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 |
|---|---|---|---|---|
17,074 | <p>My exams are nearing and I have been posting question on vector calculus which are not answered yet because I'm writing in simple English. Many people have edited my questions. Kindly guide me on how to learn that...</p>
<p>Thanks</p>
| Joel | 85,072 | <p>Many such OP's are new to the site. They would not know what that acronym means. This would just lead to confusion.</p>
|
4,315,858 | <p>When tackling the <a href="https://math.stackexchange.com/a/4313831/732917">question</a>, I found that for any <span class="math-container">$a>1$</span>,</p>
<p><span class="math-container">$$
I_1(a)=\int_{0}^{\pi} \frac{d x}{a-\cos x}=\frac{\pi}{\sqrt{a^{2}-1}}.
$$</span>
Then I started to think whether there is... | spaceman | 522,096 | <p>From what you have deduced, we can apply here the <a href="https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula" rel="nofollow noreferrer">Faà di Bruno's formula</a>
<span class="math-container">$$
\begin{align}
I_n(a) &= \frac{(-1)^{n-1}\pi}{(n-1)!} \frac{d^{n-1}}{da^{n-1}} \left(\frac{1}{\sqrt{a^2 - 1}}... |
26,313 | <p>In page 21 of <em>A Problem seminar</em>, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the <strong>failure probability formula</strong>. His formula goes like this</p>
<p>$f_{0}+f_{1}+f_{2}+\ldots$</p>
<p>where $f_{n... | S. Carnahan | 121 | <p>I'm not an expert in this area, but I'm told that the key phrase in the superalgebra world is "Weyl groupoid" rather than Weyl group. <strike>I did not look at the construction long enough to understand it.</strike> Serganova has a <a href="http://math.berkeley.edu/~serganov/papers/superk.pdf">paper</a> describing... |
187,489 | <p>I have a long expression involving different complex numbers. I want to force <em>Mathematica</em> to replace <code>z Conjugate[z]</code> by <code>Abs[z]^2</code> without using polar coordinates in the whole expression. For example,</p>
<pre><code>z w Conjugate[z] Conjugate[w] + Conjugate[z] w + z^2
</code></pre>
... | Robert Jacobson | 27,662 | <p>The <code>FullSimplify</code> function knows how to do that:</p>
<pre><code>FullSimplify[z Conjugate[z]]
</code></pre>
<blockquote>
<pre><code> Abs[z]^2
</code></pre>
</blockquote>
|
1,713,713 | <p>To my knowledge, the exponential function is the unique function satisfying</p>
<p>$f'=f$ and
$f(0)=1$</p>
<p>however, unless I've made a mistake, we have</p>
<p>$$\frac{\partial}{\partial x} (ax)^x = x (ax)^{x-1} a = ax (ax)^{x-1} = (ax)^x$$</p>
<p>and </p>
<p>$$(a0)^0 = 0^0 =1$$</p>
<p>so I feel like I must ... | Michael Hardy | 11,667 | <p>The rule that $\dfrac d{dx} x^n = nx^{n-1}$ holds when $n$ is constant, i.e. $n$ does not change as $x$ changes. In the case of $(ax)^x$, the exponent changes as $x$ changes, and the power rule is not applicable. You can use logarithmic differentiation in such a case.</p>
|
2,604,372 | <p>The formula for adding two vectors, as defined in Kells' Analytical Geometry is </p>
<p>$AB+BC=AC$</p>
<p>This makes sense since we're concerned with both the direction and magnitude of the vectors. When I got to the first exercise though, I was given the question </p>
<blockquote>
<p>A man walks east 6 miles t... | Jay Zha | 379,853 | <p>The addition "+" for vectors is defined <em>differently</em> than the "+" defined for scalar numbers.</p>
<p>Here we already have the definition for "+" of two vectors $AB$ and $BC$:</p>
<p>$$AB + BC = AC$$</p>
<p>You have $AB$ with direction EAST and magnitude $6$, $BC$ with direction NORTH and magnitude $6$, so... |
2,604,372 | <p>The formula for adding two vectors, as defined in Kells' Analytical Geometry is </p>
<p>$AB+BC=AC$</p>
<p>This makes sense since we're concerned with both the direction and magnitude of the vectors. When I got to the first exercise though, I was given the question </p>
<blockquote>
<p>A man walks east 6 miles t... | Siong Thye Goh | 306,553 | <p>We have $AB+BC=AC$ for vectors.</p>
<p>However, we do not have $|AB|+|BC|=|AC|$ in general.</p>
<p>If our goal is to travel from point $A$ to point $C$, we can make a stop at point $B$ and this would cost us more distance.</p>
<p>To minimize the distance, we can travel directly from $A$ to $C$ directly.</p>
|
1,400,436 | <p>This is a question we asked on a second semester calculus test.</p>
<p>For what values of $p$ does this series converge?
$$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$</p>
<p>I believe that it actually can be shown that $p> 0$ is a valid answer. </p>
<p>However. I am interested in finding a proof that is simple ... | Siminore | 29,672 | <p>Since
$$
\sin \frac1n \approx \frac1n,
$$
your series is equivalent to
$$
\sum_n \frac{1}{n^{p+1}}.
$$</p>
|
1,845,517 | <p>The number of primes less than $2^{2^{100}}$ is $(a)101$ $(b)100$ $(c) 2^{100}$ $(c)2^{101}$.</p>
<p>How can I solve this ? Please help.</p>
<p>Thank you.</p>
| SAJW | 346,682 | <p>$\frac{x}{ln(x)}$=number of primes until x (for large enough x, the error becomes almost nill)</p>
<p>i think the rest is easy, btw search prime number theorem :)</p>
|
37,316 | <p>I have defined a bunch of functions over several different files, and I would like a list of prototypes of all the functions I have defined in my current session (I want the mathematica equivalent of the .h file for my .c file).</p>
<p>For example, if I defined two functions,</p>
<pre><code>f[arg1_, arg2_]:= Modul... | m_goldberg | 3,066 | <p>There is something close to what you ask for built into <em>Mathematica</em>. It is the function called <a href="http://reference.wolfram.com/mathematica/ref/Information.html" rel="nofollow noreferrer"><code>Information</code></a>. In the form you would use to get what you asked about, there is a keyboard shortcut, ... |
37,316 | <p>I have defined a bunch of functions over several different files, and I would like a list of prototypes of all the functions I have defined in my current session (I want the mathematica equivalent of the .h file for my .c file).</p>
<p>For example, if I defined two functions,</p>
<pre><code>f[arg1_, arg2_]:= Modul... | cormullion | 61 | <p>How's your pattern matching? Mine's not at all good (yet), but you might be able to rustle up some kind of solution using this as a starting point:</p>
<pre><code>functionScan[p_] :=
StringCases[p,
result : (
WordCharacter .. ~~ "[" ~~ Except[ "]", _] .. ~~ "]" ~~ _ ~~
":=" ) :> result]
fil... |
37,316 | <p>I have defined a bunch of functions over several different files, and I would like a list of prototypes of all the functions I have defined in my current session (I want the mathematica equivalent of the .h file for my .c file).</p>
<p>For example, if I defined two functions,</p>
<pre><code>f[arg1_, arg2_]:= Modul... | Szabolcs | 12 | <p>Here's one solution (that needs a bit of improvement to make it useful):</p>
<pre><code>functionList[context_String] :=
TableForm@Sort[
HoldForm @@@ Flatten[
(ToExpression[#, InputForm, DownValues] & /@ Names[context <> "*"])[[All, All, 1]]]]
</code></pre>
<p>Now just do <code>functionList["Gl... |
2,214,287 | <p>My exam review states that I need to utilize the difference formula for sine to solve the equation on the interval $0 \leq \theta < 2\pi $</p>
<p>$$\sqrt3\sin \theta- \cos\theta = 1$$</p>
<p>I know that: $\sin \frac\pi3 = \frac{\sqrt3}{2}$
and $\cos\frac\pi3 = \frac12 $, so I divide each term by 2 and rewrite t... | Community | -1 | <p>What you are missing is how to match the addition formula and the given equation.</p>
<p>You have
$$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$$
vs.
