall that integration is basically summing up infinitely many small rectangles with infinitesimally smal... |
2,757,469 | <p>The set $A=\{1, \frac{1}{2}, \frac{1}{4}, \dots \}$ is obviously not closed in $\mathbb{R}$ with the Euclidean metric, as the sequence $1, \frac{1}{2}, \frac{1}{4} \dots$ converges to $0 \notin A$.</p>
<p>But if we look at the complement, $A'$ and let $p\in A'$ then there exist some $a_i$ and $a_{i+1}$ such that $a... | Tsemo Aristide | 280,301 | <p>Since $A$ is not closed, its complement cannot be open, $0$ is in the complement of $A$ and every ball which contains $0$ meets $A$.</p>
|
2,757,469 | <p>The set $A=\{1, \frac{1}{2}, \frac{1}{4}, \dots \}$ is obviously not closed in $\mathbb{R}$ with the Euclidean metric, as the sequence $1, \frac{1}{2}, \frac{1}{4} \dots$ converges to $0 \notin A$.</p>
<p>But if we look at the complement, $A'$ and let $p\in A'$ then there exist some $a_i$ and $a_{i+1}$ such that $a... | dyf | 556,562 | <p>Your complement $A'$ includes $0$ but you cannot find an open ball around it. Your mistake is tacitly assuming $p \in A'$ is in $(0,1)$ as well.</p>
|
1,158,956 | <p>To show that orthogonal complement of a set A is closed.</p>
<p>My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by Cauchy-Schwarz inequality we get,
$$|\langle x_1,y_1\rangle - \langle x_2,y_2\rangle| = |\langle x_1- x_2,y_1\rangle... | MUH | 214,956 | <p>Let's denote the orthogonal complement of <span class="math-container">$A$</span> by <span class="math-container">$A^{\perp}$</span>. Also, we denote the scalar product <span class="math-container">$\langle \cdot, y \rangle : V \rightarrow \mathbb{F}$</span> as the function <span class="math-container">$\varphi_y$</... |
2,166,075 | <blockquote>
<p>Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively.</p>
</blockquote>
<p>Where $a_i=a_{i+1}-r$.</p>
<p>I tried to start proving it inductively, but any try lead to a bad conclusion, so I ended up proving it by making $a_n$ depend on $a_i$.</p>
<p>But I didn't know how to prove it inductively,... | Siong Thye Goh | 306,553 | <p>If $n=1$, the result is trivial.</p>
<p>Suppose $$\sum_{i=1}^k a_i = \frac{(a_1+a_k)k}{2}$$</p>
<p>\begin{align}\sum_{i=1}^{k+1} a_i &= \frac{(a_1+a_k)k}{2}+a_{k+1}\\&=\frac{a_1k+a_{k+1}(k+1)-rk+a_{k+1}}{2}\\
&= \frac{a_1k+a_{k+1}(k+1)+a_1}{2}\\
&=\frac{(a_1+a_{k+1})(k+1)}{2}\end{align}</p>
|
4,047,601 | <p>I did a question <span class="math-container">$\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$</span>, and evaluating this is divergent integral yes? Then as a general form <span class="math-container">$\int_{0}^{1} \frac{1}{x^p}\,dx$</span>, <span class="math-container">$p \in \mathbb{R}$</span>, what values of <span cl... | DMcMor | 155,622 | <p>Start with the antiderivative, assuming that <span class="math-container">$p\ne 1$</span> <span class="math-container">$$\int x^{-p}\,dx = \frac{1}{1-p}x^{1-p} + C.$$</span> Then it must be the case that <span class="math-container">$$\frac{1}{1-p}x^{1-p}\bigg|_{0}^{1} = \frac{1}{1-p}\left(1-\lim_{x\to 0}x^{1-p}\ri... |
3,375,459 | <p>I am trying to study differential geometry.</p>
<p>I am confused with regards to the following function for finding the length of
a curve <span class="math-container">$\gamma$</span> connecting two points <span class="math-container">$p, q ∈ S^2$</span></p>
<p><span class="math-container">$$L(γ) = \int^1_0|\dot{γ}... | Surb | 154,545 | <p><strong>Hint</strong></p>
<p>If <span class="math-container">$X\sim \mathcal N(0,1)$</span>,</p>
<p><span class="math-container">$$\mathbb E[f(X)]=\int_{\mathbb R}f(x)e^{-\frac{x^2}{2}}\,\mathrm d x.$$</span></p>
