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 |
|---|---|---|---|---|
2,238,830 | <p>I have just come across "Foundations of Mathematics". Wikipedia describes it as </p>
<blockquote>
<p>Foundations of mathematics can be conceived as the study of the basic
mathematical concepts (number, geometrical figure, set, function,
etc.) and how they form hierarchies of more complex structures and
conc... | Mikhail Katz | 72,694 | <p>Let me put it this way: if you had to learn precisely how an engine works before you could drive a car, there will be very few drivers on the road. Mathematicians mainly drive cars rather than analyzing the functioning of the engine. For some details on the metaphor of set theory as the automobile, see <a href="http... |
216,748 | <ol>
<li><p>SO far i showed that if A matrix is left invertible (L) then in Ax = b, x has at most 1 solution.
<strong>I got that LAx = x = Lb, so x = Lb</strong></p></li>
<li><p>for right inverse (R) of A, in Ax = b, x has at least one solution.
<strong>I got that x = Rb</strong>. </p></li>
</ol>
<p>In the book it sa... | Brian M. Scott | 12,042 | <p>What you did for the left inverse is fine, but what you did for the right inverse doesn’t work: you know that $AR=I$, but you don’t know that $RA=I$, so you can’t say that $x=Rb$. (Actually, you may have seen that $ARb=Ib=b$ and realized on that basis that $Rb$ is a solution; if so, you’re right, but that doesn’t sa... |
426,264 | <ol>
<li><p>Given one sine wave in time domain, I want to find its frequency. Because I observe only a very small part of the sine wave ~1 cycle, FFT methods have a poor spectral resolution. </p></li>
<li><p>Has there been work that bounds the error on the frequency estimate? </p></li>
</ol>
<p>Thanks a ton</p>
| Ross Millikan | 1,827 | <p>If you have lots of points on the sine wave, and FFT will do very well. If you have $1024$ points and FFT them, the bin width is $0.1\%$ of the period. You can also just look at the time between two zero crossings. The fractional error in frequency will be twice the (rss of two) measurement errors divided by the ... |
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | Prahlad Vaidyanathan | 89,789 | <p>Your induction step could look like this: </p>
<p>Suppose $4^n + 1 \equiv 1\pmod{3}$, then $4^n \equiv 0\pmod{3}$, so
$$
4^{n+1} + 1 \equiv 4(4^n) + 1 \equiv 1\pmod{3}
$$
and if $4^n+1 \equiv 2\pmod{3}$, then $4^n \equiv 1\pmod{3}$, so
$$
4^{n+1} +1 \equiv 4+1\pmod{3} = 2\pmod{3}
$$</p>
|
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | 2'5 9'2 | 11,123 | <p>I think that if you need to use induction, instead of proving "$4^n+1$ is not divisible by $3$", you should prove the more specific "$4^n+1$ has remainder $2$ when divide by $3$".</p>
<p>$$4^n+1=3k+2\implies4^n=3k+1\implies4^{n+1}=12k+4$$
$$\implies4^{n+1}+1=12k+5\implies4^{n+1}+1=3(4k+1)+2$$</p>
|
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | Wojowu | 127,263 | <p>Hint: $4^n-1=4^n-1^n$ is divisible by $3$ (why?).</p>
|
3,928,429 | <p>As titled, I was considering the mimimization problem where <span class="math-container">$y(x)$</span> has two endpoints fixed. That is, minimizing <span class="math-container">$$\int_a^b L(x,y(x),y'(x)) \, dx$$</span></p>
<p>where <span class="math-container">$$\ y(a)=m, y(b)=n $$</span>for all <span class="math-co... | Mozibur Ullah | 26,254 | <p>The simplest reference on this is a discussion in section 2 of Landau & Lifschitz's <em>Classical Mechanics</em>, their first volume in their coyrse of theoretical phtsics. This uses the same setup as you do: one spatial and one temporal dimension. They show that a Lagrangian that differs from another Lagrangian... |
4,173,308 | <p><a href="https://i.stack.imgur.com/G55Op.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/G55Op.png" alt="triangle of area 0.5 on a lattice grid" /></a></p>
<p>I'm trying to find the area of this triangle using the <span class="math-container">$\frac{1}{2} \times b \times h$</span> formula, but for... | Laxmi Narayan Bhandari | 931,957 | <p>Here is an approach using Beta function.</p>
<p>We start with the substitution <span class="math-container">$ x \mapsto zx $</span>.</p>
<p><span class="math-container">$$\begin{align} \int_0^z (xz-x^2)^{1/2}\, \mathrm dx &= z^2 \int_0^1 (x-x^2)^{1/2}\, \mathrm dx\\ &= z^2 \int_0^1 x^{1/2}(1-x)^{1/2}\, \math... |
77,092 | <p>I am wondering if anyone could please post the solution to the following differential equation for the function $f(x)$:</p>
<p>$$\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$$</p>
<p>Thanks!</p>
| Agustí Roig | 664 | <p>$$
\frac{f}{f'} = \frac{f'}{f''} \qquad \Longleftrightarrow \qquad \frac{f'}{f} = \frac{f''}{f'} \qquad \Longleftrightarrow \qquad \int \frac{f'}{f} = \int\frac{f''}{f'} + C \qquad \Longleftrightarrow \qquad \ln f = \ln f' + C
$$</p>
<p>Hence, taking exponentia... |
1,207,247 | <blockquote>
<p>Prove that if $\gcd(a,b)=1$, then $\gcd(a^2+b^2, a^2b^2)=1$.</p>
</blockquote>
<p>My attempt:</p>
<p>If $a$ is prime to $b$ the $gcd(a,b)=1$. Assume that $a^2+b^2$ and $a^2b^2$ are not prime to each other. Let $d=gcd(a^2+b^2, a^2b^2)$. Then $d|a^2+b^2, ~~ \&~~d|a^2b^2$. We shall have to prove th... | Prahlad Vaidyanathan | 89,789 | <p>If $p$ is a prime such that $p\mid (a^2+b^2)$ and $p\mid a^2b^2$, then either $p\mid a$ or $p\mid b$. Assume that $p\mid a$, then $p\mid a^2$. But then $p\mid a^2+b^2$ implies that $p\mid b^2$, which means that $p\mid b$. This would imply that $(a,b) \neq 1$ - a contradiction.</p>
|
129,072 | <p>We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with weight $\mu$. I didn't find the corresponding definition for representations of finite dimensional algebras (for example... | Aaron | 28,841 | <p>In ordinary group representation theory, a character is a homomorphism $\chi:G\to \mathbb{C}^\times$. A character associated to a representation $\rho: G\to GL(V)$ is $\chi_\rho: g\mapsto trace(\rho(g))$. Character of Lie algebra arising from differentiating a Lie group is the same thing in the sense that characte... |
2,698,903 | <p>So this has caused some confusion. In one exercise one was asked to prove that</p>
<p>$$\lim_{k \rightarrow \infty} \int_A \cos(kt)dt=0$$</p>
<p>where $A \subset [-\pi, \pi]$ is a measurable set.</p>
<p>My initial idea was to take any $a,b \in A$ and then show that:</p>
<p>$$\lim_{k \rightarrow \infty} \int_a^b ... | Community | -1 | <p>Trying to use an interval as a replacement for $A$ is not a good approach. $A$ could be a very complicated set, after all; it could be a countable union of intervals, or a set like the irrationals, or some totally disconnected Cantor set with positive measure. There is a general principle that measurable sets are <e... |
1,668,487 | <p>$$2^{-1} \equiv 6\mod{11}$$</p>
<p>Sorry for very strange question. I want to understand on which algorithm there is a computation of this expression. Similarly interested in why this expression is equal to two?</p>
<p>$$6^{-1} \equiv 2\mod11$$</p>
| JMP | 210,189 | <p>It is part of the collection of integer pairs such that:</p>
<p>$$xy\equiv1\mod p$$</p>
<p>in this case with $p=11$.</p>
<p>For $p=11$, they are $1\cdot1,2\cdot6,3\cdot4,5\cdot9,7\cdot8$ and $10\cdot10$</p>
<p>Except for $1$ and $p-1$, each $x$ is paired to a unique $y$.</p>
|
847,672 | <p>I repeat the Peano Axioms:</p>
<ol>
<li><p>Zero is a number.</p></li>
<li><p>If a is a number, the successor of a is a number.</p></li>
<li><p>zero is not the successor of a number.</p></li>
<li><p>Two numbers of which the successors are equal are themselves equal.</p></li>