$$\sin \theta\cdot\frac{\sqrt3}2-\cos\theta\cdot\frac12.$$</p>
<p>Then you identify</p>
<p>$$\alpha\leftrightarrow\theta\text{ and }\beta\leftrightar... |
2,250,900 | <p>As the title states, I'm trying to determine whether the series $$\sum_{n=1}^\infty \sin(n+1)-\sin(n+2)$$ converges or diverges. My intuition is saying diverging since the sines are oscillating. How will I go about 'formally' proving this? </p>
<p>I searched online for this and I got an 'explanation' saying $s_n=\s... | zhw. | 228,045 | <p>As others have noted, the $N$th partial sum of the series in this problem is $\sin 2 - \sin (N+2).$ So to show the series diverges, it's enough to show $\sin n,n=1,2,\dots$ diverges.</p>
<p>We can prove this by noting that $e^{in}$ lands in the arc $A=\{e^{it}: \pi/4 < t < 3\pi/4\}$ infinitely many times. Thu... |
1,123,694 | <p>Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$?</p>
<p>The first one seems very obvious, but is there any way to show it very formally? </p>
| Pedro M. | 21,628 | <p>Hint: By switching $z$ and $\bar{z}$ if necessary, we may assume that $a \neq 0$. Multiply the equation by $z^n$ to get
$a z^{2n} + b |z|^{2n} = 0$, so that
$z = |z| \omega$ for some $\omega \in \mathbb{C}$ with the property that $\omega^{2n} = -b/a$.</p>
<p>Now can you see that $|\omega| \neq 1$? (hint: look at $... |
2,426,244 | <p>Suppose $u:[0,1]\to\mathbf R$ is a bounded function, show that there exist $x,y\in[0,1]$ such that $|u(x)-u(y)|<|x-y|^{1/2}.$</p>
<p>I have some stupid trials but failed. It seems that we could argue via contradiction. </p>
<p>If for each $x,y\in[0,1]$ with $x\neq y,$ there holds $$|u(x)-u(y)|\geq|x-y|^{1/2}.$$... | Michael L. | 153,693 | <p>Let's prove the problem statement by contradiction. Suppose that $\lvert u(x)-u(y)\rvert\geq \lvert x-y\rvert^{1/2}$ for all $x, y\in [0, 1]$. Then, let $x_n = \frac{6}{\pi^2}\sum_{k=1}^n \frac{1}{k^2}$ so that $\lim_{n\to \infty} x_n = 1$ (recall that $\zeta(2) := \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}... |
3,443,910 | <blockquote>
<p>Prove that <span class="math-container">$f:[0,1] \rightarrow\mathbb{R}:f(x)= \left \{\begin {array}{ll}
\sin \frac1x &, \textrm{if}~ x\in(0,1]\\
0 &, \textrm{if}~~x =0
\end{array}
\right.~~$</span> is Riemann integrable using Darboux sums.</p>
</blockquote>
<p><em>Attempt.</em> The proof... | mathcounterexamples.net | 187,663 | <p>Take <span class="math-container">$\epsilon > 0$</span> and <span class="math-container">$n \in \mathbb N$</span> such that <span class="math-container">$0 < \frac{1}{n \pi} < \frac{\epsilon}{4}$</span>.</p>
<p>As <span class="math-container">$f$</span> is continuous on <span class="math-container">$[\frac... |
370,570 | <p>In the wikipedia page (<a href="http://en.wikipedia.org/wiki/Birthday_problem" rel="nofollow">http://en.wikipedia.org/wiki/Birthday_problem</a>) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some pair of them will have the same birthday. ... | Ittay Weiss | 30,953 | <p>Random, in its every-day meaning, simply means that uncertainty is involved. In the precise mathematical meaning it means that uncertainty is potentially involved. In any case, random does not imply any uniformity over the possible outcomes. Tossing a die with six sides numbered 1,2,3,4,5,6 produces a random event w... |
2,895,481 | <p>Consider a one-dimension $10$-step random walk with step size $1$, namely $S_n=\sum_{n=1}^{10} X_n$ and $X_i = 1$ or $-1$ for all $i\in\{1,2,...,10\}$. Further we require $S_i \geq 0$ for any $i$. And we have $S_{10} = 4$. How many paths satisfy this condition?</p>
<p>Actually it is a counting problem. It troubles ... | P. Quinton | 586,757 | <p>This is a pretty big spoiler but you could try to go check up the <a href="https://en.wikipedia.org/wiki/Catalan_number" rel="nofollow noreferrer">Catalan numbers</a></p>
|
2,895,481 | <p>Consider a one-dimension $10$-step random walk with step size $1$, namely $S_n=\sum_{n=1}^{10} X_n$ and $X_i = 1$ or $-1$ for all $i\in\{1,2,...,10\}$. Further we require $S_i \geq 0$ for any $i$. And we have $S_{10} = 4$. How many paths satisfy this condition?</p>
<p>Actually it is a counting problem. It troubles ... | Muralidharan | 520,162 | <p>If there are $n$ step 1 and $m$ step $-1$ to be placed in a row such that number of 1s is always equal or more than the number of $-1$s, the number of ways is $\frac{n-m+1}{n+1}\binom{n+m}{m}$. In this case, $n=7, m= 3$ and hence equals $\frac{5}{8}\binom{10}{3} = 75$.</p>
|
1,439,337 | <blockquote>
<p>Let $(a_n)_n$ and $(\sigma_n^2)_n$ be sequences of real numbers with $\sigma_n^2>0$ for all $n\in \Bbb N$ and let $$\mathcal P = \{\Bbb P^{X_n} : X_n \sim \mathcal N (a_n,\sigma_n^2)\}.$$
Then $\mathcal P$ is tight if and only if there is a $K>0$ such that $|a_n|\leq K$ and $\sigma_n^2\leq K$ ... | UUB | 271,519 | <p>Try using Markov's Inequality and get a bound that works.</p>
|
479,249 | <p>Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a set of zero Lebesgue measure?</p>
<p>Thanks.</p>
| Dominic Michaelis | 62,278 | <p>The <a href="https://en.wikipedia.org/wiki/Cantor_function" rel="nofollow">devils staircase</a> on the <a href="https://en.wikipedia.org/wiki/Cantor_set" rel="nofollow">Cantor set</a>. The Cantor set is a null set and its image is $[0,1]$</p>
<p>The Cantor set can be written as all $x$ of $\mathbb{R}$ such that
$$... |
3,460,766 | <p>I have a function:
<span class="math-container">$$r=\sqrt{x^2+y^2+z^2}$$</span>
and wish to calculate:
<span class="math-container">$$\frac{d^2r}{dt^2}$$</span></p>
<hr>
<p>so far I have said:
<span class="math-container">$$\frac{d^2r}{dt^2}=\frac{\partial^2r}{\partial x^2}\left(\frac{dx}{dt}\right)^2+\frac{\parti... | mathsdiscussion.com | 694,428 | <p>Let <span class="math-container">$f(x)=x^x$</span>
<span class="math-container">$$\ln(f(x))=x\ln x$$</span>
<span class="math-container">$$\frac{f'(x)}{f(x)}=(1+\ln x)$$</span>
<span class="math-container">$$f'(x)=f(x)\ln(ex)$$</span>
<span class="math-container">$$\text{For }0<ex<1,f'(x)\lt0$$</span>
<span cl... |
2,038,245 | <p>So, an integral is notated like this:</p>
<p>$$\int_a^bf(x)dx$$</p>
<p>And from my understanding, it's an operator that is defined for three operands: $a$ and $b$, which can be anything, and an integrand of the form $f(x)dx$.</p>
<p>$dx$ is just an infinitesimal number, so $f(x)dx$ is simply $f(x)$ multiplied by ... | imranfat | 64,546 | <p>Since you assume that $u$ is real, there is an alternative method: Use Pythagorean Theorem we get $\cos z=\pm\sqrt{1-u^2}$=$\pm{i}\sqrt{u^2-1}$. Now use $e^{iz}=\cos z+i\sin z$, take the $\ln$ on both sides and you are done. (assuming you know how to take the complex $\ln$). A numerical problem is worked out here, i... |
833,814 | <p>Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so </p>
<ol>
<li>$z=-96+160i$</li>
<li>$z=96-160 i$</li>
<li>$z=160-96i$</li>
<li>$z=-160+96i$</li>
</ol>
<p>what is the right answer ? Thanks in advance.</p>
| Anurag A | 68,092 | <p>Let $$F(x)=\int_x^1\frac{f(t)}{t^2} \, dt.$$
Since $f$ is continuous on $[0,1]$ hence it has a minimum value on this interval. Call it $m$. Then
$$\int_x^1\frac{f(t)}{t^2} \, dt \geq m \int_x^1\frac{1}{t^2} \, dt=m\left(\frac{1}{x}-1\right).$$
But as $x \to 0^{+}$, the last expression approaches $\infty$.</p>
<p>No... |
444,517 | <p>Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.</p>
<p>Question $1)$ Are there any other solutions ?</p>
<hr>
<p>If we set $x=y$ we can conclude that if there exi... | vadim123 | 73,324 | <p>Some partial results:</p>
<p>Assume henceforth that neither $f$ nor $g$ is constant and $0$.
