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 |
|---|---|---|---|---|
485,471 | <p>Of course we are familiar with the notion that
if $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ ($p_{i}$ distinct primes;
$k_{i}>0$), then $$\varphi\left(n\right)=n\left(1-1/p_{1}\right)\left(1-1/p_{2}\right)\cdots\left(1-1/p_{r}\right),$$
where $\varphi$ is the Euler totient function.</p>
<p>The goal is to prove this u... | minar | 86,791 | <p>Let us start from the definition of Euler's Phi:
$$
\varphi(n)=|\{x\in [1,n-1] | \gcd(x,n)=1\}|,
$$
i.e., $\varphi(n)$ is the number of positive integers at most equal to $n$ and prime to $n$.</p>
<p>Now, if $n$ and $m$ are coprime, we want to show that $\varphi(mn)=\varphi(n)\varphi(m)$. For this, let us denote:
$... |
485,471 | <p>Of course we are familiar with the notion that
if $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ ($p_{i}$ distinct primes;
$k_{i}>0$), then $$\varphi\left(n\right)=n\left(1-1/p_{1}\right)\left(1-1/p_{2}\right)\cdots\left(1-1/p_{r}\right),$$
where $\varphi$ is the Euler totient function.</p>
<p>The goal is to prove this u... | Dietrich Burde | 83,966 | <p>The group $(\mathbb{Z}/n\mathbb{Z})^*$ has order $\phi(n)$. Because $(\mathbb{Z}/nm\mathbb{Z})^*\simeq (\mathbb{Z}/n\mathbb{Z})^*\times (\mathbb{Z}/m\mathbb{Z})^*$ for coprime $n$ and $m$ it follows $\phi(nm)=\phi(n)\phi(m)$.</p>
|
1,457,478 | <p>I'm trying to teach myself some number theory. In my textbook, this proof is given:</p>
<blockquote>
<p><strong>Example (2.3.1)</strong> Show that an integer is divisible by 3 if and only if the sum of its digits is a multiple of 3.</p>
<p>Let <span class="math-container">$n=a_0a_1\ldots a_k$</span> be the decimal r... | Paul Sinclair | 258,282 | <p>A simple way to see this (that actually generalizes nicely to Fermat's little theorem): </p>
<p>$$10 - 1 = 9 = 9 \times 1$$
$$100 - 1 = 99 = 9 \times 11$$
$$1000 - 1 = 999 = 9 \times 111$$
In general $$10^n - 1 = 9 \times \underbrace{111...111}_\mbox{$n$ times}.$$ This is just the algebraic identity
$$x^n - 1 = (x ... |
730,357 | <p>A group of order 48 must have a normal subgroup of order 8 or 16 .<br>
Solution:Let G be a group of order n.<br>
Let H be a normal subgroup of G.<br>
Then G/H is a group.<br>
Then by Lagrange's Theorem o(G/H)=o(G)/o(H)<br>
So in this case order of G is 48 and divisors of 48 are 8 and 16.<br>
so a group of order 48 ... | Mark | 136,155 | <p>Lagrange's theorem states if $H$ is a subgroup of $G$, then the order of $H$ divides the order of $G$. The converse is not true at all.</p>
<p>Have you learned about the Sylow theorems? </p>
|
162,360 | <p>First, the definition of a connected set:</p>
<blockquote>
<p><strong>Definition:</strong> A topological space is connected if, and only if, it cannot be divided in two
nonempty, open and disjoint subsets, or, similarly, if the empty set and the whole set are the
only subsets that are open and closed at the ... | Amitesh Datta | 10,467 | <p>The following steps lead to a solution:</p>
<p><em><strong>Exercise 1</em></strong>: Let us <em>assume</em> that we have proved $I=[0,1]$ is connected. Prove that as a <em>consequence</em> $(a,b),[a,b),(a,b],[a,b]$ are connected for any real numbers $a$ and $b$. (<em>Hint</em>: A homeomorphism preserves connectedne... |
1,549,920 | <p>Prove that
$$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$
where e is exponenial ,
can any one give a proof...I tried with series expansion i could not get it.</p>
| Kelenner | 159,886 | <p>You can also compute $f^{\prime}(x)$. As $f(x)=\exp(\frac{\log(1+x)}{x})$, you get
$$f^{\prime}(x)=\frac{\frac{x}{1+x}-\log(1+x)}{x^2}f(x)=\frac{\sum_{n\geq 1}(-1)^{n-1}x^n-\sum_{n\geq 1}(-1)^{n-1}\frac{x^n}{n}}{x^2} f(x)$$</p>
<p>Thus:</p>
<p>$$f^{\prime}(x)=(\sum_{n\geq 2}(-1)^{n-1}\frac{n-1}{n}x^{n-2})f(x)=(\su... |
4,212,850 | <p>I have an ellipsoid centered at <span class="math-container">$0$</span> (the contour of a Gaussian distribution centered at <span class="math-container">$0$</span> with covariance matrix <span class="math-container">$\Sigma=\Lambda^{-1}$</span>)
<span class="math-container">$$
x^\top \Lambda x = \gamma
$$</span>
and... | TonyK | 1,508 | <p><em>Almost no</em> functions from <span class="math-container">$\Bbb R^2$</span> to <span class="math-container">$\Bbb R$</span> are the product of two functions in the way that you describe.</p>
<p>Consider this: suppose <span class="math-container">$f$</span> is non-zero on the <span class="math-container">$x-$</s... |
3,600,868 | <p>Ellipse can be
<a href="https://math.stackexchange.com/q/3594700/122782">perfectly packed with <span class="math-container">$n$</span> circles</a>
if </p>
<p><span class="math-container">\begin{align}
b&=a\,\sin\frac{\pi}{2\,n}
\quad
\text{or equivalently, }\quad
e=\cos\frac{\pi}{2\,n}
,
\end{align}</span> <... | Blue | 409 | <p>Here's a way to proceed with the investigation:</p>
<ol>
<li>Determine the minimal polynomial of <span class="math-container">$e = \cos\frac{\pi}{2n}$</span> for a desired chain-length <span class="math-container">$n$</span>.</li>
<li>Determine a formula for the eccentricity <span class="math-container">$e$</span> ... |
2,653,401 | <p>I have a question about the following partial fraction:</p>
<p>$$\frac{x^4+2x^3+6x^2+20x+6}{x^3+2x^2+x}$$
After long division you get:
$$x+\frac{5x^2+20x+6}{x^3+2x^2+x}$$
So the factored form of the denominator is
$$x(x+1)^2$$
So
$$\frac{5x^2+20x+6}{x(x+1)^2}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^2}$$</p>
<p>Wh... | Doug M | 317,162 | <p>You need an $(x+1)^2$ in the denominator of at least one of the partial fractions, or when you sum them they would not have an $(x+1)^2$ term in the denominator of the sum.</p>
<p>But you might find it easier to solve:</p>
<p>$\frac{5x^2+20x+6}{x(x+1)^2}=\frac{A}{x}+\frac{Bx + C}{(x+1)^2}$</p>
<p>And that is comp... |
1,571,706 | <p>I have found the solutions by a little calculation $(2,3,5,7)$ and $(2,3,4,5)$. But I don't know if there's any other solutions or not?</p>
| André Nicolas | 6,312 | <p>Let's grind it out. We want all ordered quadruples, but to get them it is enough to find all ordered triples $(a,b,c)$ where $a\le b\le c$. Then any triple of this kind we find can be permuted arbitrarily.</p>
<p>There cannot be triples $(a,b,c)$ with $a\le b\le c$ and $a\gt 2$, since for these we get that $3$ divi... |
1,571,706 | <p>I have found the solutions by a little calculation $(2,3,5,7)$ and $(2,3,4,5)$. But I don't know if there's any other solutions or not?</p>
| fleablood | 280,126 | <p>Let $a \le b \le c$.</p>
<p>If $m \ge 2$ m! is even so if $a = 1$ (odd), $b = 1$ and $c! = 2^n - 2 = 2(2^{n-1} - 1)$ so $c = 2; n=1$ or $c = 3; n = 2$. Two answers so far. (1,1,2,1)(1,1,3,2)</p>
<p>If $a > 1$ then $a! + b! + c! = a!(1 + b!/a! + c!/a!) = 2^n$ If $a \ge 3$ $3|a!$ and so $3|2^n$ which is impossib... |