<li><p>If a set S of numbers contains zer... | Asaf Karagila | 622 | <p>Note that from the fifth axiom it follows that if $x\neq 0$ then $x=S(y)$ for some $y$. </p>
<p>Therefore if $x,y$ are both $0$'s (and are distinct) then at least one of them is a successor. Which means that one of them is not a $0$.</p>
<hr>
<p>I think that the source of the confusion here is the fact that $0$ i... |
2,442,705 | <p>Like the title says, I'm wondering if there's a non-brute-force way to determine $x$ such that
$$45\le x<200,$$
$$x\bmod5\equiv0,$$
$$x\bmod8\equiv1,$$
$$x\bmod12\equiv1$$
I know I can simply enumerate all integers in $[45, 200)$ to get the answer (145), but I'm wondering if there's a more elegant solution.</p>
| Parcly Taxel | 357,390 | <p>Just use the Chinese remainder theorem:
$$x\equiv1\bmod12\implies x\equiv1\bmod3$$
$$x\equiv1\bmod8\land x\equiv1\bmod3\implies x\equiv1\bmod24$$
$$x\equiv1\bmod24\land x\equiv0\bmod5\implies x\equiv 25\bmod120$$
Then add 120 repeatedly to 25 until getting a number within the desired range, yielding $x=145$.</p>
|
3,911,297 | <p>Chapter 12 - Problem 26)</p>
<blockquote>
<p>Suppose that <span class="math-container">$f(x) > 0$</span> for all <span class="math-container">$x$</span>, and that <span class="math-container">$f$</span> is decreasing. Prove that there is a <em>continuous</em> decreasing function <span class="math-container">$g$</... | Calvin Lin | 54,563 | <p>Hint: <span class="math-container">$\gcd(7, 12) = 1$</span></p>
<p>so done.</p>
|
3,911,297 | <p>Chapter 12 - Problem 26)</p>
<blockquote>
<p>Suppose that <span class="math-container">$f(x) > 0$</span> for all <span class="math-container">$x$</span>, and that <span class="math-container">$f$</span> is decreasing. Prove that there is a <em>continuous</em> decreasing function <span class="math-container">$g$</... | lone student | 460,967 | <p>Here is general way. You need to solve this linear equation</p>
<blockquote>
<p><span class="math-container">$$\dfrac {a^{12x}}{a^{7y}}=a \Longrightarrow 12x-7y=1$$</span>
where <span class="math-container">$x,y\in \mathbb {Z^+}$</span></p>
</blockquote>
<p>Now, <span class="math-container">$\gcd (12,7) =1$</span> ,... |
1,139,723 | <p>Given the differential equation $\frac{dy}{dt}+a(t)y=f(t)$ </p>
<p>with a(t) and f(t) continuous for:</p>
<p>$-\infty<t<\infty$</p>
<p>$a(t) \ge c>0$</p>
<p>$lim_{t_->0}f(t)=0$</p>
<p>Show that every solution tends to 0 as t approaches infinity.</p>
<p>$$\frac{dy}{dt}+a(t)y=f(t)$$</p>
<p>$$μ(t)=e^... | abel | 9,252 | <p>here is way to do this. we will set the lower arbitrarily to $t_0.$ you can change it to anything more convenient if you like. suppose
$$A = exp\left(\int_{t_0}^t a(s)\,ds\right), \dfrac{1}{A}= exp\left(-\int_{t_0}^t a(s)\,ds\right)$$ is the unique solution of $$ \frac{dA}{dt} - aA = 0, \, A(t_0) = 1.$$ $A$ is cal... |
3,100,420 | <p>So my previous question states that: We are given the sequence <span class="math-container">$_{n}= 6^{({2}^n)} + 1$</span>. We must prove that the elements of this sequence are pairwise co-prime, i.e prove that if m ≠ n then <span class="math-container">$(_{m},_{n}) = 1$</span>.</p>
<p>I previously proved that <spa... | Don Fanucci | 329,831 | <p>I don't see why the same approach doesn't work. If <span class="math-container">$m>n$</span></p>
<p><span class="math-container">$$k_m-2=6^{2^m}-1=\big(6^{2^n}\big)^{2^{m-n}}-1\equiv(-1)^{2^{m-n}}-1\equiv1-1\equiv0\pmod{k_n}$$</span></p>
<p>So, indeed <span class="math-container">$k_n\mid k_m-2$</span> ...</p>
... |
272,588 | <p>I am trying to do the partition of the list variable called
<strong>test</strong> so that answer will look like the variable called <strong>goal</strong>. The <strong>test</strong> list is actually a list of long data. In this example I have a short version example according to the code below which also includes m... | cvgmt | 72,111 | <p><strong>Edit(reply to comment)</strong></p>
<p>For the pattern <code>{{#,#,#},{#,#,#},{#,#,#},{#,#,#},#}</code>,there are <code>3*4+1</code> elements, so we use</p>
<pre><code>test = Range[39];
FlattenAt[#, {-1}] & /@ (Partition[#, UpTo[3]] & /@
Partition[test, UpTo[3*4 + 1]])
</code></pre>
<p><a href="... |
13,922 | <p>How to sum up this series :</p>
<p>$$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$
</p>
<p>Any hint that will lead me to the correct solution will be highly appreciated.</p>
<p>EDIT: Here $C_i = ^nC_i $</p>
| Andrés E. Caicedo | 462 | <p>Let's assume $C_i=\binom ni$. I'll give a solution that is not precalculus level. Consider first the equality
$$ (1+x)^n=C_0+xC_1+x^2C_2+\dots+x^nC_n. $$
This is the <em>binomial theorem</em>.</p>
<p>Integrate from 0 to t. On the left hand side we get $\frac{(1+t)^{n+1}-1}{n+1}$ and on the right hand side $\sum \f... |
348,395 | <p>If <span class="math-container">$f$</span> a distribution with compact support then they exist <span class="math-container">$m$</span> and measures <span class="math-container">$f_\beta$</span>,<span class="math-container">$|\beta|\leq m$</span> such that
<span class="math-container">$$f=\sum_{|\beta|\leq m}\fra... | Johannes Hahn | 3,041 | <p>That's an easy consequence of the Hahn-Banach theorem and the Riesz theorem characterising the dual of <span class="math-container">$C^0(\Omega)$</span>.</p>
<p>"of order m" means that the distribution is linear w.r.t. to the <span class="math-container">$C^m$</span>-norm. It is defined on a dense subspace of <span... |
3,441,779 | <p>Given that <span class="math-container">$(b_k)_{k \in \mathbb{N}} \rightarrow m$</span> and that <span class="math-container">$b_k > 0$</span> for every <span class="math-container">$k \in \mathbb{N}$</span> and that <span class="math-container">$m > 0$</span>, how do I show that <span class="math-container">$... | N.N. | 469,634 | <p>You can prove a more general statement.</p>
<p>In order that there exist integers <span class="math-container">$i$</span> and <span class="math-container">$j$</span> satisfying the equation
<span class="math-container">$$
ai+bj = c,
$$</span>
it is both necessary and sufficient that <span class="math-container">$\g... |
3,441,779 | <p>Given that <span class="math-container">$(b_k)_{k \in \mathbb{N}} \rightarrow m$</span> and that <span class="math-container">$b_k > 0$</span> for every <span class="math-container">$k \in \mathbb{N}$</span> and that <span class="math-container">$m > 0$</span>, how do I show that <span class="math-container">$... | D_S | 28,556 | <p>Let <span class="math-container">$R$</span> be a commutative ring with identity. Let's say that two elements <span class="math-container">$a, b \in R$</span> have "GCD equal to <span class="math-container">$1$</span>" if there is no irreducible element <span class="math-container">$p$</span> of <span class="math-co... |
1,133,581 | <p>$\lim\limits_{(x,y) \to (0,0)} \frac{\log(1-x^2y^2)}{x^2y^2}$</p>
<p>in polar coordinates:</p>
<p>$\lim\limits_{r \to 0} \frac{\log(1-r^2\cos^2t \sin^2t)}{r^2\cos^2t \sin^2t}$</p>
<p>The first limit exists if the second is independent from $t$. But if $t$ is $k \pi/2$ that fraction does not exist. So I could argu... | layman | 131,740 | <p>Actually, if you haven't already learned this, you will learn that in <span class="math-container">$\Bbb R^{n}$</span>, a set is compact <em>if any only if</em> it is both closed and bounded.</p>
<p>The set <span class="math-container">$(0,1)$</span> is bounded, but it is not closed, so it can't be compact.</p>
<p>T... |
1,593,433 | <p>How do I define the integral of this Riemann sum?