Take $y=0$, we get $$f(x)=f(x)g(0)+f(0)g(x),\textrm{ or }f(x)(1-g(0))=f(0)g(x)$$
then either $g(0)=1$ and $f(0)=0$, or </p>
<blockquote>
<p>$$f(x)=\frac{f(0)}{1-g(0)}g(x)$$
Substituting $x=0$ gives $g(0)=1-g(0)$ so $g... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| SuperDave | 32,522 | <p>Here are my thoughts on the matter. However, this is not too much more than what is done above. I think...</p>
<p>$\quad$ We seek to show that a homotopy from the identity map of $S^{\infty}$ ($id_{S^{\infty}}$) to a constant map can be constructed and thus it must be null-homotopic, $i.e.$, contractible.
Let $T: ... |
3,073,189 | <p>Evaluate </p>
<p><span class="math-container">$$
\int_0^{+\infty} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx \, \, \, (\phi>0)
$$</span></p>
<p>See my answer below for a solution using a nice substitution.</p>
| NetUser5y62 | 252,948 | <p><strong>Solution</strong> </p>
<p><span class="math-container">$$ \small
\int_0^{+\infty} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx = \int_0^{1} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx + \int_1^{+\infty} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx
$$</span></p>
<p>Notice that with the substitution <span class="math-container">$y=\frac... |
3,384,793 | <p>I have the following sequence,
<span class="math-container">\begin{align*}
P_n=\displaystyle \cfrac{1}{n^2}{\prod_{k=1}^{n}(n^2+k^2)^\frac{1}{n}} \:\:\:\: \:\: n\geq 1
\end{align*}</span>
The sequence seems to converge toward zero. But I have a hard time proving it. My strategy is to use a the following theorem.... | Jack D'Aurizio | 44,121 | <p>Further hint:
<span class="math-container">$$ \log P_n = \frac{1}{n}\sum_{k=1}^{n}\log\left(1+\frac{k^2}{n^2}\right)\stackrel{\text{Riemann sums}}{\longrightarrow}\int_{0}^{1}\log(1+x^2)\,dx =-2+\frac{\pi}{2}+\log 2$$</span>
gives that <span class="math-container">$P_n$</span> <strong>does not</strong> converge towa... |
3,384,793 | <p>I have the following sequence,
<span class="math-container">\begin{align*}
P_n=\displaystyle \cfrac{1}{n^2}{\prod_{k=1}^{n}(n^2+k^2)^\frac{1}{n}} \:\:\:\: \:\: n\geq 1
\end{align*}</span>
The sequence seems to converge toward zero. But I have a hard time proving it. My strategy is to use a the following theorem.... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,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{\exp... |
2,734,257 | <p>I have tried to show that this limit :
$$\lim\limits_{n\to \infty }\frac{n}{n!^{\frac 1 n}}=e$$</p>
<p>using $ \lim (1+\frac 1 n)^{\frac 1 n} , n \to \infty $ , I don't find any equivalence , however wolfram alpha says that is $e$ as shown <a href="https://www.wolframalpha.com/input/?i=lim+(n+%2F+((n!)%5E(1%2Fn))... | DeepSea | 101,504 | <p>We have: $\displaystyle \lim_{n \to \infty} -\dfrac{1}{n}\displaystyle \sum_{k=1}^n \ln \left(\dfrac{k}{n}\right)= -\displaystyle \int_{0}^1\ln xdx=1\implies \displaystyle \lim_{ n\to \infty} \ln\left(\dfrac{n}{n!^{1/n}}\right)=1\implies \displaystyle \lim_{n \to \infty} \dfrac{n}{n!^{1/n}}=e$. </p>
|
3,847,369 | <p>I was given the following definition of the cross product:</p>
<blockquote>
<p>The vector product <span class="math-container">$\underline{a}\times\underline{b}$</span> is defined as the vector with magnitude <span class="math-container">$\lvert\underline{a} \times \underline{b}\rvert = \vert\underline{a}\rvert\lver... | Gyro Gearloose | 290,307 | <p>The cross product can also be defined as a mapping of <span class="math-container">$\vec a$</span> and <span class="math-container">$\vec b$</span> to a <em>function</em> <span class="math-container">$\vec x\mapsto \det(\vec a,\vec b, \vec x)$</span>.</p>
<p>This function has properties that make it a <em>homomorphi... |
357,102 | <p>If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$?</p>
<p>It has been established that this is $m^n$. </p>
<p>How many of these are one-to-one?
I think this means that each element of $|A|$ is fixed to a unique $|B|$.</p>
| Xiaolang | 71,857 | <p>$m^n$ is right
if we let element in A be $a1,a2...an$
in B be $b1,b2...bm$</p>
<p>then every $ai$ has $m$ kinds of choice to be destination.</p>
<p>so in sum $m^n$ </p>
<p>if one to one
there must be $m>n$
then for $a1$ he has m kinds of choice and $ai$ has m-i+1
so in sum $m(m-1)...(m-i+1)...(m-n+1)=m!/(m-n... |
357,102 | <p>If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$?</p>
<p>It has been established that this is $m^n$. </p>
<p>How many of these are one-to-one?
I think this means that each element of $|A|$ is fixed to a unique $|B|$.</p>
| WakeUpDonnie | 71,907 | <p>Think what it means for a function to map from A into B. It is pairing an element of A with one and only one element of B. How many ways can we do that? Think similarly for one-to-one functions. (The number of surjective functions are a lot trickier to count.)</p>
|
684,639 | <p>Prove that in the inverse function theorem, the hypothesis that $f$ is $C^1$ cannot be weakened to the hypothesis that $f$ is differentiable. I read an example of my teacher, but I can't have any analysis argument for the fact that $f$ is not one to one in any neighborhood of $0$. Here it is, </p>
<p>$$f(0)=0, \qq... | Zatrapilla | 293,077 | <p>It seems to me that your teacher is wrong. Look here for a version of the inverse function theorem when the function is only everywhere differentiable but not $C^1$ (see for example <a href="https://mathoverflow.net/questions/75049/does-the-inverse-function-theorem-hold-for-everywhere-differentiable-maps">this MO q... |
2,698,775 | <p>Let E be the cylindrical frame field
<span class="math-container">$E_1 = \cos\theta U_1 + \sin\theta U_2, E_2 = − \sin\theta U_1 + \cos\theta U_2, E_3 = U_3$</span></p>
<p>(a) Starting from the basic cylindrical equations <span class="math-container">$x = r \cos\theta, y = r \sin\theta, z = z$</span>, show that the... | Ivo Terek | 118,056 | <p>One way to do it is just recalling from linear algebra that the transformation between dual basis uses the inverse of the transition matrix between the original bases. Using columns whose entries are the vectors themselves requires an extra transposition. Meaning that if <span class="math-container">$$\begin{pmatrix... |
1,120,737 | <p>Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$</p>
<p>After a bit of algebra I get that</p>
<p>$$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$</p>
<p>So now how do I justify that it's divisible by $3$? Can I show that both expressions are divisible by $3$ ... | user26486 | 107,671 | <p>$$3\mid a+b\mid (a+b)((a+b)^2-3ab)=(a+b)^3-3ab(a+b)=a^3+b^3$$</p>
<p>or</p>
<p>$$3\mid a+b\mid (a+b)(a^2-ab+b^2)=a^3+b^3$$</p>
<hr>
<p>This uses the fact that $a\mid b\mid c\implies a \mid c$, where $a\neq 0$. It can be proved <a href="http://en.wikipedia.org/wiki/Divisor" rel="nofollow">by definition</a> as fol... |
1,026,066 | <p>Problem:
It is researched that 60% of people in city goes to cinema on daily basis, 40% of people goes to theater on daily basis. It is also known that 20% simultaneously goes to both theater and cinema.</p>
<p>What is</p>
<ol>
<li>
Probability that chosen person does not attend both.</li>
<li>Probability that ch... | Alistair | 168,361 | <p>For question <strong>3</strong>: You are looking for a probability that person attends theater <strong>from those</strong> that attend cinema. Therefore;</p>
<p>$$
P(person \space attends \space theater \space | \space person \space attends \space cinema)=P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{0.24}{0.6}=\frac{2}{5}
... |
1,721,584 | <p>The image attached below is a problem on induction, the proof has been included.