396,717 | <p>It's not difficult to see that <span class="math-container">$S^{2n}$</span> doesn't admit a Lie group structure. Since if <span class="math-container">$S^{2n}$</span> admit a Lie group structure, then there exists a left invariant vector field. While the Hairy ball theorem says that there exists no continuous tangen... | Pierre PC | 129,074 | <p>The function <span class="math-container">$g\mapsto\|(-g)\cdot g^{-1}-e\|$</span> is continuous and never zero on the sphere, so it admits a minimum <span class="math-container">$\varepsilon>0$</span>. Let <span class="math-container">$v$</span> be an element of the sphere with <span class="math-container">$0<... |
3,900,679 | <p>So there's this one thing I can't wrap my head around.</p>
<p>The problem is:</p>
<p>Use the change of variables <span class="math-container">$x = \sinh u$</span> to compute the indefinite integral
<span class="math-container">\begin{equation*}
\int \frac{dx}{\sqrt{1+x^2}}
\end{equation*}</span></p>
<p>Solution:... | reuns | 276,986 | <p>I doubt there is any formula telling immediately that it is non-negative. What I mean is that sometimes you need numerical checks to know the sign of a given series.</p>
<p>For <span class="math-container">$s > 0$</span></p>
<p><span class="math-container">$$\sum_{n=1}^\infty (-1)^{n+1} n^{-s}=
\sum_{n=1}^\infty ... |
3,494,820 | <p>Yesterday I learned about dual spaces when reading about spaces of linear maps. The concept of a linear map and why linear maps form a vector space is clear to me. But there are some details about the dual space and its basis that I could not fully understand.</p>
<p>The text I am reading states the following:</p>
... | Kavi Rama Murthy | 142,385 | <p>The derivative of <span class="math-container">$\ln (1+x)-x+\frac {x^{2}} 2$</span> is <span class="math-container">$\frac 1 {1+x} -1+x=\frac {x^{2}} {1+x}$</span> which is positive. Hence this functinn is increasing. Since its value when <span class="math-container">$x=0$</span> is <span class="math-container">$0$<... |
338,155 | <p>I've problem with this surface integral:
$$
\iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS}
$$, where
$$
S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}= 1\}
$$</p>
| Christian Blatter | 1,303 | <p>Let the ellipsoid $S$ be given by
$${\bf x}(\theta,\phi)=(a\cos\theta\cos\phi,b\cos\theta\sin\phi,c\sin\theta)\ .$$
Then for all points $(x,y,z)\in S$ one has
$$Q^2:={x^2\over a^4}+{y^2\over b^4}+{z^2\over c^4}={1\over a^2b^2c^2}\left(\cos^2\theta(b^2c^2\cos^2\phi+a^2c^2\sin^2\phi)+a^2b^2\sin^2\theta\right)\ .$$
On ... |
338,155 | <p>I've problem with this surface integral:
$$
\iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS}
$$, where
$$
S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}= 1\}
$$</p>
| Zarrax | 3,035 | <p>Note that the outward unit normal ${\bf n}$ to the ellipsoid is given by ${\displaystyle {({x \over a^2}, {y \over b^2}, {z \over c^2}) \over \sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}}$. </p>
<p>So if ${\bf F} = ({x \over a^2}, {y \over b^2}, {z \over c^2})$, your integral is $\int_S {\b... |
39,654 | <p>$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$</p>
<p>I've let $u=x^2+3$ but can't seem to get the right answer.</p>
<p>Really not sure what to do.</p>
| Américo Tavares | 752 | <p><em>Hint</em>: Note that $$\frac{1}{x^{2}+3}=\frac{1}{3}\cdot \frac{1}{\left( \frac{x}{\sqrt{%
3}}\right) ^{2}+1}$$</p>
<p>and use the substitution $u=\frac{x}{\sqrt{3}}$.</p>
|
709,815 | <p>Let $R$ be a commutative ring with identity $1$.</p>
<p>If for some $a \in R$ there exists $b \in R$ such that $ab = 1$, then we say that $a$ is a <strong>unit</strong> and that $b$ is a <strong>multiplicative inverse</strong> or <strong>reciprocal</strong> of $a$.</p>
<p>If for some $c \in R$ there exists nonzero... | Cyllindra | 122,907 | <p>If $c=0$, then we get noting interesting, e.g. $d$ is any nonzero element in our ring. If $c\ne0$, then $d$ would also be a zero divisor, since $\exists{c}\in R$, with $c\ne0$ such that $cd=0$.</p>
<p>I am not sure what else you are looking for.</p>
|
709,815 | <p>Let $R$ be a commutative ring with identity $1$.</p>
<p>If for some $a \in R$ there exists $b \in R$ such that $ab = 1$, then we say that $a$ is a <strong>unit</strong> and that $b$ is a <strong>multiplicative inverse</strong> or <strong>reciprocal</strong> of $a$.</p>
<p>If for some $c \in R$ there exists nonzero... | Nate Eldredge | 822 | <p>I may have an answer to my own question. The set $\mathrm{Ann}(c) := \{d \in R : cd = 0\}$ is known as <strong>the annihilator</strong> of $c$ (it is actually an ideal of $R$). So it might not be too great an abuse of terminology to also say that each element $d \in \mathrm{Ann}(c)$ is <strong>an annihilator</stro... |
335,295 | <p>How do you show that a deduction exist in the Hilbert Proof System, as used in Herbert Enderton, <em>A Mathematical Introduction to Logic</em>.</p>
<p>L is a FOL (First Order Language) which contains R, where R is a single binary predicate symbol.</p>
<p>a1, a2, a3 are defined as:</p>
<p>a1 = $∀x∀y∀z(Rxy → (Ryz →... | Mauro ALLEGRANZA | 108,274 | <p>I'll complete the answer above with a proof using the axioms and rules of Herbert Enderton, <a href="http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=Herbert%20Enderton_A%20Mathematical%20Introduction%20to%20Logic%20" rel="nofollow">A Mathematical Introduction to Logic</a> (2nd... |
4,370,574 | <p>A fixed point iteration formula for <span class="math-container">$2x^3-4x^2+x+1=0$</span> can be derived:
<span class="math-container">$$x_{r+1}=4x_r^2-2x_r^3-1$$</span></p>
<p>Starting with the initial value of 2:</p>
<p><span class="math-container">$$x_0=2$$</span>
<span class="math-container">$$x_1=-1$$</span>
<s... | VanBaffo | 263,664 | <p>You can either write <span class="math-container">$1^\top \operatorname{vec}(M)$</span> or <span class="math-container">$\operatorname{nnz}(M)$</span> to indicate the number of non-zero entries in a matrix, where <span class="math-container">$\operatorname{vec}(\cdot)$</span> is the vectorization operator, and nnz s... |
3,193,107 | <p>As the title says, I need to find the explicit form of the recursive sequence defined above, and I am very stuck on this.</p>
| pi66 | 293,251 | <p>So you have <span class="math-container">$b_k = 2\cdot 3^k + 2\cdot 3^{k-1}+\dots+2\cdot 3^2 + 2$</span>. This is the same as <span class="math-container">$2\cdot (3^k+3^{k-1}+\dots+3^0) - 2\cdot 3$</span>. </p>
<p>The term in parentheses is a <a href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofol... |
3,007,044 | <p>Sequence <span class="math-container">$a_n$</span> is defined as <span class="math-container">$a_n = \ln (1 + a_{n-1})$</span>, where <span class="math-container">$a_0 > -1$</span>. Find all <span class="math-container">$a_0$</span> for which the sequence converges.</p>
<p>Using arithmetic properties of limits, ... | Mohammad Riazi-Kermani | 514,496 | <p>You are correct in finding the limit in case of existence to be <span class="math-container">$0$</span></p>
<p>Notice that you are finding the fixed point of the function <span class="math-container">$$f(x)=\ln (1+x)$$</span> </p>
<p>The derivative of your function is <span class="math-container">$$f'(x) = \frac ... |
4,200,434 | <p>I am having difficulty with what should be a routine question, Exercise 2.2.2 (b) of <em>Understanding Analysis</em> by Stephen Abbott (2015).</p>
<blockquote>