$$\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m\cos\left({\frac{2\pi n x}{b}}\right)$$</p>
| Mark Viola | 218,419 | <p>The term of interest is not a Riemann sum. But I thought it would be instructive to present a way to evaluate the limit. </p>
<p>To that end, let $a_m$ be the sequence defined by</p>
<p>$$a_m=\frac1m \sum_{n=1}^m\cos\left(\frac{2\pi nx}{b}\right) \tag 1$$</p>
<p>We can evaluate the sum in $(1)$ in closed form b... |
4,246,461 | <p>I am an economics undergrad who wants to develop a firm grasp over mathematics, so will I be able to understand Bourbaki's books on my own without needing additional books? What about the books titled analysis released by Roger Godement?</p>
| Alex Wertheim | 73,817 | <p>I want to preface what I'm about to say by first expressing my admiration for the Bourbaki textbooks, or at least the ones I've read from (namely, Algebra and Commutative Algebra). The Bourbaki series is the distillation of countless hours of work from some of the greatest mathematicians of the 20th century on what ... |
42,016 | <p>I am looking for algorithms on how to find rational points on an <a href="http://en.wikipedia.org/wiki/Elliptic_curve">elliptic curve</a> $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominator... | James Weigandt | 4,872 | <p>A good reference to get started from the algorithmic point of view is Chapter 3 of Cremona's <a href="http://www.warwick.ac.uk/~masgaj/book/fulltext/index.html" rel="noreferrer"><em>Algorithms for Modular Elliptic Curves</em></a>. It contains a good deal of pseudocode which explains how Cremona's C++ package mwrank ... |
47,181 | <p>I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure that the connected graph has a unique topology in 3-space. More specifically, I'm interested in insuring that some ... | j.c. | 353 | <p>This seems like a question in rigidity theory. In particular it seems like part of what you want is conditions for <strong>global rigidity</strong> in 3 dimensions. </p>
<p>Let me write down some definitions and basic facts from the introduction of <a href="http://www.math.columbia.edu/~dpt/speaking/Rigidity/talk... |
2,954,991 | <p>How do I prove that <span class="math-container">$\forall p, b, c \in \mathbb{Z}^+, (\text{Prime}(p) \wedge p^2 + b^2 = c^2) \Rightarrow p^2 = c+b$</span> ?
I know this much;
<span class="math-container">$$p^2 + b^2 = c^2\\
p^2= c^2 - b^2\\
p^2=(c-b)(c+b)
$$</span>
I know the next part has something to do with the ... | Fabio Lucchini | 54,738 | <p>If <span class="math-container">$p^2=(c-b)(c+b)$</span> and <span class="math-container">$c-b\neq 1$</span>, then <span class="math-container">$c-b=c+b=p$</span> because <span class="math-container">$p $</span> is prime.
This implies <span class="math-container">$b=0$</span>, a contradiction which proves <span class... |
2,470,062 | <blockquote>
<p><span class="math-container">$$\sqrt{k-\sqrt{k+x}}-x = 0$$</span></p>
<p>Solve for <span class="math-container">$k$</span> in terms of <span class="math-container">$x$</span></p>
</blockquote>
<p>I got all the way to
<span class="math-container">$$x^{4}-2kx^{2}-x+k^{2}-x^{2}$$</span>
but could not facto... | Leucippus | 148,155 | <p>As discovered $\sqrt{k - \sqrt{k+x}} = x$ leads to the equation $k^2 - (2 \, x^2 + 1) \, k + (x^4 - x) = 0$. Now,
\begin{align}
k &= \frac{1}{2} \, \left[(2 \, x^2 + 1) \pm \sqrt{ (2 \, x^2 + 1)^2 - 4 \, (x^4 - x) } \right] \\
&= \frac{1}{2} \, [(2 \, x^2 + 1) \pm (2 x + 1)] \\
&= \begin{cases}{ x^2 + x ... |
3,165,276 | <p>I am attempting to prove if the limit for <span class="math-container">$S_n = \frac{\sqrt{n}}{\sqrt{n}+1} $</span>, exists, and if it exists, what it is?
If the limit exists, proving it in terms of the definition for <span class="math-container">${e}$</span> and <span class="math-container">${N}$</span>.</p>
<p>I g... | WaveX | 323,744 | <p>No need for the conjugate here.</p>
<p>Let <span class="math-container">$\varepsilon>0$</span>. Then by the Archimedean Property, we have that for some <span class="math-container">$N \in \mathbb{N}$</span>, <span class="math-container">$\frac1N \lt \varepsilon^2$</span> </p>
<p>See that for <span class="math-c... |
3,520,057 | <p>how would you draw <span class="math-container">$A\cup(B\cap C\cap D)$</span>
I tried with paint doesn't look good and I don't know if there are tools to make this easier would appreciate any help or tips</p>
<p><img src="https://i.stack.imgur.com/fc6Pq.png" alt="enter image description here"></p>
| Jonas Linssen | 598,157 | <p>Note that the tensor product of algebras is in fact the coproduct in the category of <span class="math-container">$A$</span>-algebras. What AM defines there should therefore satisfy the universal property of an infinite coproduct. I am not sure about the universal property with respect to multilinear maps of the und... |
273,463 | <p>I want to solve this differential equation</p>
<p><span class="math-container">$$
\frac{1}{\Phi(\phi)} \frac{d^2 \Phi}{d \phi^2}=-m^2
$$</span></p>
<p>with b.c.</p>
<p><span class="math-container">$$
\Phi(\phi+2 \pi)=\Phi(\phi)
$$</span></p>
<p>it is easy to solve by wolfram</p>
<pre><code>eq1 = D[Phi[phi], {phi, 2}... | Nasser | 70 | <p>Just change constants.</p>
<pre><code>ClearAll[Phi, phi, m];
eq1 = D[Phi[phi], {phi, 2}]/Phi[phi];
sol = Phi[phi] /. First@DSolve[eq1 == -m^2, Phi[phi], phi]
</code></pre>
<p><img src="https://i.stack.imgur.com/ve3Lz.png" alt="Mathematica graphics" /></p>
<pre><code>sol = TrigToExp[sol]
</code></pre>
<p><img src="ht... |
3,299,072 | <p>I want to prove that
<span class="math-container">$$\frac{(x-y)(x-z)}{x^2}+\frac{(y-x)(y-z)}{y^2}+\frac{(z-x)(z-y)}{z^2}\geq 0$$</span>
for positive numbers <span class="math-container">$x,y,z$</span>.