I am enquiring if anyone could explain line for line what the proof states with its notation ( the notation is new to me). (I have a bit of experience with proof by induction, but is stumped by this problem)</p>
<p><a href="https://i.s... | Matan L | 111,632 | <p>Lets denote the following:</p>
<p>$y(x) = (x^2-x)^3$</p>
<p>$f(x)=x^2-x$</p>
<p>$g(x)=x^3$</p>
<p>than $y(x)=g(f(x))$
so by the chain rule you get that $y_x = g_x (f(x)) f_x (x)$</p>
<p>thus in your case $y_x = 3f(x)^2f_x(x)=2(x^2-x)^2(2x-1)$</p>
|
4,144,159 | <p>I was reading <a href="https://rads.stackoverflow.com/amzn/click/com/188652923X" rel="nofollow noreferrer" rel="nofollow noreferrer">Introduction to Probability, 2nd Edition</a>, and the following question appears as exercise <span class="math-container">$23 (b)$</span> in the 2nd chapter:</p>
<blockquote>
<p>A fair... | Doug M | 317,162 | <p>One approach is to enumerate potential sequences and look for a pattern.</p>
<p>HH, TT<br />
HTT, THH<br />
HTHH, THTT</p>
<p><span class="math-container">$E[X] = \frac 12\cdot 2 + \frac 14\cdot 3+\frac 18\cdot 4+\cdots$</span></p>
<p>And resolve this infinite series.</p>
<p><span class="math-container">$E[X] = 3$</... |
4,144,159 | <p>I was reading <a href="https://rads.stackoverflow.com/amzn/click/com/188652923X" rel="nofollow noreferrer" rel="nofollow noreferrer">Introduction to Probability, 2nd Edition</a>, and the following question appears as exercise <span class="math-container">$23 (b)$</span> in the 2nd chapter:</p>
<blockquote>
<p>A fair... | Graham Kemp | 135,106 | <blockquote>
<p>such that Y is the number of tossed until a head appears, and Z is the number of tosses until a tail appears (starting from the moment after the head appears).</p>
</blockquote>
<ul>
<li>None of the tails <em>before</em> the <em>first head</em> can follow a head.</li>
<li>All of the results <em>between<... |
259,156 | <p>I am using a piecewise function to define the height of columns. My goal is to make a nice picture that illustrates the different heights of the columns using color. Below in the picture I have described what I have generated using code on the left and on the right I have illustrated what I am trying to achieve:</p>... | kglr | 125 | <pre><code>Manipulate[Plot[f[[1]][a x], {x, 0, f[[2]] Pi}],
{a, 1, 10},
{{f, {Sin, 1}}, {{Sin, 1} -> Sin, {Cos, 2} -> Cos, {Tan, 3} -> Tan}, PopupMenu}]
</code></pre>
<p><a href="https://i.stack.imgur.com/wkNNj.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/wkNNj.gif" alt="enter image descripti... |
2,410,940 | <p>The question states that if $f (x)$ is a polynomial such that $x-1|f(x^n)$ prove that $f(x^n)$ is divisible by $x^n-1$</p>
<p>This is how I proceeded since$x-1|f(x^n)$ </p>
<p>$f(1)=0$ </p>
<p>$\frac {f (x^n)-f(1)}{x^n-1}=g(x)$ </p>
<p>since $f(1)=0$ </p>
<p>$\frac{f(x^n)}{x^n-1}=g(x)$
hence $x^n-1|f(x^n)$</p>... | Jyrki Lahtonen | 11,619 | <p>José's solution is nice. And I agree with his assessment that an error you made is that you didn't explain why $g(x)$ is a polynomial. To see the error consider the following piece of faulty reasoning:</p>
<p><strong>A False Fact.</strong> If $f(1)=0$ then $(x-1)^2\mid f(x)$.</p>
<p><strong>Proof.</strong>
Write
$... |
2,410,940 | <p>The question states that if $f (x)$ is a polynomial such that $x-1|f(x^n)$ prove that $f(x^n)$ is divisible by $x^n-1$</p>
<p>This is how I proceeded since$x-1|f(x^n)$ </p>
<p>$f(1)=0$ </p>
<p>$\frac {f (x^n)-f(1)}{x^n-1}=g(x)$ </p>
<p>since $f(1)=0$ </p>
<p>$\frac{f(x^n)}{x^n-1}=g(x)$
hence $x^n-1|f(x^n)$</p>... | A.Γ. | 253,273 | <p>$x-1|f(x^n)$ $\quad\Rightarrow\quad$ $f(1)=0$ $\quad\Rightarrow\quad$ $x-1|f(x)$ $\quad\Rightarrow\quad$ $x^n-1|f(x^n)$. </p>
|
22,794 | <p>I have 2d list which is upper triangular. I would like to interpolate it, but I cannot unless I set the interpolation order to 1. I tried making the list rectangular by filling the bottom half with <code>Null</code>. This does allow me to interpolate with higher order. However, the interpolation function ends too ea... | andre314 | 5,467 | <p>Here are the interpolation along row N°10 of your table :</p>
<p><img src="https://i.stack.imgur.com/k4dmK.jpg" alt="enter image description here"></p>
<p>One see on the first graphic that the end of data are at position 21. </p>
<p>On the other graphics, one see that if you use <code>InterpolationOrder</code> s... |
1,664,385 | <blockquote>
<p>A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x axis and end point B lies on y axis. A point P(x,y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.</p>
</blockquote>
<p>I understand the definitions of... | peter.petrov | 116,591 | <p>The locus of P is the set of points which P covers/forms, when A and B are being moved/varied while still satisfying the conditions from the problem statement. </p>
|
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| colormegone | 71,645 | <p>The discussion on this question has been an education to me, as it is clear that a fair amount of "sloppiness" has crept into the typical presentation of fractional exponents in introductory courses or general practice. I, for one, was shown a long time ago that $ \ \sqrt{x^2} \ = \ |x| \ . $ [In fact, in the earl... |
2,650,454 | <p>I am able to prove the binomial theorem (see $(1))$ by induction. I have also seen the binomial theorem written in another way (see $(2)$) where the summation changes a little bit. I understand how to compute both theorems, but I am having a difficult time proving the second theorem from the first. </p>
<p>Obviousl... | BallBoy | 512,865 | <p>First, we rename $i$ to $i_1$, giving us
$$ (x_1+x_2)^n = \sum_{i_1=0}^n \frac{n!}{(n-i_1)!i_1!}x_1^{n-i_1}x_2^{i_1}$$
Now we introduce the index $i_2$ and define $i_2=n-i_1$. A substitution like this
is equivalent to summing one term -- just the $i_2=n-i_1$ term, no other values
of $i_2$ are included -- so I'll ... |
2,650,454 | <p>I am able to prove the binomial theorem (see $(1))$ by induction. I have also seen the binomial theorem written in another way (see $(2)$) where the summation changes a little bit. I understand how to compute both theorems, but I am having a difficult time proving the second theorem from the first. </p>
<p>Obviousl... | epi163sqrt | 132,007 | <blockquote>
<p>We obtain
\begin{align*}
\color{blue}{\sum_{i=0}^n\frac{n!}{(n-i)!i!}x_1^{n-i}x_2^i}
&=\sum_{{i+j=n}\atop{i,j\geq 0}}\frac{n!}{j!i!}x_1^jx_2^i\tag{1}\\
&\color{blue}{=\sum_{{i_1+i_2=n}\atop{i_1,i_2\geq 0}}\frac{n!}{i_1!i_2!}x_1^{i_1}x_2^{i_2}}\tag{2}
\end{align*}</p>
</blockquote>
<p><em>Co... |
101,805 | <p>Let $m$ be a positive integer. Let $a,b$ be integers with $0 \leq a,b < m$, $a,b$ not both zero, $\gcd(a,b,m)=1$.</p>