<p><strong>Exercise 2.2.2.</strong> Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.</... | Kman3 | 641,945 | <p>As mentioned in the comments, based on the way the question is written, the limit should be <span class="math-container">$2$</span> and not <span class="math-container">$0$</span>.</p>
<p>Assuming the question is correct as written, we have</p>
<p><span class="math-container">$$\left| \frac{2n^2}{n^2+3} - 2 \right| ... |
1,063,334 | <p>I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. Everywhere I look on the internet it says that |x| is not differentiable at 0 even there is nothing on absolute value of funct... | Blah | 6,721 | <p>$g(x) \geq 0$ for all $x$ and $g(c)=0$, so if $g$ is differentiable at $c$ there is a local minimum at $c$ and therefore $g'(c)=0$</p>
<p>now
$$
\dfrac{g(c+h)-g(c)}{h}=\dfrac{g(c+h)}{h}=\dfrac{|f(c+h)|}{h}
\rightarrow 0 \text{ for } h\rightarrow 0
$$
and note that (for example) a sequence $a_n \rightarrow 0$ iff $... |
449,457 | <p>A desk has three drawers. The first contains two gold coins, the second has two silver coins and the third has one gold and one silver coin. A drawer is selected at random and a coin is drawn at random from the drawer. Suppose that the coin selected was silver. Use Bayes's Theorem to find the probability that the ot... | Deepak | 151,732 | <p>Let the choice of picking a drawer amongst 3 represent 3 events called $gg, gs, ss$ for both gold, one gold one silver and 2 silver drawer respectively. These probabilities are $\frac{1}{3}$ each.</p>
<p>From the $2g$ drawer, probability of picking a silver coin $p(s|gg) = 0$.</p>
<p>Similarly, $p(s|gs) = \frac{1}... |
3,738,719 | <p>We got</p>
<p><span class="math-container">$$x'=\frac{t^2x^2+1}{2t^2}$$</span></p>
<p><span class="math-container">$$\frac{dx}{dt}=\frac{t^2x^2+1}{2t^2}$$</span></p>
<p><span class="math-container">$$2t^2dx=\left(t^2x^2+1\right)dt$$</span></p>
<p><span class="math-container">$$0=\left(t^2x^2+1\right)dt+\left(-2t^2dx... | Ninad Munshi | 698,724 | <p>Use the substitution <span class="math-container">$tx = v$</span>:</p>
<p><span class="math-container">$$\frac{v'}{t}-\frac{v}{t^2} = \frac{v^2+1}{2t^2} \implies 2tv' = (v+1)^2$$</span></p>
<p>from here it is separable with solution</p>
<p><span class="math-container">$$x = \frac{2}{Ct-t\log t} - \frac{1}{t}$$</span... |
302,599 | <p><strong>Problem:</strong></p>
<p>For every $n$ in $\mathbb{N}$, we consider the sets $A_{n}:=\left \{ (2n+1)\lambda :\ \lambda \in \mathbb{N}\right \}$. The question is to find $\bigcap_{n=1}^{\infty }A_{n}$, and $\bigcup_{n=1}^{\infty }A_{n}$.</p>
<p>For the intersection, I think it is the empty set, because for... | Brian M. Scott | 12,042 | <p>I’m assuming that you’re using $\Bbb N$ for $\Bbb Z^+$, the set of positive integers, rather than for the set of non-negative integers. In that case $A_1$ is the set of positive multiples of $3$, $A_2$ is the set of positive multiples of $5$, $A_3$ is the set of positive multiples of $7$, and so on. Clearly no posit... |
992,125 | <p>If I rolled $3$ dice how many combinations are there that result in sum of dots appeared on those dice be $13$?</p>
| Henno Brandsma | 4,280 | <p>It's the coefficient of $x^{13}$ in the product $(x+x^2 + x^3 + x^4+ x^5 + x^6)^3$.
To see this, note that to compute that coefficient we have to identify all ways we can form $x^{13}$ by picking one term from each of the three terms $(x + x^2 + x^3 + x^4 + x^5 + x^6)$ we have; we could have $x$ from the first, $x^6... |
3,625,886 | <blockquote>
<p>Show that if <span class="math-container">$a$</span> and <span class="math-container">$b$</span> belong to <span class="math-container">$\mathbb{Z}_+$</span> then there are divisors <span class="math-container">$c$</span> of <span class="math-container">$a$</span> and <span class="math-container">$d$<... | Cookiemaster | 714,130 | <p>Using prime factorization formula for a and b we have:
a= <span class="math-container">$p_1$$^{a1}$</span> <span class="math-container">$p_2$$^{a2}$</span> ... <span class="math-container">$p_n$$^{an}$</span>
b=<span class="math-container">$p_1$$^{b1}$</span> <span class="math-container">$p_2$$^{b2}$</span> ... <spa... |
3,625,886 | <blockquote>
<p>Show that if <span class="math-container">$a$</span> and <span class="math-container">$b$</span> belong to <span class="math-container">$\mathbb{Z}_+$</span> then there are divisors <span class="math-container">$c$</span> of <span class="math-container">$a$</span> and <span class="math-container">$d$<... | fleablood | 280,126 | <p>List the prime factors of <span class="math-container">$a$</span> and/or <span class="math-container">$b$</span> as <span class="math-container">$p_1, p_2, .....,p_n$</span>.</p>
<p>Let <span class="math-container">$a= \prod_{i=1}^n p_i^{k_i}$</span> for some non-negative integer values of <span class="math-contain... |
2,832,025 | <p>$$\sum_{k=1}^{\infty}\ln\left[ \frac{(4k+1)^{1/(4k+1)^{n}}}{(4k-1)^{1/(4k-1)^{n}}} \right] = -\beta'(n)$$.
Where $\beta$ is the Dirichlet Beta Function and $n$ is a positive integer. </p>
<p>I cannot find this cited anywhere nor values of the beta function derivative apart from at $-1,0,1$. How can I go about find... | marty cohen | 13,079 | <p>$\sum_{k=1}^{\infty}\ln\left[ \frac{(4k+1)^{1/(4k+1)^{n}}}{(4k-1)^{1/(4k-1)^{n}}} \right] = -\beta(n)
$</p>
<p>I'll naively play with
the left side
and see what I can get.</p>
<p>The result is
complicated expressions
of dubious value,
but here they are anyway.</p>
<p>$\begin{array}\\
t_k
&=\ln\left[ \dfrac{(4... |
1,584,933 | <p>Let's have $f_n(x)$ defined on $\mathbb{R}$ by:</p>
<ul>
<li>$f_n(0)=0$</li>
<li>$f_n(x)=\frac{1-e^{-nx^2x^2}}{x}$if $x\neq 0$</li>
</ul>
<p>$f_n(x)\rightarrow \frac{1}{x}$</p>
<p>therefore, $f_n(x)$ converges weakly to $\frac{1}{x}$
\begin{align}
\lim\limits_{x\rightarrow +\infty}\sup\limits_{n\in I}{|f_n(x)-f(x... | C. Falcon | 285,416 | <blockquote>
<p><strong>Definition.</strong> Let $\gamma:[0,1]\rightarrow\mathbb{C}$ be a loop and let $a\in\mathbb{C}\setminus\gamma([0,1])$. Let define: $$\textrm{Wnd}(\gamma,a):=\frac{1}{2i\pi}\int_{\gamma}\frac{\mathrm{d}z}{z-a}.$$
$\textrm{Wnd}(\gamma,a)$ is the winding number of $\gamma$ around $a$.</p>
</blo... |
3,259,073 | <p>Let <span class="math-container">$A$</span> be a commutative ring with <span class="math-container">$1$</span>, and let <span class="math-container">$a, b, c \in A$</span>. Suppose there exist <span class="math-container">$x, y, z ∈ A$</span> such that <span class="math-container">$ax+by +cz = 1$</span>. Then there ... | Ehsaan | 78,996 | <p>If <span class="math-container">$R$</span> is commutative, the statement is completely elementary --- not homological at all.</p>
<p>Suppose <span class="math-container">$I+J=R$</span>, so that <span class="math-container">$a+b=1$</span> for some <span class="math-container">$a\in I$</span>, <span class="math-conta... |
1,023,013 | <blockquote>
<p>Prove:$\forall a_1,b_1,a_2,b_2: \left|\max(a_1,b_1) - \max(a_2,b_2)\right| \le \max(\left|a_1-a_2\right|, \left| b_1-b_2 \right|) $</p>
</blockquote>
<p>How can I prove it easily? Or should I just go over each case?