I don't know how to even begin. I must say I'm not 100% certain the inequality always holds.</p>
<p>I tried the so... | Michael Rozenberg | 190,319 | <p>After replacing <span class="math-container">$x$</span> on <span class="math-container">$\frac{1}{x}$</span> and similar we need to prove that
<span class="math-container">$$\sum_{cyc}x(x-y)(x-z)\geq0,$$</span> which is Schur.</p>
|
16,158 | <p>I have a Manipulate control, the main element of which is a single slider. Moving the <em>one</em> slider triggers calls to <em>two</em> functions - one which is evaluated very quickly, the other more slowly. I'd like to see 'live' update ( i.e. ContinuousAction->True ) for the fast one, but would only like to updat... | Nasser | 70 | <p>Yes. Each control can have its own <code>ContinuousAction</code> option. As follows:</p>
<pre><code>Manipulate[{x, y, z},
{{x, 0, "x"}, 0, 10, 1, ContinuousAction -> True},
{{y, 0, "y"}, 0, 10, 1, ContinuousAction -> False},
{{z, 0, "z"}, 0, 10, 1},
ContinuousAction -> True
]
</code></pre>
<p>The inn... |
16,158 | <p>I have a Manipulate control, the main element of which is a single slider. Moving the <em>one</em> slider triggers calls to <em>two</em> functions - one which is evaluated very quickly, the other more slowly. I'd like to see 'live' update ( i.e. ContinuousAction->True ) for the fast one, but would only like to updat... | jVincent | 1,194 | <p>This is not a full answer to your question, however I'd just like to add how this can be solved outside of Manipulate, perhaps someone can translate this into using Manipulate.
As I understand it you problem is that you dynamically display both <code>fast[a]</code> and <code>slow[a]</code> while interactively updati... |
1,602,078 | <p>There are $n$ prisoners and $n$ hats. Each hat is colored with one of $k$ given colors. Each prisoner is assigned a random hat, but the number of each color hat is not known to the prisoners. The prisoners will be lined up single file where each can see the hats in front of him but not behind. Starting with the pris... | Parnad Bhattacharjee | 778,200 | <p>Since, I don't have the permission to comment yet, so I am adding here... we should label the k colours with number from 0 to k-1 : {0,1,2,3,....k-1} and then use the concept of modular arithmetic.</p>
|
3,422,861 | <p>The problem is as follows:</p>
<blockquote>
<p>The acceleration of an oscillating sphere is defined by the equation
<span class="math-container">$a=-ks$</span>. Find the value of <span class="math-container">$k$</span> such as <span class="math-container">$v=10\,\frac{cm}{s}$</span> when <span class="math-conta... | David K | 139,123 | <p>The first thing to realize is that <span class="math-container">$s$</span> is <em>not</em> a "weird notation for time". It's a fairly standard notation for distance along a path.
It is used much the same way as <span class="math-container">$x$</span> is used in other problems, except that we tend to use <span class=... |
1,358,574 | <p>When I searched for the derivative of the Gamma function I got something of the form:</p>
<p>$$\Gamma'(x)=\Gamma(x) \psi(x)$$</p>
<p>But from the definition of the Digamma function to me it's like writing:</p>
<p>$$\Gamma'(x)=\Gamma'(x)$$</p>
<p>And this doesn't seem very useful to me (if I'm wrong feel free to ... | Community | -1 | <p>$4^n = \sum_{k=0}^{2n} \binom{2n}{k} \ge \binom{2n}{n}$, so the terms of this series are all at least one. (In particular, they don't converge to zero.)</p>
|
2,940,287 | <p>My ring theory teacher has a very broad definition of rings: he doesn't require them to be associative. As such, he told us to work out a definition of a product operation <span class="math-container">$\cdot$</span> on <span class="math-container">$R = \mathbb{Z}_2 \times \mathbb{Z}_2$</span> that is distributive ov... | egreg | 62,967 | <p>The statement is true when <span class="math-container">$T\colon V\to V$</span> is a linear transformation on the vector space <span class="math-container">$V$</span>, not just when <span class="math-container">$V=\mathbb{R}^{2\times2}$</span>.</p>
<p>Suppose <span class="math-container">$T^2=T$</span>; we want to ... |
609,744 | <p>I am not sure about the last step of my proof:</p>
<p>$(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). $C[a,b]$ is the space of continuous functions on [a,b].</p>
<p>Main goal in order to prove the claim is that the in... | Stefan4024 | 67,746 | <p>You can use this:</p>
<p>$$x^y > y^x \iff \sqrt[x]{x} > \sqrt[y]{y}$$</p>
<p>which is true because the function $f(x) = \sqrt[x]{x}$ is strictly decreasing from $e$ to $\infty$, i.e it's convex</p>
|
2,548,254 | <p>Consider this problem:</p>
<p><em>Prove $x$ and $y$ in a complex vector space is orthogonal if and only if $||\alpha x + \beta y|| ^2 = ||\alpha x||^2 + ||\beta y||^2$ for all scalars $\alpha$ and $\beta$.</em></p>
<hr>
<p>I have solved a similar problem that asks:</p>
<p><em>Prove $x$ and $y$ in a real vector s... | Jbag1212 | 509,460 | <p>First we show that if $x$ and $y$ are orthogonal then
$$||\alpha x + \beta y||^2 = ||\alpha x||^2 + || \beta y||^2.$$</p>
<p>Assume $x$ and $y$ are orthogonal. Then by definition $\langle x, y \rangle =0.$ </p>
<p>Therefore, $ \alpha \beta^{*} \langle x, y \rangle = \alpha^{*} \beta \langle y, x \rangle =0.$</p>
... |
2,548,254 | <p>Consider this problem:</p>
<p><em>Prove $x$ and $y$ in a complex vector space is orthogonal if and only if $||\alpha x + \beta y|| ^2 = ||\alpha x||^2 + ||\beta y||^2$ for all scalars $\alpha$ and $\beta$.</em></p>
<hr>
<p>I have solved a similar problem that asks:</p>
<p><em>Prove $x$ and $y$ in a real vector s... | Math Lover | 348,257 | <p>First, it is easy to show that if $x$ and $y$ are orthogonal then
$$\|\alpha x + \beta y\|^2 = \|\alpha x\|^2 + \|\beta y\|^2. \tag{1}$$
Now, let's try to prove that if $(1)$ is true <strong>for all</strong> scalars $\alpha$ and $\beta$ then $x$ and $y$ are orthogonal. To this end, we first observe that
$$\|\alpha x... |
1,770,508 | <p>I am confronted with the following definition:</p>
<blockquote>
<p>Let <span class="math-container">$K$</span> be a field and <span class="math-container">$e_1,e_2,\ldots,e_n$</span> the standard basis of the <span class="math-container">$K$</span> vector space <span class="math-container">$K^n$</span>.</p>
<p>For <... | user5713492 | 316,404 | <p>My solution is another way of stating @Jack D'Aurizio's solution. We want to rotate coordinates so that the plane $x+4z=a$ is horizontal. This can be achieved through
$$\begin{array}{rl}x^{\prime}&=\frac4{\sqrt{17}}x-\frac1{\sqrt{17}}z\\
z^{\prime}&=\frac1{\sqrt{17}}x+\frac4{\sqrt{17}}z\end{array}$$
The inve... |
2,590,537 | <p>Let $A$ be a proper subset of of $X$, $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X\times Y)-(A\times B)$ is connected.</p>
<p>I have already seen the solution of this problem here on math.stackexchange, but my confusion is this.
Suppose $X$=(a,b) & $Y$=(c,d) and let $A$={m} &... | Michael Hardy | 11,667 | <p>Here's your list of $k$-vectors:
$$
\underbrace{ \begin{bmatrix} \vdots \\ \vdots \\ \vdots \end{bmatrix}, \ldots\ldots\ldots, \begin{bmatrix} \vdots \\ \vdots \\ \vdots \end{bmatrix} }_{\large \text{This list has $2^k$ items.}}
$$
Here's your list of $(k+1)$-vectors:
$$
\underbrace{ \begin{bmatrix} \vdots \\ \vdots... |
2,070,574 | <p>I had a question </p>
<p>$$\sum_{a=1}^n \frac{1}{a^2}=?$$</p>
<p>I had learned newton's method of undetermined method
,but that doesn't work here because of negative power</p>
<p>than I saw another question </p>
<p>$$\sum_{a=1}^n \frac{1}{a(a+1)}$$</p>
<p>here we transformed it as</p>
<p>$$\sum_{a=1}^n (\frac... | Simply Beautiful Art | 272,831 | <p>There isn't such a beautiful solution as the one you've presented, since it is the case that as $n\to\infty$, you get the Basel problem.</p>
<p>Notice that</p>
<p>$$\frac1{a(a+h)}=\frac1h\left(\frac1a-\frac1{a+h}\right)$$</p>
<p>but since $h$ is not a whole number, we will not be seeing any cancellations like wit... |
3,072,274 | <p><a href="https://i.stack.imgur.com/WaOF8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WaOF8.png" alt="enter image description here"></a></p>
<blockquote>
<p>Let <span class="math-container">$n\in\mathbb N$</span> be fixed. For <span class="math-container">$0\le k \le n$</span>, et <span clas... | Community | -1 | <p>The determinants of similar matrices are the same: <span class="math-container">$\operatorname{det}A=\operatorname{det}PBP^{-1}=\operatorname{det}P\cdot
\operatorname{det}B\cdot\operatorname{det}P^{-1}=\operatorname{det}P\cdot \frac1{\operatorname{det}P}\cdot \operatorname{det}B=\operatorname{det}B$</span>.
