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 |
|---|---|---|---|---|
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Community | -1 | <p>I propose the following word</p>
<p><strong>question</strong> </p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with the inclusion. Down-vote if you disagree with the inclusion.</p>
|
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Community | -1 | <p><a href="/questions/tagged/status-completed" class="post-tag moderator-tag" title="show questions tagged 'status-completed'" rel="tag">status-completed</a></p>
<p>I propose the following word</p>
<p><strong>pls</strong></p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with the i... |
2,890,207 | <p>Consider the following system of equations where $x,y,z$ are variables and for a constant $\mathrm C$, $y \times z = \mathrm C \neq 0$</p>
<p>\begin{equation}
\left\{
\begin{array}{lcl}
a_1x + b_1y + c_1z &= 0\\
a_2x + b_2y + c_2z &= 0\\
\end{array}
\right.
\end{equation}</p>
<p>The most straightforward so... | Trurl | 72,915 | <p>I suggest you start with Spivak's Calculus, which is meant to be an encounter with real math, and then look at books on Real Analysis, such a Rudin, Royden and the like.</p>
|
2,328,068 | <p>I've came up with a proof to the above problem, but I'm stuck on whether or not this should be done with induction instead. I get stuck in this pit a lot (the "is or isn't this enough for a proof") so I wanted to see what others think.</p>
<p>Here's my proof:</p>
<blockquote>
<p>Let $d \leq n$. Consider the the ... | Maxime Ramzi | 408,637 | <p>"Obviously $x$ is of order $d$" is clearly enough for a homework assignment or a paper or an exercise.
However, to prove it formally, I see no other way to do so than describing $x^k$ by induction, and showing, with this description, that $x^k$ isn't the identity unless $k=0$ mod $d$.</p>
|
2,567,559 | <blockquote>
<p>Find absolute extrema of $f(x,y) = 2\sin(x) + 5 \cos(y)$ in the
rectangle with vertices $(0,0), \; (3,0), \; (3,-3), \; (0,-3)$</p>
</blockquote>
<p>I am stuck on this problem.</p>
<p>First of all for critical points</p>
<p>$f_x = 2 \cos(x) = 0 \Rightarrow x \in \{ \frac{(2n+1)\pi}{2} \; |\; n \... | Dylan | 135,643 | <p>You're looking to optimize the function with $0 \le x \le 3$ and $-3 \le y \le 0$.</p>
<p>The first step is to locate the zeros of the derivatives which are $x = \left(n + \frac12\right)\pi, y = m\pi$. While there are infinitely many of these, you only need to consider the ones inside your region. Given the boundar... |
4,073,814 | <p><strong>Given</strong></p>
<p><span class="math-container">$X \in M_{nn}(\mathbb{F})$</span> is a static matrix</p>
<p><span class="math-container">$U = \{A \in M_{nn}(\mathbb{F})\ | AX = XA\}$</span></p>
<p>Proof that <span class="math-container">$U$</span> is a subspace of <span class="math-container">$M_{nn}(\mat... | S.Farr | 683,130 | <p>Your teacher stated that <span class="math-container">$A\in U$</span>, so by definition <span class="math-container">$AX=XA$</span>. That also <span class="math-container">$aA\in U$</span> is what you want to show!</p>
|
4,073,814 | <p><strong>Given</strong></p>
<p><span class="math-container">$X \in M_{nn}(\mathbb{F})$</span> is a static matrix</p>
<p><span class="math-container">$U = \{A \in M_{nn}(\mathbb{F})\ | AX = XA\}$</span></p>
<p>Proof that <span class="math-container">$U$</span> is a subspace of <span class="math-container">$M_{nn}(\mat... | Community | -1 | <blockquote>
<p>Show that for <span class="math-container">$a \in \mathbb{F}, A \in U: aA \in U$</span></p>
</blockquote>
<p>The string of symbols above does not quite read as a valid mathematical satatement.</p>
<p>What you professor was saying, be it as it may in the amount of predicate logic you'll want to use, is t... |
28,463 | <p>The <a href="http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations" rel="nofollow noreferrer">Wikipedia article on numerical differentiation</a> mentions the formula</p>
<p>$$
h=\sqrt \epsilon \times x
$$</p>
<p>where $\epsilon$ is the machine epsilon (approx. $2.2\times 10^{-16}$ for 64-... | J. M. isn't a mathematician | 7,934 | <p>I use the technique given by J.C. Nash in the book <em><a href="http://www.crcpress.com/product/isbn/9780852743195" rel="nofollow noreferrer">Compact Numerical Methods for Computers</a></em>.</p>
<p>On <a href="HTTPS://books.google.com/books?hl=en&id=M9hTn3UAheQC&pg=PA219" rel="nofollow noreferrer">page 219... |
1,030,177 | <p>Suppose you have 4 stamps with probabilities 0.2, 0.4, 0.3, and 0.1, respectively. Each time you pick a stamp is independent of every other pick. Suppose you pick your 10th stamp. What is the probability this stamp is one you haven't seen before?</p>
<p>How should I approach these kinds of problems? Should I do 1-p... | Robert Israel | 8,508 | <p>It holds if $a_n \to 0$ as $n \to \infty$ and both $\sum_n \max(a_n, 0) = \infty$ and $\sum_n \max(-a_n,0) = \infty$.</p>
|
2,970,108 | <p>This is a question about understanding a proof. Here is what my professor did.</p>
<p><a href="https://i.stack.imgur.com/pYdww.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pYdww.png" alt="enter image description here"></a></p>
<p>However, I do not know how he reached the final conclusion wher... | Fred | 380,717 | <p><span class="math-container">$-c_k^2+(c_k-a_k)^2=-c_k^2+c_k^2-2c_ka_k+a_k^2=-2a_kc_k+a_k^2$</span>.</p>
|
2,970,108 | <p>This is a question about understanding a proof. Here is what my professor did.</p>
<p><a href="https://i.stack.imgur.com/pYdww.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pYdww.png" alt="enter image description here"></a></p>
<p>However, I do not know how he reached the final conclusion wher... | Robert Lewis | 67,071 | <p><strong><em>Observe:</em></strong></p>
<p><span class="math-container">$\displaystyle \sum_{k= 1}^n (c_k - a_k)^2 - \sum_{k = 1}^n c_k^2 = \sum_{k= 1}^n (c_k^2 - 2c_k a_k + a_k^2) - \sum_{k = 1}^n c_k^2 = \sum_{k = 1}^n a_k^2 - 2 \sum_{k = 1}^n a_k c_k. \tag 1$</span></p>
|
496,769 | <p>I'm currently taking a course in Probability Theory and was hoping someone could point me in the direction of a useful supplementary textbook. </p>
<p>Our course currently uses <em>A Modern Approach to Probability Theory</em> by Fristedt and Gray. </p>
<p>A link to the book is here: <a href="http://rads.stackov... | Don Shanil | 103,948 | <p>A really comprehensive, easy to read book is "An Introduction to measure and probability" by J.C Taylor (<a href="http://rads.stackoverflow.com/amzn/click/0387948309" rel="nofollow">Amazon</a>).