<p>Moreover, <span class="math-container">$$\int_{\mathbb R}e^{-x^2}\,\mathrm d x=\pi.$$</span></p>
... |
103,397 | <p>Is there functionality in <em>Mathematica</em> to expand a function into a series with Chebyshev polynomials? </p>
<p>The <code>Series</code> function only approximates with Taylor series.</p>
| J. M.'s persistent exhaustion | 50 | <p>One slick way to derive the analytic Chebyshev series of a function is to use the relationship between the Chebyshev polynomials and the cosine, and then use the built-in <code>FourierCosSeries[]</code>. As an example:</p>
<pre><code>f[x_] := Exp[x];
n = 5; (* degree of approximation *)
approx[x_] = FourierCosSerie... |
1,232,363 | <p>I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). </p>
<p>Suppose that we have a box with 100 balls and I take a random sample of size 10</p>
<p>Is a random sample of size 10 if</p>
<ul>
<li>I take... | Mark Viola | 218,419 | <p>Let's use the "brute force" approach here. We have the matrix equation $Av=\lambda v$. Thus, the eigenvalue equation becomes</p>
<p>$$(A-\lambda I)v=0$$</p>
<p>which implies that the determinant of $A-\lambda I$ is zero. Thus, we have </p>
<p>$$(\lambda -a)(\lambda -d)-bc=0$$</p>
<p>which implies that </p>
<... |
870,030 | <p>Q: Prove that the relation given by $a\sim b\Leftrightarrow a-b\in\mathbb{Z}$ is a congruence relation on the additive group $\mathbb{Q}$.</p>
<p>A: Maybe...
<ul>
<li>$a\sim a\Leftrightarrow a-a=0\in \mathbb{Z}$ ✓
<li>$a\sim b\Leftrightarrow a-b\in \mathbb{Z}$. $a\in \mathbb{Z}\Rightarrow -a\in \mathbb{Z}$ ... | Lee Mosher | 26,501 | <p>The second bullet is wrong. It should be $a \sim b \iff a-b \in \mathbb{Z} \iff b-a \in \mathbb{Z} \iff b \sim a$.</p>
|
2,619,638 | <p>I know a function which is not equal a.e to a continuous function is the step function or the characteristic of any interval and I also know the Dirichlet function is not an a.e continuous function but I want an example of a function with both properties.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>The first derivative of $\arccos$ is given by $$-\frac{1}{\sqrt{1-x^2}}$$, so the first derivative of $$\arccos\left(\frac{z}{B}\right)$$ is given by $$-\frac{1}{B \sqrt{1-\frac{z^2}{B^2}}}$$</p>
|
7,130 | <p>I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).</p>
<p>Thank you.</p>
| richard | 395,150 | <p>Just remove one edge from G, this will construct a G'. There's a Hamiltonian path in G' iff There's a Hamiltonian cycle in G.</p>
|
712,736 | <p>For any continuous function $f(z)$ of period $1$. Show that $\varphi'=2\pi \varphi+f(t)$ has a unique solution of period $1$.</p>
<p>Is this problem wrong with the counter example $\varphi(t)=e^{2\pi t}$. Shall we change it into $\varphi'=2\pi i\varphi+f(t)$</p>
| user127096 | 127,096 | <p>What you want is a solution of the boundary value problem on $[0,1]$ with periodic boundary condition: $\varphi(1)=\varphi(0)$. It helps to focus on the map $\mathbb R\to\mathbb R$ defined by $T: \varphi(0) \mapsto \varphi(1) $. You should check that: </p>
<ol>
<li>$T$ is injective. This follows from the unique... |
77,744 | <p>Hopefully a simple one for you (or at least seemingly)!</p>
<p>I import a .txt file, from which i make a ListLinePlot. I simply want to read in the name of the file, store it in a variable so I can use it to tag my plots later.</p>
<pre><code> Data = Import["C:\\Users\\Name\\Folder\\test2.txt", "CSV"];
ListLine... | Basheer Algohi | 13,548 | <p>I don't know if I understand you correctly. but here is what I got for you:</p>
<pre><code>FileNameTake["C:\\Users\\Name\\Folder\\test2.txt"]
(*"test2.txt"*)
</code></pre>
