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 |
|---|---|---|---|---|
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Leox | 97,339 | <p>Let $p$ be the last prime. Then according to Bertrand's postulate the interval $(p,2p)$ contains a prime number. We get a contradiction.</p>
|
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Mathelogician | 386,128 | <p>There is a very clever one-line proof which I can't understand why it's not here. The Proof: </p>
<p>$$
0
< \prod\limits_{p} \sin\left(\frac{\pi}{p}\right)
= \prod\limits_{p} \sin\left(\frac{\pi(1+2\prod_{p'}p')}{p}\right)
=0
.
$$</p>
<p>This proof has created by Sam Northshield in 2015 and published in Ameri... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | LPenguin | 586,419 | <p>(Another take on Euler's proof, I believe)</p>
<p>Assume there are finitely many primes $\left \{ p_{0}, p_{1}, ... p_{k} \right \}$. Since every natural number admits a unique prime factorization up to reordering:</p>
<p>$$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+... = \sum_{n_{0}=0}^{\infty}p_{0}^{-n_{0}... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Maged Saeed | 492,661 | <p>Here is another idea to prove this statement. Our instructor attributed it to the French mathematician <em><strong>Hermite</strong></em>. The proof goes as follows:</p>
<blockquote>
<p><em>for a given integer <span class="math-container">$n$</span>. Let <span class="math-container">$q_n$</span> be the greatest prime... |
1,341,130 | <p>I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from <a href="http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/" rel="nofollow">this</a> lecture.</p>
<p>In the lecture... | EditPiAf | 418,542 | <p>The PDE <span class="math-container">$u_y + (1-2u)u_x = 0$</span> is the Lighthill-Whitham-Richards traffic flow equation. The proposed initial-value problem may be interpreted as a one-way road with the car density <span class="math-container">$u=1/4$</span> before the abscissa <span class="math-container">$x=0$</s... |
81,805 | <blockquote>
<p>The binomial sum
$$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$
satisfies the Fibonacci relation.</p>
</blockquote>
<p>I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions?</p>
| Marc van Leeuwen | 18,880 | <p>The problem with proving $\binom{n-k+1}k=\binom{n-k}k+\binom{n-k-1}k$ is that it is not true (try $k=0$ for starters). However for $n\geq1$ one has
$$\begin{aligned}
s_{n-1}&=\sum_{k=0}^{\lfloor\frac n2\rfloor}\binom{n-k}k\\
&= 1+\sum_{k=1}^{\lfloor\frac n2\rfloor}(\binom{n-1-k}k+\binom{n-1-k}{k-1})\\
&a... |
81,805 | <blockquote>
<p>The binomial sum
$$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$
satisfies the Fibonacci relation.</p>
</blockquote>
<p>I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions?</p>
| TheRealFakeNews | 43,823 | <p>We begin with the identity:
$$
\binom{n+1}{k+1} = \binom{n}{k+1} + \binom{n}{k}.
$$
We apply the identity to
$$
s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots
$$
to get
$$
s_n= \underbrace{\color{blue}{\binom{n}{0}} + \color{grey}{\binom{n}{-1}}}_{\text{first term}} + \underbrace{\color{blue}{\binom{n-1}{1}}... |
120,675 | <p>Given these condition, I am seeking a proof:</p>
<p>Define a sequence of real numbers by $x_1 = 3$ and then, for $n \geq 2$,</p>
<p>$x_n = \sqrt{2 \; x_{n-1} + 1 }$</p>
<p>Prove that for all positive integers $n$, $x_n \geq x_{n+1}$.</p>
<p>I began a proof by induction, but ran out of steam. I tested the base ca... | J126 | 2,838 | <p>For the induction, assume that $x_n\leq x_{n-1}$. Then</p>
<p>$$
\begin{align*}
x_n &\leq x_{n-1}\\
2x_n &\leq 2x_{n-1}\\
2x_n+1 &\leq 2x_{n-1}+1\\
\sqrt{2x_n+1} &\leq \sqrt{2x_{n-1}+1}\\
\end{align*}
$$</p>
|
3,067,021 | <p>I've been working through a derivation and have arrived at the following exprssion: </p>
<p><span class="math-container">$$E = 1 - \frac{x}y \left( \bigg\lfloor \dfrac{2x+yx-2}{2y} \bigg\rfloor \right)^{-1}$$</span></p>
<p>where <span class="math-container">$x,y \in \mathbb{R^+}$</span>.</p>
<p>I would like to kn... | Dan Uznanski | 167,895 | <p>In general, a floor function cannot be simplified away: it creates an often-infinite series of discontinuities that make it impossible to characterize the function without a way to get all the discontinuities in. For this, the floor function is ideal, and so we use it.</p>
<p>In this particular case, we have a lar... |
81,279 | <p>I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there?</p>
<p>I'm aware of two examples: <a href="https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem" rel="nofollow noreferrer">Szemerédi's theorem</a> and <a href="https://arxiv.org/abs/0910.3926" rel="... | Kevin O'Bryant | 935 | <p>Fix <span class="math-container">$\epsilon>0$</span>. Every subset <span class="math-container">$A$</span> of <span class="math-container">$\{1,2,\dots,n\}$</span> with <span class="math-container">$|A|\geq \epsilon n$</span> contains a centrally symmetric subset <span class="math-container">$E$</span> (in other ... |
761,072 | <p>A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure).
The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = \{q_1, q_2, \dots\}$. Then:</p>
<p>$$
\lambda(\mathbb{Q})
\le \lambda\left(\bigcup_{n \in \mathbb{N}} B_{r_n}(q_n)\r... | leonbloy | 312 | <p>This is not exactly your scenario, but it might help to gain intuition. Inside the interval $[0,1]$, consider the set of rationals of the form $m/2^n$ for $m,n \in \mathbb{N}$. These are the numbers that, expressed in binary, have finite fractional part. This set is dense in $[0,1]$. However, if we place over each o... |
785,087 | <p>Off the top of my head, I can't think of a non-separable compact space. </p>
<p>Can you provide a good example?</p>
| Thomas Lumley | 847,299 | <p>Based on <a href="https://math.stackexchange.com/questions/3903794/a-one-point-compactification-of-a-non-separable-metric-space-separable-or-not/3906738#3906738">this question I asked recently</a>, the one-point compactification of a non-separable metric space.</p>
<p>I specifically asked about the space of cadlag r... |
466,603 | <p>How to go about evaluating the following integral?</p>
<p>$$ I = \int \sqrt{ \dfrac {\sin(x-\alpha)} {\sin(x+\alpha)} }\,\operatorname d\!x$$</p>
<p>What I have done so far:</p>
<p>$$ I = \int \sqrt{ 1-\tan\alpha\cdot\cot x }\,\operatorname d\!x$$
Let $ t^2 = 1-\tan\alpha\cdot\cot x $</p>
<p>$$ \begin{align}
2... | dajoker | 72,504 | <p>Given that,</p>
<p>$$I = \int \sqrt{ \dfrac {\sin(x-\alpha)} {\sin(x+\alpha)} }\,dx$$</p>
<p>multiplying and dividing by $\sqrt{\sin(x-\alpha)}$.</p>
<p>we get,</p>
<p>$$I = \int { \dfrac {\sin(x-\alpha)} {\sqrt{\sin(x+\alpha)\cdot \sin (x-\alpha) }}}\,dx$$</p>
<p>$$I=\int \dfrac{ \sin x\cdot \cos{\alpha }}{\sq... |
1,301,701 | <p>What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$? Also, since $\mathbb{Z}[i]$ is a PID, we should be able to write this $\mathbb{Z}[i]$-module as a direct sum of cyclic $\mathbb{Z}[i]$-modules. Can we?</p>