Lots of examples, exercises, and really nice geometric view of conditional expectation via Hilbert spaces. All the results... |
1,154,680 | <p>I was playing around with series recently and asymptotics of $\sum \sqrt{k}$ and $\sum (-1)^k\sqrt{k}$ were required to solve another problem.</p>
<p>I have dealt with the first one using an integral estimate: $$\forall n\in \mathbb N,\;\; \frac23 n^{3/2}\leq \sum_{k=1}^n\sqrt{k}\leq \frac23 (n+1)^{3/2}$$</p>
<p>H... | Timbuc | 118,527 | <p>Hints:</p>
<p>The point $\;(x,y)\;$ is on the unit circle iff </p>
<p>$$\;x^2+y^2=1\;\iff \begin{cases}x=\cos t\\y=\sin t\end{cases}\;,\;\;t\in[0,2\pi)$$</p>
<p>So now just substitute $\;t=\frac\pi4\;$ to obtain your point (it has the same first and second entry, by the way)</p>
|
2,418,997 | <blockquote>
<p>Given the series $\displaystyle{\sum_{n=1}^{\infty}} \frac{1}{n! z^n}$ I have to show that it defines a holomorphic function in $\mathbb{C}-\{ 0 \}$ and then I have to calculate the integral of this function over the unit circle.</p>
</blockquote>
<hr>
<p>For the first point, it is sufficient to not... | NewBee | 473,224 | <p>kingW3 answer is absolutely correct. I'm not debating on his answer, just answering in the way the question has been asked. </p>
<ol>
<li><p>Selecting either 2 or 4 from the Set for the unit place as the no. should be even: 2C1</p></li>
<li><p>Selecting the digit from the left over digits for the tens place: 4C1</p... |
2,592,869 | <p>We continuously use that a second order anti-symmetric tensor remains anti-symmetric in any coordinate system, but could not find the proof anywhere.</p>
| edm | 356,114 | <p>Let's say we have two local coordinates $x^\mu$ and $y^\mu$. Say we have an anti-symmetric contravariant tensor $t^{\mu\nu}$ in coordinates $x^\mu$. We are to show that in coordinates $y^\mu$, the same tensor $$t'^{\mu\nu}=\frac{\partial y^\mu}{\partial x^\rho}\frac{\partial y^\nu}{\partial x^\lambda}t^{\rho\lambda}... |
1,786,403 | <p>Consider an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ with $O$ as the origin. $n$ points denoted as $P_1,P_2,\cdots$ are taken on the ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$ where $i\in(1,n-1)$. Find the value of: $$\sum_{i=1}^n\frac{1}{OP_i^2}$$</p>
<p>I took $P_1$ as $(3,0)$. Let other points be $P_i(3\... | choco_addicted | 310,026 | <p>Suppose that $x<-1$, then
$$
7=\frac{-x((-x^2)^8-1)}{-x^2-1}=\frac{x(x^{16}-1)}{x^2+1}
$$
and right side is negative, which is a contradiction. Also, if $-1\le x \le 0$, $\left|\frac{x(x^{16}-1)}{x^2+1}\right|\le 1$. Thus $x>0$. By AM-GM inequality,
$$
x^{16}=\frac{7x^2+7}{x}+1= 7x+\frac{7}{x}+1 \ge 2\cdot 7 +... |
439,820 | <blockquote>
<p>Is $\large i^{i^i}$ real ? How to find it?</p>
</blockquote>
<p>Thank You!</p>
| pritam | 33,736 | <p>We have $i^i=(e^{i\pi /2})^i=e^{-\pi /2}$. Then, $$i^{i^i}=i^{e^{-\pi /2}}=(e^{-i\pi /2})^{e^{-\pi /2}}=e^{-i\pi e^{-\pi /2}/2}=\cos (\pi e^{-\pi /2}/2)-i\sin(\pi e^{-\pi /2}/2) $$ Now $\sin(\pi e^{-\pi /2}/2)$ is non-zero, since $e^{-\pi /2}/2$ is not an integer.</p>
|
2,658,754 | <p>This is what I have to show:</p>
<blockquote>
<p>Let $f: G\to H$ be a homomorphism and let $p$ be prime. If range of $f$ has an element
of order $p$, then $G$ has an element of order $p$.</p>
</blockquote>
<p>I start off by proving the lemma:</p>
<blockquote>
<p><strong>Lemma</strong>. For each element $a\i... | Nick L | 532,949 | <p>Once you know that $p$ divides the order of $b$ you are done. The element $b^{(o(b)/p)}$ has order $p$.</p>
|
1,080,847 | <p>Let $s(x)=\int_0^xa(t)\,dt$. I wonder:</p>
<p>If $\displaystyle \lim_{x\to\infty}s(x)=L$ is it true that $\displaystyle\lim_{x\to\infty}a(x)=0?$</p>
| Henrik supports the community | 193,386 | <p>No. $\lim_{x\to\infty} a(x)$ need not even exist, $a$ could be $1$ in a set of measure zero.</p>
|
2,580,496 | <p>I've come across this problem trying to find an integral representation to a PDE (damped wave equation with initial conditions).</p>
<p>What I would like to do is compute $$f(t) = \mathcal{L}^{-1}\left[e^{-\sqrt{s^2+s}}\right](t)$$</p>
<p>via the bromwich integral,</p>
<p>$$f(t) = \frac1{2 \pi i} \int_{c- i \inft... | Ron Gordon | 53,268 | <p>I will illustrate a slightly different, more rigorous derivation of the ILT result here, which is</p>
<blockquote>
<p>$$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{-\sqrt{s (1+s)}} \, e^{s t} = \frac12 e^{-t/2} \frac{I_1 \left ( \frac12 \sqrt{t^2-1} \right )}{\sqrt{t^2-1}} H(t-1) $$</p>
</blockquote>
... |
1,241,748 | <p>Evaluate $\gamma$ expressed, involving Lambert function, by
$$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for $\gamma$ in closed form. Numerical solutions or estimates are welcome.</p>
| giorgiomugnaini | 215,560 | <p>Exact expression for the solution is: $$\gamma=\frac{\pi/4}{e^{\pi/4}-1}=0.6581842...$$</p>
<p><strong>EDIT</strong>
The two branches are related in a trascendental way. Their difference can be used to solve other trascendental equations.</p>
<p>If $y(x)=\frac{x}{1-e^x}$</p>
<p>then :</p>
<p>$x=W_0(ye^y)-W_{-1}(... |
2,899,959 | <p>I try to use Mathematica to find such $n$, and I believe that only $n=2,3,4,5,6,7$ meet the requirement but I don't know how to prove that for $n\ge 8$, there is always some root lying outside the unit ball by using the basic complex analysis knowledge. </p>
<p>In fact, I do have a dirty way to deal with this probl... | Mathillda | 548,136 | <p>It is clear that the result is true for $n=2$. Now we consider $n\ge 3$.</p>
<p>Note that $(z+1)^n-z^n-1=\sum_{k=0}^n\binom{n}{k} z^k-z^n-1 = \sum_{k=1}^{n-1}\binom{n}{k} z^k= nz\sum_{k=1}^{n-1}\frac{\binom{n}{k}}{n}z^{k-1}$. Suppose that $\sum_{k=1}^{n-1}\frac{\binom{n}{k}}{n}z^{k-1}=\prod_{k=1}^{n-2}(z-z_k)$. The... |
2,075,664 | <p>In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits.</p>
<p><strong>My attempt</strong>:We should find number of two pairs that can be digits of this number for choosing the place of digits we have $*8$ (We should notice we cant have $0$ in the beginning.But the biggest ... | Sil | 290,240 | <p>First let's not bother with numbers starting with $0$ for now, we can subtract those cases later. Now consider what possible value can the sum $s$ of two digits be? We see that $s \in \{0,1,\dots,18\}$. Let's look at possible ways to get individual values of $s$:</p>
<p>\begin{array}{ccc}
s& &\#\\
0 & 0... |
21,048 | <p>When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.</p>
<p>Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" ve... | Jochen Glueck | 16,250 | <p>Some more (rather general) examples of vector spaces over any field <span class="math-container">$\mathbb{F}$</span>:</p>
<ul>
<li><p>For any set <span class="math-container">$S$</span>, the space <span class="math-container">$\mathbb{F}^S$</span> of mappings from <span class="math-container">$S$</span> to <span cla... |
293,444 | <p>I have a Boolean expression. we'll call it <code>F</code>.</p>
<p>for instance, <code>F = ab' + ad + c'd + d'</code>.</p>
<p>Assuming I did all the necessary steps too get <code>F</code> complement , i.e. <code>F'</code>.</p>
<p>I got: <code>F' = b'd + ac'd'</code>.</p>
<p>How do I get the <code>Product of sums<... | Masiha | 102,892 | <p>F = ab' + ad + c'd + d'</p>
<p>F'= (ab' + ad + c'd + d')'
= (ab')' . (ad)' . (c'd)' . (d')' ---> a'.b' =(a'+b') De Morgans law
= (a'+b)(a'+b')(c+d')(d)</p>
<p>Hence Product of sums.</p>
|
1,145,378 | <p>I believe a function has to be bijective to be invertible, in fact Wikipedia says that an invertible function is another name for a bijection.