<p>If you want without extension, then:</p>
<pre><code>FileBaseName["C:\\Users\\Name\\Folder\\test2.txt"]
(*test2*)
</code></pre>
|
2,825,652 | <p>I have a magnetometer sensor on each vertices of an isosceles triangle. I also have a magnet that can be anywhere on the triangle (inside, on edges, etc). I have the magnitude reading from each sensor (essentially giving me the distance the magnet is from each vertices of the triangle). I'd like to calculate the x,y... | amd | 265,466 | <p>In theory, you just compute the common intersection of three circles. In practice, noisy real-world data make it likely that the resulting system of equations is inconsistent: the three measured circles don’t have a common intersection point. You will in all likelihood need to find an approximate solution. </p>
<p... |
1,328,909 | <p>I know how to find for which $n$ $\phi(n)=n/2$ or $\phi(n)=n/3$, my method for finding those was simply to find primes $p$ that satisfy $\Pi_p$$_|$$_n$$1-1/p$ $ = 1/2$ or $1/3$.</p>
<p>However, I don't know how to find $\Pi_p$$_|$$_n$$1-1/p = n/6$. Intuitively it seems that if I combine results for both $\phi(n) =... | Ian | 83,396 | <p>In general, if <span class="math-container">$X$</span> is a topological space, <span class="math-container">$Y$</span> is a metric space, and <span class="math-container">$f : X \to Y$</span> is a function, then the set of continuity points of <span class="math-container">$f$</span> must be a <span class="math-conta... |
313,437 | <p>I have to find out the convergence of the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
Any help? Thanks</p>
| Julien | 38,053 | <p>The only problem is at $0$.</p>
<p>We have, as $x$ approaches $0$,
$$
\sin x\sim x
$$
so
$$
\frac{\log(\sin x)}{\sqrt{x}}\sim\frac{\log x}{\sqrt{x}}
$$
and the integral of the latter converges at $0$.</p>
<p>If you did not know that, compare for instance with $\frac{1}{x^{2/3}}$.
Since
$$
x^{2/3}\frac{\log x}{\sq... |
1,617,890 | <blockquote>
<p>Question: Solve $\sin(3x)=\cos(2x)$ for $0≤x≤2\pi$.</p>
</blockquote>
<p>My knowledge on the subject; I know the general identities, compound angle formulas and double angle formulas so I can only apply those.</p>
<p>With that in mind</p>
<p>\begin{align}
\cos(2x)=&~ \sin(3x)\\
\cos(2x)=&~... | lab bhattacharjee | 33,337 | <p>$$\cos2x=\sin3x=\cos\left(\dfrac\pi2-3x\right)$$</p>
<p>$$\iff2x=2m\pi\pm\left(\dfrac\pi2-3x\right)$$ where $m$ is any integer</p>
<p>Alternatively, $$\sin3x=\cos2x=\sin\left(\dfrac\pi2-2x\right)$$</p>
<p>$$3x=n\pi+(-1)^n\left(\dfrac\pi2-2x\right)$$ where $n$ is any integer</p>
|
3,391,280 | <p>Prove by Induction on n that <span class="math-container">$\exists x,y,z \in Z$</span> s.t. <span class="math-container">$x\ge 2, y\ge 2, z\ge 2$</span> satisfies <span class="math-container">$x^2+y^2=z^{2n+1}$</span> </p>
<p>I'm a lot more comfortable with proving induction with <span class="math-container">$\for... | lhf | 589 | <p><em>Hint:</em> The <a href="https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity" rel="nofollow noreferrer">Brahmagupta–Fibonacci identity</a> implies that if <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span> are sums of squares, then so is their product <span c... |
700,004 | <p>I have been working on this proof for a few hours and I can not make it work out.</p>
<p>$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$</p>
<p>i need to get to $1-\frac{1}{k+2}$</p>
<p>I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$
then I have tried $1-\frac{(k+2)+1}{(k+1)(k+2)}$ which got me exac... | ljfa | 127,600 | <p>Hint: Write $\frac 1{i(i+1)}$ as $\frac 1i - \frac 1{i+1}$.