All those cases make me confused!</p>
| Mirko | 188,367 | <p>So simplify further and factor the difference:
$a_{n+2}-a_{n+1}= \frac{1}{2}(a_n^2-a_n-a_{n+1}^2+a_{n+1}) =
\frac{1}{2}(a_{n+1}-a_n-(a_{n+1}^2-a_n^2)) =
\frac{1}{2}(a_{n+1}-a_n-(a_{n+1}-a_n)(a_{n+1}+a_n)) =
\frac{1}{2}(a_{n+1}-a_n)(1-(a_{n+1}+a_n))$. </p>
<p>Note $a_{n+1}-1=\frac{a_n(1-a_n)}2$. Also, if $a>1... |
1,340,888 | <p>I came across the picture below through random means.</p>
<p><img src="https://i.stack.imgur.com/7POpd.gif" alt="ellipse magic"></p>
<p>What is being demonstrated? All I could think of is <em>maybe</em> the center of the triangle is moving back and forth between the focii of the ellipse, but even if that's true (w... | Robert | 201,696 | <p>You are given a circle with two diameters that are not perpendicular to each other. Two points go back and forth along the diameters. A third point creates a triangle. As the two points move along the diameters, the third point of the triangle traces an ellipse. </p>
|
1,708,420 | <p>Here's what I am trying to figure out: Find all vectors $\vec{b}$ that are in $span \{\vec{u},\vec{v},\vec{w}\}$ where $\vec{u},\vec{v},\vec{w}$ are vectors.</p>
<p>I'm given specific vectors in $\mathbb{R}^3$ for $\vec{u},\vec{v},\vec{w}$, but I really just want to ask about concepts. I can do the algebra, certain... | KR136 | 186,017 | <p>There is a solution in special functions, achieved via use of substitutions. </p>
<p>$\int_{-2}^2 \frac{x^2}{1+5^x} \, dx$</p>
<p>Let $t=5^x$</p>
<p>$x= \log_5(t)$</p>
<p>$dx = \frac{dt}{\log_5(t) \ln(5)}$</p>
<p>$t(-2)=\frac{1}{25}; t(2)=25$</p>
<p>$\int_{-2}^2 \frac{x^2}{1+5^x} \, dx = $$\int_{\frac{1}{25}}^... |
1,143,893 | <p>Struggling with basic AS Statistical maths, Any help would be much appreciated.</p>
<p>The two events $A,B$ are such that $P(A)= 0.65, P(A\cup B)= 0.93$</p>
<p>Evaluate $P(B)$ given that $A$ and $B$ are independent. (4 Marks) </p>
<p>Thank you.</p>
| user66081 | 66,081 | <p>General formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Now, $A$ and $B$ are independent, that means $P(A \cap B) = P(A) P(B)$. It does not mean $P(A \cap B) = 0$. Therefore
$$
P(A \cup B) = P(A) + P(B) (1 - P(A)),
$$
which you can solve for $P(B)$.</p>
|
961,304 | <p>I am studying a book and I am stagnating on a what should be a straightfoward proof:</p>
<p>Show that if $X$ is compact, $V\subset X$ is open and $x\in V$, then there exists an open set $U$ in $X$ with $x\in U\subset \bar U\subset V$</p>
<p>I don't know how to find the appropriate set $U$. I am guessing you need t... | meh | 70,191 | <p>Sure a continuous function can have a median, as for if it always has a median I am unsure of. </p>
<p>Let's suppose we have a some function $f$ and it is continuous on $[a,b]$. We want to find its median $m$. We know that if $m$ is a median than if we take any value $c \in [a,b] $ at random then the $P(c \leq m) =... |
3,267,153 | <p>I want to understand an equation I saw in a paper. If <span class="math-container">$\theta$</span> is some angle, and <span class="math-container">$\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$</span>, then it is stated that <span class="math-container">$e^{-i \theta \sigma_y } = R_{\theta}$</sp... | JinSeok | 680,072 | <p><span class="math-container">$f(x) = (x^2 + 4)(ax +b) +7x - 20$</span></p>
<p><span class="math-container">$f(-3)=-2 = 13(-3a+b)-41 $</span></p>
<p>therefore <span class="math-container">$b= 3a+3$</span> </p>
<p>so, <span class="math-container">$f(x)= (x^2+4)(ax +3a+3)+ 7x-20 $</span> (non-zero <span class="math... |
3,267,153 | <p>I want to understand an equation I saw in a paper. If <span class="math-container">$\theta$</span> is some angle, and <span class="math-container">$\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$</span>, then it is stated that <span class="math-container">$e^{-i \theta \sigma_y } = R_{\theta}$</sp... | Dmitry Ezhov | 602,207 | <p>Solving in <strong>pari/gp</strong>:</p>
<pre><code>? chinese(Mod(7*x-20,x^2+4),Mod(-2,x+3))
%1 = Mod(3*x^2 + 7*x - 8, x^3 + 3*x^2 + 4*x + 12)
?
? Mod(a*x^3+b*x^2+c*x+d,(x^2+4)*(x+3))
%2 = Mod((-3*a + b)*x^2 + (-4*a + c)*x + (-12*a + d), x^3 + 3*x^2 + 4*x + 12)
</code></pre>
<p>I.e. <span class="math-container">$3... |
1,557,359 | <p>I've been doing Project-Euler just as a way to increase my competency in computer science. I'm currently a Pure and Applied Math major who recently adopted computer science as a minor in order to apply to grad schools in comp. sci.</p>
<p>Some built-in Sage functions that are extremely fast and I do not have compa... | Ian Miller | 278,461 | <p>It depends upon what your goal is for doing Project Euler. Many of the later problems need creative problem solving skills rather than brute force for their approach so for me I use it as a means to involve my problem solving and algorithm design skills rather than my programming skills in a specific language. If yo... |
4,586,330 | <p>How do I prove the following equality?
<span class="math-container">$$\sqrt[5]{\frac{5\sqrt5+11}{2}}+\sqrt[5]{\frac{5\sqrt5-11}{2}}=\sqrt{5} $$</span></p>
<p>My approach was to notice that the first term equals the golden ratio and the second term equals the reciprocal of the golden ratio, and adding them up would g... | Aryan Arora | 1,068,387 | <p>Hint : Use Binet's formula for <span class="math-container">$n=5$</span> <span class="math-container">$ \displaystyle F_5 = \frac{1}{\sqrt5}((\frac{1+\sqrt5}{2})^5 - (\frac{1-\sqrt5}{2})^5) $</span>.</p>
|
2,347,995 | <p>Not sure what I'm doing wrong.</p>
<p>Here's my work:</p>
<p>Expressing the first part $A \setminus (B\setminus C)$ using logical symbols:</p>
<p>$A \land \neg(B \land \neg C)$ becomes</p>
<p>$A \land \neg B\lor C$ (De Morgan's law)</p>
<p>While the second expression $(A \setminus B) \cup (A \cap C)$ is </p>
<... | Jack D'Aurizio | 44,121 | <p>By the concavity of the sine function over $(0,1)$ we have
$$\forall x\in(0,1),\qquad \sin(1)x\leq \sin(x) \leq x \tag{1} $$
and by termwise integration it follows that
$$ \forall x\in(0,1),\qquad \sin(1)\frac{x^2}{2}\leq 1-\cos(x) \leq \frac{x^2}{2}\tag{2} $$
so
$$ \forall x\in(0,1),\qquad 0.42 \leq \frac{\sin(1)}... |
2,106,983 | <p><strong>Question</strong></p>
<p>how to evaluate $\tan x-\cot x=2.$</p>
<p>Given that it lies between on $\left[\frac{-\pi} 2,\frac \pi 2 \right]$.</p>
<p><em>My Steps so far</em></p>
<p>I converted cot into tan to devolve into $\frac{\tan^2 x-1}{\tan x}=2$.</p>
<p>Then I multiply $\tan{x}$ on both sides and th... | Michael Hardy | 11,667 | <p>$\tan x$ and $\cot x$ are each other's reciprocals. Let $u=\tan x,$ so that $\dfrac 1 u = \cot x.$</p>
<p>Then $\tan x - \cot x = 2$ becomes $u - \dfrac 1 u = 2,$ and multiplying both sides by $u$ yields $u^2 - 1 = 2u,$ a quadratic equation. You get $u=1\pm\sqrt2.$</p>
<p>Next, if $\tan x= 1+\sqrt2,$ then what is ... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Zhanxiong | 192,408 | <p>Hint: Do the algebra and use the inequality that $|1 - x| > 1/2$ for $x > 3/2$:
$$\left|\frac{1}{1 - x} - (-1)\right| = \left|\frac{2 - x}{1 - x}\right| < 2|x - 2| < \cdots.$$</p>
|
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Andrey Rekalo | 5,371 | <p><img src="https://upload.wikimedia.org/wikipedia/commons/b/b4/Dirac_function_approximation.gif" alt="alt text"></p>
|
2,886,502 | <p>In the book of <em>Function of One Complex Variable</em> by Conway, at page 31, it is given that</p>
<p><a href="https://i.stack.imgur.com/Ff0nB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ff0nB.png" alt="enter image description here"></a></p>
<p>However, normally, if $z$ was a real number, ... | xbh | 514,490 | <p>$z^{n+1} \to 0 \iff \lim |z^{n+1} - 0| = 0 \iff |z|^{n+1} \to 0.$</p>
<p>First iff: let $z^n = x_n +\mathrm i y_n$, then by
$$
\max(|x_n|, |y_n|) \leqslant |z^n| \leqslant |x_n| + |y_n|,
$$
we conclude the first iff. </p>
<p>Example: $z^2 = (a^2 - b^2) + \mathrm i 2ab$, then $a^2 -b^2 \leqslant a^2 + b^2$, $2ab \... |
3,014,144 | <p>Say we have a closed interval <span class="math-container">$[a,b]$</span>. How can I prove <span class="math-container">$|[a,b]|$</span> equal to <span class="math-container">$c$</span>, where <span class="math-container">$c$</span> is the continuum.</p>
<p>I already proved <span class="math-container">$|(a,b)| = |... | RRL | 148,510 | <p>For <span class="math-container">$0 < y < x < 1$</span>, by Taylor's theorem there exists <span class="math-container">$\theta \in (0,1)$</span> such that</p>
<p><span class="math-container">$$f(y) = f(x) + f'(x)(y-x) + \frac{1}{2} f''(x - \theta(x-y)) (y-x)^2$$</span></p>