<a href="https://en.wikipedia.org/wiki/Bijection" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Bijection</a></p>
<p>But in answers to the question <a href="https:/... | GFauxPas | 173,170 | <p>A function has an inverse <em>if and only if</em> it is both surjective and injective. (You can say "bijective" to mean "surjective and injective".)</p>
<p>Khan Academy has a <a href="https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_transformations/v/proof-invertibility-implies-a-uniqu... |
1,145,378 | <p>I believe a function has to be bijective to be invertible, in fact Wikipedia says that an invertible function is another name for a bijection.
<a href="https://en.wikipedia.org/wiki/Bijection" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Bijection</a></p>
<p>But in answers to the question <a href="https:/... | fgp | 42,986 | <p>That depends on subtleties of your definition of <em>inverse function</em>. If you expect a function $f\,:\, A \to B$ to have an inverse $f^{-1} \,:\, B \to A$, then $f$ needs to be surjective. If it is not, there are some $b \in B$ that aren't <em>reached</em> by $f$ at all, so how would you define $f^{-1}$ for suc... |
2,230,226 | <p>This is a contest math problem which I was not able to solve. A hint toward the solution would be helpful as well. </p>
<p>Problem: Show that there are infinitely many powers of 2 starting with the digit 7.</p>
<p>Thanks in advance. </p>
| Misha Lavrov | 383,078 | <p>A general idea for how to approach this problem would be to take logs. If $7 \cdot 10^k < 2^n < 8 \cdot 10^k$, then we have $$k + \log_{10} 7 < n \log_{10} 2 < k + \log_{10} 8.$$
Since $k$ is just an arbitrary integer, we can rephrase this as $$\log_{10} 7 < \{n \log_{10} 2\} < \log_{10} 8,$$ where... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | Biro | 463,410 | <p>From the diagram below, you can see that triangle GHK = 3. You also see that triangle FGK = 0.5. Finally, you can see that triangle AHE = 4, but we have to subtract out triangle AFE (since it is red). Triangle AFE = 0.5. That means the blue area is 3 (GHK) + 0.5 (FGK) + 4 (AHE) - 0.5 (AFE) = 7. I believe this is "ou... |
16,946 | <p>Polya Enumeration Formula gives us 6 equivalence classes of 2-colorings of a square. But, in the Polya coloring, the following 2 colorings belong to 2 different equivalence classes:</p>
<pre><code>00
01
</code></pre>
<p>and</p>
<pre><code>11
10
</code></pre>
<p>(0 and 1 are the 2 colors.)</p>
<p>What is the the... | Qiaochu Yuan | 290 | <p>The PET (if you're referring to the same thing I'm thinking of) is a special case of Burnside's lemma, which still applies here - but the group is slightly larger. Instead of the group $C_4$ of rotations of a square (which I assume is the group of symmetries you care about - it might be $D_4$ instead, so this is am... |
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... | Qiaochu Yuan | 232 | <p>Proof 3 is <a href="https://doi.org/10.2307%2F2307043" rel="nofollow noreferrer">due to Fürstenberg</a> (see also <a href="https://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes" rel="nofollow noreferrer">the Wikipedia page</a>) and is honestly not that different from Euclid's proof. See <a ... |
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... | Aryabhata | 1,102 | <p>One proof approach is to construct an infinite set of numbers, any two of which are relatively prime. The proof using Fermat numbers/Euclid's proof can be considered to follow that approach (so I am not sure if I should even be adding this answer!).</p>
<p>We construct a set explicitly as follows.</p>
<p>Start wit... |
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... | Keith Backman | 29,783 | <p>Consider Euler's totient function applied to primorials. Assume there is a greatest prime number <span class="math-container">$p_n$</span>, with the associated primorial <span class="math-container">$p_n\#$</span> (that is, the product of all <span class="math-container">$n$</span> prime numbers).</p>
<p>The totien... |
1,551,124 | <p>Lebniz's notation for ordinary derivatives as quotients of differentials is a convenient abuse of notation, since it lets you express things like the chain rule and the derivative of the inverse function in a suggestive form:</p>
<p>$$\frac{dz}{dx} = \frac{dy}{dx} \cdot \frac{dz}{dy}$$</p>
<p>$$\frac{dx}{dy} = \fr... | johnnyb | 298,360 | <p>I think you just did the math wrong. When using Leibniz notation, always treat it as two operations - differential followed by division. So, we are going to take the differential of <span class="math-container">$\frac{1}{\frac{dy}{dx}}$</span> and then divide it by <span class="math-container">$dy$</span>:</p>
<p>... |
2,502,533 | <p>For example, when solving second order differential equation with constant coefficients, we start with guessing the solution is the form of the linear combination of two independent exponential functions. I read some explanations saying it's for the convenience of integral and derivation. Is there any more reason? ... | eepperly16 | 239,046 | <p>Imagine I want to solve the first order ODE $y' = ky$. By seperation of variables, the solution is easily deduced to be $y = Ce^{kt}$.</p>
<p>For reasons that will become clear later, I will introduce the notation $Dy = y'$. The symbol $D$ can be thought of as a differential operator. Thus, saying $y_0 = Ce^{kt}$ i... |
