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 |
|---|---|---|---|---|
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| mathreadler | 213,607 | <p>$$(x-a)^2 = (x+a)^2$$
Expand squares:
$$x^2 - 2ax + a^2 = x^2 + 2ax + a^2$$
Subtract $x^2+a^2$ from both sides:
$$-2ax = 2ax$$
add $2ax$ and divide by $4$:
$$0 = ax$$</p>
<p>So either 1) $a = 0$ or 2) $x = 0$.</p>
<ol>
<li>$$(x-0)^2 = (x+0)^2 \Leftrightarrow x^2 = x^2$$
Which is true for all x.</li>
<li>$$(0-a)^2 ... |
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| Narasimham | 95,860 | <p>Simplifying, after expansion of squares you get</p>
<p>$$ a \cdot x = 0 $$</p>
<p>either or both of them can be zero.</p>
<p>However you are specifically given that <em>for all values of $x$</em>... So <em>do not put</em> a particular $ x=0. $</p>
<p>Only choice is $ a=0. $</p>
<p>Also for any function if $ f(... |
102,304 | <p>I have here a complex equation:</p>
<p>$$z^2 - (7+j)z + 24 +j7 = 0$$</p>
<p>How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much complexity on it. Is there any way to directly attack this? Thanks.</p>
| Did | 6,179 | <p>One can <a href="http://en.wikipedia.org/wiki/Completing_the_square" rel="nofollow">complete the square</a>, that is, write $z^2-(7+j)z$ as the beginning of the expansion of
$$
\left(z-\frac12(7+j)\right)^2.
$$
This yields
$$
z^2-(7+j)z+24+7j=\left(z-\tfrac12(7+j)\right)^2-u,
$$
with
$$
u=\tfrac14(7+j)^2-24-7j.... |
235,661 | <p>Is this sufficient? Also, any good books/other suggestions regarding the subject will be very helpful.</p>
<p>Find min, max, inf, sup (if they exist):</p>
<p>$$B=\left\{\frac{m}{m+n}:m,n\in\mathbb{N}\right\}$$</p>
<p>Showing B has an upper bound:
Let $M=1$, we need to find $m,n$ fulfilling:$$\frac{m}{m+n}>1$$
... | DonAntonio | 31,254 | <p>If you take $\,\Bbb N=\{1,2,3,...\}\,$, then I think you'll agree with</p>
<p>$$\forall\,\,m,n,\in\Bbb N\,\,\,,\,\,\frac{m}{m+n}>0\Longrightarrow 0\,\,\text{is a lower bound for}\,\,M\,...$$</p>
<p>I think it'd be a good idea to try to prove that zero is actually the infimum of $\,M\,$</p>
|
235,661 | <p>Is this sufficient? Also, any good books/other suggestions regarding the subject will be very helpful.</p>
<p>Find min, max, inf, sup (if they exist):</p>
<p>$$B=\left\{\frac{m}{m+n}:m,n\in\mathbb{N}\right\}$$</p>
<p>Showing B has an upper bound:
Let $M=1$, we need to find $m,n$ fulfilling:$$\frac{m}{m+n}>1$$
... | ackshooairy | 47,570 | <p>Consider first case where the value m is much larger than the value of n. $n << m$</p>
<p>Then consider the case where the value n is much larger than the value m. $m <<n$ </p>
<p>Write out a few iterations and you'll see where each one is headed. That will give you the supremum and infimum.</p>
<p>I ... |
1,085,279 | <p>There is a <a href="https://math.stackexchange.com/questions/265619/meaning-of-normalization">question</a> already asked here about this. But I know almost nothing of algebraic geometry, nothing fancy to understand the answer. So I would highly appreciate an elementary explanation to my question.</p>
<p>I encounter... | Georges Elencwajg | 3,217 | <p>0) Recall that a domain $A$ is said to be normal if it is integrally closed in its fraction field $K=Frac(A)$.<br>
This means that any element $q\in K$ killed by a monic polynomial in $A[T]$, i.e. such that for some $n\gt 0, a_i\in A$ one has $$q^n+a_1q^{n-1}+\cdots+a_n=0$$ already satisfies $ q\in A$ .<br>
A v... |
1,085,279 | <p>There is a <a href="https://math.stackexchange.com/questions/265619/meaning-of-normalization">question</a> already asked here about this. But I know almost nothing of algebraic geometry, nothing fancy to understand the answer. So I would highly appreciate an elementary explanation to my question.</p>
<p>I encounter... | Takumi Murayama | 116,766 | <p>I like Georges Elencwajg's answer, but I think it's useful to see some topological intuition for what normalization does over <span class="math-container">$\mathbf{C}$</span>.</p>
<p>Note we say a variety is <strong>normal</strong> if its local rings are integrally closed in their fraction field.</p>
<h2>Riemann Ext... |
2,275,951 | <p>The parabola y=x² is parameterized by x(t) = t and y(t) = t². At the point <strong>A</strong> (t,t²) a line segment <strong>AP</strong> 1 unit long is drawn normal to the parabola extending inward. Find the parametric equations of the curve traced by the point <strong>P</strong> as <strong>A</strong> moves along the... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
4,146,629 | <p>I'm reading D. E. Knuth's book "Surreal Numbers". And I'm completely stuck in chap. 6 (The Third Day) because there is a proof I don't understand. Alice says</p>
<blockquote>
<p>Suppose at the end of <span class="math-container">$n$</span> days, the numbers are <span class="math-container">$$x_1<x_2<... | peawormsworth | 603,214 | <p>An ordered list of numbers in the universe after each day:</p>
<pre><code>Day 0: empty
Day 1: 0
Day 2: -1, 0, 1
Day 3: -2, -1, -1/2, 0, 1/2, 1, 2
</code></pre>
<p>New numbers:</p>
<pre><code>Day 1: 0
Day 2: -1, 1
Day 3: -2, -1/2, 1/2, 2
</code></pre>
<p>On any given day the universe can be sorted:</p>
<p><span class... |
2,946,379 | <p>The question posed is the following: Let <span class="math-container">$X$</span> be a Banach Space and let <span class="math-container">$T:X\to X$</span> be a Lipschitz-Continuous map. Show that, for <span class="math-container">$\mu$</span> sufficiently large, the equation
<span class="math-container">\begin{equat... | J.G. | 56,861 | <p>Since <span class="math-container">$dx=dt/t$</span>, you need to divide the whole integrand by <span class="math-container">$t$</span>.</p>
|
2,946,379 | <p>The question posed is the following: Let <span class="math-container">$X$</span> be a Banach Space and let <span class="math-container">$T:X\to X$</span> be a Lipschitz-Continuous map. Show that, for <span class="math-container">$\mu$</span> sufficiently large, the equation
<span class="math-container">\begin{equat... | Community | -1 | <p>There are a ton of mistakes here, unfortunately. The key issue is that you've got something like</p>
<p><span class="math-container">$$\frac{t + t^3}{t - t^5} = 1 + t^2 - \frac{1}{t^4} - \frac{1}{t^2}$$</span></p>
<p>where you've just mixed-and-matched all four terms. This is a (very) incorrect manipulation of the... |
2,264,791 | <p>I have a problem that I'm having trouble figuring out the distribution with given condition.</p>
<p>It is given that 1/(<span class="math-container">$X$</span>+1), where <span class="math-container">$X$</span> is an exponentially distributed random variable with parameter 1.</p>
<blockquote>
<p><strong>Original Prob... | Graham Kemp | 135,106 | <p>$X$ is <em>not</em> "writen" as $e^{-x}$. The probability density function of $X$, called $f_X(x)$, is equal to $e^{-x}~\big[x\geqslant 0\big]$.</p>
<p>The cummulative distribution function of $X$ is: $$\begin{align}F_X(x) ~&=~ \mathsf P(X\leqslant x) \\[1ex] &=~ (1-e^{-x}) ~\big[x\geqslant 0\big]\en... |
345,844 | <p>Should be simple enough, yet I can't show that there are no monomorphisms $\mathbb{Z}^3\rightarrow \mathbb{Z}^2$. (It is true, right?)</p>
| Jim | 56,747 | <p><strong>Hint:</strong> Show that given any $x, y, z \in \mathbb Z^2$ there exist <em>non-zero</em> integers $a, b, c \in \mathbb Z$ such that
$$ax + by + cz = (0, 0)$$
(You should be able to give an explicit formula for $a, b, c$ in terms of the entries of $x, y, z$)</p>
<p>Next show that if you had a monomorphism ... |
936,525 | <p>I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension $n\times 1$, $$\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $I$ is an $n\times n$ matrix, and $\|.||$ is a matrix norm.... | dineshdileep | 41,541 | <p>I think you should check the notations of that book first. The authors are probably talking about the spectral-norm. Note that spectral norm is completely different from the frobenius norm. For any square matrix $P$