| Slade | 33,433 | <p>If $R=\mathbb{Z}[2i]$, then $\mathbb{Z}[i] \cong R[X]/I$, where $I=(X^2+1,2X-2i)$.</p>
<p>So we have $\mathbb{Z}[i] \otimes_R R[X]/I \cong \mathbb{Z}[i][X]/I$.</p>
<p>Substituting $Y=X-i$, this is $\mathbb{Z}[i][Y]/(Y^2+2iY,2Y) \cong \mathbb{Z}[i][Y]/(Y^2,2Y) \cong \mathbb{Z}[i] \oplus \mathbb{Z}[i]/2\mathbb{Z}[i]... |
1,301,701 | <p>What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$? Also, since $\mathbb{Z}[i]$ is a PID, we should be able to write this $\mathbb{Z}[i]$-module as a direct sum of cyclic $\mathbb{Z}[i]$-modules. Can we?</p>
| Qiaochu Yuan | 232 | <p>The simpler tensor product $\mathbb{Z}[i] \otimes \mathbb{Z}[i]$ is free abelian on the generators $1 \otimes 1, 1 \otimes i, i \otimes 1, i \otimes i$. As either a left or a right $\mathbb{Z}[i]$-module it is free on two generators $1 \otimes 1, i \otimes i$.</p>
<p>This tensor product is the quotient of the above... |
1,905,891 | <p><a href="https://i.stack.imgur.com/IDrI5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IDrI5.png" alt="enter image description here"></a></p>
<p>Doubt is that If $\gamma < \alpha$ then how does it follow that $\gamma$ doesnot belong to B and how does it follows that $\alpha \leq x $ for ever... | Sathasivam K | 355,833 | <p>L is the set of all lower bound of B and $\alpha$ is the supreme of L.if an element $\gamma$ <$\alpha$ ,it mean that $\gamma$ is belong to L and not belong to B.</p>
<p>Also,since,L is thethe set of lower bound of B,which means x$\in B$ is the upper bound of L.but $\alpha $ is greatest lower bound ,so x must be ... |
2,016,775 | <blockquote>
Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and that $f(r) = 0$ for every rational number $r$. Prove that $f(x) = 0$ $\forall$ $x \in \mathbb{R}$
</blockquote>
<p>Ok, here's what I've done: </p>
<p>Let $c \in \mathbb{Q}$.<br>
Since, $f$ is continuous so $\forall \ep... | MPW | 113,214 | <p>Let $x\in\mathbb R$. By continuity, $f(x)=\lim_{y\to x}f(y)$. Since this limit exists, it is independent of the manner of approach $y\to x$; specifically, we may assume that $y\to x$ through rational values since $\mathbb Q$ is dense in $\mathbb R$. But then $f(x) = \lim_{y\to x} 0 = 0$.</p>
|
3,059,572 | <p>I've been practicing horizontal asymptotes and I came across a problem that I do not understand.</p>
<p>I understood why <span class="math-container">$\lim_{x \rightarrow \infty}\frac{x - 2}{\sqrt{x^2 + 1}} = 1$</span>, but i couldn't understand why <span class="math-container">$\lim_{x \rightarrow -\infty}\frac{x... | Peter Szilas | 408,605 | <p>Set <span class="math-container">$y=-x$</span>, where <span class="math-container">$y>0$</span>(why?).</p>
<p><span class="math-container">$-\dfrac{y}{\sqrt{y^2}} = - \dfrac{y}{|y|}=-\dfrac{y}{y}=-1.$</span></p>
|
3,059,572 | <p>I've been practicing horizontal asymptotes and I came across a problem that I do not understand.</p>
<p>I understood why <span class="math-container">$\lim_{x \rightarrow \infty}\frac{x - 2}{\sqrt{x^2 + 1}} = 1$</span>, but i couldn't understand why <span class="math-container">$\lim_{x \rightarrow -\infty}\frac{x... | Ryan Goulden | 398,056 | <p>If you consider that <span class="math-container">$\frac{-2}{(x^2 +1)^{\frac{1}{2}}}$</span> goes to <span class="math-container">$0$</span>, then let us just observe <span class="math-container">$\frac{x}{(x^2 +1)^{\frac{1}{2}}}$</span>
<span class="math-container">$$= \frac{1}{\frac{1}{x} (x^2 +1)^{\frac{1}{2}}}$$... |
183,022 | <p>can the following proposition be proved? If so please suggest a method. Can Kempe’s Argument be used for proof ?<br>
Proposition: A normal map has a colouring of countries by 4 colours iff the edges of the map can be properly coloured by 3 colours. </p>
| Christian Blatter | 8,050 | <p>This was proven already in the $19^{\rm th}$ century. In the following paper it is shown that the "eessentially different" $3$-colorings of the edges are in one-one correspondence with the "essentially different" $4$-colorings of the countries:</p>
<p><a href="http://retro.seals.ch/digbib/view?rid=ensmat-001:1965:1... |
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Will Jagy | 3,324 | <p>This one is pretty good. Kaplansky wanted squarefree numbers $x$ such that
$$ \sigma ( x^3) = y^2, $$ where $\sigma$ is the sum of divisors function. Somewhere around here I have a short note of his. He referred to this (and some very similar problems) as Ozanam's problem, from page 56 of Dickson's History, volume 1... |
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Timothy Chow | 3,106 | <p>There is a Martin Gardner problem, reprinted in his <i>Unexpected Hanging</i> collection, that goes like this:</p>
<blockquote>
<p>Miranda beat Rosemary in a set of tennis, winning 6–3. There were five service breaks. Who served first?</p>
</blockquote>
<p>One solution is as follows. The wins by the pla... |
1,186,424 | <p>The "divisibility rule" to test whether a given integer is divisible by $7$ (or, more generally, to find the remainder when an integer is divided by $7$) is in my opinion, ridiculous. The method is so convoluted that I can't even remember it off the top of my head, and it is almost certainly slower than simply subtr... | Bamboo | 179,361 | <p>I ran through your two examples using the method you linked (assuming you're talking about the $10x+y$, $x-2y$ method). </p>
<p>1234<br>
123-2*4=115<br>
11-2*5=1</p>
<p>314159<br>
31415-2*9=31397<br>
3139-2*7=3125<br>
312-2*5=302<br>
30-2*2=26 </p>
<p>So there are 2 steps in the first instance and 4 steps in th... |
970,160 | <p>Suppose the set </p>
<p>$$S := \left\lbrace x+i \sin \left( \frac{1}{x} \right) \Bigg\vert x \in (0,1]\right\rbrace \subseteq \mathbb{C}. $$</p>
<p>I want to show that $S$ is path-connected but $\overline{S}$ is not path connected.</p>
<p>To show that $S$ is path-connected I have to show that for all points $p,q... | Haha | 94,689 | <p>Hint:$f(x)=\sin(\frac {1}{x})$ is continuous on $(0,1]$. So intermediate value theorem.</p>
<p>Write is as $S=\{(x,f(x)):x\in(0,1]\}$. $f$ is continuous , and $S$ is the graph of $f$.</p>
|
1,599,025 | <p>I am working through some problems on induction and I have been stuck on this one for a while. If anyone has any hints. I can show it is true for the $n=1$ and $n=2$ case but I am having difficulty on the induction step.
This is what I have so far,</p>
<p>$ (1+\frac{1}{n} )^n = 1+ \sum\limits_{k=1}^n[ \frac{1... | spaghetti | 302,297 | <p>s(n+1)
Hints:
1.(1 + 1/n+1)^n+1 = (1 + 1/n)^n(1+ 1/n)...</p>
<p>2.Split up the sums</p>
<p>3.<strong>Use the binomial Theorem</strong></p>
|
1,599,025 | <p>I am working through some problems on induction and I have been stuck on this one for a while. If anyone has any hints. I can show it is true for the $n=1$ and $n=2$ case but I am having difficulty on the induction step.