284,848 | <p>I have this recurrence relation $T(n)=4T(\frac{n}{5})+n$ with the base case $T(x)=1$ when $x\leq5$. I want to solve it and find it's $\theta$. I think i have solved it correctly but I can't get the theta because of this term $\frac{5}{5^{log_{4}n}}$ . Any help?</p>
<p>$T(n)=4(4T(\frac{n}{5^{2}})+\frac{n}{5})+n$</... | Did | 6,179 | <p>An alternative approach is to prove that $T(n)\leqslant5n$ for every $n$. This holds for every $n\leqslant5$ and, if $T(n/5)\leqslant5(n/5)=n$, then $T(n)\leqslant4n+n=5n$. By induction, the claim holds.</p>
<p>On the other hand, $T(n)\geqslant n$ for every $n\gt5$, hence $T(n)=\Theta(n)$.</p>
|
1,393,634 | <p>Can anyone here help me out to rearrange the following formula and solve for $n$?</p>
<p>$$16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$$</p>
| alkabary | 96,332 | <p>take $\frac{1}{n}$ as a common factor
and so $$16 = \frac{1}{n} \big(25 + (n−1)218.75\big)$$</p>
<p>now multiply by $n$ each side to get $$16n = 25 + (n-1)218.75$$</p>
<p>now expand $$(n-1)218.75 = n218.75 - 218.75$$</p>
<p>So now you have $$16n = 25 + n218.75 - 218.75$$</p>
<p>and subtract $n218.75$ each side ... |
167,765 | <p>civil engineers belive that $W$, the amount of weight (in units of 1000 pounds) that a certain
span of a bridge can withstand without structural damage resulting, is normally distributed
with mean 400 and standard deviation 40. suppose that the weight (again,in units of 1000 pounds) of a car is a random variable wit... | André Nicolas | 6,312 | <p>Make the obvious independence assumptions, and proceed along the lines described by Michael Chernick. Since everything has normal distribution, there is no need to make a normal <em>approximation</em>. </p>
<p>Let $X_i$ be the weight of the $i$-th car, and let $S=\sum X_i$. Then $S$ has normal distribution with mea... |
4,282,679 | <p>The equation is: <span class="math-container">$$
2^x3^{x-1}=y\cdot3^{x-1}\cdot2+z\cdot2^{x-1}$$</span> for natural numbers.</p>
<p>I’ve tried to divide this expression or try various substitutions, but nothing is working.</p>
| Just a user | 977,740 | <p>I'm gonna assume natural numbers don't include <span class="math-container">$0$</span> for simplicity.</p>
<p>If <span class="math-container">$x=1$</span>, then <span class="math-container">$2=2y+z$</span>, which has no solution, as <span class="math-container">$2y+z\ge 2 + 1 = 3$</span>.</p>
<p>Assume <span class="... |
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="... | Juris Steprans | 13,878 | <p>Kostas Tyros spoke in the Toronto set theory seminar just yesterday about a density version of the Halpern Lauchli Theorem. Actually he spoke about a finite version of the density theorem, and the density version is due to P. Dodos et. al. from 2010 I believe.</p>
|
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... | Learnmore | 294,365 | <p>Only thing that remains to show is that $f(r)=0\forall r\in \Bbb Q^c$</p>
<p>Since $\Bbb Q$ is dense in $\Bbb R$ so given $r\in \Bbb Q^c;\exists r_n\in \Bbb Q$ such that $r_n\to r$</p>
<p>$f$ is continuous so $r_n\to r\implies f(r_n)\to f(r)\implies f(r)=0$</p>
|
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... | Lonidard | 206,444 | <p>A slightly more topological approach:</p>
<blockquote>
<blockquote>
<p><strong>Theorem</strong>: Let $(X, \tau_X)$, $(Y, \tau_Y)$ be two topological spaces. Suppose $Y$ is Hausdorff and that $D \subset X$ is a dense subset in $X$. Let $f,g: X \to Y$ be two continuous functions such that $f_{|D}=g_{|D}$. Then ... |
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">$\lim_{x \rightarrow -\infty}\frac{x... | Boris Valderrama | 628,721 | <p>Because when <span class="math-container">$x\to\infty^{+}$</span>, <span class="math-container">$x$</span> tends to numbers positives and when <span class="math-container">$x\to\infty^{-}$</span>, <span class="math-container">$x$</span> tends to numbers negatives. If you develop this:
<span class="math-container">\b... |
183,022 | <p>can the following proposition be proved? If so please suggest a method. Can Kempe’s Argument be used for proof ?<br>
Proposition: A normal map has a colouring of countries by 4 colours iff the edges of the map can be properly coloured by 3 colours. </p>
| Riyafa Abdul Hameed | 59,292 | <p>I have found a proper proof in these places:</p>
<p><a href="http://books.google.lk/books?id=sMSOSgbCx3kC&pg=PA72&lpg=PA72&dq=Every%20bridgeless%20cubic%20planar%20graph%20is%203-edge%20colourable.&source=bl&ots=PoYmhUn8Ad&sig=5Y3h8-JadWRagoXZdbqf_lvZTKU&hl=en&sa=X&ei=yvY0VK6fOob... |
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Anthony Quas | 11,054 | <p>How about binary linear codes? You can "see" the Hamming distance between codewords, and use linear transformations to encode/decode</p>
|
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Lior Bary-Soroker | 2,042 | <p>A trivial application: The group $(\mathbb{Z}/2\mathbb{Z})^n$ is generated by no less than $n$ elements. </p>
<p>[Probably this is the easiest way to prove that the free profinite group on $n$ letters cannot be topologically generated by less elements. And more abstractly, it is in general hard to find the minimal ... |
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... | Noam D. Elkies | 14,830 | <p>The game of <a href="http://ugcs.caltech.edu/~timblack/pro/" rel="nofollow">Projective Set</a>
is tantamount to finding a linear dependence on
$7$ (distinct, nonzero) vectors in $\mathbb{F}_2^6$.