\begin{align}
||P||_{\text{spectral norm}}=||P||_2=\max_{||x||_2 = 1}||Px||_2=\sigma_{1} &&\{... |
200,658 | <p>What is the value of :</p>
<p>$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$</p>
| Ayush Khemka | 42,108 | <p>I don't know the exact steps of how to get that, but i figured out that this comes out to be
$\frac 11+\frac 79+\frac {15}{27}+\ldots$</p>
<p>and this link here, gives the solution to be $\frac {11}4$. Refer
<a href="http://www.wolframalpha.com/input/?i=sum%20from%201%20to%20infinity%20%28%28n%5E2%2bn%2b1%29/3%5En... |
200,658 | <p>What is the value of :</p>
<p>$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$</p>
| Jakub Konieczny | 10,674 | <p>In most practical applications, you can just ask Mathematica, and it will tell you it's $\frac{11}{4}$.</p>
<p>If you want to arrive at the formula in a more rigorous way, you can do the following: </p>
<p>Consider the function $f$: $$f(x) = \frac{1}{1 - x} = \sum_{n=0}^\infty x^n$$</p>
<p>An initial observation ... |
1,712,457 | <blockquote>
<p>Assume $f$ is differentiable over an open interval $I$. Suppose $a<b$ are two numbers in $I$ with $f'(a) < f'(b)$. Show that if $f'(a) < 0 <f'(b)$, then neither $f(a)$ nor $f(b)$ can be the minimum value of $f$ over $[a,b]$.</p>
</blockquote>
<p>Intuitively this makes sense: $f$ must ch... | Jared | 138,018 | <p>You have the following picture where $a$ is the left and $b$ is the right:
<a href="https://i.stack.imgur.com/Vcpzz.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Vcpzz.jpg" alt="enter image description here"></a></p>
<p>$f'(a) < 0$ (given) therefore the function decreases as you move <em>imm... |
359,742 | <p>I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. </p>
<p>So consider a generic function $f : \mathbb{R} \mapsto \mathbb{R}$ and consider these hypothesis:</p>
<ul>
<li>$f$ is continuos in $\mathbb{R}$.... | Bragadeesh | 203,169 | <p>I had a similar question for years. Only recently I was able to solve. So here you go.</p>
<p>As you have mentioned, you can assume $f$ as a pdf of a random variable multiplied by a scaling factor, since it satisfies all the required properties you've mentioned.</p>
<p>So following the approach, let me first consi... |
2,520,768 | <p>How would I approach this problem? </p>
<p>Let $(a, b, c) \in \mathbb{Z^3}$ with $a^2 + b^2 = c^2$. Show that:
$$
60 \,\mid\, abc
$$</p>
| John Lou | 404,782 | <p>\begin{align}
n^2 \mod 3 &\equiv 0 \text{ or } 1\\
a^2 + b^2 &\equiv c^2 \mod 3 \quad\text{the following two cases are the only possibilities}\\
0+0 &\equiv 0 \mod 3 \\
0+1 &\equiv 1 \mod 3\\
\end{align}
Regardless, at least one number in the triple is divisible by $3$.</p>
<p>\begin{align}
n^2 \mod... |
3,190,828 | <p>Let <span class="math-container">$A\in M_n(\mathbb{C})$</span> be a matrix such that <span class="math-container">$A^n=aA$</span>,where <span class="math-container">$a\in \mathbb{R}-\{0,1\}$</span>.<br>
I wanted to find <span class="math-container">$A$</span>'s eigenvalues and I thought that they are the roots of th... | Fred | 380,717 | <p>It is correct: if <span class="math-container">$ \mu $</span> is an eigenvalue of <span class="math-container">$A$</span> with corresponding eigenvector <span class="math-container">$x$</span>, then <span class="math-container">$A^nx= \mu^n x$</span>, hence <span class="math-container">$\mu^nx=a Ax=a \mu x$</span>. ... |
3,190,828 | <p>Let <span class="math-container">$A\in M_n(\mathbb{C})$</span> be a matrix such that <span class="math-container">$A^n=aA$</span>,where <span class="math-container">$a\in \mathbb{R}-\{0,1\}$</span>.<br>
I wanted to find <span class="math-container">$A$</span>'s eigenvalues and I thought that they are the roots of th... | trancelocation | 467,003 | <p>You have</p>
<ul>
<li><span class="math-container">$A^n-aA=O_{n\times n}$</span></li>
</ul>
<p>So, based on <a href="https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem" rel="nofollow noreferrer">Cayley-Hamilton theorem</a> you can conclude that any matrix <span class="math-container">$A$</span> satisfie... |
2,080,716 | <p>I have the quadratic form
$$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$</p>
<p>I want to diagonalize the matrix of Q. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.</p>
| Atul Mishra | 396,163 | <p>If you are familiar with unions and intersections of sets , then it is not a difficult problem.</p>
<p><strong>Your answer should be:</strong> Total sum-(sum of multiple of $3 +$ sum of multiple of $7 -$ sum of multiple of $21)$</p>
<p>Since $21$ is LCM of $3$ & $7$</p>
|
125,116 | <p>Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?</p>
<p>In 2D, a simple angle satisfies this, as it can have additional multiples of $2\pi$. For example, rotating by a turn and a half would be $3\pi$.</p>
<p>Is there someth... | P Vanchinathan | 22,878 | <p>The resolution in 2D that you suggested may also be viewed as
going from the circle to its universal covering space:
$\mathbb{R}\to S^1$.</p>
<p>SO the the same trick should work: take the universal cover of SO(3).</p>
|
203,464 | <p>I would like to exclude the point <code>{x=0,y=0}</code> in the function definition</p>
<pre><code>f = Function[{x, y}, {x/(x^2 + y^2), -(y/(x^2 + y^2))}]
</code></pre>
<p>So far I tried <code>ConditionalExpression</code>and <code>/;</code> without success.</p>
<p>Thanks!</p>
| N.J.Evans | 11,777 | <p>As with other solutions, you have to do some cleaning up afterword, but you can use <code>Table</code> with lists defining the iterators:</p>
<pre><code>Flatten[Table[{i + j}, {i, {a, b, c}}, {j, {d, e, f}}], 1]
</code></pre>
<p>If you really want to map it onto the lists, you can use the following, but the resul... |
12,949 | <p>Let $\kappa$ be an infinite cardinal. Then there exists at least one <a href="http://en.wikipedia.org/wiki/Real-closed_field">real-closed field</a> of cardinality $\kappa$ (e.g. <a href="http://en.wikipedia.org/wiki/Lowenheim-Skolem">Lowenheim-Skolem</a>; or, start with a function field over $\mathbb{Q}$ in $\kappa... | Joel David Hamkins | 1,946 | <p>In the countable case, the bound of 2<sup>ω</sup> is realized, since any countable real-closed field will contain the rational numbers and fill at most countably many cuts in the rationals with LUBs. But we can arrange that any given cut is filled by a real closed subfield of R containing that real. So there ... |
2,301,198 | <p>Solve the initial value problem for the sequence $\left \{ u_{n}| n \in \mathbb{N} \right \}$ satisfying the recurrence relation:
$u_n − 5u_{n-1} + 6u_{n−2} = 0 $ with $u_0 = 1$ and $u_1 = 1$.</p>
<p>Ive gotten the general solution to be $u_n = A(2)^n + B(3)^n$. </p>
<p>Once I sub the initial values: </p>
<p>$u_0... | Michael Rozenberg | 190,319 | <p>$$u_n-2u_{n-1}=3(u_{n-1}-2u_{n-2}),$$
which says that
$u_{n}-2u_{n-1}=-3^{n-1}$.</p>
<p>Thus,
$$u_n-2u_{n-1}=-3^{n-1}$$
$$2^1u_{n-1}-2^2u_{n-2}=-2^13^{n-2}...$$
$$2^{n-1}u_1-2^nu_0=-2^{n-1}3^0,$$
which after summing gives:
$$u_n-2^nu_0=-3^{n-1}-2\cdot3^{n-2}-...-2^{n-1}$$ or
$$u_n=2^n-\frac{3^n-2^n}{3-2}$$ or
$$u_... |
188,336 | <p>Let $|\cdot|_1$ and $|\cdot|_2$ be two norms on a field $\mathbb F$. We call the two norms equivalent if every Cauchy-sequence with respect to $|\cdot|_1$ is also a Cauchy-sequence with respect to $|\cdot|_2$. Prove the following statement:</p>
<p>$$|\cdot|_1\sim|\cdot|_2\quad\Leftrightarrow\quad\exists \alpha\in\m... | user29999 | 29,999 | <p>Hint:
\begin{equation}
|a_m-a_n| < \varepsilon \Leftrightarrow |a_m-a_n|_{2}^{\alpha}<\varepsilon \Leftrightarrow |a_m-a_n|_{2}< \varepsilon^{1/\alpha}.