<p>Do there necessarily exist integers $x,y$ such that<br>
$x \equiv a \pmod{m}$<br>
$y \equiv b \pmod{m}$<br>
$(x,y)=1$?</p>
<p>Equivalently, are there integers $c,d,k,l$ such that
$$
c(a+mk)... | André Nicolas | 6,312 | <p>Suppose that $a$ and $m$ are relatively prime, and also $b$ and $m$. (This is a more stringent condition than what you want.)</p>
<p>By Dirichlet's Theorem on primes in arithmetic progressions, there are infinitely many primes of the form $a+sm$, also infinitely many primes of the form $b+tm$. So in fact there are ... |
340,417 | <p>How to calculate following integration?</p>
<p><span class="math-container">$$\int 5^{x+1}e^{2x-1}dx$$</span></p>
| Dennis Gulko | 6,948 | <p>$$\int 5^{x+1}e^{2x-1}dx=\int e^{(x+1)\ln 5}e^{2x-1}dx=\int e^{(2+\ln 5)x+\ln5-1}dx=5e^{-1}\int e^{(2+\ln 5)x}dx$$
Can you continue?</p>
|
340,417 | <p>How to calculate following integration?</p>
<p><span class="math-container">$$\int 5^{x+1}e^{2x-1}dx$$</span></p>
| Abhijit | 27,809 | <p>$$\displaystyle \int 5^{x+1}e^{2x-1}dx$$
$$=\int e^{(x+1)ln(5)}e^{2x-1}dx$$
$$=\int e^{(x+1)ln(5)+2x-1}dx$$
$$=\int e^{(\ln(5)+2)x+\ln(5)-1}dx$$
$$=\int e^{(\ln(5)+2)x}\cdot e^{\ln(5)-1}dx$$
$$=\frac{5}{e}\int e^{(\ln(5)+2)x}dx$$
$$=\frac{5}{e}\int e^{(\ln(5)+2)x}dx$$
$$=\frac{5^{x} e^{2 x}}{e \log{\left (5 \right )... |
2,378,004 | <p>Let $n\ge 1$ be an integer and let $\vec{x} := \left( x_j \right)_{j=1}^n$ be normal variables with zero mean and with a correlation matrix ${\bf C}$. The question is to compute the following expectation value:
\begin{equation}
\mu_T(n):=E\left[ x_1 \cdot \prod\limits_{\xi=2}^n \theta_{T}(x_\xi) \right] = ?
\end{equ... | Przemo | 99,778 | <p>This is not a full answer to this question however I want to post it because I see it as a necessary milestone in order to achieve the full answer. Let us consider a slightly different expectation value. We define:
\begin{equation}
{\bar \mu}_T(n) := E\left[ \prod\limits_{\xi=1}^n \theta_T(x_\xi) \right]
\end{equat... |
2,615,825 | <p>What I did was:
I tested for $\lim_\limits{n\to\infty}u_n$ by taking log</p>
<p>$$\lim_\limits{n\to\infty} \frac{\ln\ \left(4 - \frac{1}{n}\right)} {\frac{n}{(-1)^n}}$$</p>
<p>Applying L'hopital's rule,</p>
<p>$$\lim_\limits{n\to\infty} \frac{\left(\frac{1} {4-\frac{1}{n}}\right)\left(\frac{1}{n^2}\right)}{(-1)^n... | Barry Cipra | 86,747 | <p>Rewriting the equation in polar coordinates, with $x=r\cos\theta$ and $y=r\sin\theta$, we get</p>
<p>$$r^2(10\cos^2\theta+14\cos\theta\sin\theta+10\sin^2\theta)-7=0$$</p>
<p>Using the trig identities $\sin^2\theta+\cos^2\theta=1$ and $2\sin\theta\cos\theta=\sin2\theta$, we find this simplifies to</p>
<p>$$r^2(10+... |
1,374,676 | <p>The series is as follows:</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=0}^\infty \frac{(-1)^nx^n}{(n!)^2}$$</span></p>
</blockquote>
<p>I tried working on it. The square in the denominator is breaking me. Please If any one could help.
And I need to find the sum of this series.</p>
<p>Edit:</p>
<blockqu... | gogurt | 29,568 | <p>If $X_1, \ldots, X_{19}$ are the 19 students you sample from the class, then you can assume they're iid with distribution $N(112, 14)$ as stated in the problem. Then write $\bar{X} = (X_1 + \ldots + X_{19})/19$ for the sample mean you're dealing with.</p>
<p>Just from rules for means and variances of sums of indepe... |
3,119,573 | <blockquote>
<p>Let <span class="math-container">$R$</span> be the equivalence relation on the real numbers given by
<span class="math-container">$$R = \{(x, y) \in \Bbb R^2: (x−y)(x+y) = 0 \} $$</span>
What are the equivalence classes of <span class="math-container">$R$</span>?</p>
</blockquote>
<p>So I wrote t... | PrincessEev | 597,568 | <p>Yes, I believe you're correct.</p>
<hr>
<p>We know</p>
<p><span class="math-container">$$(x-y)(x+y) = x^2 - y^2$$</span></p>
<p>as it is the difference of two squares. From that,</p>
<p><span class="math-container">$$(x-y)(x+y) = 0 \iff x^2 = y^2 \iff |x| = |y| \iff x = \pm y$$</span></p>
<p>Thus,</p>
<p><spa... |
1,212,185 | <p>I have the following equation to solve, I think is just the case of founding a suitable change of variables, but I couldn't think of anything: $$y'' + (y')^2 = 2e^{(-y)}.$$</p>
<p>Any suggestions?</p>
| Chappers | 221,811 | <p>Set $u=e^{y}$. Then $y=\log{u}$, and $y'=u'/u$, $y''=u''/u-u'^2/u^2$. Substituting in,
$$ \left( \frac{u''}{u}-\frac{u'^2}{u^2} \right) + \left( \frac{u'}{u} \right)^2 = \frac{2}{u}. $$
Then
$u''=2,$
and no doubt you can carry on easily enough.</p>
|
1,607,395 | <blockquote>
<p>Consider quadratic equations <span class="math-container">$Ax^2 + Bx + C = 0$</span> in which <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, and <span class="math-container">$C$</span> are
independently distributed <span class="math-container">$\mathsf{Unif}(0,1)$<... | BruceET | 221,800 | <p><strong>Sketch of analytic solution.</strong> An analytic solution is based on noting that the density of $Q = B^2$ is $f(q) = \frac{1}{2\sqrt{q}},$ for $q \in (0,1),$ the density of $X = 4AC$ is $g(x) = \frac{-\log(x/4)}{4},$ for $x \in (0,4).$ </p>
<p>An appropriate double integration of the joint density $h(q, x... |
1,607,395 | <blockquote>
<p>Consider quadratic equations <span class="math-container">$Ax^2 + Bx + C = 0$</span> in which <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, and <span class="math-container">$C$</span> are
independently distributed <span class="math-container">$\mathsf{Unif}(0,1)$<... | Will Jagy | 10,400 | <p>Someone asked almost this a few days ago. I did want to point out that a cube is not the most natural shape to consider for this problem, although it was the one chosen. Better, in some ways, to consider the ball $A^2 + B^2 + C^2 \leq R^2.$ In that case we take rotated coordinates
$$ u = B; v = (A - C)/ \sqrt 2; w =... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | lhf | 532 | <p>Also <a href="https://sites.math.washington.edu/~morrow/336_14/fta.pdf" rel="nofollow noreferrer">The Fundamental Theorem of Algebra: A Visual Approach</a> by Velleman.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Amy Pang | 712 | <p>Two (or three) more complex analysis approaches, the first is "essentially the same" as the proof in Alfors I think, but the second is different (I'm afraid I don't have a reference, but I can type up the full proofs if you want):</p>
<p>Let p be a polynomial and n be its degree.</p>
<ul>
<li><p>apply the residue ... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Margaret Friedland | 14,493 | <p>Another probabilistic proof:</p>
<p>Pascu, Mihai N.