Linear algebra over $\mathbb{F}_2$ shows that there's always a solution,
and moreover that the number of solutions is alw... |
3,570,186 | <p><strong>The question is shown below:</strong></p>
<p>Suppose <span class="math-container">$S=\{0,1,2,3,4,5,6,7,8,9,10\}$</span> and that the function <span class="math-container">$f:S\rightarrow S$</span> is given by:</p>
<p><span class="math-container">$f(x)=6x^2+3x+8$</span> (mod 11)</p>
<p>Let <span class="mat... | lonza leggiera | 632,373 | <p>If <span class="math-container">$\ a\equiv \alpha^2\pmod{11}\ $</span> is a quadratic residue <span class="math-container">$\mod{11}\ $</span>, then
<span class="math-container">$$
a^6 \equiv \alpha^{12}\equiv \alpha^2\equiv a\pmod{11}\ ,
$$</span>
so the square roots of <span class="math-container">$\ a\pmod{11}\ $... |
2,629,164 | <p>I'm working on a calculus question, where we're asked to find $g'(z)$ for
$$g(z)=\frac{z^2-2z-8}{z-3}\cdot\frac{z^2-9}{z-4}$$
So I was thinking that first you have to use the quotient rule on each side individually and then multiply them, but I seem to keep getting stuck and it's becoming pretty frustrating.</p>
| dxiv | 291,201 | <blockquote>
<p>I have proven there exists no rational solutions $a,b$, and $c$ such that [...] $a + b\sqrt2 + c\sqrt3 = \sqrt6$</p>
</blockquote>
<p>Hint: if you square $a + b\sqrt2 + c\sqrt3 + d\sqrt6 = \sqrt5$ then you'll get an equality of precisely that form.</p>
|
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-... | Thomas Strelecky | 8,624 | <p>I believe <a href="https://www.whitman.edu/mathematics/multivariable/" rel="nofollow noreferrer" title="Community Calculus">Community Calculus</a> is exactly what you're looking for. It is under a CC BY-NC-SA 3.0 license.</p>
|
3,321,963 | <blockquote>
<p>Let <span class="math-container">$x\geq 1, n\in N$</span>
Show that
<span class="math-container">$$
\prod_{i=1}^n(x+n-i)\geq x^{n+1}
$$</span></p>
</blockquote>
<p>I can see that the product is greater or equal to <span class="math-container">$x^n$</span>, but I cannot understand where power <spa... | Community | -1 | <p>if <span class="math-container">$n\mid ab-1$</span>, then there is some integer <span class="math-container">$c$</span> such that <span class="math-container">$cn=ab-1$</span>, or <span class="math-container">$ab-cn=1$</span>. Since 1 is a linear combination of <span class="math-container">$a$</span> and <span clas... |
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^... | John Wayland Bales | 246,513 | <p>\begin{eqnarray}
a\sin\theta+b\cos\theta&=&\sqrt{a^2+b^2}\left[\frac{a}{\sqrt{a^2+b^2}}\sin\theta+\frac{b}{\sqrt{a^2+b^2}}\cos\theta\right]\\
&=&\sqrt{a^2+b^2}\left[\sin\phi\sin\theta+\cos\phi\cos\theta\right]\\
&=&\sqrt{a^2+b^2}\cos(\theta-\phi)
\end{eqnarray}</p>
<p>Where $$\phi=\arcsin\fr... |
1,030,536 | <p>how can I solve following equation analytically </p>
<blockquote>
<p>$$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$</p>
</blockquote>
| Aditya Hase | 190,645 | <p>For quadratic equation in form of
$$ax^2+bx+c=0$$
Using quadrtic formula we get</p>
<p>$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$</p>
<p>Then for
$$x^2+(3+5i)x+(7+11i)=0$$
we get
$$x=\frac{-(3+5i)\pm\sqrt{(3+5i)^2-4(7+11i)}}{2}$$</p>
<p>$$x=\frac{-(3+5i)\pm\sqrt{-44-14i}}{2}$$</p>
<p>$$-44-14i=2\sqrt{533}\exp\left(\a... |
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 ... | ConMan | 82,793 | <p>If the domain is <span class="math-container">$-\pi/2 < \alpha < \pi/2$</span>, then the two functions are identical. However, if the domain contains any values for which <span class="math-container">$\cos(\alpha) < 0$</span>, then as you've shown the functions differ by a factor of <span class="math-contai... |
2,426,562 | <p>I'm aware of other questions regarding "increasing" mathematical intuition, but I wanted to explain my own context, since that way any advice posted on this question feed will be specific to myself.</p>
<p>I'm 16. I'm a (new) Sixth Former (roughly college level) student and has just today had my fifth Further Pure ... | G Tony Jacobs | 92,129 | <p>I don't think there's an answer to this question that doesn't involve doing lots of mathematics.</p>
<p>What I find helpful is, in addition to learning new math, to go back and re-learn or somehow review math that I already know. This can be done by teaching, or by reading over texts a little bit below my own level... |
3,616,484 | <p>Let <span class="math-container">$I = \{f(x) \in \mathbb{Z}[x] : \text{$f(n) \equiv 0 \pmod2$ for all $n \in \mathbb{Z}$}\}$</span>.
Prove that <span class="math-container">$I$</span> is a finitely-generated ideal of <span class="math-container">$\mathbb{Z}[x]$</span> by finding a finite set of generators for <span... | amd | 265,466 | <p>You don’t need to compute <span class="math-container">$P$</span> explicitly. As the problem states, “use the geometric meaning.” </p>
<p>The first part of the problem asks you what the projection of a vector onto itself is. The answer to that ought to be self-evident. For the second half, notice that <span class=... |
914,349 | <p>Find the equation of the line that satisfies the given conditions:</p>
<p>Through $(-1,2)$; parallel to the line $x=5$</p>
<p>I know that the equation of this line is $x=-1$ because $x=5$ is a vertical line and any line parallel to it would be vertical as well. So the line runs vertically along $x=-1$. </p>
<p>B... | Rory Daulton | 161,807 | <p>The reasoning you gave in your question <strong>is</strong> algebraic: I see nothing wrong with it. You can't use the standard "parallel lines have equal slopes" since your lines have undefined slopes.</p>
<hr>
<p>However, here is another approach. All lines can be put in general linear form:</p>
<p>$$Ax+By=E$$</... |
1,618,789 | <p>Solving the homogeneous equation
$$y^{''}-6y^{'}+8=0\Rightarrow \lambda^2-6\lambda+8=0\Rightarrow \lambda_1=2,\lambda_2=4$$</p>
<p>General solution to this homogeneous equation is $y_h=c_1e^{2x}+c_2e^{4x}$.</p>
<p>How to find one particular solution to the non-homogeneous equation?</p>
| Bernard | 202,857 | <p>With <em>superposition of solutions:</em></p>
<p>Find a particular solution $y_0(x)$ for $ y''-6y'+8=x\mathrm e^{2x} $, a particular solution $ z_0(x) $ for $y''-6y'+8=x\mathrm e^{4x}\sin x$. A particular solution for $y''-6y'+8=5x\mathrm e^{2x}+2\mathrm e^{4x}\sin x$ will be $5y_0(x)+2z_0(x)$. </p>
<p>For a ri... |
2,531,089 | <p>So I have the following values:</p>
<p>$X_i ~ N(\mu=9.6, \sigma=1.6)$</p>
<p>Samples = 8.4, 7.8, 10.3, 8.7, 10.7, 10.0, 8.8, 12.1</p>
<p>X̄ = 9.6</p>
<p>α = 0.05</p>
<p>I should also mentioned that I have the following hypothesis test: </p>
<p>$H_0: μ \le 8$</p>
<p>$H_1: μ > 8$</p>
<p>I want to find the n... | Zaharyas | 457,607 | <p>A method like this can work:</p>
<blockquote>
<p>$$\left( x+1-x\right)×\left(x+1+x \right)×\left((x+1)^2+x^2 \right)×\left( (x+1)^4+x^4\right)×...×\left((x+1)^{64}+x^{64}\right)=(x+1)^{128}-x^{128}$$</p>
</blockquote>
|
3,999,029 | <p>Hi guys I came across a few neat formulas to calculate the number of intergers in an interval divisable by another integer!