This is what I have so far,</p>
<p>$ (1+\frac{1}{n} )^n = 1+ \sum\limits_{k=1}^n[ \frac{1... | Mark Viola | 218,419 | <p>While the following development is not an induction proof, I thought it would be instructive to present a direct proof of the equality of interest. To that end, we have from the binomial theorem </p>
<p>$$\begin{align}
\left(1+\frac1n\right)^n&=\sum_{k=0}^n\binom{n}{k}\frac1{n^k}\\\\
&=\sum_{k=0}^n\frac{n!... |
1,696,535 | <p>I am trying to find if $f(x)=x^3-117x+53$ is irreducible or reducible. </p>
<p>I tried Eisenstein's criterion but this fails as $53$ does not divide $117$</p>
<p>After some thinking, if f(x) has any roots then they will be negative, but I have had no luck finding any.</p>
<p>Would really appreciate some help, th... | André Nicolas | 6,312 | <p>A cubic (or quadratic) with rational coefficients is irreducible over $\mathbb{Q}$ if and only if it has no rational roots.</p>
<p>By the <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">Rational Roots Theorem,</a> the only candidates for rational roots of our cubic are $\pm 1$ and $\pm ... |
15,086 | <p>I'm looking around for a text that covers vector calculus and multivariable calculus, and that is also "free as in speech," not just "free as in beer." In other words, I'm looking for texts that are not just available for downloading free of charge but that are also under a license such as GFDL or Wikipedia's CC-BY-... | Daniel Hast | 3,505 | <p><a href="https://openstax.org/subjects/math" rel="nofollow noreferrer">OpenStax</a> textbooks are freely available under a Creative Commons license. They have a series of three calculus books, including multivariable calculus. I haven't used that book, but I have taught using their Calculus 2 book, which I found to ... |
3,576,301 | <p>Suppose <span class="math-container">$a, m, n \in \mathbb{R}$</span>. In real-number arithmetic, is <span class="math-container">$a^m \in \mathbb{R}$</span> and <span class="math-container">$a^n \in \mathbb{R}$</span> sufficient for <span class="math-container">$(a^m)^n=a^{mn}$</span>?</p>
<hr>
<p><strong>Edit:</... | user62498 | 135,451 | <p>The "rule" <span class="math-container">$(a^b)^c = a^{bc}$</span> doesn't necessarily hold when <span class="math-container">$a < 0$</span>.</p>
|
1,283,065 | <p>I am asking a sample of men to compare the perceived sharpness of two blades (A and B). One blade may be only sharper, duller, or similar to the other. How do I make sense of the following examples:</p>
<ul>
<li>0% rate A sharper, 100% rate both similar, 0% rate B sharper</li>
<li>50% rate A sharper, 50% rate B sha... | Henry | 6,460 | <p>It depends on what you mean by equivalent. You can certainly say:</p>
<ul>
<li>In the first example, both times as many men rated A shaper as rated B sharper</li>
<li>In the second example, more men rated A sharper than rated B sharper and the difference represented $25\%$ of the men</li>
</ul>
<p>but you could a... |
2,642,291 | <p><strong><em>Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus. I am deeply grateful to the members of this community for their time.</em></strong></p>
<p>I am not sure why you need to use B vs. Bx+C. I tried both, and got 2 different answers. In the p... | bames | 464,998 | <p>A simple sanity check shows that your second solution is incorrect:
$$\frac{1}{x^2+x+1} - \frac{1}{x-1} = \frac{x-1 - x^2 -x - 1}{(x-1)(x^2+x+1)} = \frac{-x^2-2}{x^3-1}\neq \frac{x^2-2x-2}{x^3-1}$$</p>
<p>It is easiest to explain why your second approach does not work using the language of linear algebra. Let $P_2(... |
2,642,291 | <p><strong><em>Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus. I am deeply grateful to the members of this community for their time.</em></strong></p>
<p>I am not sure why you need to use B vs. Bx+C. I tried both, and got 2 different answers. In the p... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <blockquote>
<p>Why you need to use $Bx+C$ instead of just using $B$?</p>
</blockquote>
<p>Since you know the trick of "let $x=1$" to find coefficient $A$ in $\dfrac{A}{x-1}$, I'll give you another answer (for fun) using the <em>same</em> substitution trick, but applied on each complex root $r_i$ of the denominator... |
530 | <p>In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove th... | lhf | 589 | <p>See the Wikipedia article <a href="https://en.wikipedia.org/wiki/B%C3%A9zout_domain" rel="nofollow noreferrer">Bézout domain</a>.</p>
|
530 | <p>In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove th... | Pete L. Clark | 299 | <p>Note: for more on Bézout domains, see e.g.</p>
<blockquote>
<p>Section 8.2 of <a href="http://alpha.math.uga.edu/%7Epete/factorization2010.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/factorization2010.pdf</a></p>
</blockquote>
<p>or</p>
<blockquote>
<p>Section 12.4 of <a href="http://alpha.math.ug... |
2,149,723 | <h2>Binomial Theorem Formula:</h2>
<ul>
<li>Sigma Notation:
$\sum_{i=0}^{n} $ $n\choose i $$ x^i $ $y^{(n-i)}$</li>
</ul>
<p>Note: $(n-i)$ is the exponent of $y$.</p>
<h2>Logical Question:</h2>
<ul>
<li>Can the place of exponents for $x$ and $y$ be switched ?</li>
</ul>
<p>Ex.
$\sum_{i=0}^{n} $ $n\choose i $$ x^... | Community | -1 | <p>$$\cos x+\sin x=2\implies(\cos x+\sin x)^2=4\implies\cos^2+2\cos x\sin x+\sin^2x=4\\\implies\sin2x=3\ (!)$$</p>
|
1,030,536 | <p>how can I solve following equation analytically </p>
<blockquote>
<p>$$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$</p>
</blockquote>
| Claude Leibovici | 82,404 | <p>If you have to solve $$x^2+(3+5i)x+(7+11i)=0$$ and search for $x=a+ib$, just replace in the equation which, after developing and simplifications, leads to $$a^2+3 a-b^2-5 b+7+i (2 a b+5 a+3 b+11)=0$$ So, you have to set to $0$ the real and imaginary parts and you so have two equations for two unknowns $a$ and $b$ $$... |
1,035,983 | <p>The identity is: </p>
<p>$$\sum_{k=0}^{m} (-1)^{k} {{n} \choose {k}}{{n-k}\choose{m-k}} = 0$$</p>
<p>I'm not even sure where to begin. Does anyone have any suggestions?</p>
| Brian M. Scott | 12,042 | <p>Let $A$ be an $n$-element set, and assume that $0<m\le n$; we’ll count the $m$-element subsets of $A$ that don’t contain any element of $A$ in two different ways. First, of course, it’s obvious that there are no such subsets. (This is why we have to take $m>0$: if $m=0$ there actually is one such subset, $\var... |
376,036 | <p>Let <span class="math-container">$X$</span> be a riemannian surface. Suppose <span class="math-container">$f:X\to X$</span> is a Teihmuller map with respect to a quadratic differential <span class="math-container">$q$</span> on <span class="math-container">$X$</span>. This means that, if <span class="math-container"... | Andrey Ryabichev | 76,500 | <p>The map <span class="math-container">$f$</span> is a Teichmuller map with respect to <strong>a pair</strong> of quadratic differentials <span class="math-container">$q_1,q_1$</span> on <span class="math-container">$X$</span> --- initial differential and terminal differential. More precisely, <span class="math-contai... |
197,019 | <p>Suppose <span class="math-container">$X$</span> can take integer values from 0 to 10. Suppose further that <span class="math-container">$Y$</span> is continuous from 0 to 1. Finally, let us define <span class="math-container">$Z = X\,Y$</span>. </p>
<p>I now want to create a table that contains values of <span clas... | m_goldberg | 3,066 | <p>The easy way to make your table is:</p>
<pre><code>pts = Table[{x, y, x y}, {x, 0, 10}, {y, 0, 1, .1}] // Catenate;
</code></pre>
<p>Then you can plot it like so:</p>
<pre><code>ListPlot3D[pts]
</code></pre>
<p><a href="https://i.stack.imgur.com/MbAiA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgu... |
3,595,184 | <p>I'm being asked to prove the following equality</p>
<p><span class="math-container">$$\sin(\alpha)=\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$$</span></p>
<p>and I support the idea that they are <strong>not</strong> equal (the rest of my class seems to desagree with me which made me doubt a little).</p>