This m... |
92,492 | <p>If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the <strong>projective model structure</strong>. The fibrations and weak equivalences are defined point wise. </p>
<p>There is also the one called <strong>inject... | Igor Rivin | 11,142 | <p>The magic words are: <a href="http://dl.dropbox.com/u/5188175/langweil.pdf" rel="nofollow">Lang-Weil</a></p>
|
92,492 | <p>If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the <strong>projective model structure</strong>. The fibrations and weak equivalences are defined point wise. </p>
<p>There is also the one called <strong>inject... | David E Speyer | 297 | <p>This is a big field and improvements are being made all the time. The key phrases to search on are "zeta function" and "hypersurface" -- the $\zeta$-function of $F=0$ is $\exp(\sum \frac{N_n}{n} t^n )$ where $N_n$ is the number of solutions to $F=0$ over $\mathbb{F}_{p^n}$. </p>
<p>I started learning the current ap... |
4,227,776 | <p>Let <span class="math-container">$R$</span> be a commutative unital ring, and let <span class="math-container">$p$</span> be a prime number.</p>
<p><strong>Question 1</strong>: If <span class="math-container">$R$</span> is a <em>local</em> ring (with unique maximal ideal <span class="math-container">$m$</span>) of c... | Minseon Shin | 7,719 | <p>(Sorry this might be incorrect, see below.)</p>
<p>(Set <span class="math-container">$\mathbb{N} := \mathbb{Z}_{\ge 0}$</span>.) Let <span class="math-container">$\mathbb{Z}_{p}[[\{x_{n}\}_{n \in \mathbb{N}}]]$</span> be the <span class="math-container">$\langle \{x_{n}\}_{n \in \mathbb{N}} \rangle$</span>-adic comp... |
2,959,628 | <p>I'm struggling to find the density function of <span class="math-container">$Y=X^3$</span> where <span class="math-container">$X \sim \mathcal{N}(0,1)$</span> with density function <span class="math-container">$\phi(\cdot)$</span>. </p>
<p>Since <span class="math-container">$g(X) = X^3$</span> is monotonic I suppos... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$1/x=h^2,h>0$</span> to find</p>
<p><span class="math-container">$$\lim_{h\to0}\dfrac{\sqrt{1+4h^4}-\sqrt{1+h^4}}{h^3}$$</span></p>
<p><span class="math-container">$$=\lim...\dfrac{1+4h^4-(1+h^4)}{h^3}\cdot\lim...\dfrac1{\sqrt{1+4h^4}+\sqrt{1+h^4}}=?$$</span></p>
|
2,806,164 | <p>I recently came across a question that asked for the derivative of $e^x$ with respect to $y$. I answered $\frac{d}{dy}e^x$ but the answer was $e^x\frac{dx}{dy}$. How is that the answer? I am confused.</p>
| Richard Ambler | 65,729 | <p>If $x$ is a function of $y$ then the given answer follows by the chain rule:</p>
<p>$$\frac {\text d}{\text dy} \left(e^x\right) = e^x \cdot \frac {\text d}{\text dy}(x)$$</p>
|
531,080 | <blockquote>
<p>Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane. Let $\Omega$ be a closed, bounded set in the $xy$-plane with $2$-volume $B$. Pick a point $Q$ in $P$ and make a pyramid by joining each point in $\Omega$ to $Q$ with a straight line segment. Find the $3$-volume of this pyramid.</p>
</blo... | Daniel Robert-Nicoud | 60,713 | <p>Let's answer to a bit more ambitious question: <em>Let $\Omega$ be an open, bounded and connected subset of an $(n-1)$-dimensional hyperplane $P$ of $\mathbb{R}^n$, and let $x_0\in\mathbb{R}^n$ be a point not contained in $P$. What is the volume of the pyramid with basis $\Omega$ and vertex $x_0$?</em></p>
<p>By a ... |
4,480,276 | <p>[This question is rather a very easy one which I found to be a little bit tough for me to grasp. If there is any other question that has been asked earlier which addresses the same topic then kindly link this here as I am unable to find such questions by far.]</p>
<p>let <span class="math-container">$f(x)=x.$</span... | CyclotomicField | 464,974 | <p>I think the <span class="math-container">$\epsilon$</span>-<span class="math-container">$\delta$</span> definition is best understood as trying to find the bounds on the input that will ensure the output is correct to within some tolerance. So <span class="math-container">$\epsilon$</span> is the acceptable error in... |
2,253,676 | <p>$f(x)$ continuous in $[0,\infty)$. Both $\int^\infty_0f^4(x)dx, \int^\infty_0|f(x)|dx$ converge. </p>
<p>I need to prove that $\int^\infty_0f^2(x)dx$ converges.</p>
<p>Knowing that $\int^\infty_0|f(x)|dx$ converges, I can tell that $\int^\infty_0f(x)dx$ converges.</p>
<p>Other than that, I don't know anything. I ... | Jack D'Aurizio | 44,121 | <p>Improved version of Lord Shark's answer: for any $K>0$ we have, by AM-GM,
$$ 3 f(x)^2 \leq K^2 f(x)^4 + \frac{1}{K}|f(x)|+ \frac{1}{K}|f(x)| \tag{1}$$
from which it follows that
$$ \int_{0}^{+\infty}f(x)^2\,dx \leq \frac{K^2}{3}\int_{0}^{+\infty} f(x)^4\,dx + \frac{2}{3K}\int_{0}^{+\infty}|f(x)|\,dx \tag{2} $$
an... |
3,447,872 | <p>Subtracting both equations
<span class="math-container">$$x(a-b)+b-a=0$$</span>
<span class="math-container">$$(x-1)(a-b)=0$$</span>
Since <span class="math-container">$a\not = b$</span>
<span class="math-container">$$x=1$$</span>
Substitution of x gives
<span class="math-container">$$1+a+b=0$$</span> which is con... | Alexey Burdin | 233,398 | <p>Let <span class="math-container">$c$</span> be the difference between roots of <span class="math-container">$x^2+ax+b$</span> and <span class="math-container">$x^2+bx+a$</span><br>
(in other words, let <span class="math-container">$x_{1,2}^2+ax_{1,2}+b=0$</span>,<br>
<span class="math-container">$x_{3,4}^2+bx_{3,4}+... |
3,301,387 | <p>I don't understand what it is about complex numbers that allows this to be true.</p>
<p>I mean, if I root both sides I end up with Z=Z+1. Why is this a bad first step when dealing with complex numbers?</p>
| José Carlos Santos | 446,262 | <p>Because it is not true (even in <span class="math-container">$\mathbb R$</span>) that <span class="math-container">$a^6=b^6\implies a=b$</span>, which is what you are assuming.</p>
<p>In <span class="math-container">$\mathbb C$</span>,<span class="math-container">$$a^6=b^6\iff a=b\times\left(\cos\theta+i\sin\theta\... |
286,082 | <p><a href="https://en.wikipedia.org/wiki/Budan's_theorem" rel="nofollow noreferrer">Budan's theorem</a> gives an upper bound for the number of real roots of a real polynomial in a given interval $(a,b)$. This bound is not sharp (see the example in Wikipedia).</p>
<p>My question is the following: let us suppose th... | Alexandre Eremenko | 25,510 | <p>If this were so, real parts of non-real roots will form a subset of
the real critical points which is absurd. So almost any random example is a counterexample.