Then you have a telescoping sum.</p>
|
2,705,794 | <p>I ran across this problem on a practice Putnam worksheet. Completely stumped.</p>
<p>Is $$\large \frac{m^{6} + 3m^{4} + 12m^{3} + 8m^{2}}{24}$$ an integer for all $m \in \mathbb{N}$?</p>
<p>I suspect it is an integer for any $m$. It checks out for small cases.</p>
<p>Any hints for proving the general case?</p>
| Raghib | 462,749 | <p>It is always an integer. A standard "brute force" approach is simply to show that each factor of 24 divides the expression in the numerator. This is equivalent to showing that 3 and 8 both divide it in all cases.</p>
<p>The case of the factor 3 splits into two sub cases: one where m is divisible by 3 and one where ... |
2,705,794 | <p>I ran across this problem on a practice Putnam worksheet. Completely stumped.</p>
<p>Is $$\large \frac{m^{6} + 3m^{4} + 12m^{3} + 8m^{2}}{24}$$ an integer for all $m \in \mathbb{N}$?</p>
<p>I suspect it is an integer for any $m$. It checks out for small cases.</p>
<p>Any hints for proving the general case?</p>
| robjohn | 13,854 | <p>By noting the values at $m=0$, $m=1$, ... $m=6$, it is easy to compute
$$
\tfrac{m^6+3m^4+12m^3+8m^2}{24}
=\textstyle30\binom{m}{6}+75\binom{m}{5}+68\binom{m}{4}+30\binom{m}{3}+8\binom{m}{2}+\binom{m}{1}
$$</p>
|
1,875,351 | <p>I saw this problem in a book (not homework),</p>
<p>Assuming $L(n) = F(n)$ for$ n = 1,2,\cdots, k$
where $L(n)$ is Lucas Number and $F(n)$ is Fibonacci number.</p>
<p>$$\qquad L(k+1) = L(k) + L(k-1) \qquad \tag{by definiton}$$</p>
<p>$$ \qquad\qquad= F(k) + F(k-1) \tag{assumption}$$</p>
<p>$$ \ =F(k+1)... | Bill Dubuque | 242 | <p>Your induction step is well-defined only for $\,k\ge 2\,$ since for $\,k\le 1\,$ the term $\,L(k-1)\,$ is not defined. The recurrence lifts equality at the prior two indices to the current index. It cannot be used to deduce equality at the <em>initial</em> indices $\,n = 1,2$ (the "initial conditions" = base cases).... |
129,788 | <blockquote>
<p>Let be A and B two events from the same sample set. If $\space P(A)+P(B)=1$, can one say that they are opposite events?</p>
</blockquote>
<p>In my thought:</p>
<p>$\space P(A)+P(B)=1$</p>
<p>$\space P(A)=1-P(B)$</p>
<p>So they are opposite events. But my book says no! It says that is not necessary... | Michael Hardy | 11,667 | <p>Doesn't work. Throw a die. The sample space is $\{1,2,3,4,5,6\}$.</p>
<p>Let $A=\{1,2,3,4\}$ and $B=\{1,2\}$.</p>
<p>Then $P(A)+P(B)=1$ but $A$ and $B$ are not complementary events.</p>
|
3,995,119 | <p>I've difficulties calculating the following integral</p>
<p><span class="math-container">$$\int_z^\infty\mu\mathrm e^{-\mu y}(\mathrm e^{-\lambda z}-\mathrm e^{-\lambda y})\ \mathrm{d}y$$</span></p>
<p>I'm gonna use her to find a joint distribution of two random variables. I've try to apply the following substitutio... | Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\int_u^\infty e^{-at}dt=-\left.\frac{e^{-at}}a\right|_u^\infty=\frac{e^{-au}}a.$$</span></p>
<p>Your integral is</p>
<p><span class="math-container">$$\mu e^{-\lambda z} \int_z^\infty e^{-\mu y}dy-\mu\int_z^\infty e^{-(\mu+\lambda) y}dy.$$</span></p>
|
3,059,571 | <p><span class="math-container">$$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$</span></p>
<p>I only know of L'hopital method but that is very long. Is there a shorter method to solve this?</p>
| klirk | 385,702 | <p>Another approach is to use that <span class="math-container">$\tan x=\frac{\sin x}{\cos x}$</span>.</p>
<p>Then <span class="math-container">$$\frac{1-\tan(x/2)}{1+\tan(x/2)}=\frac{\cos(x/2)-\sin(x/2)}{\cos(x/2)+\sin(x/2)}=\frac{(\cos(x/2)-\sin(x/2))^2}{\cos^2(x/2)-\sin^2(x/2)}=\frac{1-\sin(x)}{\cos(x)} .$$</span><... |
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