<p>The ... | robert timmer-arends | 468,626 | <p>First,<span class="math-container">$$\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)}$$</span>
Pythagorean identity: <span class="math-container">$1+\tan^2(\alpha)=\sec^2(\alpha)$</span>
<span class="math-container">$$ \therefore \sqrt{1+\tan^2(\alpha)}=\sec(\alpha)$$</span>
Putting it all together,
<span class="math-... |
421,844 | <p>I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ .</p>
<p>Help. </p>
<p>Thanks.</p>
| Narasimham | 95,860 | <p>Let $AC,BD$ intersect at $O$. Let $ (AO,BO,CO,DO)= (p,q,r,s) $ respectively.</p>
<p>$ p^2 + q^2= AB^2 , r^2 + s^2 = CD^2$</p>
<p>$ q^2 + r^2 = BC^2 , p^2 + s^2= AD^2 $</p>
<p>and add up each line.</p>
|
1,155,705 | <p>Find the last digit of $$77777^{77777}$$</p>
<p>I got a pattern going for $77777^n$ for $n=1, 2, ....$ to be:</p>
<p>$$7, 9, 3, 1$$ for $n = 1, 2, 3, 4$ respectively. </p>
<p>The idea is:</p>
<p>$$77777^{77777} \pmod{10}$$</p>
<p>I see that:</p>
<p>$$77777^n \equiv 77777^{n + 4} \pmod{10}$$</p>
<p>Using the ... | barak manos | 131,263 | <p><strong>Observe that:</strong></p>
<p>$$77777\equiv1\pmod{4}\implies77777^{77777}\equiv77777^1\equiv7\pmod{10}$$</p>
<p><strong>In other words:</strong></p>
<p>$$77777^{77777}\equiv77777^{(77777\bmod{4})}\equiv77777^{1}\equiv7\pmod{10}$$</p>
<p><strong>Some clarifications:</strong></p>
<ul>
<li>You've already n... |
1,968,108 | <p>Is there a name for the follow function $\frac{x}{1+x^2}$?</p>
<p>Or the integral of $\frac{x}{1+x^2}$ is a typical function?</p>
| DonAntonio | 31,254 | <p>The function is a <em>rational funcion</em> defined on the whole real line, and its integral is a rather well known function ( I don't know what "typical function" may mean ):</p>
<p>$$\int\frac x{1+x^2}dx=\frac12\int\frac{d(x^2+1)}{1+x^2}=\frac12\log(1+x^2)+C$$</p>
|
2,588,967 | <p>A solution to the system of equations </p>
<blockquote>
<p>$ax + by +cz = 0$</p>
<p>$a^2x+b^2y+c^2z=0$ is</p>
<p>Ans: $x=\frac{k (b-c)}{a}$ $y= \frac{k (c-a)}{b}$ $z=\frac{k (a-b)}
{c}$ where $k$ is an arbitrary constant.</p>
</blockquote>
<p>A system of two equations in three variables is suppos... | Matthew Leingang | 2,785 | <p>Starting from $e^{i\pi} = -1$, we have $\log(-1) = i\pi \log(e)$.</p>
<p>What is $\log(e)$? It depends on who you ask. Many mathematicians use log as an abbreviation for the natural logarithm (base $e$), while others (including most calculators) use it for the common logarithm. Indeed
$$
\pi \log_{10}(e) \ap... |
612,461 | <p>How do you determine what solving method to use, for example what is the reason you solve by the quadratic formula, or by factoring, or by completing the square, or by taking a square root.</p>
| Newb | 98,587 | <p>To find $x$, each approach can be considered more specific to a certain form of problem, or less general, than the next:</p>
<p>Taking the square root is only really useful when you have $ax^2 = n$. </p>
<p>Completing the square works well when you have $(x+c)^2 = n$, or a statement that can easily be arranged to ... |
673,939 | <blockquote>
<p>$\alpha$ and $\beta$ are two different values of $\theta$ lying between $0$ and $2\pi$ which satisfy the equation $\begin{equation}6\cos\theta + 8\sin\theta = 9\end{equation}$. Find $\sin(\alpha+\beta).$ </p>
</blockquote>
<p>I solved and got $\cos(\alpha+\beta) = \frac{-28}{100}$.<br>
So $\sin(\alp... | lsp | 64,509 | <p>$$\sin \theta=\frac{2\tan \frac{\theta}{2}}{1+\tan^2 \frac{\theta}{2}}$$
$$\cos \theta=\frac{1-\tan^2 \frac{\theta}{2}}{1+\tan^2 \frac{\theta}{2}}$$
Substitute the above two equations in the main equation and let $\tan \frac{\theta}{2} = x$. Now you get a quadratic equation $x$ from where you can find $x=\tan \frac{... |
2,988,298 | <p>For which polynomials <span class="math-container">$p(x)$</span> is <span class="math-container">$p(p(x))+p(x)$</span>=<span class="math-container">$x^4+3x^2+3$</span>, for all <span class="math-container">$x \in \mathbb{R}$</span></p>
<p>Since the power of the right hand side is 4, <span class="math-container">$p(... | Anna Petrova | 421,944 | <p>Identifying the coefficient of <span class="math-container">$x^4$</span> one gets <span class="math-container">$a=1$</span> and then the coefficient of <span class="math-container">$x^3$</span> you get <span class="math-container">$b=0$</span> hence <span class="math-container">$c=1$</span> or <span class="math-cont... |
27,879 | <p>Let $(x_{min},x_{max})$ and $(y_{min},y_{max})$ represent lowerbound and upperbound acceptable values for some coordinate $c_i$. </p>
<p>Setting $x_{min} = y_{min} = 1$ & $x_{max} = y_{max} = 9$, how can I take a list of values, e.g.: </p>
<pre><code>val = {{0, 10}, {2, 10}, {4, 4}, {5, 6}, {10, 4}};
</code>... | DavidC | 173 | <p>You can achieve the same end with <code>DeleteCases</code> or <code>Cases</code>.</p>
<p><code>f</code> signals a coordinate that is out of bounds.</p>
<p><code>g</code> signals that a coordinate is in bounds.</p>
<pre><code>f[n_] := n < 1 \[Or] n > 9
g[n_] := 1 <= n <= 9
</code></pre>
<p><strong>Fou... |
3,216,832 | <p><span class="math-container">$B$</span> is an <span class="math-container">$n\times n$</span> invertible matrix.</p>
<p>Define <span class="math-container">$\mathcal T :M_{n\times n}(\mathbf F)\to M_{n\times n}(\mathbf F)$</span>
by </p>
<p><span class="math-container">$\mathcal T (A)=B^{-1}AB$</span>.
Prove that... | José Carlos Santos | 446,262 | <p>Since it is a linear map between finite-dimensional vector spaces with the same dimension, all you need to prove is that <span class="math-container">$\mathcal T$</span> is injective <em>or</em> surjective. But it <em>is</em> injective:<span class="math-container">\begin{align}\mathcal T(A)=0&\iff B^{-1}AB=0\\&a... |
3,257,250 | <p>The precise source is <a href="https://pi.math.cornell.edu/%7Ehatcher/AT/AT.pdf" rel="nofollow noreferrer">Chapter 3, page 235, - line 9</a>.</p>
<p>So what Hatcher has stated is:</p>
<blockquote>
<p><span class="math-container">$M$</span> is an <span class="math-container">$n$</span> manifold, and <span class="math... | Minus One-Twelfth | 643,882 | <p><span class="math-container">$\newcommand{\L}{\mathscr{L}}$</span><strong>Hint:</strong> </p>
<p>Note that <span class="math-container">$$\L^{-1}\left(\frac{2s}{(s+1)^2+4}\right) = 2 \L^{-1}\left(\frac{s+1}{(s+1)^2+4}-\frac{1}{(s+1)^2+4}\right).$$</span> Using linearity, it is thus enough to find the inverse Laplac... |
138,924 | <p>Typically, one doesn't just write down lists of axioms and then sees
if there are enough interesting examples that satisfy them; they evolve
over time, usually from a couple of very important/interesting examples
that are then generalized.</p>
<p>For the vector space axioms, for example, it is pretty easy to motiva... | Community | -1 | <p>Symmetry, bilinearity, and positive definiteness are exactly the properties used to prove the Cauchy-Schwarz inequality. Well, there are a zillion proofs of the Cauchy-Schwarz inequality; I mean the one that proceeds by observing $0\le \|x-ty\|^2$ for all $t\in\mathbb R$, expanding to obtain a quadratic in $t$, and... |
4,314,547 | <p>Edit: the problem would've gone something like (I saw it years ago): "find the speed you must run on a second lap around the same track in order to double your average speed for the 2 laps combined."</p>
<p>I have 2 approaches to this problem that give different answers, in both cases we let</p>
<p><span c... | Dstarred | 955,900 | <p><strong>EDIT</strong>:</p>
<p>I changed the previous answer completely as it was incorrect...</p>
<p>The second approach still stands correct than the first approach.</p>
<p>However, <span class="math-container">$t_2 = 0$</span> is incorrect. This is due to the fact that the first average (first lap) cannot be compa... |
16,742 | <p>I have been stuck on this one for hours. </p>
<p>Let $x$, $y$, $z$ be non-negative real numbers.</p>
<p>Also we know $x + z \leq 2$.</p>
<p>Prove the following:</p>
<blockquote>
<p>$(x - 2y + z)^2 \geq 4xz - 8y$.</p>
</blockquote>
<p>Apparently this can be proven with or without AGM, which is $xy \leq \left(\... | JDH | 413 | <p>This is the solution to the old version of the problem:</p>