\end{equation}</p>
|
188,336 | <p>Let $|\cdot|_1$ and $|\cdot|_2$ be two norms on a field $\mathbb F$. We call the two norms equivalent if every Cauchy-sequence with respect to $|\cdot|_1$ is also a Cauchy-sequence with respect to $|\cdot|_2$. Prove the following statement:</p>
<p>$$|\cdot|_1\sim|\cdot|_2\quad\Leftrightarrow\quad\exists \alpha\in\m... | Sangchul Lee | 9,340 | <p>To prove the direction in question, assume otherwise. That is, there exists $x, y \in \Bbb{F}^{\times}$ such that</p>
<p>$$ \frac{\log|x|_1}{\log|x|_2} = \alpha(x) \neq \alpha(y) = \frac{\log|y|_1}{\log|y|_2}. \tag{1} $$</p>
<p>The naive idea of the proof is to exaggerate this difference in a deliberate way.</p>
... |
295,517 | <p>My math is not incredibly strong and perhaps I have just not been searching for the right terms, but I have a summation that is part of an algorithm I've been working on and would really like to reduce it to just a formula, but am really struggling to find a solution (if one exists).</p>
<p>$\sum_{i=1}^{n}\frac{5}{... | GEdgar | 442 | <p>If you are interested in how it behaves for large $n$, you could try an approximation like
$$
\sum_{i=1}^n \frac{5}{i^{0.35}} \approx \int_{1/2}^{n+1/2}\frac{5\;dx}{x^{0.35}}
$$
For example,
$$
\sum_{i=1}^{100} \frac{5}{i^{0.35}} \approx 148.93,\qquad
\int_{1/2}^{100.5}\frac{5\;dx}{x^{0.35}}\approx 149.08 .
$$</p>
|
1,120,013 | <p>Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true:</p>
<p>E [ X | E [ X | Y ] ] = E [ X | Y]</p>
<p>Note that E [ X | Y ] is a random variable in it's own right. Also note that equality here is point-wise, for every point in the s... | pre-kidney | 34,662 | <p>Let $Z=E[X\ | \ Y]$. Your equation states: $E[X \ | \ Z]=Z$. This follows from the following fact.</p>
<p><strong>Tower Property of Conditional Expectation:</strong></p>
<p>$$E[E[X\ | \ \mathcal{F}]\ | \ \mathcal{G}]=E[X\ | \ \mathcal{G}],\text{ whenever }\mathcal{G}\subset \mathcal{F}.$$</p>
<p><strong>Proof of ... |
308,520 | <p>The DE is $y' = -y + ty^{\frac{1}{2}}$. </p>
<p>$2 \le t \le 3$</p>
<p>$y(2) = 2$</p>
<p>I tried to see if it was in the <a href="http://www.sosmath.com/diffeq/first/lineareq/lineareq.html" rel="nofollow">linear form</a>. I got:</p>
<p>$$\frac{dy}{dt} + y = ty^{\frac{1}{2}}$$</p>
<p>The RHS was not a function o... | Ron Gordon | 53,268 | <p>Let $y=u^2$, then you can cancel a factor of $u$ and get</p>
<p>$$2 u' + u = t$$</p>
<p>for which you can apply an integrating factor of $e^{t/2}$ to both sides and get</p>
<p>$$\frac{d}{dt} [u e^{t/2}] = \frac{t}{2} e^{t/2}$$</p>
<p>Integrating both sides, we get the general solution:</p>
<p>$$u(t) = t-2 + C e... |
308,520 | <p>The DE is $y' = -y + ty^{\frac{1}{2}}$. </p>
<p>$2 \le t \le 3$</p>
<p>$y(2) = 2$</p>
<p>I tried to see if it was in the <a href="http://www.sosmath.com/diffeq/first/lineareq/lineareq.html" rel="nofollow">linear form</a>. I got:</p>
<p>$$\frac{dy}{dt} + y = ty^{\frac{1}{2}}$$</p>
<p>The RHS was not a function o... | Kaster | 49,333 | <p>I'll just add that integration factor is not necessary. From the equations
$$
2z'+z = t
$$
you can assume that particular solution is linear $z^p = At+B$ and substitute it to ODE
$$
2A+At+B=t
$$
from which you can easily find that $A = 1, B = -2$, so $z^p = t - 2$. General solution of inhomogeneous problem is a sum ... |
490,064 | <p>Solve the Cauchy problem, $\forall t \in \mathbb{R}$,
$$ \begin{cases}
u''(t) + u(t) = |t|\\
u(0)=1, \quad u'(0) = -1
\end{cases} $$</p>
<p>The solution to the homogeneous equation is $A\cos(t) + B \sin(t)$. Empirically, $|t|$ is "more or less" a particular solution, however it is not differentiable in $0$... What ... | Anthony Carapetis | 28,513 | <p>This isn't the easiest or most systematic way to get a solution, but I had a bit of fun finding it so I'll post it anyway. </p>
<p>Let's look for solutions of the form $u(t) = |t|f(t)$ where $f$ is twice differentiable with $f(0)=0$. Such a function is twice differentiable everywhere, with second derivative $$ u''(... |
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | José Carlos Santos | 446,262 | <p>Use the fact that<span class="math-container">\begin{align}x^3-4x^2-4x+16=0&\iff x(x^2-4)-4(x^2-4)=0\\&\iff(x-4)(x^2-4)=0.\end{align}</span></p>
|
3,008,162 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be well-ordered sets, and suppose <span class="math-container">$f:A\to B$</span> is an
order-reversing function. Prove that the image of <span class="math-container">$f$</span> is finite.</p>
<p>I started by supposing not. Then... | Dante Grevino | 616,680 | <p>Let <span class="math-container">$C=f(A)$</span> the image of <span class="math-container">$f$</span> with the order induced by <span class="math-container">$B$</span>. Every non-empty subset of <span class="math-container">$C$</span> has minimum and maximum. This implies that every element in <span class="math-cont... |
300,163 | <p>I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is $|z|=2$. </p>
<p><strong>My method:</strong></p>
<p>$z/\bar z = e^{2i\theta}$ - which is entire over the complex plan... | Daniel Mckenzie | 60,074 | <p>1) $e^{2i\theta}$ is not holomorphic, and therefore not entire. There are many way to check this, but it suffices to observe that $\frac{\partial}{\partial \bar{z}}\frac{z}{\bar{z}} = -\frac{z}{\bar{z}^2}\neq 0$. See the discussion of the Wirtinger derivative in the definition section here: <a href="http://en.wikipe... |
3,102,905 | <p>I have the following sequence <span class="math-container">$$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$</span>
Also I know that <span class="math-container">$a,b,c\in \mathbb{R}$</span> and <span class="math-container">$|c|<1,\ b\neq 1, \ |bc|<1$</span>
I need to find the limit of... | John Omielan | 602,049 | <p>You seem to be doing everything correctly. Using your final value for <span class="math-container">$x_n$</span>, and taking the limit as <span class="math-container">$n \to \infty$</span>, I get, using the sum of an infinite geometric series being <span class="math-container">$\frac{a}{1-r}$</span>, where <span clas... |
2,292,324 | <p>I know what the answer to this question is, but I am not sure how the answer was reached and I would really like to understand it! I am omitting the base case because it is not relevant for my question.</p>
<p>Inductive hypothesis:</p>
<p>$$\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \dotsb + \frac... | John | 7,163 | <p>You want to show</p>
<p>$$\sum_{j=1}^n \frac{1}{j(j+1)} = \frac{n}{n+1}$$</p>
<p>The inductive step involves assuming it holds for $n=k$ and then showing that it also holds for $n=k+1$. So you assume</p>
<p>$$\sum_{j=1}^k \frac{1}{j(j+1)} = \frac{k}{k+1}$$</p>
<p>and show</p>
<p>$$\sum_{j=1}^{k+1} \frac{1}{j(j... |
2,292,324 | <p>I know what the answer to this question is, but I am not sure how the answer was reached and I would really like to understand it! I am omitting the base case because it is not relevant for my question.</p>
<p>Inductive hypothesis:</p>
<p>$$\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \dotsb + \frac... | Angina Seng | 436,618 | <p>The inductive hypothesis is
$$\frac1{1\cdot2}+\frac1{2\cdot3}+\cdots+\frac{1}{k(k+1)}
=\frac{k}{k+1}.\tag1$$
You need to prove that (1) implies the statement got from (1) by replacing
$k$ by $k+1$. This is
$$\frac1{1\cdot2}+\frac1{2\cdot3}+\cdots+\frac{1}{(k+1)(k+2)}
=\frac{k+1}{k+2}\tag2$$