A probabilistic proof of the fundamental theorem of algebra. (English summary)
Proc. Amer. Math. Soc. 133 (2005), no. 6, 1707–1711 (electronic).</p>
<p>Summary: "We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support th... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Joe Shipman | 25,424 | <p>Thanks to Tim Chow for citing me. Technically, you don't need to show every polynomial of prime degree in F[x] has a root, you just need to show that there is a field G such that every polynomial of odd prime degree in G[x] has a root and every element or its additive inverse has a square root; then G[i] will be alg... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Yuri Sulyma | 18,702 | <p>I don't have it on hand, but Ronald Solomon's <em>Abstract Algebra</em> has an interesting proof using symmetric polynomials and induction on the 2-adic valuation of the degree.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Johannes Huisman | 85,592 | <p>In <a href="https://mathoverflow.net/q/237856/85592">this post</a> the OP wonders how one can prove the FTA using the hairy ball theorem, after having read a book of Ian Stewart containing an allusion to such a proof. The <a href="https://mathoverflow.net/a/237900/85592">answer</a> I gave could be added to the curre... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Uri Bader | 89,334 | <p>I believe the following proof had not appeared before, though it is motivated by <a href="https://mathoverflow.net/a/42499/89334">this previous answer</a> and a comment by Benjamin Steinberg to <a href="https://mathoverflow.net/a/10685/89334">this answer</a>.</p>
<hr>
<p>Consider a field extension $F$ of $\mathb... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Ali Taghavi | 36,688 | <p>There is an alternative proof for FTA using "Fredholm operators on Hilbert spaces":</p>
<p>Assume that $P(z)=z^n+a_{n-1}z^{n-1}+\ldots+a_1 z+a_0$ has no root in $\mathbb{C}$. Then for every $\epsilon$ the polynomial $Q(z)=\epsilon^nP(z/\epsilon)=z^n+\epsilon a_{n-1} z^{n-1} +\ldots+\epsilon^{n-1}a_1z+\epsilon^n a... |
1,053,738 | <p>I've been asked to prove that:
$$
\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}
$$
When $0<x<2\pi$.</p>
<p>I know there are many similar posts on this site, but using $\cos(kx)$ instead, that's why I created this post, I can't get to the $1/2\cot$ in this one.
Thanks!</p... | IAmNoOne | 117,818 | <p>You asked to prove this by DeMovire's formula, which states $$e^{ikx} = (\cos x + \sin x)^k = \cos kx + \sin kx.$$</p>
<p>Everything you did until the final line is correct. </p>
<p>The final move is to do the following, $$\sin \left ( \frac{nx}{2} \right ) \frac{\sin\frac{(n+1)x}2}{\sin{\frac{x}2}} = \frac{\cos(x... |
1,452,559 | <p>This is a problem from a linear algebra textbook. Given a finite dimensional inner product space $V$ with orthonormal basis $e_1, \ldots, e_n$, show that if a list of vectors $v_1, \ldots, v_n$ satisfies $\|e_j - v_j\| < \frac{1}{\sqrt{n}}$ for all $j$ in $\{1, \ldots, n\}$, then $v_j$'s form a basis of $V$.</p>
... | metic | 274,426 | <p>May be just use the definition of linear dependence of vectors. Let $\alpha=(\alpha_1,\cdots,\alpha_m)$ such that $\sum_{i=1}^n\alpha_iv_i=0$
then
$\sum_{i=1}^n|\alpha_i|^2=\|\sum_{i=1}^n\alpha_ie_i-\sum_{i=1}^n\alpha_iv_i\|^2=\|\sum_{i=1}^n\alpha_i(e_i-v_i)\|^2$
$\leq (\sum_{i=1}^n|\alpha_i|\|(e_i-v_i)\|)^2<(\s... |
2,494,160 | <p>I've been having trouble using the definition of a limit to prove limits, and at the moment I am trying to prove that
$$\lim_{n\to\infty} \frac{n^x}{n!}=0$$</p>
<p>for all $x$ which are elements of natural numbers.
I'm able to start the usual setup, namely let $0<\epsilon$ and attempt to obtain $\left\lvert\dfr... | Magneto | 460,436 | <p>we have following theorem:</p>
<p>if $<a_n>$ is a sequence then if $ \lim_{n \to \infty} \frac {a_{n+1}}{a_n} = 0$ then $<a_n> \rightarrow 0$</p>
<p>Using the above theorem</p>
<p>$ \lim_{n \to \infty} \frac {a_{n+1}}{an} = \lim_{n \to \infty} \frac {1}{n+1} (1+\frac{1}{n})^x = 0 \implies $ given lim... |
956,490 | <p>A 5 card hand is dealt from a well-shuffled deck of 52 poker cards. If the first two cards are the 10 of diamonds and the 10 of hearts, what is the probability of having been dealt a full?</p>
<p>This is what I did: </p>
<p>Let $A$ be the event the first two cards are the 10 of diamonds and the 10 of hearts.
We w... | Elaine | 178,914 | <p>Given that you have already been dealt the two tens, you have two potential scenarios for a full house. The first scenario would be to get one more ten and a pair, and the second scenario would be to get three of a kind. We can compute these the probabilities of these two scenarios separately and sum them.</p>
<p>F... |
3,730,256 | <p>I have a real orthogonal matrix so the column vectors form an orthogonal system and thus the vectors have length one.</p>
<p>I now want to show that for an arbitrary column vector <span class="math-container">$v_k \in \mathbb{R^n}$</span> the absolute value of the greatest entry <span class="math-container">$|v_{k_i... | José Carlos Santos | 446,262 | <p>Every column has norm <span class="math-container">$1$</span>. So, the sum of the squares of its entries is equal to <span class="math-container">$1$</span> and therefore no entry can have absolute value greater than <span class="math-container">$1$</span>.</p>
<p>But if all of them had absolute value smaller than <... |
4,512,068 | <p>In an <a href="https://arxiv.org/abs/math/0506319" rel="nofollow noreferrer">article</a> by Guillera and Sondow, one of the <a href="https://mathworld.wolfram.com/UnitSquareIntegral.html" rel="nofollow noreferrer">unit square integral</a> identities that is proved (on p. 9) is: <span class="math-container">$$\int_{0... | Zacky | 515,527 | <p>We can start by considering the following integral:
<span class="math-container">$$\int_0^1\int_0^1 \frac{(xy)^a}{z-xy}dxdy=\frac{1}{z}\sum_{n=0}^\infty \int_0^1\int_0^1(xy)^a\left(\frac{xy}{z}\right)^ndxdy$$</span>
<span class="math-container">$$=\sum_{n=0}^\infty \frac{1}{z^{n+1}}\int_0^1x^{n+a}dx\int_0^1y^{n+a}dy... |
3,664,795 | <p>I have two binomial expression: <span class="math-container">$S1= \sum_{k=0}^{\frac{n}{2}}{{n}\choose{2k}} $</span> and <span class="math-container">$ S2 =\sum_{k=0}^{\frac{n-1}{2}}{{n}\choose{2k + 1}} $</span>. I have to prove those two expression are equal, and then find their shared value.</p>
<p>So far I'm tryi... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>For natural number <span class="math-container">$n,$</span></p>
<p><span class="math-container">$$(1+x)^n+(1-x)^n=?$$</span></p>
<p><span class="math-container">$$(1+x)^n-(1-x)^n=?$$</span></p>
<p>Then set <span class="math-container">$x=1$</span></p>
|
3,695,971 | <p>The way I understand it currently, saying "<em>only if <span class="math-container">$P$</span>, then <span class="math-container">$Q$</span></em>" is like saying that "<em>only if <span class="math-container">$P$</span> happens, <span class="math-container">$Q$</span> happens.</em>" To me, it seems to say the same t... | Peter Smith | 35,151 | <p>A number greater than two is prime <em>only if</em> it is odd. True. [If it were even it would be divisible by two and not a prime.]</p>