<p>The right hand side is $4xy-8y=4y(x-2)$ and hence non-positive, since $y$ is non-negative $x$ is at most $2$. But the left hand side is a square and hence non-negative. So the inequality follows. </p>
<hr>
<p>For the revised version:</p>
<p>Let $u=x+... |
1,329,892 | <p>I'm trying to reproduce the splines from the program, and I have the correct point data, but my representation of Bezier Splines using the same anchor point data fails to give me a correct curvature.</p>
<p>Which kind are those?</p>
<p><img src="https://i.stack.imgur.com/abOUR.jpg" alt="3ds max graph editor. The s... | bubba | 31,744 | <p>It would have been easier if you had given us some data. </p>
<p>Specifically: create a curve whose four control points have nice simple coordinates. Then put points on the curve at $t= \tfrac13$ and $t=\tfrac23$, and tell us the coordinates of those points, too. </p>
<p>But, even without that data, it's possible ... |
1,329,892 | <p>I'm trying to reproduce the splines from the program, and I have the correct point data, but my representation of Bezier Splines using the same anchor point data fails to give me a correct curvature.</p>
<p>Which kind are those?</p>
<p><img src="https://i.stack.imgur.com/abOUR.jpg" alt="3ds max graph editor. The s... | fang | 180,792 | <p>Judging from the white dots, black dots and black lines in the graph, the spline looks like a G1 cubic Hermite spline. The white dots are the points the spline will interpolate. The black lines are used to control the tangent direction at the points and the black dots are used to control the tangent magnitudes at th... |
3,278,430 | <p>Can someone help me with this question. I've found the maximum which is really easy. However, the minimum gives me some trouble. I want to solve it algebraically, rather than solving it through trial and error (the problem is done without calculator so I can just select values of "x" that yield sin(special angle, ex... | Kavi Rama Murthy | 142,385 | <p>To show that <span class="math-container">$\frac {\sqrt3} 2$</span> is the minimum value on <span class="math-container">$[0,\frac {2\pi} 3]$</span> you can observe that the only point where the derivative is <span class="math-container">$0$</span> is <span class="math-container">$x =\frac {\pi} 3$</span>. This impl... |
3,278,430 | <p>Can someone help me with this question. I've found the maximum which is really easy. However, the minimum gives me some trouble. I want to solve it algebraically, rather than solving it through trial and error (the problem is done without calculator so I can just select values of "x" that yield sin(special angle, ex... | Etotheipi | 589,939 | <p>As <span class="math-container">$f(x)=f(\pi-x)$</span>, rewrite the wave as follows:- <span class="math-container">$$sin(\pi-x) +sin(\frac 23\pi -(\pi-x))$$</span></p>
<p><span class="math-container">$$=\sin(x)+\sin(x+\frac \pi3)$$</span>
<span class="math-container">$$=\sin(x)+\frac {\sin(x)}2+\frac{\sqrt3}2\cos(x... |
144,395 | <p>Suppose I have an image containing several morphological components and I wish to extract them as individual images (with sizes equal to their bounding boxes; these individual images shouldn't contain parts of neighboring components), then apply some image-processing functions to each of them <em>individually</em>, ... | yode | 21,532 | <p>An <code>ArrayPad</code> version of <a href="https://mathematica.stackexchange.com/users/27331/ali-hashmi">Ali</a>'s <a href="https://mathematica.stackexchange.com/a/144402/280">second</a> method:</p>
<pre><code>data = KeyValueMap[
Round[Transpose[#2]] -> ImageData[#1] &, <|Thread[newComps -> cm[[All... |
145,146 | <p>I want to zoom/exaggerate a portion of the following plot for <code>t</code> between <code>1</code> and <code>2</code>, such that the zoomed portion is also within the main plot.</p>
<pre><code>x[t_] = Sin[50 t]/t + Cos[t];
y[t_] = Sin[30 t]/t^2 + Cos[t^2];
P1 = Plot[x[t], {t, 0, 2 \[Pi]}, PlotStyle -> Blue];
P... | george2079 | 2,079 | <pre><code>x[t_] = Sin[50 t]/t + Cos[t];
y[t_] = Sin[30 t]/t^2 + Cos[t^2];
sub = Plot[{x[t], y[t]}, {t, 1, 2}, PlotStyle -> {Blue, Red},
PlotRange -> All]
Plot[{x[t], y[t]}, {t, 0, 2 Pi}, PlotStyle -> {Blue, Red},
Epilog -> Inset[sub, Scaled[{.3, .55}], Scaled[{0, 0}], 3],
PlotRange -> {-12, 12}, ... |
2,704,575 | <p>I tried this:</p>
<p>$$\biggl (\frac{1}{8}\biggr)^{-\frac{1}{3}}= \frac{1^{-\frac{1}{3}}}{8^{-{\frac{1}{3}}}}=\frac{-\sqrt[3]{1}}{-\sqrt[3]{8}}=\frac{-1}{-2}$$</p>
<p>Also, is it possible to see what I don't understand here or are there maybe several things..</p>
| Francisco | 355,916 | <p>there are some simple algebraic errors there, for instance
$$a^{-\frac{1}{b}} = \frac{1}{\sqrt[b]{a}}$$
not
$$ a^{-\frac{1}{b}} = -\sqrt[b]{a} $$</p>
<p>then</p>
<p>$$ \left(\frac{1}{8}\right)^{-\frac{1}{3}} = \sqrt[3]8 = 2 $$</p>
|
1,599,002 | <p>As I am no mathematician, I have been struggling to find an equation to accurately predict points spaced along a curve separated by distance d. Given two points, assume a string with a length equal to 125% of the distance between the two points. If the string is "hooked" at its ends to the two points, what is the e... | marty cohen | 13,079 | <p>The difficult part
of fitting a catenary
of a specified length
between two specified points
is solving
$y = \dfrac{\sinh(x)}{x}
$
for $x$
in terms of $y$.</p>
<p>This can only be done numerically;
there is no analytic solution.</p>
<p>Here is question
with answers
that tell you how:</p>
<p><a href="https://math.s... |
3,705,362 | <p>For real numbers <span class="math-container">$x,y,z$</span> solve the system of equations:
<span class="math-container">$$\begin{align} \sin x = \cos y,\\
\sin y = \cos z,\\
\sin z = \cos x\end{align}$$</span>
Source: high school olympiads, from a collection of problems for systems of equations, no unusual tricks i... | lab bhattacharjee | 33,337 | <p>Please avoid squaring whenever possible as it almost all cases introduces <a href="https://math.stackexchange.com/questions/1513842/extraneous-roots">Extraneous Roots</a> </p>
<p><span class="math-container">$$\cos y=\sin x=\cos\left(\dfrac\pi2-x\right)$$</span></p>
<p><span class="math-container">$$\implies y=2m... |
2,184,667 | <p>I'm reading a book called <em>Linear Algebra</em> by Friedberg, Insel, and Spence. In their chapter on diagonalization, they provide the following definition:</p>
<blockquote>
<p>Let $\lambda$ be an eigenvalue of a linear operator or matrix with the characteristic polynomial $f(t)$. The <strong>(algebraic) multip... | hunter | 108,129 | <p>In a polynomial ring, the variable $t$ is a formal symbol, not a function. </p>
<p>Thus, for example, in the polynomial ring with coefficients in the field with two elements, the polynomials $t^2$ and $t$ are different (they are the same function, but different polynomials). </p>
<p>When we say a polynomial $f(t)$... |
852,476 | <p>Evaluation of $\displaystyle \int\frac{1}{\prod_{k=0}^{n}(x+k)}dx$</p>
<p>$\bf{My\; Trial::}$We can write</p>
<p>$\displaystyle \int\frac{1}{\prod_{k=0}^{n}(x+k)}dx = \int \frac{1}{(x+0)\cdot(x+1)\cdot (x+2)..........(x+n)}dx$</p>
<p>Now Using Partial fraction.......</p>
<p>$\Rightarrow \displaystyle \frac{1}{(x... | robjohn | 13,854 | <p>Partial Fractions gives
<span class="math-container">$$
\begin{align}
\int\frac1{\prod_{k=0}^n(x+k)}\,\mathrm{d}x
&=\int\sum_{k=0}^n\frac{(-1)^k}{k!(n-k)!}\frac1{x+k}\,\mathrm{d}x\\
&=\frac1{n!}\sum_{k=0}^n(-1)^k\binom{n}{k}\log(x+k)+C
\end{align}
$$</span></p>
|
273,459 | <p>I have the polynomial</p>
<p><code>5x^4 - 6x^3*y - 3x^2*y^2 - 3x*y^3 - y^4 + x^8 + x^7 + x^7*y + 4x^6*y^2 + x^6*y^2 + x^6*y + x^6 + x^5*y^3 + 4x^5*y^3 + 4x^5*y^2 - x^5*y^2 + x^5*y + x^4*y^4 + x^4*y^4 + x^4*y^4 + x^4*y^3 - x^4*y^3 + x^4*y^2 + 3x^4*y^2 + 3x^3*y^5 + x^3*y^5 + x^3*y^4 - x^3*y^4 + x^3*y^3 + x^3*y^3 + ... | Michel | 62,210 | <p>I assume that the polynomial does not need to be held in the form you list? It simplifies to:</p>
<p>-x^3 + 8 x^4 + x^5 + 6 x^6 + x^7 + x^8 - 6 x^3 y + x^5 y + x^6 y +
x^7 y + x y^2 - 2 x^2 y^2 - 7 x^3 y^2 + 3 x^5 y^2 + 5 x^6 y^2 -