but instead of (2) you ha... |
2,241,100 | <p>Please someone help me solve the following equation in terms of $y$:</p>
<blockquote>
<p><strong>$\frac{y^2}{2}+y = \frac{x^3}{3}+\frac{x^2}{2}+c_1$</strong></p>
</blockquote>
<p>The calculator gives me:</p>
<blockquote>
<p>$y = \frac{1}{3}(\sqrt{3}\sqrt{c_1+2x^3+3x^2+3}-3), -\frac{1}{3}(\sqrt{3}\sqrt{c_1+2x^... | Ahmed S. Attaalla | 229,023 | <p>Multiply both sides by $6$ to make things a little nicer.</p>
<p>$$3y^2+6y=2x^3+3x^3+6c_1$$</p>
<p>$$3y^2+6y-2x^3-3x^3-6c_1=0$$</p>
<p>At this point you should realize that the variable $y$, what we are trying to solve for, is quadratic in the above equation. Although it looks quite messy, the equation is really... |
1,255,311 | <p><img src="https://i.stack.imgur.com/5V9e0.png" alt="enter image description here"></p>
<p>I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something with notation here. Any help/... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Write</p>
<p>$z=x+iy$</p>
<p>and use Euler Formula $e^{iw}=\cos w+i\sin w$</p>
<p>and equate the real & the imaginary parts </p>
|
1,662,398 | <p>I am currently studying for my upcoming midterm and I am stumped on this example provided in the slides. Basically here is the question:</p>
<blockquote>
<p>Given 35 computers, what is the probability that more than 10 computers are in use(active)? We are told that each computer is only active 10% of the time. Th... | Graham Kemp | 135,106 | <p>You have <em>correctly</em> identified this count as having a <strong>Binomial Distribution</strong>.</p>
<p>So far, so good. However, what happened next was not okay.</p>
<p>The complement of having more than $10$ computers active is <strong>not</strong> of having <em>exactly</em> 10 computers active. &nbs... |
3,392,871 | <blockquote>
<p>Let <span class="math-container">$k>1$</span> and define a sequence <span class="math-container">$\left\{a_{n}\right\}$</span> by <span class="math-container">$a_{1}=1$</span> and <span class="math-container">$$a_{n+1}=\frac{k\left(1+a_{n}\right) }{\left(k+a_{n}\right)}$$</span>
(a) Show that <s... | YiFan | 496,634 | <p>In general, to show that a sequence defined by the recurrence <span class="math-container">$a_{n+1}=f(a_n)$</span> is monotonically increasing, what you want to do is to show <span class="math-container">$a_{n+1}>a_n$</span>, which converts to <span class="math-container">$f(a_n)>a_n$</span>. Then you consider... |
618,986 | <p>I'm having trouble with this question, I'd like someone to point me in the right direction.</p>
<p>let $A$ be a n by n matrix with real values.
show that there is another n by n real matrix $B$ such that $B^3=A$, and that $B$ is symmetric. Are there more matrices like this $B$ or is it the only one?</p>
<p>What I... | Oria Gruber | 76,802 | <p>I would first like to thank Mariano Suarez-Alvarez in advance for pointing me in the right direction.</p>
<p>if $A$ is symmetric over $\mathbb R$, then it is diagonalizable:</p>
<p>$A=PDP^{-1}$ such that $D$ is diagonal.</p>
<p>let $B = PD_2P^{-1}$ such that $D_2$ is a diagonal matrix whos values are the third ro... |
618,986 | <p>I'm having trouble with this question, I'd like someone to point me in the right direction.</p>
<p>let $A$ be a n by n matrix with real values.
show that there is another n by n real matrix $B$ such that $B^3=A$, and that $B$ is symmetric. Are there more matrices like this $B$ or is it the only one?</p>
<p>What I... | Yiorgos S. Smyrlis | 57,021 | <p>Note that, every symmetric $A\in\mathbb R^{n\times n}$ matrix is diagonalisable, it has real eigenvalues $d_1,\ldots,d_n$, and its diagonalization is realised with an orthogonal matrix $U$, i.e.,
$$
A=U^TDU,
$$
where $D=\mathrm{diag}(d_1,\ldots,d_n)$, and $U^TU=I$. Now let
$$
B=U^T\mathrm{diag}(d_1^{1/3},\ldots,d_... |
618,986 | <p>I'm having trouble with this question, I'd like someone to point me in the right direction.</p>
<p>let $A$ be a n by n matrix with real values.
show that there is another n by n real matrix $B$ such that $B^3=A$, and that $B$ is symmetric. Are there more matrices like this $B$ or is it the only one?</p>
<p>What I... | mjw | 655,367 | <p>If <span class="math-container">$B$</span> satisfies <span class="math-container">$B^3=A$</span>, then <span class="math-container">$\{ B, \alpha B, \overline{\alpha} B \}$</span> are solutions, where <span class="math-container">$\alpha = \exp \left(\frac{2\pi i}{3}\right)$</span> and where <span class="math-contai... |
830,977 | <p>I'm having some real trouble with lebesgue integration this evening and help is very much appreciated.</p>
<p>I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over $(0,\infty)$.</p>
<p>My first thought was to write the integral as $f(x) = \frac{\cosh(x)}{\cosh(2x)}$ and then no... | Zarrax | 3,035 | <p>For the numerator, observe that $e^x$ dominates $e^{-x}$ as $x \rightarrow \infty$. </p>
<p>For the denominator, observe that $e^{2x}$ dominates $e^{-2x} $ as $x \rightarrow \infty$. </p>
<p>So the integrand will decay like ${e^x \over e^{2x}}$ = $e^{-x}$ as $x \rightarrow \infty$ and the integral will converge. T... |
293,245 | <p>Most true statements independent of PA that I know of is equivalent to some consistency statement. For example</p>
<ul>
<li>Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$</li>
<li>Goodstein's theorem is equivalent to Con(PA)</li>
<li>Any conjunction or disjunction of the above.</li>
</ul... | Payam Seraji | 65,878 | <p>$1$-consistency of $PA$ is a true $\Pi_3$ sentence which is not provable in $PA$+{all true $\Pi_1$ sentences} (see this <a href="https://academic.oup.com/jigpal/advance-article-abstract/doi/10.1093/jigpal/jzx061/4792773?redirectedFrom=fulltext" rel="noreferrer">article</a>). Simple (iterated) consistency statements ... |
1,401,516 | <p>Given is the unit circle in the plane. Choose randomly point in it, such that $P(\left(x,y\right)\in A)$ is proportional to area of $A$, where $A$ is measurable set in plane. Find density function of random variable $X$ which represents the $x$ coordinate of this point.</p>
<p>My idea was to find $P(X\leq x)$ and t... | georg | 144,937 | <p>I would say, if you do not integrate, then from the surface of the unit circle subtract the area of a circle segment a radius of 1 and a angle $\alpha$:</p>
<p><a href="https://i.stack.imgur.com/NFUkn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NFUkn.png" alt="enter image description here">... |
244,214 | <p>One major approach to the theory of forcing is to assume that ZFC has a countable <em>transitive</em> model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, uses the fact that $M$ is <em>countable</em> to prove that there exists a generic set $G \in V$, then defin... | Joel David Hamkins | 1,946 | <p>Yes, one can undertake forcing without the transitivity assumption,
and even the countability of the model is not important.</p>
<p>One of the standard ways to do this is with the Boolean-valued
model quotient construction, which has been described in many places. Basically, given a forcing notion $B$,
a complete B... |
3,830,204 | <p>Working through <em>Spivak's Calculus</em> and using old assignments from the course offered at my school I'm working on the following problem, asking me to find the integral <span class="math-container">$$\int \frac{1}{x^{2}+x+1} dx$$</span></p>