<p>Taking <span class="math-con... |
1,100,266 | <p>I begin with this:</p>
<blockquote>
<p>If $f:[a,b]\rightarrow [a,b]$ is continuous and $|f(x)-f(y)|<|x-y|$ for all $x\neq y$, then the equation $f(x)=x$ has exactly one root.</p>
</blockquote>
<p>The problem is not very hard and I solved it. Then I wonder if this problem is true or not:</p>
<blockquote>
<p... | hmakholm left over Monica | 14,366 | <ol>
<li><p>Some counterexamples include:</p>
<p>$$ f_1(x) = \begin{cases} 2 & x\le 1 \\
x+1/x & x \ge 1 \end{cases} $$
or
$$ f_2(x) = \sqrt{1+x^2} $$</p>
<p>In both cases there is no $x$ such that $f(x)=x$.</p></li>
<li><p>You need to add hypotheses that ensure that $f(x)=x$ has at least one solution. In the... |
918,510 | <p>I am trying to solve the following problem :
Find all the positive integers $n$ and $k$ such that it is possible to write integers in an $n \times n$ grid so that the sum of all elements in the grid is negative but the sum of elements of each $k \times k$ grid contained in it is positive. I am only looking for a sm... | almagest | 172,006 | <p>Here is a way of putting numbers on an $n\times n$ board to satisfy the conditions if $n$ is not a multiple of $k$.</p>
<p>Suppose $n=qk+r$ with $0<r<k$. The idea is to put $-N$ into every cell in row 1, then to have $k-1$ rows with $+m$ in every cell. Then repeat, so a row of $-N$ followed by $k-1$ rows with... |
1,606,709 | <p>I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The cotangent sheaf is later defined in terms of the diagonal map. Why is this geometrically interesting? I tried to write a s... | Community | -1 | <p>Let $$x = \prod_{n = 1}^{\infty} y_n \qquad \text{and} \qquad y_n = \sqrt {\frac {1} {2} + \sqrt {\frac {1} {2} + \cdots + \sqrt {\frac {1} {2}}}},$$ and $$y = \lim_{n \to \infty} y_n.$$ We have $$y_{n + 1} = \sqrt {\frac {1} {2} + y_n},$$ which, passing to limit, gives $$y = \sqrt {\frac {1} {2} + y}.$$ We have $y ... |
1,022,184 | <p>A game is played as follows: A random number $X$ is chosen uniformly from $[0,
1]$. Then a sequence $Y_1, Y_2,\ldots$ of random numbers is chosen independently
and uniformly from $[0, 1]$. The game ends the first time that $Y_i > X$. You are
then paid $(i-1)$ dollars. What is a fair entrance fee for this game?</p... | Ross Millikan | 1,827 | <p>Hint: If I told you $X$, can you calculate the value $V(X)$? Then since $X$ is selected uniformly, the overall value is $\int_0^1V(X)\ dX$</p>
|
3,565,632 | <blockquote>
<p>Given <span class="math-container">$x_1 := a > 0$</span> and <span class="math-container">$x_{n+1} := x_n + \frac{1}{x_n}$</span> for <span class="math-container">$n \in \mathbb{N}$</span>, determine whether <span class="math-container">$(x_n)$</span> converges or diverges.</p>
</blockquote>
<p>Si... | LHF | 744,207 | <p>I will assume (since it's not specified) that <span class="math-container">$n$</span> is a non-negative integer.</p>
<p><strong>Claim:</strong> For <span class="math-container">$f:(0,\infty)\to \mathbb{R}$</span>, defined by <span class="math-container">$f(x)=\dfrac{\sin x}{x}$</span> and any non-negative integer <... |
2,069,849 | <p>For a given $x\in\left\{ 0,1\right\} ^{n}$, define $x_{\setminus k}=\left(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{n}\right)$. [Is there standard notation for this?]</p>
<p><strong>Prove or disprove the following claim:</strong></p>
<p>Let $X$ be some random variable with values in $\left\{ 0,1\right\} ^{n}$. There ex... | setholopolus | 397,375 | <p>$$\sum_{i = 1}^{12} \sum_{j = 1}^i jf(j)$$</p>
<p>I believe that gives you everything that you would end up with over the twelve days. </p>
<p>To get the lyrics, you would want </p>
<p>$\sum_{i = 1}^{12}$"On the $i$-th day of Christmas my true love gave to me" + $\left( \sum_{j = -i}^{-1} -j + f(-j) \right)$</p>... |
2,069,849 | <p>For a given $x\in\left\{ 0,1\right\} ^{n}$, define $x_{\setminus k}=\left(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{n}\right)$. [Is there standard notation for this?]</p>
<p><strong>Prove or disprove the following claim:</strong></p>
<p>Let $X$ be some random variable with values in $\left\{ 0,1\right\} ^{n}$. There ex... | Hypergeometricx | 168,053 | <p>Define
$$g(n)=\text{On the }n\text{-th day of Christmas my true love gave to me}$$
and the operator $\circ$ as the concatenation of two text strings. </p>
<p>The lyrics of the song can then be represented as
$$\color{red}{\boxed{\sum_{n=1}^{12} g(n)\circ \sum_{m=1}^n (n+1-m)\circ f(n+1-m)}}$$</p>
<p><em>Note</em>... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | Calum Gilhooley | 213,690 | <p>The three normals <span class="math-container">$n_1, n_2, n_3$</span> all lie in a plane <span class="math-container">$P$</span> through the origin, because <span class="math-container">$n_1 - n_2 = n_3.$</span> The three given planes are orthogonal to <span class="math-container">$P.$</span> If their lines of inter... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | Carsten S | 90,962 | <p>Good job looking at the normals instead of blindly computing intersections! Indeed the point is that the three vectors <span class="math-container">$(1, -3, 2)$</span>, <span class="math-container">$(1, 3, -2)$</span>, <span class="math-container">$(0,-6, 4)$</span> are linearly dependent but the vectors <span class... |
3,243,328 | <p>X is a random variable with values from <span class="math-container">$\Bbb N\setminus{0}$</span></p>
<p>I am trying to show that <span class="math-container">$E[X^2]$</span> = <span class="math-container">$\sum_{n=1}^\infty (2n-1) P(X\ge n)$</span> iff <span class="math-container">$E[X^2]$</span> < <span class="... | Sri-Amirthan Theivendran | 302,692 | <p>Using a bit more machinery.</p>
<p>First note that
<span class="math-container">$$
X^2=\sum_{n=1}^\infty(2n-1)I(X\geq n)\tag{0}
$$</span>
with probability one where <span class="math-container">$I$</span> is the indicator function. To see that this identity is true note that when <span class="math-container">$X^2=k... |
1,344,690 | <p>I was wondering how to find the vertices of an equilateral triangle given its center point?</p>
<p>Such as:</p>
<pre><code> A
/\
/ \
/ \
/ M \
B /________\ C
</code></pre>
<p>Provided that <code>AB, AC, BC = x</code> and <code>M = (50,50)</code> and <code>M</code> is the ... | Ashkay | 246,967 | <p>Note that we are assuming that $BC$ is parallel to the coordinate axis. Now, we can get all the horizontal and vertical distances we need to solve this problem.</p>
<p>Let $P$ be the midpoint of $BC$. Note that we have $PC = BP = x/2$ and $MP = (x/2)/\sqrt{3}$. Finally, $MA = x/\sqrt{3}$. This should be enough to g... |
1,344,690 | <p>I was wondering how to find the vertices of an equilateral triangle given its center point?</p>
<p>Such as:</p>
<pre><code> A
/\
/ \
/ \
/ M \
B /________\ C
</code></pre>
<p>Provided that <code>AB, AC, BC = x</code> and <code>M = (50,50)</code> and <code>M</code> is the ... | Rory Daulton | 161,807 | <p>Let's assume that <span class="math-container">$x$</span>, the side of the equilateral triangle, is a known positive quantity and that side <span class="math-container">$BC$</span> is horizontal (or that point <span class="math-container">$A$</span> is directly above point <span class="math-container">$M$</span>). L... |
2,456,561 | <p>A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour.</p>
<p>Why is the answer not the following:</p>
<p><a href="https://i.stack.imgur.com/25ImM.png" rel="nofollow nore... | Graham Kemp | 135,106 | <p>Take a simpler example. Six chips in the bowel, 2 of each color. There are $6$ distinct ways to pick five chips (in no particular order). Count ways to pick five chips of at least one in each colour.</p>
<p>Your method would have $\binom 21\binom 21\binom 21\binom 32 = 24$, which is clearly mor... |
1,448,062 | <p>Find $\overline{\lim}\limits_{n\to\infty}\left(\frac{1}{n}-\frac{2}{n}+\frac{3}{n}-...+(-1)^{n-1}\frac{n}{n}\right)$</p>
<p>$\frac{1}{n}-\frac{2}{n}+\frac{3}{n}-...+(-1)^{n-1}\frac{n}{n}=\frac{1}{n}\sum\limits_{k=1}^{n}(-1)^{k-1}k$</p>
<p>How to evaluate sum $\sum\limits_{k=1}^{n}(-1)^{k-1}k$?</p>
| thanasissdr | 124,031 | <p>We see that we have $3 E$ 's, $2S$ 's, $2Y$ 's and all the other letters appear once. If we don't take account the restriction all the possible permutations are:
$$\frac{12!}{3!\cdot 2!\cdot 2!\cdot \underbrace{1!\cdots 1!}_{5 \text{ times }}}.$$</p>
<p>Now, consider the string $"YS"$ as one letter. Then we have $... |
2,527,754 | <p>Denote the Borel sets in $\mathbb R^d$ as $\mathcal B^d$.