<p>A number great than two is prime <em>if and only if</em> it is odd. False. [9 is odd and greater than two but not prime.]</p>
<p>A two-way conditional requires more for its tr... |
1,548,314 | <p>Does there exist a continuous function $f:\mathbb R^2\to \mathbb R$ such that $\displaystyle \frac{\partial f}{\partial x}$ <strong>does not exists</strong> but $\displaystyle \frac{\partial^2 f}{\partial x\partial y}$ exists.</p>
<p>I think yes. But I am unable to find an example of such function. Can anyone help ... | MooS | 211,913 | <p>Let $f$ only depend on $x$, being continuous but not differentiable. Then $f_y=0$, hence $f_{xy}=0$. (Assuming that this means first differentiating with respect to $y$)</p>
|
1,432,003 | <p>A is a $n\times k $ matrix.</p>
<p>I have to show that $\|A\|_2\leq \sqrt{\|A\|_1\cdot \|A\|_\infty}$. </p>
<p>I know that $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| $ for every $\| \cdot \|$ submultiplicative matrix norm, but I don't know how to conclude.</p>
<p>Any idea? </p>
| R.N | 253,742 | <p>or use $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| _\infty\leq\|A^H\|_\infty \cdot \|A\|_\infty \leq \|A\|_1.\|A\|_\infty$</p>
<p>Because $\|A^H\|_\infty=\|A\|_1 $ </p>
|
4,098,179 | <p>Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an elegant path is not always app... | Mark S. | 26,369 | <p>It all depends on the context and the goal.</p>
<p>If your goal is just to <em>find</em> answers and verify a result, then by all means use a <a href="https://en.wikipedia.org/wiki/Computer_algebra_system" rel="nofollow noreferrer">CAS</a> to do the algebra for you. (For example, some computation-heavy papers just s... |
948,819 | <blockquote>
<p>Find the domain of the function $$g(x)=\log_3(x^2-1)$$</p>
</blockquote>
<p>This is what I got so far:</p>
<p>$$\{ x\mid x^2-1>0\} =$$
$$\{ x\mid x^2>1\} =$$
$$\{ x\mid x>\sqrt { 1 } \}= $$</p>
<p>I don't know where to go from here to arrive at the correct answer... I would like a nudge in... | Chinny84 | 92,628 | <p>the set of equations to solve
$$
\dfrac{1}{3}\left(\dfrac{S_c}{100}+\dfrac{S_t}{200} + \dfrac{S_a}{300}\right) = 0.85,\\
\dfrac{S_c +S_t + S_a}{600} = 0.86,\\
\dfrac{1}{2}\left(\dfrac{S_c}{100}+\dfrac{S_t}{200}\right) = \dfrac{S_a}{300} + 4
$$</p>
|
948,819 | <blockquote>
<p>Find the domain of the function $$g(x)=\log_3(x^2-1)$$</p>
</blockquote>
<p>This is what I got so far:</p>
<p>$$\{ x\mid x^2-1>0\} =$$
$$\{ x\mid x^2>1\} =$$
$$\{ x\mid x>\sqrt { 1 } \}= $$</p>
<p>I don't know where to go from here to arrive at the correct answer... I would like a nudge in... | Sfarla | 178,341 | <p>Let $A_c, A_t, A_a$ be the average for children, teens and adults respectively.
The condition that "The average of these three averages is 85%" is given by the equation
$$ \dfrac{A_c+A_t+A_a}{3}=0.85.$$
The condition "The overall average of the 600 people is 86%"
can be written as
$$ \dfrac{100A_c+200A_t+300A_a}{600... |
932,596 | <p>Evaluation of $\displaystyle \int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx$</p>
<p>$\bf{My\; Solution::}$ Given $\displaystyle \int\frac{1}{\sin^2 x\cdot (5+4\cos x)}dx = \int \frac{1}{(1-\cos x)\cdot (1+\cos x)\cdot (5+4\cos x)}dx$</p>
<p>Now Using Partial fraction for $\displaystyle \frac{1}{(1-\cos x)\cd... | Claude Leibovici | 82,404 | <p>Doing the same as Aditya (Weierstrass substitution), you arrive to $$\int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx=\int \frac{\left(t^2+1\right)^2}{2 t^2 \left(t^2+9\right)}dt$$ Now, using partial fraction decomposition $$\frac{\left(t^2+1\right)^2}{2 t^2 \left(t^2+9\right)}=\frac{1}{18 t^2}-\frac{32}{9 \left... |
3,992,896 | <p>I had this question, and I was wondering how to do it. The question was what is the probability of not drawing a pair out of a deck of cards, and I wasn't sure how to do it could someone help me out</p>
| tommik | 791,458 | <p>the requested joint distribution can be represented by the following table</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th></th>
<th style="text-align: center;">Y=0</th>
<th style="text-align: center;">Y=1</th>
<th style="text-align: center;">Total</th>
</tr>
</thead>
<tbody>
<tr>
<td><s... |
4,618,382 | <p>An answer to <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics">this question</a> claimed that an interesting mathematical mistake is believing that</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is a smooth function with <span class="math-container">$f... | Matija | 1,096,797 | <p>Yes, this is correct. You can also use the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Formal_statements" rel="nofollow noreferrer">fundamental theorem of calculus</a>. You have that <span class="math-container">$F(x)=\int_a^xf'(y)\mathrm dy=0$</span> and for any function <span class="math... |
593,418 | <p>Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$</p>
<hr>
<p>By computing some partial sums, the answers are 0. It seems an inductive argument is possible.</p>
| Igor Rivin | 109,865 | <p>There is a heavy machinery approach. First, the sum is obviously a hypergeometric sum, and can be fed to Mathematica, which uses the W-Z method to produce:</p>
<p>$$
2^{-n} \left(2 n \,_3F_2\left(\frac{1}{2},\frac{1}{2}-\frac{n}{2},1-\frac{n}{2}; \frac{3}{2},\fra
c{3}{2};1\right)+2^{n+1} \Phi (2,1,n+1)+i \pi \ri... |
2,416,718 | <p>Prove that for all real numbers $x,y,z$:<br>
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\implies \frac{1}{x^5}+\frac{1}{y^5}+\frac{1}{z^5}=\frac{1}{(x+y+z)^5}.$$</p>
| Michael Rozenberg | 190,319 | <p>The condition gives
$$(x+y+z)(xy+xz+yz)=xyz$$ or
$$(x+y)(x+z)(y+z)=0$$
and since for $y=-x$ it's true and we need to prove something symmetry, we are done!</p>
|
244,429 | <p>If I have the following data (x value is composition and y value is temperature):</p>
<pre><code>data = {{0, 54.61`}, {100, 57.26243979492134`}, {80,53.839874154239816`}, {50, 54.09456572258326`}, {24, 56.15393883162748`}}
</code></pre>
<p>Which plotted like this gives:</p>
<pre><code>ListPlot[List /@ data, Frame -&... | JimB | 19,758 | <p>I think that the functions you're trying to fit are inappropriate for the data (or at best no theoretical justification for the models is presented) and you only have 5 data points. (There's an old saying about restaurant review: The food was bad and the portions too small.)</p>
<p>But you are right in that you ne... |
79,041 | <p>This maybe a simple question, but I am just stuck with it.
I want to do some simulation, say with 0.9 probability, I get a 1, and 0.1 probability get a 0.</p>
<p>How would I do that? Where should I start?</p>
<p>Thanks!</p>
| kirma | 3,056 | <p><code>BernoulliDistribution</code> is a perfect fit for this.</p>
<pre><code>RandomVariate[BernoulliDistribution[1 - 0.1], {50}]
</code></pre>
<blockquote>
<p>{1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1}</p>... |
79,041 | <p>This maybe a simple question, but I am just stuck with it.