<p>Looking through Spivak and previous exercises I worked on, I though... | jasmine | 557,708 | <p><span class="math-container">$\int \frac{1}{(x+1/2)^2 + 3/4} dx= \frac{2}{\sqrt3}\tan^{-1}\frac{2x+1}{\sqrt3} +c$</span></p>
|
3,830,204 | <p>Working through <em>Spivak's Calculus</em> and using old assignments from the course offered at my school I'm working on the following problem, asking me to find the integral <span class="math-container">$$\int \frac{1}{x^{2}+x+1} dx$$</span></p>
<p>Looking through Spivak and previous exercises I worked on, I though... | Robby the Belgian | 19,298 | <p>I'll take a different approach from the answers so far. For this approach, you'll have to be at least a bit comfortable working with complex numbers.</p>
<p>We can factor <span class="math-container">$x^2 + x + 1$</span> as <span class="math-container">$\left(x - \frac{1 + \sqrt{3}i}{2}\right)\left(x - \frac{1 - \sq... |
3,552,915 | <p>Determine the point on the plane <span class="math-container">$4x-2y+z=1$</span> that is closest to the point <span class="math-container">$(-2, -1, 5)$</span>. This question is from Pauls's Online Math Notes. He starts by defining a distance function: </p>
<p><span class="math-container">$z = 1 - 4x + 2y$</span></... | Bernard | 202,857 | <p>Here is a simple solution using tools from middle school for the computation:</p>
<p>Denote <span class="math-container">$x,y,z$</span> the coordinates of the orthogonal projection of the point <span class="math-container">$(-2,-1,5)$</span>
It satisfies the equations of proportionality:
<span class="math-contai... |
1,284,039 | <p>What function satisfies $f(x)+f(−x)=f(x^2)$?</p>
<p>$f(x)=0$ is obviously a solution to the above functional equation.</p>
<p>We can assume f is continuous or differentiable or similar (if needed).</p>
| grube300 | 240,897 | <p>Give f(x) = ln(|x|) a try in your equation</p>
|
423,159 | <p>What do you call a linear map of the form <span class="math-container">$\alpha X$</span>, where <span class="math-container">$\alpha\in\Bbb R$</span> and <span class="math-container">$X\in\mathrm O(V)$</span> is an orthogonal map (<span class="math-container">$V$</span> being some linear space with inner product)? A... | Vladimir Dotsenko | 1,306 | <p>Wikipedia suggests "conformal orthogonal group" for the group of all such maps; see the articles</p>
<p><a href="https://en.wikipedia.org/wiki/Conformal_group" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Conformal_group</a>
<a href="https://en.wikipedia.org/wiki/Orthogonal_group#Conformal_group... |
1,943,351 | <p>Good day,</p>
<p>In class we said that if a random variable <span class="math-container">$X-Y$</span> is independent of random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> then <span class="math-container">$X-Y$</span> is almost sure constant, i.e. there exists a <spa... | Landon Carter | 136,523 | <p>Let $\phi_A(t)$ be the characteristic function of random variable $A$.Then you know that if $A,B$ are independent, then $\phi_{A+B}(t)=\phi_A(t)\phi_B(t)$.</p>
<p>You have that $X-Y$ is independent of $X$ and $Y$. Noting that $X=(X-Y)+Y$, you have that $\phi_X(t)=\phi_{(X-Y)+Y}(t)=\phi_{X-Y}(t)\phi_Y(t)$ [using the... |
803,335 | <p>Note: this is particularly aimed at high-school/entry level college problems </p>
<p>When I'm learning a new topic:</p>
<p>1) I read the theory given in the textbook at the start of each topic</p>
<p>2) proceed to read the solved example problems which the textbook provides (usually 3-5 with full solutions)</p>
... | Dan Christensen | 3,515 | <p>You don't say anything about writing notes or summaries. For each type of problem, while it is still fresh in your mind, write a detailed reminder to yourself on how to do it -- maybe a particularly good example. Include key definitions, theorems with examples and non-examples. Keep these summary notes separate from... |
2,438,362 | <p>We have, $\rho(A) \leq \|A\|$
where $\rho(A)$ denotes the spectral radius of $A$.</p>
<p>Now there is a corollary
that $\rho(A) < 1$ iff $\|A\|<1$
it is clear that when $\|A\|<1$ then $\rho(A)<1$</p>
<p>but how to show that if $\rho(A)<1$ then $\|A\|<1$,
perhaps it is because of this one</p>
<p>... | user1551 | 1,551 | <p>The "iff", or your so-called "corollary", are wrong. Counterexample: when $A$ is the $2\times2$ Jordan block for the eigenvalue $1-\epsilon$ for some small $\epsilon>0$ we have $\rho(A)=1-\epsilon<1$ but $\|A\|_2\ge\|(1,0)\,A\|_2=\|(1-\epsilon,1)\|_2>1$.</p>
|
1,473,513 | <p>The motion of a pendulum is described by the differential equation</p>
<p><span class="math-container">$$ \ddot\theta +\frac gl \sin \theta = 0$$</span></p>
<p>if we integrate this equation with respect to <span class="math-container">$\theta$</span> we obtain</p>
<p><span class="math-container">$$ \frac 12 \dot... | Spencer | 71,045 | <p>It follows from the chain rule,</p>
<p>$$ \frac{d}{dt} = \frac{d\theta}{dt} \frac{d}{d\theta} = \dot{\theta} \frac{d}{d\theta},$$</p>
<p>$$ \ddot{\theta} = \dot{\theta}\frac{d}{d\theta} \dot{\theta} = \frac12 \frac{d}{d\theta} \left( \dot{\theta}^2 \right). $$</p>
<p>I didn't like the above too much as an undergr... |
1,981,928 | <p>While I was studying properties of limit and sequences, I found a theorm that says 'if {$s_n$}, {$t_n$} are convergent sequences, then $s_n \le t_n$ for all $n \in \mathbb{N}$ implies that $$\lim_{n\rightarrow \infty} s_n \le \lim_{n\rightarrow \infty} t_n$$
this proof is quite easy to construct, as you can say </p>... | kobe | 190,421 | <p>No, it's not true. Let $s_n = 1 $ and $t_n = 1 + 1/n$, for all $n\in \Bbb N$. Then $s_n < t_n$ for every $n$, but $\lim_{n\to \infty} s_n = 1 = \lim_{n\to \infty} t_n$.</p>
|
141,484 | <p><strong>Bug introduced in 10.4.1 or earlier and fixed in 11.1.1</strong></p>
<hr>
<p>I recently installed MMA v11.1 and encountered an issue with the memory usage of the LinearModelFit[] command. It appears than when mixing numeric and nominal variables, the LinearModelFit[] command uses a very large block of memo... | Szabolcs | 12 | <p>You could use something like this if your data is not too large:</p>
<pre><code>terp = Interpreter[
DelimitedSequence[
DelimitedSequence["Number", {"[", Whitespace, "]"}],
{"[", Whitespace, "]"}
]
]
terp["[[1 2] [3 4]]"]
(* {{1, 2}, {3, 4}} *)
</code></pre>
<p>You can add another layer of <code>Delim... |
141,484 | <p><strong>Bug introduced in 10.4.1 or earlier and fixed in 11.1.1</strong></p>
<hr>
<p>I recently installed MMA v11.1 and encountered an issue with the memory usage of the LinearModelFit[] command. It appears than when mixing numeric and nominal variables, the LinearModelFit[] command uses a very large block of memo... | Ali Hashmi | 27,331 | <pre><code>ToExpression[
StringReplace["[[1 4 5 6 2] [9 8 7 4 7]]", {" " -> ",", "[" -> "{", "]" -> "}"}]]
(* {{1, 4, 5, 6, 2}, {9, 8, 7, 4, 7}} *)
</code></pre>
|
267,971 | <p>I want to keep inside of a integral evaluated after some replacement inside it, but at the same time the integral itself unevaluated.</p>
<p>I start with:</p>
<pre><code>int=HoldForm[Integrate[x^n/(x + 1)^(n + 1), {x, 0, 1}]]
</code></pre>
<p>Output as desired:
<span class="math-container">$$\int_0^1 \frac{x^n}{(x+1... | azerbajdzan | 53,172 | <p>I found a way, but does the code really have to be so ridiculous for such a simple task?</p>
<pre><code>int = HoldForm[Integrate[x^n/(x + 1)^(n + 1), {x, 0, 1}]]
HoldForm[a] /.
HoldPattern[
a] -> (int /. n -> 3 /. Integrate -> Evaluate /.