Is there a proof for the rotation invariance of the Lebesgue measure that doesn't use already that one has
$$ \lambda(A^{-1}(B)) = \vert \operatorname{det} A \vert ^{-1} \lambda (B) \qquad \text{for all } A \in \operatorname{GL}(\mathbb R^d), \ B \in \mathca... | Aloizio Macedo | 59,234 | <p>You can prove that associated to every linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ there exists a number $\alpha(T)$ such that
$$\lambda(T(E))=\alpha(T)\cdot \lambda(E)$$
for all $E \in \mathcal{B}^d$. You don't need to prove that $\alpha(T)=\mathrm{det}(T)$, and the proof of the statement above is quit... |
629,340 | <p>Is it true that if for each partition of a graph G's vertices into two non empty sets there is an edge with end points in both sides then G is connected? Intuitively this seems true to me. But I cannot prove this. I would very much appreciate some assistance. Thanks </p>
| Kuai | 39,515 | <p>The statement is correct and one can show its contraposition as follows. Let $G(V,E)$ be a disconnected graph. Then by definition there exists a pair of vertices $i,j\in V$ such that there is no edge between them: $(i,j)\notin E$. Then we let $V_1$ and $V_2$ be a partition of $V$ such that $i\in V_1$ and $j\in V_2$ ... |
1,273,477 | <blockquote>
<p>What is the unit normal vector of the curve $y + x^2 = 1$, $-1 \leq x \leq 1$? </p>
</blockquote>
<p>I need this to calculate the flux integral of a vector field over that curve.</p>
| Ángel Mario Gallegos | 67,622 | <p>A parametric equation for the curve is $$\mathbf{r}(t)=t\mathbf{i}+(1-t^2)\mathbf{j}\qquad t\in[-1,1]$$
We can find the unit tangent vector as $$\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\left|\mathbf{r}'(t)\right|}$$
After that, the unit normal vector can be find as
$$\mathbf{N}(t)=\frac{\mathbf{T}'(t)}{\left|\mathbf{T}'... |
54,393 | <p>I used to think that in any Vector space the space spanned by a set of orthogonal
basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ and the Fourier basis which spans this vector space, the same is not true ! I'd like to get clarified on possible mi... | Qiaochu Yuan | 232 | <p>If by "the Fourier basis" you mean the functions $e^{2 \pi i n x}, n \in \mathbb{Z}$, then these functions do not lie in $L^2(\mathbb{R})$ as they are not square-integrable over $\mathbb{R}$, so in particular they can't span that space in any reasonable sense. The functions $e^{2 \pi i n x}$ <em>do</em> span $L^2(S^... |
14,568 | <p>The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable normalization). <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Here</a> is the wikipedia link.</p>
<p>For m... | MBN | 927 | <p>It is a special case of the trace formula. Both sides are the trace of the same operator.</p>
|
1,850,892 | <p>Suppose we need to prove a statement of the form $$\forall n\in\mathbb{N}(P(n)\to Q(n))$$ where $P(n)$ and $Q(n)$ are propositions using mathematical induction. Say for the base case $n=1$ it is true. Let $n=k>1$ and assume the statement is true (inductive hypothesis), that is $P(k)\to Q(k)$ is valid. If we are a... | joriki | 6,622 | <p>The slides are a bit unclear on your first point, but from the formulation "since $3$ of the $5$ individuals in the system are Species A" we can infer that birth and death are considered to occur simultaneously, and the individual selected for death has the same chance of giving birth as all others.</p>
<p>On your ... |
1,850,892 | <p>Suppose we need to prove a statement of the form $$\forall n\in\mathbb{N}(P(n)\to Q(n))$$ where $P(n)$ and $Q(n)$ are propositions using mathematical induction. Say for the base case $n=1$ it is true. Let $n=k>1$ and assume the statement is true (inductive hypothesis), that is $P(k)\to Q(k)$ is valid. If we are a... | Satish Ramanathan | 99,745 | <p>$$P = \begin{bmatrix}\\Row&5A0B & 4A1B & 3A2B & 2A3B & 1A4B & 0A5B \\ 5A0B & 1 & 0 & 0 &0&0 &0\\4A1B& 0.16 & 0.68 & 0.16 & 0 & 0 &0\\3A2B &0&0.24&0.52&0.24&0&0\\2A3B&0&0&0.24&0.52&0.24&0\\1A4B&... |
4,036,903 | <p>Suppose <span class="math-container">$\sigma_1:\Delta^k \rightarrow X$</span> is a singular <span class="math-container">$k$</span>-simplex and <span class="math-container">$\sigma_2:\Delta^l \rightarrow X$</span> is a singular <span class="math-container">$l$</span>-simplex. Is there a singular <span class="math-co... | JohannesPauling | 876,486 | <p>Well, since you want a singular simplex you only require the mapping <span class="math-container">$\sigma: \Delta^{k+l}\to X$</span> to be continuous. So you can certainly, given <span class="math-container">$\sigma_1$</span> and <span class="math-container">$\sigma_2$</span>, identify <span class="math-container">$... |
3,814,502 | <p>I came across the following question:</p>
<blockquote>
<p>Show that <span class="math-container">$(a, b:a^3 = 1, b^2= 1, ba=a^2b)$</span> gives a group of order <span class="math-container">$6$</span>. Show that it is non abelian. Is it the only non abelian group of order <span class="math-container">$6$</span> up t... | user-492177 | 492,177 | <p>Let <span class="math-container">$G$</span> be a group of order <span class="math-container">$6$</span></p>
<p>Then there are elements of order <span class="math-container">$3$</span> and <span class="math-container">$2$</span>, say <span class="math-container">$a$</span> and <span class="math-container">$b$</span> ... |
924,177 | <p>I'm trying to draw quadratic bezier curve (as line).