3 x y^3 + 2 x^3 y^3 + 5 x^5 y^3 - x y^4 + 3 x^2 y^4 + 3 x^4 y^4 +
6 x y^5 - x^2 y^5 +... |
127,359 | <p>Consider the following code:</p>
<pre><code>ListPointPlot3D[
Flatten[#, 1] &@
Table[{x, y, 1}, {x, 0, 5, 5/60}, {y, Sin[x], Cos[x] + 3, (
Cos[x] + 3 - Sin[x])/60}], PlotRange -> All]
</code></pre>
<p><a href="https://i.stack.imgur.com/6f8We.png" rel="nofollow noreferrer"><img src="https://i.stack.img... | corey979 | 22,013 | <p><code>ListPlot3D</code> accepts <code>RegionFunction</code>, so let's exploit this fact. The idea is to create a region.</p>
<pre><code>data = Flatten[#, 1] &@
Table[{x, y, 1}, {x, 0, 5, 5/60}, {y, Sin[x],
Cos[x] + 3, (Cos[x] + 3 - Sin[x])/60}]
data2D = data[[All, 1 ;; 2]]
range = {{0, 5}, {-1, 4}}
</c... |
1,843,059 | <p>I would prove that for any complex number $z \in \mathbb C^*$ such that $z = x + \mathbb i y$ with $(x,y)\in\mathbb R^2$ and $x+\vert z\vert \neq 0$:
$$ \arg z = 2\arctan\left(\dfrac{y}{x+\vert z\vert}\right)$$
How to begin a proof of this formula?</p>
<p>Thanks in advance.</p>
| reuns | 276,986 | <p>for $x>0,y>0, z = x+iy$ : $$\text{arg}(z) = \text{atan}(y/x)$$</p>
<p>with $s = x+ |z|+iy$ you get $$\text{arg}(s) = \text{atan}\left(\frac{y}{x+|z|}\right)$$ but $$s^2 = (x+ |z|)^2-y^2+2iy(x+ |z|)=2(x+ |z|)(x+iy) = 2(x+ |z|) z$$
so that
$$\text{arg}(z) = \text{arg}(s^2) = 2 \text{arg}(s)$$</p>
<p>I let yo... |
57,350 | <p>Given $A$ and $B$, $n\times n$ complex matrices. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n}$, then the following are equivalent:</p>
<p>(1) $\langle Ax,y\rangle=\langle Bx,y\rangle$, for all $x,y\in \mathbb C^{n}$.</p>
<p>(2) $\langle Ax,x\rangle=\langle Bx,x\rangle$, for all $x,y\in \mathbb C... | tparker | 268,333 | <p>A very similar but slightly easier approach is to move everyone over to one side of the equations, use linearity to combine the operators, and consider the difference $C := A - B$. Then your first equation implies that $\forall x, y \in \mathbb{C}^n, \langle C x, y \rangle = 0$, which is clearly equivalent to the o... |
2,585,259 | <p>I started with transforming the expression to $$\sqrt{1+\frac{72}{x-36}}$$ and I realised that $\frac{72}{x-36}$ must be one less than a square. Then I don't know what to do. Do I just try some number one less than a square and check if it's divisible by 72? Is there a smarter method I'm missing? I know there isn't ... | lab bhattacharjee | 33,337 | <p>Let $$\sqrt{\dfrac{x+36}{x-36}}=n\implies x=\dfrac{36(n^2-1+2)}{n^2-1}=36+\dfrac{72}{n^2-1}$$</p>
<p>So $n^2-1(\ge-1)$ must divide $72$ and</p>
<p>$$n^2-1\le72\implies2\le n\le8$$</p>
|
961,008 | <p>I should know this, but when simplifying $$-2A - 2B = 2$$ When I divide the LHS by 2, do I divide -2A AND -2B or just one of them? I always thought you did one of them, and then the next. So order of operations would be $$-2A/-2$$ then RHS $$2/-2$$ then LHS $$-2B/-2$$ and lastly RHS $$-1/-2$$</p>
| MPW | 113,214 | <p>Multiply both sides of the original equation by $\frac12$. The RHS obviously becomes $1$, and this is how you simplify the LHS:
$$\tfrac12\cdot(-2A-2B)$$
$$\tfrac12\cdot[(-2)A+ (-2)B]$$
$$\tfrac12\cdot[(-2)A]+ \tfrac12\cdot[(-2)B]$$
$$[\tfrac12\cdot(-2)]A+ [\tfrac12\cdot(-2)]B$$
$$(-1)A + (-1)B$$
$$-A-B$$</p>
|
131,386 | <p>I would just like to know if this:</p>
<pre><code>((a - b)/(b*Log[b])) - Log[Log[a]] + Log[Log[b]]
</code></pre>
<p>exceeds 0 for any combination of <code>a</code> and <code>b</code>, being both variables positive integers.</p>
<p>I have tried to compute it on Mathematica but I only get hundreds of errors. </p>
... | Jorge | 36,648 | <p>For this case you can try a graphical method. For example:</p>
<pre><code>RegionPlot[((a - b)/(b*Log[b])) - Log[Log[a]] + Log[Log[b]] > 0, {a,
0, 10}, {b, 0, 10}]
</code></pre>
<p>It shows the region on the <code>ab</code> plane where the inequality is satisfied. Of course you can change the limits for <code... |
3,516,840 | <p>Let <span class="math-container">$N(A^\top)$</span> be the left null space of <span class="math-container">$A$</span></p>
<p><span class="math-container">$N(A^\top) = \{x| A^\top x = 0\}$</span></p>
<p>Let <span class="math-container">$C(A)^\perp$</span> be the orthogonal complement of the column space of <span cl... | Peter Szilas | 408,605 | <p>Set <span class="math-container">$y:=x^{1/3}$</span>, and consider <span class="math-container">$y \rightarrow 2$</span>.</p>
<p>We have <span class="math-container">$\dfrac{y-2}{y^3-8}$</span>.</p>
<p><span class="math-container">$y-2$</span> is a factor of <span class="math-container">$y^3 -8$</span>, since <spa... |
205,405 | <p>Let $a,b,c>0$ with $a\leq b\leq c$. Let $E$ be the ellipsoid determined by $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$</p>
<p>Is there a function $f:E\rightarrow \mathbb{R}$ such that $f\in C^{\infty}(E)$ and all level curves of $f$ are circles?</p>
<p>Thanks</p>
| Community | -1 | <p>Elaborating <a href="https://math.stackexchange.com/questions/205405/level-curves-on-ellipsoid#comment467276_205452">my comment</a> on joriki's answer: The intersection of a plane with an ellipsoid is an ellipse. Consider the plane passing through the origin and spanned by the two orthonormal vectors
$$\begin{align}... |
3,266,619 | <p>Consider the following problem:-</p>
<p>If <span class="math-container">$a,b,c,d,p$</span> are distinct real numbers such that </p>
<p><span class="math-container">$(a^2+b^2+c^2)p^2 -2p(ab+bc+cd)+(b^2+c^2+d^2)≤0$</span> then prove that <span class="math-container">$a,b,c,d$</span> are in geometric progression.</p>... | CopyPasteIt | 432,081 | <p>Reading over the OP's question and their comments, it becomes evident that they are interested in 'scramble-it-up' bijections and @AlexanderGeldhof's comment. Here we want to offer some food for thought to the OP's curiosity; we'll only be examining bijections <span class="math-container">$f: \Bbb R \to \Bbb R$</spa... |
2,654,271 | <p>A certain amount is invested in scheme A for 6 years which offers simple interest at the rate of x% per annum. The same amount is invested in scheme B for 2 years which offers compound interest compounded annually at the rate of 10% per annum. Interest earned from scheme A is twice of that earned in scheme B. If the... | drhab | 75,923 | <p>$B$ is correct here. It has probability $\frac12$ in contrast to the other options that all have probability $\frac14$.</p>
<p>A) TT has probability $\frac14$</p>
<p>B) HT or TH has probability $\frac14+\frac14$ (summation of two probabilities of mutually exclusive events)</p>
<p>C) HH has probability $\frac14$</... |
273,188 | <p>This is my first time asking a question on the Mathematica StackExchange. As the title says I'm looking for a way to add a horizontal line across my bar graph. The code I currently have is as follows:</p>
<pre><code>BarPi = BarChart[dataExperimentalPi,
ChartLabels -> {A, B, C, D, E, F}, PlotRange -> {3, 3.2... | Nasser | 70 | <p>There might be better ways, but you can try <code>Epilog</code></p>
<pre><code>BarChart[{1, 2, 3}, ChartLabels -> {"A", "B", "C"},
Epilog -> {Red, Thick, Line[{{-0.4, 0.5}, {3.45, 0.5}}]}]
</code></pre>
<p><img src="https://i.stack.imgur.com/C76lK.png" alt="Mathematica graphics"... |
4,622,161 | <p>I want to solve the system:<span class="math-container">$$\begin{align}
\dfrac{dx}{dt}&= -y \sqrt{x^2+y^2} \\ \\ \dfrac{dy}{dt}&= x \sqrt{x^2+y^2}
\end{align}$$</span>I basically have no idea about how to solve this since this system is not linear. At first glance, I saw <span class="math-container">$x^2+y^... | user577215664 | 475,762 | <p><span class="math-container">$$\dfrac{dx}{dt}= -y \sqrt{x^2+y^2} \\ \\ \dfrac{dy}{dt}= x \sqrt{x^2+y^2}$$</span>
<span class="math-container">$$x\dfrac{dx}{dt}= -xy \sqrt{x^2+y^2} \\ \\ y\dfrac{dy}{dt}= xy \sqrt{x^2+y^2}$$</span>
Add both DE and integrate:
<span class="math-container">$$x\dfrac{dx}{dt}+y \dfrac{dy... |
1,042,507 | <p>$f(x)\in C[a,b]$.For any $g(x) \in C[a,b]$ ,which has the property that $\int_a^b g(x) dx=0$,$\int_a^b f(x)g(x) dx=0$.</p>
<p>Prove:$f(x)\equiv C$,$C$ is a constant.</p>
<p>I haven't any ideas yet.