I want to do some simulation, say with 0.9 probability, I get a 1, and 0.1 probability get a 0.</p>
<p>How would I do that? Where should I start?</p>
<p>Thanks!</p>
| David G. Stork | 9,735 | <pre><code>RandomChoice[{.9, .1} -> {1, 0}, 10]
</code></pre>
<p>(*
{0, 1, 1, 1, 1, 1, 1, 0, 1, 1}
*)</p>
<p><strong>Timing results</strong></p>
<ul>
<li><code>Timing[RandomVariate[BernoulliDistribution[.9], {10^8}];]</code> (* {3.38014, Null} *)</li>
<li><code>Timing[RandomChoice[{.9, .1} -> {1, 0}, 10^8];]</... |
2,833,085 | <p>This is basic, I know, but it's been a long time since I've done equations. I'm watching a tutorial video on circuits. </p>
<p>Let's say I have this equation:</p>
<p><a href="https://i.stack.imgur.com/jhU7S.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jhU7S.jpg" alt="enter image description h... | dxiv | 291,201 | <p>Hint: divide by $\,2^n\,$, then $\dfrac{a_n}{2^n} = \dfrac{a_{n−1}}{2^{n-1}} + 3\,$, which shows that $\,\dfrac{a_n}{2^n}\,$ is an arithmetic progression.</p>
|
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| Hawk | 105,577 | <p><strong>Hint:</strong> Do you know the property $a^{\log_bc}=c^{\log_ba}$. Clearly this applies here. Try to think of the proof for the above expression.</p>
|
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| vonbrand | 43,946 | <p>What you want is that $x^y$ be continuous both in $x$ and $y$, and to coincide with the (real branch of) $x^y$ for rational $y$. For simplicity, apply natural logarithms to the relevant equations.</p>
|
3,014,113 | <blockquote>
<p>A function <span class="math-container">$f$</span> defined on interval <span class="math-container">$(0,1)$</span> with a continuous twice derivation <span class="math-container">$(f\in{C^2(0,1)})$</span> satisfies <span class="math-container">$\lim_{x\to0^+}f(x)=0$</span> and <span class="math-contai... | Marco | 582,590 | <p>Let <span class="math-container">$g(x)=f(1/x)$</span> for <span class="math-container">$x\in (1,\infty)$</span>. The hypotheses on <span class="math-container">$f(x)$</span> imply that
<span class="math-container">$$\lim_{x \rightarrow \infty} g(x)=0~\mbox{and}~|(x^2g'(x))'|<C,$$</span>
since <span class="math-c... |
60,697 | <p>Let's see the following test case</p>
<pre><code>data = Table[{RandomReal[{-10, 10}], RandomReal[{-10, 10}]}, {i, 1, 50}];
l0 = ListPlot[data, PlotStyle -> {Blue, PointSize[0.01]}];
p0 = Plot[x*Sin[x], {x, -10, 10}, PlotStyle -> {Red, Thick}];
s0=Show[{l0, p0}, Frame -> True, FrameLabel -> {"x"... | eldo | 14,254 | <pre><code>Show[{l0, p0},
Frame -> True,
FrameLabel -> {"x", "y"},
FrameTicks -> {{Automatic, None},
{{0, {1.5, Style["P1\n1.5", Red, 14]}, 2, 4,
6, {7, Style["P2\n7.0", Red, 14]}, 8, 10}, None}},
Axes -> False,
GridLines -> {{1.5, 7}, {}},
GridLinesStyle -> Directive[Black, Thickness[0.... |
221,500 | <p>I have a list of lists similar to this:</p>
<pre><code>L = {{"a", "b", "c"}, {"x", "c", "y"}, {"i", "j", "h"}, {"x", "b", "z"}}
</code></pre>
<p>Each list within <code>L</code> happens to be of length 3. Suppose I need to find the position of the lists that have a particular element (say, "b") at the <span class="... | KennyColnago | 3,246 | <p>I tried a solution with <code>Pick</code>.</p>
<pre><code>f[v_,p_,e_]:=Pick[Range[Length[v]], v[[All, p]], e]
</code></pre>
<p>where <code>v</code> is your <code>L</code>, <code>p</code> is your <code>queriedPosition</code> and <code>e</code> is your <code>queriedElement</code>.</p>
<p>Here is a timing test of <c... |
1,482,253 | <p>Prove the following. What would be the summation formula be for the first part?</p>
<p><a href="https://i.stack.imgur.com/nkO0A.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nkO0A.jpg" alt="enter image description here"></a></p>
| RRL | 148,510 | <p>Hint: Binomial theorem and </p>
<p>$$1-\frac{k}{n+1} > 1-\frac{k}{n}$$</p>
|
1,482,253 | <p>Prove the following. What would be the summation formula be for the first part?</p>
<p><a href="https://i.stack.imgur.com/nkO0A.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nkO0A.jpg" alt="enter image description here"></a></p>
| Mark Viola | 218,419 | <p>From the <a href="https://en.wikipedia.org/wiki/Binomial_theorem#Statement_of_the_theorem" rel="nofollow">Binomial Theorem</a>, we have</p>
<p>$$\begin{align}
e_n&=\left(1+\frac1n\right)^n\\\\
&=\sum_{k=0}^n\binom{n}{k}\left(\frac1n\right)^k\\\\
&=1+n\left(\frac1n\right)+\frac{n(n-1)}{2!}\left(\frac1n\r... |
3,869,945 | <p>I have a problem that is formulated as this:
<span class="math-container">$$\begin{matrix}\min\\x \in \mathbb{R}^2\end{matrix} f(\mathbf{x}) := (2 x_1^2 - x_2^2)^2 + 3x_1^2-x_2$$</span>
The task is: Perform <strong>one</strong> iteration using the steepest descent algorithm when <span class="math-container">$\mathbf... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$x_1=x$</span> and <span class="math-container">$x_2=y$</span> and <span class="math-container">$y^2\leq\frac{3}{4}$</span>.</p>
<p>Thus, by AM-GM
<span class="math-container">$$(2x^2-y^2)^2+3x^2-y=4x^4-4x^2y^2+3x^2+y^4-y=$$</span>
<span class="math-container">$$\geq x^2(4x^2+3-4y^2)... |
3,103,160 | <p>I am dealing with some expressions containing combinatoric numbers. Does anybody know a formula for this?</p>
<p><span class="math-container">$$\displaystyle\sum_{k=0}^{\left\lfloor \dfrac{n}{2} \right\rfloor} \binom{n}{k}\binom{m}{k}$$</span></p>
| Ingix | 393,096 | <p>In the very last step: Multiplying <span class="math-container">$2(y−\frac12)^2$</span> by 2 gets you <strong>either</strong></p>
<p><span class="math-container">$4(y−\frac12)^2$</span></p>
<p><strong>or</strong></p>
<p><span class="math-container">$(2y−1)^2$</span></p>
<p>You incorrectly did both.</p>
|
762,762 | <p>Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$</p>
<p>What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$?</p>
<p>The answer is $a_n$, but I can't solve it.</p>
<p>See, what I've done is substitute $x$ as $-\frac{1}{x}$ and I've got:</p>
<p>${\frac{(x^2-x+1)}{... | robjohn | 13,854 | <p>Since
$$
(1+x+x^2)^n=\sum_{k=0}^{2n}a_kx^k\tag{1}
$$
we can look at the following in two ways
$$
\begin{align}
\left(1+\frac1x+\frac1{x^2}\right)^n
&=\sum_{k=0}^{2n}a_k\frac1{x^k}\\
&=\sum_{k=0}^{2n}a_kx^{-k}\tag{2}
\end{align}
$$
or as
$$
\begin{align}
\left(\frac1{x^2}+\frac1x+1\right)^n
&=\left(\frac{... |
2,836,705 | <p>Excerpt from text:</p>
<p>3.109 The range of T'</p>
<blockquote>
<p>Suppose V and W are finite-dimensional and T <span class="math-container">$\in$</span> L(V,W). Then</p>
<p>range T' = <span class="math-container">$(null\;T)^0$</span></p>
</blockquote>
<p><strong>Proof</strong></p>
<p>First suppose <span class="mat... | Will Jagy | 10,400 | <p>define
$$ w = 3^{3^n} $$</p>
<p>Your number is then
$$ w(w+1) + 3 w - 1 = w^2 +w + 3w -1 = w^2 + 4 w -1 $$</p>
<p>Now, $w$ is odd, so $w^2$ is odd, so $w^2 + 4 w -1$ is even. Also, $w \geq 3,$ so $w^2 + 4 w -1$ is bigger than 4 and even, therefore not prime. </p>
|
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