Take
$$f(x)=(1+z^2)(1-x+x^2),$$
there are no real roots, and the the real parts of the complex roots are known exactly: they are $0$ and $1/2$.
Now $$f'(x)=... |
1,782,800 | <p>I am studying for an exam and found this in the solutions:</p>
<blockquote>
<p>"... with canonical parameters $\{\log\lambda_i\}^n_1$". </p>
</blockquote>
<p>Index $I=1,2,\ldots,n$. The professor that wrote the exam has retired. Does anyone know what the $1$ and $n$ might mean?</p>
| Laars Helenius | 112,790 | <p>It is an indexed set. For example:
$$
\{x_i\}_{i=1}^n=\{x_1,x_2,\ldots,x_n\}.
$$</p>
<p>Does this help?</p>
|
1,377,927 | <p>Prove using mathematical induction that
$(x^{2n} - y^{2n})$ is divisible by $(x+y)$.</p>
<p><strong>Step 1:</strong> Proving that the equation is true for $n=1 $</p>
<p>$(x^{2\cdot 1} - y^{2\cdot 1})$ is divisible by $(x+y)$ </p>
<p><strong>Step 2:</strong> Taking $n=k$</p>
<p>$(x^{2k} - y^{2k})$ is divisible ... | Community | -1 | <p>$$x^{2(k+1)}-y^{2(k+1)}=x^2x^{2k}-y^2y^{2k}=x^2x^{2k}\color{green}{-x^2y^{2k}+x^2y^{2k}}-y^2y^{2k}\\
=x^2(x^{2k}-y^{2k})+(x^2-y^2)y^{2k}.$$</p>
<p>The first term is divisible by $x+y$ by the induction hypothesis, and the second as well, by direct factorization.</p>
|
2,233,457 | <p>I'm working on the following integral:</p>
<p>$\int_0^{\infty}\frac{y^r}{\theta}ry^{r-1}e^{\frac{-y^r}{\theta}} dy$. </p>
<p>I noticed that if I took the derivative:</p>
<p>$\frac{d}{dy}e^{\frac{-y^r}{\theta}} = \frac{-ry^{r-1}}{\theta}e^{\frac{-y^r}{\theta}}$ </p>
<p>I got most of the expression of the integran... | Pedro | 9,921 | <p>The rank of $f$ at every point is $\leq 1$. Consider the subset of points where the rank is exactly $1$. This is an open subset (since rank is upper semicontinuous or whatever). Certainly it can't be empty or else you'd have a constant map. So there's a non-empty open subset where $f$ has constant rank $1$. Then the... |
2,233,457 | <p>I'm working on the following integral:</p>
<p>$\int_0^{\infty}\frac{y^r}{\theta}ry^{r-1}e^{\frac{-y^r}{\theta}} dy$. </p>
<p>I noticed that if I took the derivative:</p>
<p>$\frac{d}{dy}e^{\frac{-y^r}{\theta}} = \frac{-ry^{r-1}}{\theta}e^{\frac{-y^r}{\theta}}$ </p>
<p>I got most of the expression of the integran... | Angina Seng | 436,618 | <p>No continuous map from $M$ to $\mathbb{R}$ can be injective.</p>
<p>As $M$ has dimension at least $2$ it contains a subspace $C$ homeomorphic to a circle. Now there is no continuous injective map $g$ from $C$ to $\mathbb{R}$. If there were, then there are $a$ and $b$ on $C$ with $g(a)<g(b)$. Then on each arc of ... |
2,629,115 | <blockquote>
<p>We will define the convolution (slightly unconventionally to match Rudin's proof) of $f$ and $g$ as follows:
$$(f\star g)(x)=\int_{-1}^1f(x+t)g(t)\,dt\qquad(0\le x\le1)$$</p>
<ol>
<li>Let $\delta_n(x)$ be defined as $\frac n2$ for $-\frac1n<x<\frac1n$ and 0 for all other $x$. Let $f(x)$... | operatorerror | 210,391 | <p>By definition,
$$
f*\delta_{10}=\int_{-1}^1f(x+t)\delta_{10}(t)\mathrm dt\\
=5\int_{-.1}^{.1}f(x+t)\mathrm dt
$$
Now we use the argument of $f$ and the definition of $f$ to determine the rest.</p>
<p>The integrand vanishes (by the definition of $f$) unless $.4<x+t<.7$, i.e.
$$
.4-x<t<.7-x
$$
Noting th... |
2,889,651 | <p>The given points are $M(3,-1,2)$ and $M1(0,1,2)$ , plane A is passing though this 2 points. Plane B $2x-y+2z-1=0$ and its normal to A. How to we find the equation of plane A? I have tried with cross product of vector $MM1$ and vector $b=(2,-1,2)$, but it didn't work.</p>
| mathcounterexamples.net | 187,663 | <p><strong>The result is wrong for real vector spaces with dimension $n>2$</strong></p>
<p>The orthogonal transformation</p>
<p>$$Q=
\begin{pmatrix}
\dfrac{\sqrt{2}}{2} & -\dfrac{\sqrt{2}}{2} & 0\\
\dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{2}}{2} & 0\\
0 & 0 &-1
\end{pmatrix}$$</p>
<p>is such tha... |
1,634 | <p>I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "ramified cover". Is there any way to make this precise? Or can anyone elaborate on the geometric meaning of it?</p>
| Andrew Critch | 84,526 | <p>I definitely agree with Peter's general intuitive description.</p>
<p>In response to some of the subsequent comments, here are some implications to keep in mind: </p>
<p><strong>Finite</strong> ==> <strong>finite fibres</strong> (1971 EGA I 6.11.1) and <strong>projective</strong> (EGA II 6.1.11), hence <strong>pro... |
758,466 | <blockquote>
<p>A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multiple of 8.</p>
</blockquote>
<p>I solved this by ... | Anant | 85,660 | <p>Here's a generalization:</p>
<p>The sample space for a six-sided die is $S = \{1,2,3,4,5,6\}$.</p>
<p><strong>Theorem:</strong> There are exactly $\color{blue}{3}^n$ n-tuples $(x_{n-1}, x_{n-2}, \cdots, x_1, x_0)$ that satisfy $f(n) = 10^{n-1}x_{n-1} + 10^{n-2}x_{n-2} + \cdots + 10x_1 + x_0 \equiv 0 \mod{2^n}$ fo... |
1,470,244 | <p>Prove that the sum of the moduli of the roots of $x^4-5x^3+6x^2-5x+1=0$ is $4$.<br></p>
<hr>
<p>I tried various methods to find its roots,but no luck.Is there any method to solve it without actually finding the roots of the equation.Please help me.<br>
Thanks.</p>
| Math-fun | 195,344 | <p>\begin{align}
x^4-5x^3+6x^2-5x+1&=(\color{red}1\times x^4-\color{red}4\times x^3+\color{red}1\times x^2)\\
&-(\color{red}1\times x^3-\color{red}4\times x^2+\color{red}1\times x)\\
&+(\color{red}1\times x^2-\color{red}4\times x+\color{red}1)\\
&=x^2(x^2-4x+1)-x(x^2-4x+1)+(x^2-4x+1)\\
&=(x^2-4x+1)(... |
1,490,828 | <p>Consider an equilateral triangle in $A \subset \mathbb{R}^2$ with vertices $x_1$, $x_2$, and $x_3$. I would like to show that the smallest circle which contains $A$ has radius $\frac{d}{\sqrt{3}}$, where $d$ is the length of each side. </p>
<p>I've been able to show that a (closed) ball centered at $(x_1 + x_2 + x_... | achille hui | 59,379 | <p>Choose coordinate system so that the centroid of the equilateral triangle of side $d$ is the origin. i.e
$$x_1 + x_2 + x_3 = 0$$
We will have $|x_1| = |x_2| = |x_3| = \frac{d}{\sqrt{3}}$.</p>
<p>Given any circle of radius $r$ centered at $x$. If it contains the three point $x_1, x_2, x_3$, it will satisfies:
$$3r^2... |
132,769 | <p>My distribution histogram looks like it is not identically distributed, as in the negative counts have a different shape than the positive counts. Here is an image:</p>
<p><img src="https://i.stack.imgur.com/VtlHd.jpg" alt="enter image description here"></p>
<p>The chart has 800 data points, and the tallest count ... | Robert Israel | 8,508 | <p>As @cardinal wrote, you probably don't mean "identically distributed": that's something you can't really tell from a histogram. If you want to test whether this is a normal distribution, you might use a D'Agostino-Pearson test, a Shapiro-Wilk test, or a Kolmogorov-Smirnov test. See e.g. <a href="http://www.graphpa... |
619,580 | <p>Please help me to solve the following problem that is in the Lebesgue integral discussion </p>
<blockquote>
<p>Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\,\,\|f_n\|_\infty \to \infty$ but $\int_0^1\lvert\, f_n\rvert\,d\lambda \to 0$ as $n\to\infty$.</p>
</block... | Yiorgos S. Smyrlis | 57,021 | <p>Let
$$
f_n(x)=\max\left\{n,n^{-1}x^{-1/2}\right\}.