I was wondering why does it work? what's the mathematics behind them?</p>
<p>If for example I want to find the number of integers between 99 and 993 that are divisable by 2
I can do the followi... | poetasis | 546,655 | <p>You showed an example but not the original formula with variables but<span class="math-container">$\cdots$</span></p>
<p>If you mean how many numbers between <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are divisible by <span class="math-container">$z$</span>, the answer is tha... |
1,523,399 | <p>Suppose we flip 10 times an unfair coin that fall a probability of $p$ on 'heads'. Knowing that we obtained 'heads' 6 times out of the 10 flips, find the conditionnal probability of the first three flips being heads, tails, tails.</p>
<p>My effort: Since we don't know the exact probability of getting heads i would ... | Graham Kemp | 135,106 | <p>The condition you have is that there is <em>some</em> arrangement of six heads, and four tails. </p>
<p>All such arrangements of those ten results are equally probable. We don't need to know what that probability is, just that all of them are identical.</p>
<p>What then is the conditional probability that t... |
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>... | bill s | 1,783 | <p>An alternative is to use <code>Select</code> to retain only those elements you desire:</p>
<pre><code>val = {{0, 10}, {2, 10}, {4, 4}, {5, 6}, {10, 4}};
Select[val, 1 <= First[#] <= 9 && 1 <= Last[#] <= 9 &]
{{4, 4}, {5, 6}}
</code></pre>
<p>In this case, all those with values outside the... |
1,022,386 | <p>This is an exercise in a book I'm reading:</p>
<p>Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by
$$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ </p>
<p>Let $A \in M^n (\mathbb C)$ and define $R_A(v) = v \cdot A$. Furthermore define an injective homomor... | John Hughes | 114,036 | <p>As HTFB says, your induction seems wrong. You might be able to get away with an induction in which on an $N \times N$ matrix, you take four overlapping $(N-1) \times (N-1)$ submatrices, each anchored in one corner. </p>
<p>But an alternative is to just write things out. I'll do half of it. </p>
<p>Fix $n > 0$. ... |
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... | 5xum | 112,884 | <p>To prove that <span class="math-container">$\mathcal T$</span> is <span class="math-container">$1-1$</span>:</p>
<p>Take <span class="math-container">$X, Y\in$</span>, and assume that <span class="math-container">$\mathcal T(X)=\mathcal T(Y)$</span>. From that, prove that <span class="math-container">$X=Y$</span>.<... |
112,538 | <p>A topological surface can be pretty strange (consider, for instance, covers of $S^{2}-K$, where $K$ is a Cantor set.) Can every orientable topological surface be topologically embedded in $\mathbb{R}^{3}$?</p>
| Alexandre Eremenko | 25,510 | <p>Here is a reference:
MR0304649 (46 #3781)
Rüedy, Reto A.
Embeddings of open Riemann surfaces.
Comment. Math. Helv. 46 (1971), 214–225.</p>
<p>He talks about Riemann surfaces, but every orientable topological surface (with a countable basis
of topology) has a structure of a Riemann surface.</p>
<p>I suppose there is ... |
4,286,907 | <blockquote>
<p>Find Distance between a Point <span class="math-container">$(-4,2)$</span> and a Line having equation <span class="math-container">$3x−y−10=0$</span></p>
</blockquote>
<p>My attempt:</p>
<p>I used a graphing calculator to graph the equation <span class="math-container">$3x−y−10=0$</span> and the point <... | Mohammad Riazi-Kermani | 514,496 | <p>Your line is <span class="math-container">$$3x−y−10=0$$</span> Therefore we have <span class="math-container">$a=3$</span> and <span class="math-container">$b=-1$</span> and <span class="math-container">$c=-10$</span></p>
<p>Your point is <span class="math-container">$$(x_1, y_1)=(-4,2)$$</span></p>
<p>You have the ... |
4,286,907 | <blockquote>
<p>Find Distance between a Point <span class="math-container">$(-4,2)$</span> and a Line having equation <span class="math-container">$3x−y−10=0$</span></p>
</blockquote>
<p>My attempt:</p>
<p>I used a graphing calculator to graph the equation <span class="math-container">$3x−y−10=0$</span> and the point <... | Deepak | 151,732 | <p>You don't need to use vectors, you can use coordinate geometry.</p>
<p>Formulate a strategy.</p>
<ol>
<li><p>First find a line <span class="math-container">$l_2$</span> perpendicular to <span class="math-container">$3x-y-10=0$</span> (call it line <span class="math-container">$l_1$</span>) that also passes through <... |
239,450 | <p>I am trying to evaluate a function, which has simplified form as below.</p>
<pre><code>f[a_] := 5*a
X[a_, b_]:=Module[{A= f[a], g[b] = 6*b, x = A + g[b]}, x];
Table[ParallelTable[X[a, b], {b, 1, 6400}], {a, 1, 40}]
</code></pre>
<p>Now, my problem is that for 6400 runs over "b", f[a] is calculated every ... | Ulrich Neumann | 53,677 | <p>Change the definition <code>X[a_,b_]</code> to</p>
<pre><code>X[a_, b_] := Module[{A = f[a], g = Function[b, 6 b]}, A + g[b]];
X[a,b] (*5 a + 6 b*)
</code></pre>
<p>That's it !</p>
|
2,063,972 | <p>Can anybody either verify or dispute the my proof for the following argument?</p>
<p>Premise 1: (E • I) v (M •U) </p>
<p>Premise 2: ~E </p>
<p>Conclusion: ~(E v ~M)</p>
<p>Proof:</p>
<p>(1) Applying DeMorgan's Second Law to the Conclusion; The Negation of a Disjunction, it is the case that ~(E v ~M) is logicall... | user21820 | 21,820 | <p>Correct, and you can present your argument concisely and cleanly as follows:</p>
<hr>
<p>$(E \land I) \lor (M \land U)$.   [Premise]</p>
<p>$\neg E$.   [Premise]</p>
<p>$\neg ( E \land I )$.   [Semantics of $\land$]</p>
<p>$M \land U$.   [Disjunctive syllogism]</p>
<p>$M$.   [Conjuncti... |
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... | Community | -1 | <p>Other than the critical points, you have the endpoints to check for extrema. Since the critical point is a max, in this case the min occurrs at the endpoints. </p>
|
1,786,382 | <blockquote>
<p>If <span class="math-container">$\cos \alpha\;,\cos \beta\;,\cos \gamma$</span> are the roots of the equation <span class="math-container">$9x^3-9x^2-x+1=0\;, \alpha,\beta,\gamma \in \left[0,\pi\right]$</span></p>
<p>Then value of <span class="math-container">$ \left(\sum \alpha\;,\sum \cos \alpha\right... | Community | -1 | <p>$x=1$ is solution, you can solve the equation</p>
|
87,114 | <p>What are some important conjectures in graph theory that have been checked by computer up to order 11?</p>
| Joseph O'Rourke | 6,094 | <blockquote>
<p>Graffiti (by S. Fajtlowicz) and Graffiti.pc (by E. DeLaViña) are computer programs that produce conjectures in graph theory.</p>
</blockquote>
<p>See Doug West's web page, "<a href="http://www.math.uiuc.edu/~west/regs/graffiti.html" rel="nofollow">Some Conjectures of Graffiti.pc (2004-07)</a>,"
and t... |
23,113 | <p>We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.</p>
<p>I am curious whether the triangle inequality is made to... | marc | 1,623 | <p>In <a href="http://en.wikipedia.org/wiki/Information_geometry" rel="noreferrer">Information Geometry</a>, the <a href="https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence" rel="noreferrer">Kullback-Liebler divergence</a> is commonly used in the manner of a metric, but it does not satisfy the triangle i... |
23,113 | <p>We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.</p>