HoldForm -> integr) /. integr -> Integrate
</code>... |
267,971 | <p>I want to keep inside of a integral evaluated after some replacement inside it, but at the same time the integral itself unevaluated.</p>
<p>I start with:</p>
<pre><code>int=HoldForm[Integrate[x^n/(x + 1)^(n + 1), {x, 0, 1}]]
</code></pre>
<p>Output as desired:
<span class="math-container">$$\int_0^1 \frac{x^n}{(x+1... | Michael E2 | 4,999 | <p>You can use Trott-Strzebonski or <code>RuleCondition</code> or controlled evaluation; see <a href="https://mathematica.stackexchange.com/questions/29317/replacement-inside-held-expression">Replacement inside held expression</a>, which might be considered a duplicate.</p>
<p>Variations:</p>
<pre><code>int = HoldForm[... |
123,918 | <p>Someone <a href="https://stackoverflow.com/questions/9851628/minimal-positive-number-divisible-to-n">asked this question</a> in SO:</p>
<blockquote>
<p><span class="math-container">$1\le N\le 1000$</span></p>
<p>How to find the minimal positive number, that is divisible by N, and
its digit sum should be equal to N.<... | marlu | 26,204 | <p>For every $N$ there is a number $X$ such that $N$ divides $X$ and the sum of digits of $X$ equals $N$. </p>
<p><strong>Proof:</strong> Write $N = RM$ where $M$ is coprime to $10$ and $R$ contains only the prime factors $2$ and $5$. Then, by Euler's theorem, $10^{\varphi(M)} \equiv 1 \pmod M$. Consider $X' := \s... |
1,970,305 | <p>I have just begun reading through Section 3.2 of Hatcher's Algebraic Topology. While I reasonably understood the computations relating to the cup product, I was unsure of the purpose of the cup product. From what I knew, it does not help us to compute cohomology groups, given that we need the cohomology groups to co... | Dean C Wills | 366,201 | <p>$x=-2,y=1,z=5$ gives $6(-2) + 15(1) + 10(5) = 53$, by inspection. I'm trying to think of the general rule. </p>
|
2,280,052 | <p>Wolfram Alpha says:
$$i\lim_{x \to \infty} x = i\infty$$</p>
<p>I'm having a bit of trouble understanding what $i\infty$ means. In the long run, it seems that whatever gets multiplied by $\infty$ doesn't really matter. $\infty$ sort of takes over, and the magnitude of whatever is being multiplied is irrelevant. I.e... | murray | 32,337 | <p>In <em>Mathematica</em>, evaluating</p>
<pre><code>Limit[x, x -> Infinity]
</code></pre>
<p>gives (the usual shorthand symbol for the build-in entity) <code>Infinity</code>. No problem there. And also in <em>Mathematica</em>, evaluating</p>
<pre><code>I Infinity
</code></pre>
<p>must returns as output the sam... |
1,427,595 | <blockquote>
<p>The <a href="https://en.wikipedia.org/wiki/Cayley_table" rel="nofollow">Cayley table</a> tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis.</p>
</blockquote>
<p>S... | Nikos M. | 139,391 | <p>The <a href="https://en.wikipedia.org/wiki/Cayley_table" rel="nofollow">Caley table</a> describes the group action ("multiplication") between elements, so if the action is commutative (abelian), then table is symmetric (along the diagonal for $a \ne b$), since $a \circ b = b \circ a$</p>
<p>if seen as a matrix of g... |
980,818 | <p>I'm working on a problem that involves the following summation:
$$y=\sum_{i=0}^{x}i2^i$$
I need to determine the largest value of $x$ such that $y$ is less than or equal to some integer K. Currently I'm using a lookup table approach which is fine, but I would really like to find and understand a solution that would ... | Peter | 82,961 | <p>Use the idendity</p>
<p>$$\sum_{i=0}^x i2^i=2(2^xx-2^x+1)$$</p>
<p>and calculate the value $x$ with binary search, for example.</p>
|
919,562 | <p>I need to prove that:</p>
<p>$$\inf\{\frac{1}{3}+\frac{3n+1}{6n^2} \Big| n\in\mathbb N\}=\frac{1}{3}$$</p>
<p>I get stuck with my proof, I'll write it down.</p>
<p>$$n\geq1$$
$$3n\geq3$$
$$3n+1\geq4$$
$$\frac{1}{3}+3n+1\geq4+\frac{1}{3}$$</p>
<p>Now, I'm having a problem with $6n^2$ if I multiply by $6n^2$, I'll... | Adriano | 76,987 | <p>Let:
$$
S =
\left\{\frac{1}{3} + \frac{3n + 1}{6n^2} ~\middle|~ n \in \mathbb N\right\}
$$
Notice that since $3n + 1 > 0$ and $6n^2 > 0$, we know that $\frac{3n + 1}{6n^2} > 0$ so that $\frac{1}{3}$ is a lower bound for $S$. It remains to show that $\frac{1}{3}$ is the <strong>greatest</strong> lower bound... |
3,478,700 | <p>As you may know we can define the equation of a tangent line of a differentiable function at any point <span class="math-container">$a$</span> is given by:
<span class="math-container">$$y = f(a) + f'(a)(x-a)$$</span></p>
<blockquote>
<p>However how can I interpret this equation?
<span class="math-container">$$... | Mohammad Riazi-Kermani | 514,496 | <p>You mean <span class="math-container">$$y = f(a) + f'(a)(x-a) + \frac {f''(a)}{2}(x-a)^2$$</span></p>
<p>which is a quadratic approximation to the function around the point <span class="math-container">$(a,f(a))$</span> instead of the linear approximation which is the tangent line.</p>
<p>This is a better approxim... |
112,021 | <p>Let $n$ be a positive integer.
The $n$ by $n$ Fourier matrix may be defined as follows:</p>
<p>$$
F^{*} = (1/\sqrt{n}) (w^{(i-1)(j-1)})
$$</p>
<p>where </p>
<p>$$
w = e^{2 i \pi /n}
$$</p>
<p>is the complex $n$-th root of unity with smaller positive argument
and $*$ means transpose -conjugate.</p>
<p>It is we... | paul garrett | 15,629 | <p>This has a little number-theoretic content, having to do with real-valued <em>characters</em> modulo $n=2k+1$. For example, for $n=p$ an odd prime number, there are exactly two such functions (up to scalar multiples), the function that is $1$ for non-zero-mod-$p$ inputs, and the quadratic character $\chi$ mod $p$, w... |
112,021 | <p>Let $n$ be a positive integer.
The $n$ by $n$ Fourier matrix may be defined as follows:</p>
<p>$$
F^{*} = (1/\sqrt{n}) (w^{(i-1)(j-1)})
$$</p>
<p>where </p>
<p>$$
w = e^{2 i \pi /n}
$$</p>
<p>is the complex $n$-th root of unity with smaller positive argument
and $*$ means transpose -conjugate.</p>
<p>It is we... | Alexey Ustinov | 5,712 | <p>There is full and simple description of all eigenvectors in the article </p>
<p>Morton, P. On the eigenvectors of Schur's matrix. J. Number Theory, 1980, 12, 122-127 <a href="http://deepblue.lib.umich.edu/bitstream/2027.42/23371/1/0000315.pdf" rel="nofollow">http://deepblue.lib.umich.edu/bitstream/2027.42/23371/1/0... |
3,324,647 | <p>Say you have the following matrix A in <span class="math-container">$R^2 \rightarrow R^2$</span>:</p>
<p><span class="math-container">$
\begin{bmatrix}
7 & -10 \\
5 & -8
\end{bmatrix}
$</span></p>
<p>Thus the eigenvalues/eigenvectors are: 2 <span class="math-container">$\begin{bmatrix} 2 \\ 1 \end{bmatrix}... | D.B. | 530,972 | <p>Keep in mind that <span class="math-container">$A(2,1) = 2(2,1)$</span> and <span class="math-container">$A(1,1) = -3(1,1)$</span> by def of eigenvalues and eigenvectors. Can you write the desired vector <span class="math-container">$(2,3)$</span> as a linear combination of <span class="math-container">$(2,1)$</spa... |
1,440,470 | <p>Given two real valued independent random variables $X$ and $Y$, write their ratio as $R = \frac{X}{Y}$</p>
<p>I know various other ways of finding a formula for the distribution of $R$, but I'm specifically interested in understanding why the following derivation does not yield the correct result.</p>
<p>$$
P(R = ... | Michael | 155,065 | <p>Your posted computations seem to blur the distinctions between $Pr[X=x]$ and $f_X(x)$. For example, $Pr[X=x]$ is a number in $[0,1]$, while $f_X(x)$ can be larger than 1 for some values of $x$. </p>
<hr>
<p>A correct way of obtaining the probability of an event by conditioning is: </p>
<p>$$ Pr[R\leq r] = \int_... |
2,751,819 | <p>I need some help solving this.