I approximate quadratic bezier curve as parabola ($y=x^2$), according to this document <a href="http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html" rel="noreferrer">http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html</a></p>
<p>There, in secti... | Bob Dobbs | 221,315 | <p>In my opinion, sd is not a distance at all, signed or not.</p>
<p>By analytic geometry, the distance of a point <span class="math-container">$P=(x_0,y_0)$</span> to the line given by the equation <span class="math-container">$f(x,y)=ax+by+c=0$</span> is <span class="math-container">$d=\frac{|ax_0+by_0+c|}{\sqrt{a^2+... |
2,620,529 | <p>Suppose $(a_n)$ is a sequence in $X$ on which there are defined two topologies $T_1,T_2$. If $(a_n)$ converges to a unique limit $x_1$ wrt $T_1$, and to a unique limit $x_2$ wrt $T_2$, is it necessary that $x_1=x_2$? If not are there any nice counter-examples?</p>
<p>I have tried to show that this is false using a ... | Tsemo Aristide | 280,301 | <p>$X=\{a,b\}$ $T_1=\{a,b\}, \{a\}$ and $T_2=\{a,b\}, \{b\}$ $a_{2n}=a, a_{2n+1}=b$.</p>
|
2,620,529 | <p>Suppose $(a_n)$ is a sequence in $X$ on which there are defined two topologies $T_1,T_2$. If $(a_n)$ converges to a unique limit $x_1$ wrt $T_1$, and to a unique limit $x_2$ wrt $T_2$, is it necessary that $x_1=x_2$? If not are there any nice counter-examples?</p>
<p>I have tried to show that this is false using a ... | freakish | 340,986 | <p>Simple example: consider $X=\mathbb{N}$ and for any $k\in\mathbb{N}$ define $T_k$ to be the smallest topology generated by</p>
<p>$$\text{the interval }[k, \infty)$$
$$\text{almost all singletons }\big\{\{s\}_{s\neq k}\big\}$$</p>
<p>Now consider the sequence $a_n=n$. It can be easily seen that the only two open s... |
3,422,505 | <p>The problem is as follows:</p>
<blockquote>
<p>In a certain shopping mall which is many stories high there is a glass
elevator in the middle plaza. One shopper ridding the elevator notices
a kid drops a spheric toy from the top of the building where is
located the toy store. The shopper riding the elevator ... | AgentS | 168,854 | <p><code>That would be the real speed of the sphere at that instant. My intuition tells me that the observer will see the ball going faster? and how about the acceleration?</code></p>
<p>No. Since both the sphere and the observer are going down, the observer will see the ball going slower. As an example, suppose you'r... |
176,033 | <p>What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know it's been done for $n \leq 7$ in 1970's. Has there been any progress since then? If not, is it because there is some... | Alexey Ustinov | 5,712 | <p>In the Lecture XV from <em>Siegel's "Lectures on the Geometry of Numbers"</em> is written that the volume of fundamental domain $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$ is something like
$$\frac{\zeta(2)\zeta(3)\ldots\zeta(n)}n.$$</p>
|
1,828,669 | <p>Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old <a href="http://math.mit.edu/classes/18.158/bourgain-restriction.pdf" rel="nofollow">paper</a> of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118)</p>
<p>$$\sum_{q=1}^{N^{1/2}-1}\sum_{{1\leq a &... | Ashwin Ganesan | 157,927 | <p>Let $G$ be a bipartite graph with bipartition $(A,B)$. Suppose $G$ satisfies Hall's condition. We need to show $G$ has a complete matching from $A$ to $B$. </p>
<p>Form a directed graph from $G$ by adding a source vertex $s$, sink vertex $t$, joining $s$ to each vertex in $A$ and joining each vertex in $B$ to $t$.... |
1,957,166 | <p>For a given set $A$,
An element such that $a \in A $ exists. </p>
<p>If $A$ is a set of all natural numbers, then:</p>
<p>$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$</p>
<p>Would maths normally be written like this, if it is correct? </p>
| ClownInTheMoon | 367,034 | <p>This question is a bit confusing and no it doesn't make a lot of "sense" overall. Especially given that $A$ being defined as the set of all natural numbers means $A\not\in\mathbb{N}$ but that $A=\mathbb{N}$.</p>
<p>As for your question, would maths normally be written like this... yes, those are all valid mathemati... |
1,429,306 | <p>As is well known Minkowski spacetime (which is four dimensional vector space with scalar product $\eta _{\mu \nu}$ of signature $-+++$) is maximally symmetric, which manifests itself in presence of ten Killing vector fields. Those are generators of one parameter groups of isometries, which can be understood as trans... | carlos arturo Hurtado | 264,995 | <p>Probably these results will serve to resolve your question</p>
<ol>
<li>In an arbitrary ring $R$, if $rad(R)\neq R$ then $rad(R)=\{r\in R| \forall x,y\in R: \;xry\mbox{ is right quasi-regular}\}$</li>
</ol>
<p>proof: Remember that $Rad(M)=\bigcap\{M|M\mbox{ maximal modular right ideal of }R\}$ and let $H=\{r\in R... |
3,180,628 | <p>The following is a quotation from Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(6.7 The Affine n-Space) .</p>
<blockquote>
<p>For any <span class="math-container">$R$</span>-algebra <span class="math-container">$R'$</span> let <span class="math-container">$\mathbb{A}^n_S(R')$</span> be the set of al... | Georges Elencwajg | 3,217 | <p>Of course I can't read Bosch's mind but his notation might be preparation for a more general situation, namely:<br>
Let <span class="math-container">$I\subset R[T_1,\cdots,T_n]$</span> be an ideal and define for any <span class="math-container">$R$</span>-algebra <span class="math-container">$R'$</span> the subse... |
2,023,500 | <p>So I need to prove that $a_{n+1}=1+1/a_n$, $a_1=1$ converges by the contraction principle.</p>
<p>That means $|a_{n+2}-a_{n+1}|\le k|a_{n+1}-a_n|$ holds for any $n$, for a certain $0<k<1$. </p>
<p>Now this question is very closely related to <a href="https://math.stackexchange.com/questions/2022897/verify-th... | Med | 261,160 | <p>We want to maximise this probability.</p>
<p>$P(X_1,X_2,...,X_n|\theta)$</p>
<p>A formula is used to get</p>
<p>$P(X_1,X_2,...,X_n|\theta)=P(X_1|\theta)P(X_2|X_1,\theta)...P(X_n|X_1,X_2,...,X_n,\theta)$</p>
<p>If the experiments are independent, then the last result can be simplified to</p>
<p>$P(X_1,X_2,...,X_... |
2,023,500 | <p>So I need to prove that $a_{n+1}=1+1/a_n$, $a_1=1$ converges by the contraction principle.</p>
<p>That means $|a_{n+2}-a_{n+1}|\le k|a_{n+1}-a_n|$ holds for any $n$, for a certain $0<k<1$. </p>
<p>Now this question is very closely related to <a href="https://math.stackexchange.com/questions/2022897/verify-th... | user365239 | 365,239 | <p>Your likelihood function is
$$
L(\theta) = \theta^n \left[ \prod_{i=1}^n x_i^{-2} \cdot \mathbb{1}_{[x_i \ge \theta]} \right]
=\theta^n \cdot \mathbb{1}_{[x_{(1)} \ge \theta]} \left[ \prod_{i=1}^n x_i^{-2} \right]
$$
Now, $L(\theta)$ is increasing in $\theta$ as long as $\theta \le x_{(1)}$, and so the MLE is $... |
3,423,840 | <p><a href="https://i.stack.imgur.com/batlm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/batlm.jpg" alt="enter image description here"></a><br>
I used the ratio test for this problem and I messed up somewhere in the algebra. Sorry for the messy writing. <a href="https://i.stack.imgur.com/hRMUF.png... | carsandpulsars | 476,417 | <p>You dropped the <span class="math-container">$n^2$</span> on the denominator!</p>
<p><span class="math-container">$$\lim_{n \to \infty} \left| \frac{(n+1)^2}{4(n+1)n^2}x \right| = \lim_{n \to \infty} \left| \frac{n+1}{4n^2} x \right| = |x|\lim_{n \to \infty} \left| \frac{n+1}{4n^2} \right| = 0$$</span></p>
<p>From... |
231,549 | <p>Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent matrix and an identity matrix? I am not looking so much for name suggestions, but rather for a generally accepted term... | Allen Knutson | 391 | <p>The closest term I know is "idempotent", though that's only for $n=1$.</p>
|
1,903,762 | <blockquote>
<p>Largest number that leaves same remainder while dividing 5958, 5430 and 5814 ?</p>
</blockquote>
<hr>
<p>$$5958 \equiv 5430 \equiv 5814 \pmod x$$
$$3\times 17\times 19 \equiv 5\times 181\equiv 3\times 331\pmod x$$
$$969 \equiv 905\equiv 993\pmod x$$</p>
<p>After a bit of playing with the calculato... | Roby5 | 243,045 | <p>Let the number leaving the same remainder be $a$ and the remainder be $b$.</p>
<p>So, $\gcd(5958-a,5430-a,5814-a)=b$</p>
<p>Note that </p>
<p>$(1)$ $$\gcd(x,y,z)=\gcd{\gcd(x,y)\gcd(y,z)}$$
$(2)$ $$\gcd(x,y) \mid x-y$$</p>
|
1,371,075 | <p>$$3^x = 3 - x$$</p>
<p>I have to prove that only one solution exists, and then find that one solution.</p>
<p>My approach has been the following:</p>