<p>I am curious whether the triangle inequality is made to... | Eric O. Korman | 4,622 | <p>I just wanted to add that in the Lorentzian case, the triangle inequality gets reversed for a certain class of vectors (those with positive time component). Since the norm of a vector corresponds to the elapsed time measured by a clock moving along the vector, a physical consequence of this is the twin's paradox: ... |
1,638,536 | <p>Compute the following integrals:</p>
<p>$I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$</p>
<p>$J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$</p>
<p>I do not know where to begin. I know I am supposed to use the substitution $z\rightarrow \frac{1}{t}$. I can see that $I$ has 13 poles inside the circle of radius 2.</p>
<p>Usi... | 3SAT | 203,577 | <p>For </p>
<blockquote>
<p>$$I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$$</p>
</blockquote>
<p>First you need to find all the roots that in $|z|<2$ then apply the <a href="https://en.wikipedia.org/wiki/Residue_(complex_analysis)" rel="nofollow">Residue formula</a>, I can't find an easy way to do this (maybe th... |
3,637,285 | <p>I have a question about how this theorem interacts with the definition of continuity in an example <span class="math-container">$f(x) = x^2$</span>.</p>
<p><strong>Theorem:</strong> If <span class="math-container">$a \in \mathbb{R}$</span> is such that <span class="math-container">$0 \leq a < \epsilon$</span> fo... | vonbrand | 43,946 | <p>Use Euler's formula: If the complex number is <span class="math-container">$z = \rho e^{i \theta} = \rho (\cos \theta + i \sin \theta)$</span> (polar coordinates; <span class="math-container">$\rho, \theta$</span> are reals), then:</p>
<p><span class="math-container">$\begin{align*}
z^\alpha
&= \rho^\alph... |
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>
| egreg | 62,967 | <p>The limitation is, of course, that $x+|z|\ne0$, that is, the formula doesn't hold for $x<0$ and $y=0$. Other than this case, we have
$$
-\pi<2\arctan\frac{y}{x+|z|}<\pi
$$
so the angle is in the required interval.</p>
<p>If $\alpha=2\arctan t$, then
$$
\cos\alpha=\cos(2\arctan t)=2\cos^2\arctan t-1=
\frac{... |
3,759,260 | <p>How do I find the inverse function of <span class="math-container">$f: X\to Y$</span> where <span class="math-container">$X,Y$</span> both are subsets of <span class="math-container">$\mathbb R^2$</span> and <span class="math-container">$f$</span> is defined as <span class="math-container">$f(x,y)=(x+y,x-y)$</span>.... | cmitch | 569,431 | <p>So this is a system of linear equations, so the core thing is to solve the system. For larger degree inputs and outputs, matrix inversion is the way to go, but for this case, solving the equations without them isn't too horrible:</p>
<p><span class="math-container">$f(x, y) = (u, v)$</span> -> <span class="math-c... |
57,350 | <p>Given $A$ and $B$, $n\times n$ complex matrices. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n}$, then the following are equivalent:</p>
<p>(1) $\langle Ax,y\rangle=\langle Bx,y\rangle$, for all $x,y\in \mathbb C^{n}$.</p>
<p>(2) $\langle Ax,x\rangle=\langle Bx,x\rangle$, for all $x,y\in \mathbb C... | Jonas Meyer | 1,424 | <p>Hint: Use the algebraic properties of the inner product to expand each side of the following equations: </p>
<ul>
<li>$\langle A(x+y),x+y\rangle=\langle B(x+y),x+y\rangle$</li>
<li>$\langle A(x+iy),x+iy\rangle=\langle B(x+iy),x+iy\rangle$</li>
</ul>
|
355,972 | <p>Show that for each $n>1$</p>
<p>$$\sum\limits_{k=1}^{n-1} \frac{(n-1)!}{(k-1)!}S(n,n-k) = (n-1)^n $$</p>
<p>where $S(n,m)$ is the <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow">Stirling number of the second kind</a>. </p>
| leonbloy | 312 | <p>Consider this:</p>
<p>$$ m^n = \sum_{j=1}^m \binom{m}{j} \, j! \, S(n,j) $$ </p>
<p>The left side is the number of ways of placing $n$ distinct balls in $m$ distinct cells. The right side is the same, each term corresponds to the number of configurations that have $j$ occupied (non-empty) cells. </p>
<p>Then,... |
355,531 | <p>Why doesn't this work?</p>
<blockquote>
<p>Find the volume of the region contained above $z=1$ and below $x^{2}+y^{2}+z^{2}=4$</p>
</blockquote>
<p>going to cylindrical this should be easy. $z=(4-r^{2})^{\frac {1}{2}}$ and $z=1$<br>
Clearly $z=1$ yields $x^{2}+y^{2}=3 \to r=(3)^{\frac {1}{2}}$</p>
<p>$$\int^{2... | ABC | 67,567 | <p><img src="https://i.stack.imgur.com/N9tIX.png" alt="enter image description here">
Here is a cute picture of the question.</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>
... | corey979 | 22,013 | <p>A mathematical proof would be best, but <em>Mathematica</em> can provide some evidence against the statement that</p>
<pre><code>f[a_, b_] := ((a - b)/(b*Log[b])) - Log[Log[a]] + Log[Log[b]]
</code></pre>
<p>is negative for any integer <code>a, b</code>.</p>
<p>A straightforward way would be to do</p>
<pre><code... |
614,270 | <p>Given the sequence of real numbers $\{x_n\}_{n \in \mathbb N}$, we define $\{y_n\}_{n \in \mathbb N}$ where $y_n=\max\{|x_1|,...,|x_n|\}$ for each $n \in \mathbb N$. Prove that if $\{x_n\}_{n \in \mathbb N}$ is bounded, then $\{y_n\}_{n \in \mathbb N}$ is a convergent sequence.</p>
<p>My attempt at a solution. </p>... | Yiorgos S. Smyrlis | 57,021 | <p>It is consequence of the following:</p>
<p><em>If a sequence is monotonic and bounded, then it converges.</em></p>
|
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... | Z Ahmed | 671,540 | <p>You may apply L-Hospital Rule
<span class="math-container">$$L=\lim_{x \rightarrow 8} \frac{x^{1/3}-2}{x-8}= \lim_{x \rightarrow 8}(1/3)(x)^{-2/3}=\frac{1`}{12}.$$</span></p>
|
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... | Fred | 380,717 | <p>Let <span class="math-container">$f(x):=\sqrt[3]{x}.$</span> Then <span class="math-container">$ \frac{\sqrt[3]{x} -2}{x-8}= \frac{f(x)-f(8)}{x-8} \to f'(8)= \frac{1}{12}$</span> as <span class="math-container">$x \to 8.$</span></p>
|
3,516,840 | <p>Let <span class="math-container">$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$</span> be the orthogonal complement of the column space of <span cl... | Claude Leibovici | 82,404 | <p><em>Just another way.</em></p>
<p>Let <span class="math-container">$x=y+8$</span> to make
<span class="math-container">$$A=\frac{\sqrt[3]{x} -2}{x-8}=\frac{\sqrt[3]{y+8} -2}{y}$$</span> Now, use the binomial expansion or Taylor series to get
<span class="math-container">$$A=\frac{1}{12}-\frac{y}{288}+O\left(y^2\ri... |
3,266,619 | <p>Consider the following problem:-</p>
<p>If <span class="math-container">$a,b,c,d,p$</span> are distinct real numbers such that </p>
<p><span class="math-container">$(a^2+b^2+c^2)p^2 -2p(ab+bc+cd)+(b^2+c^2+d^2)≤0$</span> then prove that <span class="math-container">$a,b,c,d$</span> are in geometric progression.</p>... | Thomas Andrews | 7,933 | <p>The set <span class="math-container">$\mathbb R^n\setminus\{\mathbf v\}$</span> is always open, and always goes to the open set, <span class="math-container">$\mathbb R^n\setminus\{f(\mathbf v)\}.$</span></p>
<p>More generally, if <span class="math-container">$F\subseteq \mathbb R^n$</span> is finite then <span cla... |
4,021,166 | <p>After a basis of n=1 and the induction step I have no idea where to go from here.</p>
| DMcMor | 155,622 | <p>Hint: Assume that <span class="math-container">$n(2n^2 − 3n + 1)$</span> is divisible by <span class="math-container">$6$</span> up to some <span class="math-container">$n$</span>. Then consider the expression with <span class="math-container">$n+1$</span>: <span class="math-container">$$(n+1)(2(n+1)^2 - 3(n+1) + 1... |
598,265 | <p>I have a few questions about topological spaces which I am currently studying.