I have tried:</p>
<p>$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
=\frac{1}{\operatorname{det}A}\cdot \begin{bmatrix}
d & -b \\
-c & a \\
\end{bmatrix}$$
I ended up with $$a=\frac{d}{\operatorname{det}A},$$
and
$$d=\frac{a}{\operat... | Naweed G. Seldon | 395,669 | <p>You have $A^{-1} = A \implies A^2 = I$. So we just calculate that for the matrix you have $$\begin{pmatrix} 1 & 0 \\ 0& 1 \end{pmatrix} = \begin{pmatrix} a & b \\ c& d \end{pmatrix}\cdot\begin{pmatrix} a & b \\ c& d \end{pmatrix} = \begin{pmatrix} a^2 + bc & (a+d)\cdot b \\ (a+d)\cdot c &... |
83,945 | <p>I've got a uniform random variable $X\sim\mathcal{U}(-a,a)$ and a normal random variable $Y\sim\mathcal{N}(0,\sigma^2)$. I am interested in their sum $Z=X+Y$. Using the convolution integral, one can derive the p.d.f. for $Z$:</p>
<p>$$f_Z(x)=\frac{1}{2a\sqrt{2\pi}}\int_{x-a}^{x+a}e^{-\frac{u^2}{2\sigma^2}}du=\fra... | Dilip Sarwate | 15,941 | <p>I don't have a complete answer but just a suggestion for part of your question.</p>
<p>$f_Z$ is the convolution of a uniform density and a Gaussian density. Thus,
its Fourier transform or characteristic function $\Psi_Z$ is the product of
a Gaussian function and a sinc function. Parseval's theorem then gives us
... |
83,945 | <p>I've got a uniform random variable $X\sim\mathcal{U}(-a,a)$ and a normal random variable $Y\sim\mathcal{N}(0,\sigma^2)$. I am interested in their sum $Z=X+Y$. Using the convolution integral, one can derive the p.d.f. for $Z$:</p>
<p>$$f_Z(x)=\frac{1}{2a\sqrt{2\pi}}\int_{x-a}^{x+a}e^{-\frac{u^2}{2\sigma^2}}du=\fra... | Robert Israel | 8,508 | <p>It looks to me like $$ \int_{-\infty}^\infty f_Z(x)^2\ dx = -{\frac {{\sigma}}{2{a}^{2}\sqrt {\pi }}}+\frac{1}{2a}
{{\rm erf}\left({\frac {a}{{\sigma}}}\right)}+ \frac{\sigma}{2a^2 \sqrt{\pi}} {{\rm e}^{-a^2/\sigma^2}}
$$</p>
<p>EDIT: OK, here's the proof.</p>
<p>For convenience, scale distances so that $\sigma ... |
2,779,429 | <blockquote>
<p>Evaluate $$\int \frac {dx}{\sin \frac x2\sqrt {\cos^3 \frac x2}}$$</p>
</blockquote>
<p>My try </p>
<p>Write $t=\frac x2$ and hence $dx=2dt$</p>
<p>To change the integral to $$\int \frac {\csc t dt}{\cos^{\frac 32} t}$$</p>
<p>Multiplying both bottom and top by $\csc t$ and then using $\csc^2 t=1+... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong>\begin{align}\int\frac{\mathrm dx}{\sin\left(\frac x2\right)\sqrt{\cos^3\left(\frac x2\right)}}&=\int\frac{\sin\left(\frac x2\right)}{\left(1-\cos^2\left(\frac x2\right)\right)\sqrt{\cos^3\left(\frac x2\right)}}\,\mathrm dx\\&=-2\int\frac{\mathrm dt}{(1-t^2)t\sqrt t}.\end{align}</p>
|
2,939,163 | <p>I want to find a certain <span class="math-container">$x$</span> that belongs to <span class="math-container">$\mathbb R$</span> so that </p>
<p><span class="math-container">$$\left|\begin{array}{r}1&x&1\\x&1&0\\0&1&x\end{array}\right|=1$$</span></p>
<p>This should be easy enough. I apply t... | say era | 470,734 | <p>as said before : <span class="math-container">$x=1$</span> is a solution.</p>
<p>You can then reduce you polynomial to: <span class="math-container">$x^3-2x+1=(x-1)*(x^2+x-1)$</span></p>
<p>So you can solve the second degree polynomial and get the two other solutions.</p>
|
1,077,284 | <p>I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D curve on an elliptical surface described nicely in <a href="https://math.stackexchange.com/a/1075515/62050">this post</... | JimmyK4542 | 155,509 | <p>Since $R$ is a simply connected region bounded by the curve $g$, Green's Theorem tells you that $$\displaystyle\iint\limits_{R}\left[\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right]\,dx\,dy = \oint\limits_{g}P\,dx+Q\,dy$$</p>
<p>for functions $P(x,y)$ and $Q(x,y)$. </p>
<p>You want to compute t... |
1,554,603 | <p>Let $\theta \in \mathbb R$, and let $T\in\mathcal L(\mathbb C^2)$ have canoncial matrix</p>
<p>$M(T)$ = $$
\left(
\begin{matrix}
1 & e^{i\theta} \\
e^{-i\theta} & -1 \\
\end{matrix}
\right)
$$
(a) Find the eigenvalues of $T$... | Mark | 94,840 | <p>As the other answer list, the number of ideals is actually $12$. One other way to show this is to use the Chinese Remainder Theorem, which gives an isomorphism
$$\mathbb Z\diagup60\mathbb Z \xrightarrow{\sim}
\left(\mathbb Z\diagup4\mathbb Z\right) \times
\left(\mathbb Z\diagup3\mathbb Z\right) \times
\left(\mathbb ... |
46,905 | <p>I need to draw a set of curves on one graph (characteristics equations). As you can see they have exchanged x and y axes. My goal is to plot all those curves on one graph. Are there ways to do that? </p>
<pre><code>f[t_, t0_] := -(2 - 4/Pi*ArcTan[2])*Exp[-t]*(t - t0);
g[x_, x0_] := (x - x0)/(-(2 - 4/Pi*ArcTan[x + ... | Kuba | 5,478 | <p>The problem boils down to "how to plot inverse function without explicit formula". You can use <code>ParametricPlot[{h[y],y},{y...]</code>:</p>
<pre><code>Show[
Plot[Table[f[t, t0], {t0, 0, 1, .1}], {t, 0, 1},
Evaluated -> True, PlotStyle -> Blue],
ParametricPlot[Table[{g[x, x0], x}, {x0, -0.3, 0, 0.... |
419,370 | <p>When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance:</p>
<p>Definition 1: A set $A$ is <em>open</em> in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such that $ B(x,\epsilon) \subseteq A$.</p>
<p>Since this is a definition, there are obviously no cases whe... | rschwieb | 29,335 | <p>Sidestepping the philosophical stuff that's about to ensue, let me say this. Since "if" in a definition is correct already, it would be unattractive to replace it with a more restrictive condition "iff." In mathematics, a rule of thumb is to not overcomplicate something by using a stronger thing when a weaker thing ... |
419,370 | <p>When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance:</p>
<p>Definition 1: A set $A$ is <em>open</em> in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such that $ B(x,\epsilon) \subseteq A$.</p>
<p>Since this is a definition, there are obviously no cases whe... | wendy.krieger | 78,024 | <p>The distinction between <strong>if</strong> and <strong>iff</strong> is that <strong>if</strong> can be a subset relation, while <strong>iff</strong> is a set equality relation.</p>
<p>The role of a definition is to bring things into view of a theory, so it needs to deal with failure in the theory as well. Corresp... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | Angina Seng | 436,618 | <p>The Cauchy-Riemann equations have a geometric interpretation. Let $f$
be holomorphic at $a$ and let $f'(a)\ne0$. Consider the horizontal
line through $a$ consisting of points $a+s$ for real $s$, and
also the vertical line through $a$, that is the points $a+it$ for
$t$ real.