I'm thinking about using the Fourier series,would it work?</p>
| Mher | 80,548 | <p>Let assume that, without loss of generality, $[a,b]=[0,2\pi]$. It follows from your condition, the for any $n\ge 1$</p>
<p>$$\int_0^{2\pi} f(x)\sin nx\,dx = \int_0^{2\pi} f(x)\cos nx\,dx = 0$$</p>
<p>What claims Fejer-Lebesgue theorem is that arithmetic mean of partial sums converges uniformly to a function:
$$\si... |
3,392,142 | <p>Using the recursive formula and a bit of memory, you can compute this number in linear time. </p>
<p>But there is also the direct formula. Unfortunately, it used irrational numbers, so first need to fix an adequate ring in which you do the computations (like the rationals and <span class="math-container">$\sqrt5$</... | Misha Lavrov | 383,078 | <p>There are several approaches that take <span class="math-container">$O(\log n)$</span> arithmetic operations. (Although the cost of doing operations on numbers the size of <span class="math-container">$F_n$</span> shouldn't be neglected either: the Fibonacci numbers grow exponentially, so <span class="math-container... |
3,712,349 | <p>Knowing <span class="math-container">$P(A|B)$</span> & <span class="math-container">$P(A|C)$</span>, would it be possible to infer <span class="math-container">$P(A|B,C)$</span>?<br>
I looked at this other question: <a href="https://math.stackexchange.com/questions/1120608/any-mathematical-relation-between-these... | lulu | 252,071 | <p>Just look at simple examples. </p>
<p>Suppose we are tossing a fair penny and a fair dime. Let <span class="math-container">$A$</span> be the event "the penny comes up <span class="math-container">$H$</span>". Let <span class="math-container">$B$</span> be the event "the dime comes up <span class="math-container... |
986,038 | <p>A Martingale is a discrete time stochastic process $Z_1, Z_2, ..., Z_n$ for any time $n$ that satisfies</p>
<p>$E[|Z_n|] < \infty$</p>
<p>$E[Z_{n+1}| Z_0, Z_1, ..., Z_n] = Z_n$</p>
<p><strong>By the linearity of expectation the second statement is equivalent to</strong></p>
<p>$E[Z_{n+1} - Z_n| Z_0, Z_1, ...,... | Jimmy R. | 128,037 | <p>You simply have that $$Z_n=E[Z_n|Z_n]$$ or in other words, given that at time $n$ you know $Z_n$ then the expected value of the random variable $Z_n$ will be $Z_n$. Note that the expectation on the right hand side is a random variable and not a number!</p>
<p>Obviously more knowledge does not change that so that al... |
4,111,472 | <p>The following is a standard skat game deck.
It has a total of <span class="math-container">$32$</span> cards, where <span class="math-container">$8$</span> cards each belong to one of the <span class="math-container">$4$</span> colors.
There are <span class="math-container">$4$</span> aces, kings, queens and jacks. ... | Milo Moses | 630,231 | <p><span class="math-container">$\left| e^{z}\right|=e^x$</span> is minimized when <span class="math-container">$x$</span> is chosen to be the smallest possible value it can, and if <span class="math-container">$x^2+y^2=1$</span> then then least value for <span class="math-container">$x$</span> is <span class="math-con... |
4,111,472 | <p>The following is a standard skat game deck.
It has a total of <span class="math-container">$32$</span> cards, where <span class="math-container">$8$</span> cards each belong to one of the <span class="math-container">$4$</span> colors.
There are <span class="math-container">$4$</span> aces, kings, queens and jacks. ... | Math and YuGiOh lover | 678,578 | <p>You are almost there, basically, you have shown that <span class="math-container">$|e^z| = e^x$</span>, with <span class="math-container">$x$</span> being the real part of <span class="math-container">$z$</span>, note that <span class="math-container">$x \in [-1,1]$</span>, so <span class="math-container">$$e^x \geq... |
2,270,476 | <blockquote>
<p>Knowing that $x^2+y^2 = 2 $ prove that $x+y \le 2$</p>
</blockquote>
<p>I rewrite this as
$$x^2+y^2 \ge x+y$$
Now, multiply both sides by 2
$$x^2 -2x + 1 + y^2-2y + 1 -2 +x^2 + y^2 \ge 0$$
I substitue $2$ for $x^2+y^2$
$$(x-1)^2+(y-1)^2\ge0$$
Which is true</p>
| farruhota | 425,072 | <p>Using Cauchy-Schwarz:</p>
<p>$$(1+1)(x^2+y^2)\ge(x+y)^2 \Rightarrow (x+y)^2\le4 \Rightarrow |x+y|\le2.$$</p>
|
1,819,952 | <p>Let $M$ be an $n$-dimensional smooth <strong>compact oriented</strong> manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ball in $\mathbb{R}^n$)?</p>
<p>Does anything change if we assume $M$ is simply-connected? or that ... | Hmm. | 227,501 | <p>Consider a closed annulus in $\mathbb{R}^2 $. It is not even homeomorphic to a closed disk. </p>
|
2,584,798 | <p>An usual recurrence relation can be written as:</p>
<p>$$a_{n} = \sum_k c_ka_{n-k}$$
or alternatively:
$$a_{n} - \sum_k c_ka_{n-k} = 0$$</p>
<p>for example, the famous Fibonacci sequence is the solution to the following recurrence relation:</p>
<p>$$a_n = 1\cdot a_{n-1} + 1 \cdot a_{n-2}$$</p>
<p>There exist ver... | John B | 301,742 | <p>Since $f$ is continuous, given $\epsilon>0$, we have $|f(x)-f(0)|<\epsilon$ for $x\in[0,1/n]$ and all sufficiently large $n$. Hence, since $\int_0^1g_n=1$ we have
$$
\left|\int_0^1fg_n-f(0)\right|=\left|\int_0^1(f-f(0))g_n\right|\le\epsilon\int_0^1g_n=\epsilon.
$$</p>
<p><em>PS: Note that</em>
$$
\int_0^1g_n=... |
3,491,692 | <p>A can do 50% of the job in 16 days, B can do 1/4th of the job
in 24 days. In how many days can they do 3/4 th of the job
working together?</p>
<pre><code>My approach.
A can complete the job in 32 days alone
B can complete the job in 96 days alone.
Together they can complete the job in ?