First some definitions that I am using:</p>
<p>Definition of subspace topology:
Given a topological space $(X,\tau)$ and a subset $S$ of $X$, the subspace topology on $S$ is defined by $\tau_{S} = \{S \cap U : U \in \tau \}$. </p>
<p>De... | Alex East | 639,807 | <p>As the previous responder already highlighted, there are several things at play here. Take a linear subspace of a topological vector space. By this we mean a set closed w.r.t. finite linear combinations. Then endow it with the subspace topology (this is an example of initial topology - it automatically makes it into... |
2,214,706 | <p>\begin{array}{c|lcr}
x & y\\
\hline
1 & 20\\
2 & 16\\
3 & 13\\
4 & 11\\
5 & 10
\end{array}</p>
<p>How do you start a solution when you have these type of problem? I was used to given an equation but not like all just numbers. It was state that the answer is $y=\frac{1}{2}x^2$ -$\frac{11}{2}x... | user12345 | 394,065 | <p>Since you have found out that it's a parabola (even if you've only guessed). Let $f(x)=ax^2+bx+c$. Then, any $3$ points not all colinear will determine a unique parabola. So, try some of the points and you'll be able to solve for $a,b,c$. If the $6$ points don't all satisfy the equation, then it won't actually be a ... |
4,330,793 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a relation on the integer set, where <span class="math-container">$mRn$</span> iff <span class="math-container">$3m-5n$</span> is an even number.
Prove that <span class="math-container">$R$</span> is an equivalence relation. Find equivalence class <span cla... | user3257842 | 365,433 | <p>We have <span class="math-container">$$(3m - 5n) + (3n - 5m) = -2m - 2n =-2(m+n) $$</span>
which is even. So the first term of our sum is even if and only if the second term is.</p>
|
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... | Tim | 401,695 | <p>The results from the trials are a red herring. The actual chances of 3 H and 3T out of any 6 tosses is statistically correct, but in the real world...</p>
<p>Any toss will result in 50/50 H/T. So B is closest to what 'should' happen. One more H and one more T, but in random order.</p>
<p>As already stated, the oth... |
90,694 | <p>Toss a fair coin <strong>six</strong> times. What is probability of :</p>
<ul>
<li>a- all heads</li>
<li>b- one head</li>
<li>c- two heads</li>
<li>d- an even number of heads</li>
<li>e- at least four heads.</li>
</ul>
<p>I have the answers so I ask if you could explain how you should derive rather than just givin... | Ross Millikan | 1,827 | <p>Hint: <a href="http://oeis.org/A003418" rel="nofollow">OEIS</a> says $[1,2,...,n]$ goes asymptotically as $\exp(n)$</p>
|
90,694 | <p>Toss a fair coin <strong>six</strong> times. What is probability of :</p>
<ul>
<li>a- all heads</li>
<li>b- one head</li>
<li>c- two heads</li>
<li>d- an even number of heads</li>
<li>e- at least four heads.</li>
</ul>
<p>I have the answers so I ask if you could explain how you should derive rather than just givin... | Srivatsan | 13,425 | <p>By the <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">prime number theorem</a>, the <a href="http://en.wikipedia.org/wiki/Chebyshev_function">Chebyshev function</a> has the asymptotic $\psi(n) = n(1+o(1))$ , which tells that $[1, 2, \ldots, n] = \exp(\psi(n)) = \exp(n(1+o(1))$. Therefore, $[1, 2, \ldots... |
1,042,507 | <p>$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>
| GEE20151011 | 182,407 | <p>I'm thinking about using the Fourier series,is That right?</p>
<p>Without loss of generality，prove: in $[0,l]$，$f(x)\equiv C$.</p>
<p>let $g(x)=f(\mid x\mid)$</p>
<p>use Fourier series,we get：</p>
<p>\begin{aligned}
g(x)&={a_0 \over 2}+\sum_{n=1}^{\infty} a_n \cos {n \pi x \over l}\\
a_0&=\int_{-l}^l g(x... |
984,307 | <p>I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$.</p>
<p>Now we consider the function $$f:C\to C^2 \\ 0.a_1a_2a_3... \mapsto \left( 0.a_1a_3a_5... , 0.a_2a_4a_6... \right)$$
I would ... | Henno Brandsma | 4,280 | <p>Let $h$ be the mapping between the so-called Cantor cube $K = \{0,1\}^{\mathbb{N}}$ (in the product topology, where $\{0,1\}$ is discrete) and the Cantor set (as a subset of $[0,1]$) that sends a sequence $(x_i)_{i \ge 1} \in K$ to the point in $C$ with ternary expansion $0.a_1a_2\ldots$ where $a_i = 2x_i$ for all $... |
1,947,421 | <p>I am working on a multi-part problem, which shows that $$\sqrt{2} = 1+\frac{1}{2+\frac{1}{2+...}}$$</p>
<p>I have shown that given some rational approximation $\frac{m}{n}$ to $\sqrt2$ , we can take $\frac{m'}{n'} = \frac{m+2n}{m+n}$ and get a better approximation. </p>
<p>I also know that this iteration goes back... | Will Jagy | 10,400 | <p>$$
\begin{array}{cccccccccccccccccccccccccccccc}
& & 1 & & 2 & & 2 & & 2 & & 2 & & 2 & & 2 & & 2 & & 2 & & 2 & & 2 & \\
\frac{0}{1} & \frac{1}{0} & & \frac{1}{1} & & \frac{3}{2} & & \frac{7}{... |
988,626 | <p>How to find generators to group $E(\mathbb Q) $ of following elliptic curves $E:y^2=x^3-198 $, $E:y^2=x^3-122 $. Thank you in advance.</p>
| dan_fulea | 550,003 | <p>This is a late comment, showing that sage (at least after years) provides the correct information...</p>
<pre><code>sage: E = EllipticCurve(QQ, [0, -198])
sage: E
Elliptic Curve defined by y^2 = x^3 - 198 over Rational Field
sage: E.rank(only_use_mwrank=False)
0
sage: E.torsion_order()
1
</code></pre>
<p>and</p>
<pr... |
614,285 | <p>Does this graph have a special name?</p>
<p>The vertices are arranged on a square square grid with a side length of $n$ and each inner vertices has an edge to its 8 neighbors.</p>
<p>And what about a similar graph with vertices arranged on a $n \times m$ rectangular grid?</p>
<p><img src="https://i.stack.imgur.co... | Ed Pegg | 11,955 | <p>This is the <a href="http://en.wikipedia.org/wiki/King%27s_graph">King's graph</a>. For all your Kingly needs. </p>
|
2,270,476 | <blockquote>
<p>Knowing that $x^2+y^2 = 2 $ prove that $x+y \le 2$</p>
</blockquote>
<p>I rewrite this as
$$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$$
Which is true</p>
| Hagen von Eitzen | 39,174 | <p>Or (shortened, once we know how to poceed):
$$x+y\le x+y+\tfrac{ (x-1)^2+(y-1)^2}2=x+y+\tfrac{x^2-2x+1+y^2-2y+1}2=\frac{x^2+y^2+2}2=2$$</p>
|
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