$$</p>
<p>Then $\|\,f_n\|_\infty=n$ while $\|\,f\|_1<\dfrac{1}{2n}$.</p>
|
619,580 | <p>Please help me to solve the following problem that is in the Lebesgue integral discussion </p>
<blockquote>
<p>Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\,\,\|f_n\|_\infty \to \infty$ but $\int_0^1\lvert\, f_n\rvert\,d\lambda \to 0$ as $n\to\infty$.</p>
</block... | ABIM | 83,860 | <p>Below I have given a sufficient condition for your problem to have a positive solution in a rather general measure space, afterwhich I give an example in the case of $[0,1]$ with its usual Euclidean topology to help illustrate my construction. </p>
<hr>
<p><strong>A general solution:</strong></p>
<p><strong>Prop... |
75,656 | <p>Is there a closed form for any function f(x,y) satisfying:</p>
<p>a) $\frac{df}{dx}+\frac{df}{dy}=xy$</p>
<p>b) $\frac{df}{dx}+\frac{df}{dy}+\frac{df}{dz}=xyz$</p>
| Ilya | 5,887 | <p>Forgive for using Mathematica, but the answers are:
$$
a) \quad f(x,y) = \frac16\left(3x^2y-x^3\right)+F(y-x)
$$
with $F$ - arbitrary continuously differentiable function,
$$
b)\quad f(x,y,z) = \frac1{12}(x^4-2x^3y-2x^3z+6x^2yz)+F(y-x,z-x)
$$
with $F$ - arbitrary continuously differentiable function.</p>
<p>Hop... |
9,540 | <p>I'm following <a href="http://reference.wolfram.com/mathematica/ref/FinancialData.html">http://reference.wolfram.com/mathematica/ref/FinancialData.html</a></p>
<p>I get the following:</p>
<pre><code>In[6]:= DateListLogPlot[FinancialData["^DJI", All]]
</code></pre>
<blockquote>
<p>During evaluation of In[6]:= Da... | Vitaliy Kaurov | 13 | <p>We have another option now: <a href="https://github.com/bajracha71/Quandl-Mathematica-QuandlLink" rel="nofollow noreferrer">QuandlLink</a> package. Using this package you can compute this as:</p>
<pre><code>Needs["QuandlLink`"]
dowJonesOHLCV =
QuandlFinancialData[
"YAHOO/INDEX_DJI",
startDate -> "2015-... |
1,002,191 | <p>I have a problem:</p>
<blockquote>
<p>A family has six children, A,B,C,D,E,F. The ages of these six children are added together to make an integer less than $32$ where none of the children are less than one year old.</p>
</blockquote>
<p>What is the chances of guessing the ages of all of the children.</p>
<p>I ... | RE60K | 67,609 | <p>given $$\sum_i^6x_i<32\implies \sum_i^7x_i=31(\because x_7\ge0,x_{(i,0\le i\le6)}\ge1)$$
let $$x_7'=x_7+1,x_7'\ge1$$
so $$\sum_i^6x_i+x_7'=32$$
now all $x_i\ge1$, now you can solve it maybe?</p>
|
1,419,185 | <p>I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't take a picture of the problem so I attempted to trace it.</p>
<p><a href="https://i.stack.imgur.com/FzMSf.png" rel="... | Daniel R. Collins | 266,243 | <p>The graph shows roots at $x = -3, 0, 3$, and there's a double root at zero due to the way the graph is tangent to the y-axis at that point. The equation is therefore the product of the four factors $(x + 3)(x - 3)(x + 0)(x + 0) = (x^2 - 9)(x)(x) = x^4 - 9x^2$, possibly with another constant scaling multiplier. But ... |
1,897,212 | <p>Does there exist any surjective group homomorphism from <span class="math-container">$(\mathbb R^* , .)$</span> (the multiplicative group of non-zero real numbers) onto <span class="math-container">$(\mathbb Q^* , .)$</span> (the multiplicative group of non-zero rational numbers)? </p>
| Hanul Jeon | 53,976 | <p>Suppose that $f:\Bbb{R}^*\to\Bbb{Q}^*$ is a such homomorphism. Then we have some $x$ such that $f(x)=2$. Now take a cube root $\sqrt[3]{x}$ of $x$, which always exists in $\mathbb{R}^*$. Then $(f(\sqrt[3]{x}))^3 = f(x) = 2$, i.e. $f(\sqrt[3]{x})$ is a cube root of $2$. But $2$ has no cube root in $\mathbb{Q}^*$, so ... |
698,702 | <p>Let $V$ be a vector space with finite dimension $n$ and $T:V\longrightarrow V$ is a linear transformation such that $T^{2}=0$. Then</p>
<ol>
<li><p>$rank(T)\leq\frac{n}{2}$</p></li>
<li><p>$n(T)\leq\frac{n}{2}$</p></li>
<li><p>$rank(T)\geq n(T)$</p></li>
<li><p>$rank(T)\geq \frac{n}{2}$</p></li>
</ol>
| r9m | 129,017 | <p>Sylvester's Rank Inequality $2\operatorname{rank}(T)-n\le \operatorname{rank}(T^2)=0$. $\operatorname{rank}(T)\le\frac{n}{2}$</p>
<p>and $\operatorname{rank}(T)+\operatorname{null}(T)=n$, so $\operatorname{null}(T)\ge \frac{n}{2}$.</p>
|
2,820,842 | <p>I am trying to prove that if $Q$ is an algebraic extension of $F$
($F$ is a field), if $R$ is a subring of $Q$, and we have
$F\subseteq R$, then $R$ is a subfield of $Q$.</p>
<p>My first approach was to show every element has an inverse. But there was no result.</p>
<p>Any help would be great.</p>
| Hagen von Eitzen | 39,174 | <p>We are given that $a=b+up^n$ with $u$ not a multiple of $p$.