</code></pre>
<p>A rate... | Martin R | 42,969 | <p>Building on your second approach:
<span class="math-container">$$
\cos(x+iy)=\cos(x)\cos(iy)-\sin(x)\sin(iy) =\cos(x)\cosh(y)- i\sin(x)\sinh(y)
$$</span>
so that
<span class="math-container">$$
|\cos(x+iy)| ^2 = \cos^2(x)\underbrace{\cosh^2(y)}_{1 + \sinh^2(x)} + \underbrace{\sin^2(x)}_{1 - \cos^2(x)}\sinh^2(y) = \... |
253,386 | <p>I have a function for the variable <span class="math-container">$x>0$</span> and constant <span class="math-container">$a\in\mathbb{R}$</span>. I write it in its two alternative forms</p>
<p><span class="math-container">$$f(x)=\cos a\pi-\cos x\pi =-2\sin\frac{(a-x)\pi}{2}\sin\frac{(a+x)\pi}{2} $$</span></p>
<p>Th... | Alex Trounev | 58,388 | <p>First, we can plot phase portrait for 3D system as follows (see comment @I.M.)</p>
<pre><code>Clear["Global`*"]
x0 = 0.90; y0 = 0.10; z0 = 0; p = 0.8; b = 0.5; c = 0.01; Do[
lst[i] = RecurrenceTable[{x[t + 1] == (1 - p*y[t])*x[t] +
c*(1 - x[t] - y[t]), y[t + 1] == (p*x[t] + b)*y[t],
z[t + 1]... |
253,386 | <p>I have a function for the variable <span class="math-container">$x>0$</span> and constant <span class="math-container">$a\in\mathbb{R}$</span>. I write it in its two alternative forms</p>
<p><span class="math-container">$$f(x)=\cos a\pi-\cos x\pi =-2\sin\frac{(a-x)\pi}{2}\sin\frac{(a+x)\pi}{2} $$</span></p>
<p>Th... | Chris K | 6,358 | <p>Of course trajectories are not continuous in discrete-time, so <code>StreamPlot</code> isn't totally relevant. See <a href="http://phaseportrait.blogspot.com/2013/01/discrete-time-phase-portraits.html" rel="nofollow noreferrer">this blog post</a> for some related discussion.</p>
<p>Instead, why not plot a grid of a... |
1,968,607 | <p>Twelve normal and fair dice are thrown once. What is the total number of ways in which each of the values $1,2,3,4,5,6$ occur exactly twice? e.g. $(1,1,2,2,3,3,4,4,5,5,6,6)$</p>
<p>My attempt:</p>
<p>The above problem would be similar to the to the number of arrangements of $12$ digits $(1,6)$ out of which each di... | Doug Spoonwood | 11,300 | <p>First hint: Prove ($\lnot$p$\rightarrow$(p$\rightarrow$q)) as a lemma.</p>
<p>More detailed analysis:</p>
<ol>
<li>What is the goal?</li>
</ol>
<p>The goal is to produce to (((p $\rightarrow$ q) $\rightarrow$ p) $\rightarrow$ p).</p>
<ol start="2">
<li>What evidence do you have so far?</li>
</ol>
<p>You have ev... |
2,373,731 | <p>Can anyone think of a nice example of a homomorphism whose image is not normal? I think there must be one but am having no luck.</p>
<p>Any help would be appreciated. </p>
| Sahiba Arora | 266,110 | <p>Consider the homomorphism $\phi: \langle (1,2,3,4,5) \rangle \to A_5$ defined as $\phi(x)=x$ for all $x \in \langle (1,2,3,4,5) \rangle.$ Then the image is $\langle (1,2,3,4,5) \rangle$ which is not normal in $A_5$ as $A_5$ is a simple group.</p>
|
4,281,859 | <p><a href="https://math.stackexchange.com/a/4280920/26327">Proof by another user</a> of the formula for <span class="math-container">$k = 1$</span>, that is the total number of pairs <span class="math-container">$(x, x + 2k)$</span> of odd numbers in in interval <span class="math-container">$[a,b]$</span> such that <s... | mathlove | 78,967 | <p>Let <span class="math-container">$A$</span> be the number of odd numbers <span class="math-container">$p$</span> such that <span class="math-container">$a\le p\le b-2k$</span> and <span class="math-container">$n\mid p$</span>.</p>
<p>Let <span class="math-container">$B$</span> be the number of odd numbers <span clas... |
907,620 | <p>We will call a matrix <em>positive matrix</em> if all elements in the matrix are positive, and we will denote the <em>largest eigenvalue</em> with <span class="math-container">$\lambda_{\max}$</span>, what is exist because of the <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow n... | Surb | 154,545 | <p>I recently came across the following (beautiful) result which is slightly improving the answer of @user1551. The following results come from the book <a href="http://epubs.siam.org/doi/book/10.1137/1.9781611971262" rel="nofollow">Nonnegative Matrices in the Mathematical Sciences</a> of Abraham Berman and Robert J. P... |
1,965,189 | <blockquote>
<p>If $\lim\sup$ of $\{s_n\}$ equals $M$, prove that the $\lim\sup$ of any subsequence $\{s_{n_k}\}$ is $\le M$</p>
</blockquote>
<p><strong>PROOF</strong>: Given that $\{s_n\}$ is a sequence of real numbers such that $\displaystyle \limsup_{n\to\infty}s_n = M$, then by definition we know that $\{s_n\}$... | yurnero | 178,464 | <p>You do skip something in step ($*$); I would modify it as follows.</p>
<p>Define $M_n$ as you have and define $N_{n_j}=\sup\{s_{n_j},s_{n_{j+1}},\ldots\}$. Then, $M_{n_j}\geq N_{n_j}$ because the former entails taking supremum over more elements. Moreover, $M_n\to\limsup s_n$ implies $M_{n_j}\to\limsup s_n$. It fol... |
3,097,542 | <p>So far I have showed that <span class="math-container">$K$</span> is a normal subgroup and that the Operation defined on <span class="math-container">$G/K$</span> is abelian. </p>
<blockquote>
<p>Now I have to show that if <span class="math-container">$\phi:G\rightarrow A$</span> is a group
homomorphism and <sp... | Keatinge | 547,657 | <p>Rewrite <span class="math-container">$$\frac{(n+1)^s}{e \cdot n^s}$$</span> as <span class="math-container">$$\left(\frac{n+1}{n}\right)^s \cdot \frac{1}{e}$$</span></p>
<p>If you need the next step, divide the top and bottom of the fraction by n to get</p>
<p><span class="math-container">$$\left(1+\frac{1}{n}\rig... |
2,635,972 | <blockquote>
<p>Let $p \in \mathbb{N}, p \geq 2$ even. Prove that $$\det(X^{p}+X^{p-1}+\dots+X+I)\geq 0$$
for all invertible matrices $X$ with real entries.</p>
</blockquote>
<p>The solution I found simply uses the fact that the polynomial $x^p+x^{p-1}+\dots+x+1$ doesn't have any real roots. Why is that enough?</p... | Simon Segert | 497,898 | <p>Case 1: X is diagonal</p>
<p>The determinant is equal to $\prod_i f(d_i)$ where $f$ is the polynomial in the question and $d_i$ are the diagonal entries.</p>
<p>Case 2: X is diagonalizable</p>
<p>In this case, f(X) is also diagonalizable, since $PX^kP^{-1}=(PXP^{-1})^k$, so the statement follows from case 1. </p>... |
281,870 | <p>can you help me with this question
find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$ </p>
| MathematicalPhysicist | 13,374 | <p>You've got:</p>
<p>$$F(w) = \int \frac{\sin x}{ x} exp(-iw x) dx$$</p>
<p>Differentiate wrt $w$ to get:</p>
<p>$$F'(w) = -i \int \sin x exp(- iwx)dx $$</p>
<p>You know how to find fourier transform of sine and then you should integrate your result.</p>
|
281,870 | <p>can you help me with this question
find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$ </p>
| Christian Blatter | 1,303 | <p>Note that the sinc function is not in $L^1$; therefore you cannot compute its Fourier transform in the "obvious" way by means of the residue theorem or something like that. But this function is in $L^2$, so it is indeed Fourier related to some other function $f\in L^2$. If we can guess an $f$ such that its inverse ... |
1,229,647 | <p>Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ?
<em>Just wanted to share a special type of equation and the fastest way to solve it.I am not asking for an answer and i have solved it in my answer given below.Thank You for viewing.</em></p>
| Community | -1 | <p><em>A shortcut formula for such sums is if $|f(x)|+|g(x)|=|f(x)+g(x)|$ then $f(x).g(x)>0$ then $[\frac{x}{x-1}][x]>=0$ which implies $x^2(x-1)>=0$.But $x^2$ is always >=0.Hence <strong>$x>1$</strong> is the solution as well as x=0.</em></p>
|
1,229,647 | <p>Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ?
<em>Just wanted to share a special type of equation and the fastest way to solve it.I am not asking for an answer and i have solved it in my answer given below.Thank You for viewing.</em></p>
| Jared | 138,018 | <p>To me the easiest and most systematic way to solve it is to explicitly write out the absolute value which means breaking the equation into regions:</p>
<p>$$
\frac{|x|}{|x - 1|} + |x| = \frac{x^2}{|x - 1|} \rightarrow \begin{cases}
\left(\frac{1}{x - 1} + 1\right)x = \frac{x^2}{x - 1} & 1 < x < \infty \\ ... |
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