Then these are mapped by $f$ into two curv... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | Didier | 788,724 | <p>Here is an idea that uses a weaker version of the open mapping theorem:</p>
<ul>
<li>if <span class="math-container">$f$</span> is constant, then there is nothing to prove</li>
<li>if <span class="math-container">$f$</span> is non-constant, <span class="math-container">$f'$</span> is not identically zero: let <span ... |
2,994,296 | <p>I'm trying to figure out how to prove, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{4n}}{(4n)!} = 0$$</span>
The problem is, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{n}}{n!} = \infty$$</span>
and I have no idea how to prove the first limit equals <span class="math-container"... | Anurag A | 68,092 | <p>Using Stirling's approximation: <span class="math-container">$n! \approx c \sqrt{n}(n/e)^n$</span>, we get
<span class="math-container">$$\frac{n^{4n}}{(4n)!} \approx \frac{n^{4n}}{c\sqrt{4n}(4n/e)^{4n}} \approx \left(\frac{e}{4}\right)^{4n}\frac{1}{c\sqrt{4n}} \overbrace{\longrightarrow}^{\because \frac{e}{4}<1}... |
3,844,448 | <p>Find all values of <span class="math-container">$h$</span> such that rank(<span class="math-container">$A$</span>) = <span class="math-container">$2$</span>.</p>
<p><span class="math-container">$A$</span> = <span class="math-container">$\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\en... | Cade Reinberger | 450,991 | <p>Well, you know that the column space has at least two linearly independent vectors. So, you just want the middle vector in your transformed matrix to be in the span of the other 2. Well, you can always add the left vector in the column space (<span class="math-container">$\hat{i}$</span>, you might call it) so that ... |
2,771,034 | <p>$\frac{a_n}{b_n} \rightarrow 1$ and $\sum_{n=1}^\infty b_n$ converges, can it be concluded that $\sum_{n=1}^\infty a_n$ converges?<br>
My attempt at an answer to this question: since $\sum_{n=1}^\infty b_n$ converges, $b_n \rightarrow 0$. Because of this, $a_n \rightarrow 0$ equally fast. However, I'm well aware tha... | SK19 | 509,159 | <p>$$a_n = \frac{a_n}{b_n}\cdot b_n$$</p>
<p>(given that $b_n\neq 0$) The intuition now tells us, that from a certain $N$ on, $\frac{a_n}{b_n}$ will be so close to $1$ that $a_n$ basically add not much more than $b_n$, so if $\sum b_n$ converges, so will $\sum a_n$. </p>
<p>But as the other answers have raised concer... |
888,319 | <p><span class="math-container">$ABC$</span> is an acute angled triangle, where <span class="math-container">$P$</span> is the orthocenter, and <span class="math-container">$R$</span> is the circumradius. I want to show that <span class="math-container">$PA+PB+PC\le 3R$</span> geometrically, that is without using trig... | DeepSea | 101,504 | <p><strong>Hint:</strong> the identity $a^2 + b^2 + c^2 + PA^2 + PB^2 + PC^2 = 12R^2$ is useful.</p>
|
479,594 | <p>I was wandering which is the best way to generate various combinations of $x_i$ such that $$\sum\limits_{i=1}^7 x_i = 1.0$$</p>
<p>where $ x_i \in \{0.0, 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$</p>
<p>I can generate these using brute-force, i.e checking through all $ 11^7$ combinations and only taking those whi... | bkarpuz | 53,441 | <p>Let $b_{n}:=\sum_{k=1}^{n-1}a_{k}$ for $n=2,3,\cdots$,
then $b_{n+1}-b_{n}=a_{n}$ for $n=2,3,\cdots$.
Then, the equation reads as $b_{n+1}-b_{n}=-\frac{1}{2}b_{n}+\frac{n}{4}$ for $n=2,3,\cdots$ with $b_{2}=a_{1}=\frac{1}{4}$. Rearraging the terms, we get $$\begin{cases}b_{n+1}-\frac{1}{2}b_{n}=\frac{n}{4},{\quad}n=... |
1,812,675 | <p>Is there a recurrence solution to $a_n=\frac{n}{a_{n-1}}$? I'm wondering if it could be done in the form of an alternating series partial to $n$ or as a trigonometric function.</p>
| Clement C. | 75,808 | <p><strong>Hint:</strong></p>
<p>Set $b_n = \ln a_n$. Then
$b_n = - b_{n-1} + \ln n$, and we can write
$$\begin{align}
b_n &= - b_{n-1} + \ln n
= b_{n-2} - \ln (n-1) + \ln n\\
&= -b_{n-3} + \ln(n-2) - \ln (n-1) + \ln n \\
&\vdots \\
&= b_1 + \sum_{k=1}^n (-1)^{n-k} \ln k
\end{align}$$</p>
<p>Can you c... |
2,458,863 | <p>I tried to find the critical points of the function</p>
<p>$$f(x,y) = x^2y-2xy + \arctan y $$</p>
<p>And I found that is $P(1,0)$, the problem is that the Hessian is null, and I don't know how to procede to determine the nature of that point.
Can you help me ?</p>
<p><strong>Update:</strong> Thanks you all, and I... | Tsemo Aristide | 280,301 | <p>If the class of of $[p]$ is invertible mod $m$, $a'p=am+1$ or $am-1$, you have
$(am+1)(bm+1)=m(mab+a+b)+1, (am+1)(bm-1)=m(abm-a+b)-1$</p>
<p>$ (am-1)(bm-1)=m(abm-a-b)+1$. This shows the product of two invertible numbers mod $m$ is $1$ mod $m$ or $-1$ mod $m$.</p>
|
2,278,431 | <p>"Apply Green's Theorem to evaluate the line integral of F around positively oriented boundary"</p>
<p>$$F(x,y)=x^2yi+xyj$$</p>
<p>C: The region bounded by y=$x^2$ and y=4x+5</p>
| farruhota | 425,072 | <p>If $n!\sim \left(\frac{n}{e}\right)^n\sqrt{2\pi n}$, then:</p>
<p>$$\frac{n^n}{n!e^n}=\frac{1}{n!} \cdot \left(\frac{n}{e}\right)^n\sim \left(\frac{e}{n}\right)^n \frac{1}{\sqrt{2\pi n}} \cdot \left(\frac{n}{e}\right)^n = \frac{1}{\sqrt{2\pi n}}$$
which is decreasing.</p>
|
3,443,094 | <blockquote>
<p>If <span class="math-container">$$\lim_{x\to 0}\frac{ae^x-b}{x}=2$$</span> the find <span class="math-container">$a,b$</span></p>
</blockquote>
<p><span class="math-container">$$
\lim_{x\to 0}\frac{ae^x-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)+a-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)}{x}+\lim_{x\to 0}\frac{a-b}{... | Ross Millikan | 1,827 | <p>You can use the Taylor series, <span class="math-container">$e^x=1+x+$</span> terms of order <span class="math-container">$x^2$</span> and higher. Plug that in. The fact that <span class="math-container">$a=b$</span> cancels the <span class="math-container">$1$</span> and you will be working with the <span class="... |
3,443,094 | <blockquote>
<p>If <span class="math-container">$$\lim_{x\to 0}\frac{ae^x-b}{x}=2$$</span> the find <span class="math-container">$a,b$</span></p>
</blockquote>
<p><span class="math-container">$$
\lim_{x\to 0}\frac{ae^x-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)+a-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)}{x}+\lim_{x\to 0}\frac{a-b}{... | Peter Szilas | 408,605 | <p>Option:</p>
<p>1) <span class="math-container">$\lim_{x \rightarrow 0}\dfrac{ae^x-b}{x}=2$</span>;</p>
<p><span class="math-container">$\lim_{x \rightarrow 0}(ae^x-b)=$</span></p>
<p><span class="math-container">$\lim_{x \rightarrow \infty}((\dfrac {ae^x-b}{x})\cdot x)=$</span></p>
<p><span class="math-container... |
1,246,356 | <p>Let $A,B \in {M_n}$ . suppose $A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?</p>
| robjohn | 13,854 | <p>The <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow">Euler-Maclaurin Summation Formula</a> says
$$
\begin{align}
\sum_{k=5}^m\frac{\log(\log(k))}{k\log(k)}
&=\frac12\log(\log(m))^2+C+O\left(\frac{\log(\log(m))}{m\log(m)}\right)
\end{align}
$$
Therefore,
$$
\begin{align}
&... |
4,480,905 | <p>When <span class="math-container">$T$</span> is any linear operator acting on a vector space <span class="math-container">$V$</span>, and <span class="math-container">$n$</span> is a natural number, <span class="math-container">$T^n$</span> means <span class="math-container">$T$</span> applied <span class="math-cont... | Gerald | 167,701 | <p><span class="math-container">$0^0$</span> is an indeterminate form. Consider the two limits:</p>
<p><span class="math-container">$$\lim_{x \rightarrow 0}\lim_{y \rightarrow 0}\ x^y
$$</span></p>
<p><span class="math-container">$$\lim_{y \rightarrow 0}\lim_{x \rightarrow 0}\ x^y
$$</span></p>
<p>For the first limit, ... |
3,583,879 | <blockquote>
<p>a) $P_5=11$$</p>
<p>b) <span class="math-container">$P_1+P_2+P_3+P_4+P_5 =26$</span></p>
</blockquote>
<p>For the first part
<span class="math-container">$$\alpha^5+\beta ^5$$</span>
<span class="math-container">$$=(\alpha^3+\beta ^3)^2-2(\alpha \beta )^3$$</span></p>
<p>I found the value of <... | trancelocation | 467,003 | <p>You can make life easier by realizing that</p>
<p><span class="math-container">$$P_k = \alpha^k+\beta^k$$</span> </p>
<p>is the solution of the linear recurrence</p>
<p><span class="math-container">$$a_{k+2}=a_{k+1}+a_k \text{ with } a_1 = \alpha + \beta = 1 $$</span>
<span class="math-container">$$\text{ and } ... |
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