<p>$$\log 3^x = \log (3 - x)$$</p>
<p>$$x\log 3 = \log (3 - x)$$</p>
<p>$$\log 3 = \frac{\log (3 - x)}{x}$$</p>
<p>And this is where I get stuck. Any help will b... | Mick | 42,351 | <p>Edited version</p>
<p>After $x \cdot \log 3 = \log (3 – x)$ is $x = \frac {\log (3 – x)}{\log 3} $</p>
<p>which is equivalent to $y = x$ and $y = \frac {\log (3 – x)}{\log 3} $</p>
<p>Plotting the above TWO curves on the same graph, you will find they intersect at only one point (i.e. one solution).</p>
<p>Note ... |
6,534 | <p>I apologize if this isn't the right place to ask this question.</p>
<p>Two features of stackexchange would be very useful for a personal math blog -- Latex works great, and comments and replies can be voted upon.</p>
<p>Is there any way to use the stackexchange functionality in a personal math blog?</p>
| ncmathsadist | 4,154 | <p>I use mathJax on plain old XHTML pages, and it displays math beautifully.</p>
|
6,534 | <p>I apologize if this isn't the right place to ask this question.</p>
<p>Two features of stackexchange would be very useful for a personal math blog -- Latex works great, and comments and replies can be voted upon.</p>
<p>Is there any way to use the stackexchange functionality in a personal math blog?</p>
| GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Use <a href="https://docs.gitlab.com/ee/user/markdown.html#math" rel="nofollow noreferrer"><span class="math-container">$\rm\LaTeX$</span> on GitLab</a> directly, thanks to GitLab's support for <a href="https://katex.org" rel="nofollow noreferrer">KaTeX</a>.</p>
<p><a href="https://gitlab.com/gitlab-org/gitlab/blob/... |
2,174,979 | <p>I am trying to prove that if a function $f:\mathbb{R} \to \mathbb{R}$ is bounded and the second derivative $f''$ is bounded, then the first derivative $f'$ is also bounded. My hint is to use Taylor's theorem.</p>
<p>I know for any $x$ and $y$, $f(y) = f(x) + f'(x)(y-x) + \frac{1}{2} f''(c)(y-x)^2,$ is the Taylor a... | Drinzjeng Triang | 736,809 | <p>Here is a solution with Tyler's expansion:
<span class="math-container">$$
\forall x\in\mathbb{R},\forall h>0,\\\
f(x+h)=f(x)+hf'(x)+\frac{h^2}{2}f''(\eta_1)\text{ ,where $\eta_1\in(x,x+h)$}\\
f(x-h)=f(x)-hf'(x)+\frac{h^2}{2}f''(\eta_2)\text{ ,where $\eta_2\in(x-h,x)$}\\
\Rightarrow f(x+h)-f(x-h)=2hf'(x)+\frac{h^... |
262,655 | <p>This question arose from the recent one, <a href="https://mathoverflow.net/q/262380/41291">roots of a polynomial linked to mock theta function?</a>. Let
$$
g(x):=\sum_{k=0}^\infty x^k\prod_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+21 x^8+30 x^9+43 x^{10}+59 x^{11}+...;
$$
the sequence $1,1,1,3... | Nemo | 82,588 | <p>The conjectured identity
$$
f(q)=(q;q)_\infty\left(1+\sum_{k=1}^\infty q^k(-q;q)^2_{k-1}\right)=\sum_{\substack{m,n\geqslant0\\n\ne1}}(-1)^mq^{\frac{(m+n)(3m+n+1)}2},\tag{1}
$$
using Euler's pentagonal number theorem $(q;q)_\infty=\sum _{m=-\infty}^\infty (-1)^m q^{\frac{1}{2} m (3 m+1)}$ can be brought to an equiva... |
1,131,092 | <p>I've come across many different versions of this question on here, but not any that map the $[0,1]$ to $(1, \infty)$. </p>
<p>I was thinking that it must be piece-wise defined, since the endpoints 0 and 1 will be the trickiest part of defining the bijection... The only method of doing this that I could come up with... | Paul Vithayathil | 211,283 | <p>You can consider a piece-wise function defined where 0 maps to some value y1 not equal to 1 (say 2). Then you can manipulate the period and height of tan(x) such that tan(0) = 1 and tan(1) is infinite.</p>
|
72,209 | <p>Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$.</p>
<p>(a) Determine in which of the senses (a.s., m.s., p., d.) the sequence $\{X_n\}$ converges as $n\to\infty$, and identify the... | Greg Martin | 16,078 | <p>Let $T_k = \log U_k$ and $Y_n = \log X_n$. Then the $T_k$ are iid random variables with a particular distribution over the ray $(-\infty,\log 2]$ that can be worked out. You can now prove whatever you want about the convergence of $Y_n$ using what you know about sums of random variables; the answers can be translate... |
235,145 | <p>I have the following 3D-model:</p>
<pre><code>RevolutionPlot3D[Sqrt[E^-x (1 + E^x)^2], {x, 0, 4},
RevolutionAxis -> {1, 0, 0}]
</code></pre>
<p>Is there a way to make this model into a solid so that I can export it to let it be printed by a 3D printer. Now it is kind of hollow from the inside but I want it to b... | Tim Laska | 61,809 | <p>Here's another way to do it using <a href="https://reference.wolfram.com/language/ref/ImplicitRegion.html" rel="nofollow noreferrer"><code>ImplicitRegion</code></a> and the finite element method package.</p>
<pre><code>Needs["NDSolve`FEM`"]
ℛ =
ImplicitRegion[y^2 + z^2 <= (Sqrt[E^-x (1 + E^x)^2])^2, ... |
105,413 | <p>I know: I'm going to make a poor showing, but really I can't understand this:</p>
<p><code>a</code> is an expressione whose FullForm is</p>
<pre><code>Power[Plus[Subscript[u,x],Times[Complex[0,-1],Subscript[u,y]],Times[Complex[0,1],Subscript[v,x]],Subscript[v,y]],2]
</code></pre>
<p>Why does the following code re... | Daniel Lichtblau | 51 | <p>The original formulation was close, and the level spec of <code>Infinity</code> "almost" worked. As was noted in comments, it does work if <code>Rule</code> is replaced by <code>RuleDelayed</code>. The reason it otherwise causes trouble is from a "variable capture" in scoping. The pattern variables, <code>a_</code> ... |
105,158 | <p>Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also the case that these matrices (for all positive $j$ multiple of $1/2$) are a representation of the rotation group $SU(2)... | heisenBug | 154,067 | <h1>Cheat Code: spherical Bessel roots <em>are</em> their respective half-integer Bessel roots</h1>
<p>According to <a href="http://people.math.sfu.ca/~cbm/aands/page_440.htm" rel="nofollow">Abramowitz, 1964, Ch9, pp 440,"Zeros and Their Asymptotic Expansions"</a> </p>
<blockquote>
<p>The zeros of $j_{n}(x)$ and $y... |
2,242,865 | <p>I have seen proofs using the delta-epsilon definition of continuity, and they make perfect sense, but I have not found one proof using the sequential definition of continuity. </p>
<p>For example, when given functions, $f$ and $g$ that are continuous on [$a,b$], prove that the function $h=f+g$ is also continuous on... | Yes | 155,328 | <p>Yes a proof using the sequential definition exists. In my opinion the reason that most textbooks do not prove the results using the sequential definition is because usually the definition of convergence of a sequence comes <strong>after</strong> the concept of limit. If the definition of convergence of a sequence co... |
108,209 | <p>A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global information, or alternatively, lack a complete, instantaneous understanding of the structure of the problem? </p>
<p... | Dima Pasechnik | 11,100 | <p>An $n$-variate polynomial of degree 4 can have exponentially many local minima. Indeed, they can be "written down" as solutions of the the corresponding systems of cubic equations, but this doesn't really help you to find a global minimum. </p>
<p><b>Edit</b>: e.g. take $f(x_1,\dots x_n)=\sum_{k=1}^n (1-x_k^2)^2.$ ... |
219,165 | <p>I have the following equation which I want to solve it for r:</p>
<pre><code>TGB5 = (-((2 q^2)/r^5) + 2/r + (16 P \[Pi] r)/3)/(4 \[Pi]);
rlarge5 = Last[r /. Solve[TGB5 == T, r, Reals]] // Normal
</code></pre>
<p>The answer is a root object:</p>
<blockquote>
<p>Root[-3 q^2 + 3 #1^4 - 6 [Pi] T #1^5 + 8 P [Pi] #1... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "12.1.0 for Mac OS X x86 (64-bit) (March 14, 2020)" *)
Clear["Global`*"]
TGB5 = (-((2 q^2)/r^5) + 2/r + (16 P π r)/3)/(4 π);
rlarge5 = Last[r /. Solve[TGB5 == T, r, Reals]] // Normal
(* Root[-3 q^2 + 3 #1^4 - 6 π T #1^5 + 8 P π #1^6 &, 4] *)
</code></pre>
<p>Since all variables are real... |
3,189,462 | <p>Question:
<span class="math-container">$$2x^{2}y''-6xy'+6y=x^{3} $$</span></p>
<p>I have tried solving it, and obtained the roots 3 and 1. Apparently this is a case of resonance. Even so, I wasn't able to get the particular solution through variation of parameters.</p>
<p>Perhaps there are other methods to solve t... | parsiad | 64,601 | <p><strong>Hint</strong>:
<span class="math-container">\begin{align*}
\mathbb{P}(W_{1}\geq0,W_{2}\geq0) & =\mathbb{P}(W_{1}\geq0,(W_{2}-W_{1})+W_{1}\geq0)\\
& =\mathbb{P}(W_{1}\geq0,(W_{2}-W_{1})\geq-W_{1})\\
& =\mathbb{P}(X\geq0,Y\geq-X)
\end{align*}</span>
where <span class="math-container">$X$</span> a... |
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