Then $$ a^p-b^p=\sum_{k=1}^p{p\choose k}b^{p-k}u^kp^{nk}$$
Using the fact that $p\mid {p\choose k}$ for $0<k<p$, we see:
The summand for $k=1$ is divisible precisely by $p^{n+1}$, the summand for $k=p$ is divisible by $p^{np}>p^{n+1}$, and all in-... |
1,823,023 | <h2>1°</h2>
<p>$n_0=1$</p>
<p>$(1!)^2 \ge 1^1$</p>
<p>$1\ge1$</p>
<h2>2°</h2>
<p>$k \ge n_0$</p>
<p>assumption: $$(k!)^2 \ge k^k$$</p>
<p>and for k+1: $$((k+1)!)^2 \ge (k+1)^{k+1}$$</p>
<p>I also noticed that: $$((k+1)!)^2 = (k!)^2 * (k+1)^2$$</p>
| Daniel McLaury | 3,296 | <p>Well, the simplest interpretation is that they take a vector and recover its coordinates. So, for instance, if you have a vector (2, 3, 1), and you want to express the fact that the second coordinate is 3, the most effective way of doing this is by writing something like</p>
<p>$$(2, 3, 1) \cdot (0, 1, 0) = 3$$</p... |
1,823,023 | <h2>1°</h2>
<p>$n_0=1$</p>
<p>$(1!)^2 \ge 1^1$</p>
<p>$1\ge1$</p>
<h2>2°</h2>
<p>$k \ge n_0$</p>
<p>assumption: $$(k!)^2 \ge k^k$$</p>
<p>and for k+1: $$((k+1)!)^2 \ge (k+1)^{k+1}$$</p>
<p>I also noticed that: $$((k+1)!)^2 = (k!)^2 * (k+1)^2$$</p>
| Christian Blatter | 1,303 | <p>Well, it's a <em>symmetric bilinear function</em> $G:\>V\times V\to{\mathbb R}$ such that the associated quadratic form $q(x):=G(x,x)$ is positive definite. This means that
$$G(x,y)=G(y,x),\quad G(x+x',y)=G(x,y)+G(x',y),\quad G(\alpha x,y)=\alpha G(x,y), $$
and $G(x,x)>0$ for all $x\ne0$. If such a $G$ has bee... |
3,137,681 | <p>I'm trying to see if there is any way to solve for x? I'm told Option 1 is correct and I think I understand it but why is Option 2 incorrect?</p>
<p>Option 1: subtract both sides by y
xy + y -y = y - y
xy = 0
No solution as either x or y could be zero</p>
<p>Option 2: factor out y
y * (x + 1) = y
(1/y) * y * (x + ... | Michael Rozenberg | 190,319 | <p>Take <span class="math-container">$x=1$</span> and <span class="math-container">$y=0$</span>.</p>
<p>I think the right reasoning is the following.</p>
<p>We need to solve <span class="math-container">$xy=0$</span>.</p>
<p>If <span class="math-container">$y=0$</span> so we see that all real number is a root of the... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | user84413 | 84,413 | <p>Let <span class="math-container">$a_n=|r|^n$</span> for <span class="math-container">$n\ge1$</span>,
so <span class="math-container">$(a_n)$</span> is a decreasing sequence which is bounded below by zero </p>
<p>and therefore converges, so let <span class="math-container">$\displaystyle\lim_{n\to\infty}a_n=L$</spa... |
2,076,910 | <p>proving $\displaystyle \binom{n}{0}-\binom{n}{1}+\binom{n}{2}+\cdots \cdots +(-1)^{\color{red}{m}-1}\binom{n}{m-1}=(-1)^{m-1}\binom{n-1}{m-1}.$</p>
<p>$\displaystyle \Rightarrow 1-n+\frac{n(n-1)}{2}+\cdots \cdots (-1)^{n-1}\frac{n.(n-1)\cdot (n-2)\cdots(n-m+2)}{(m-1)!}$</p>
<p><strong>Added</strong></p>
<p>writti... | Brian M. Scott | 12,042 | <p>Extended HINT: The result is incorrect as originally stated; it should read</p>
<p>$$\binom{n}0-\binom{n}1+\binom{n}2-\ldots+(-1)^{\color{crimson}m-1}\binom{n}{m-1}=(-1)^{m-1}\binom{n-1}{m-1}\;,$$</p>
<p>or, more compactly,</p>
<p>$$\sum_{k=0}^{m-1}(-1)^k\binom{n}k=(-1)^{m-1}\binom{n-1}{m-1}\;.\tag{1}$$</p>
<p>F... |
47,188 | <p>I am an amateur mathematician, and I had an idea which I worked out a bit and sent to an expert. He urged me to write it up for publication. So I did, and put it on arXiv. There were a couple of rounds of constructive criticism, after which he tells me he thinks it ought to go a "top" journal (his phrase, and he wen... | Andrew D. King | 4,580 | <p>It certainly does happen that amateur mathematicians publish articles in good journals. Unfortunately the good amateur mathematicians have the burden of distinguishing themselves from the crackpots.</p>
<p>Without more information it is hard to give advice. Everybody's ideas of a good paper and a top journal are ... |
2,940,649 | <p>solve <span class="math-container">$$\frac{|x|-1}{|x|-3} \ge 0$$</span></p>
<p>here <span class="math-container">$x$</span> is not equal to <span class="math-container">$3$</span> and <span class="math-container">$-3$</span>.</p>
| Andrei | 331,661 | <p>You are oscillating between the <strong>meaning</strong> of having some particular color of stripes. You have the following categories: a) blue stripes only, b) red stripes only, c) both blue and red. In addition, in the problem appear d) blue stripes which is a+c, and e) red stripes, which is b+c.</p>
<p>Now look ... |
81,348 | <p>There is a well-known proof of the Compactness Theorem in propositional logic which uses the compactness of the space $\{0,1\}^P$, where $P$ is the set of propositional variables in consideration. In general, this compactness relies on the Tychonoff theorem which in turn requires the Axiom of Choice. Let me sketch i... | Buschi Sergio | 6,262 | <p>In P. Johnstone "Notes on Logic and Set theory" there is a proof of completeness (and then compactness) of propositional logic for a countable set of base atomic proposition, avoiding the choise axiom (in the exercise Johnstone prove it in general by Zorn Lemma).
If you want I send you the outline of the proof. \</p... |
1,912,628 | <p>Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!</p>
| Reveillark | 122,262 | <p>First consider the set:</p>
<p>$$A=\left\{ \frac{1}{n}:n\in\mathbb{Z}^+\right\}\cup\{0\}$$</p>
<p>For each $n>1$, take a sequence $\{\varepsilon_k^n\}_{k=1}^\infty$ such that $\frac{1}{n}<\varepsilon^n_k<\frac{1}{n-1}$, and $\varepsilon_k^1>1$ such that </p>
<p>$$\lim_{k\to\infty}\varepsilon_k^n=\frac... |
1,912,628 | <p>Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!</p>
| Brian M. Scott | 12,042 | <p>HINT: Start with $\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$, and figure out how to add for each $n\in\Bbb Z^+$ a sequence between $\frac1{n+1}$ and $\frac1n$ converging downwards to $\frac1{n+1}$.</p>
|
1,912,628 | <p>Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!</p>
| DanielWainfleet | 254,665 | <p>Let $a_0=0$ and let $(a_n)_{n\in N}$ be a strictly increasing sequence of members of $(0,1)$ with $\lim_{n\to \infty}a_n=1.$</p>
<p>For each $n\in N$ (i.e. $n>0$) let $(b_{n,j})_{j\in N}$ be a strictly increasing sequence of members of $[a_{n-1},a_n)$ with $b_{n,1}=a_{n-1}$ and $\lim_{j\to \infty}b_{n,j}=a_n.$ <... |
535,226 | <p>I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq N$ the sequence is monotonically increasing or decreasing). However, I'm having a lot of trouble finding a solution th... | Noor | 495,811 | <p>To show that <span class="math-container">$a_n$</span> is unbounded, we can show that, <span class="math-container">$a_n > n$</span> for all <span class="math-container">$n$</span>. And since <span class="math-container">$n$</span> is unbounded by Archimedes, <span class="math-container">$a_n$</span> must also be... |
323,781 | <p>In the <a href="https://arxiv.org/abs/1902.07321" rel="noreferrer">paper</a> by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in <a href="https://www.pnas.org/content/early/2019/05/20/1902572116" rel="noreferrer">PNAS</a>) the abstract includes</p>
<blockquote>
<p>In the... | Michael Griffin | 61,910 | <p>Just for clarification, what do you mean by the horizontal distribution? Are you asking about the distribution of the real parts of the zeros of <span class="math-container">$\zeta'(z):=\frac{\operatorname{d}}{\operatorname{d}s}\zeta(s)$</span>? In that case, the answer is no. It is known that <span class="math-cont... |
868,384 | <p>Let's assume I have a set like $S = \{2,1,3,4,8,10\}$. </p>
<p>What's the math notation for the smallest number in the set? </p>
| DiegoMath | 64,437 | <p>The notation you looking for is:
$$\min$$</p>
<p>Suppose you have a ordinary finite set $A=\{a_1,\ldots,a_k\}$, then you can write the minimum notation as follows:</p>
<p>$$\min\{a_1,\ldots,a_k\}$$</p>
<p>In your case,</p>
<p>$$\min\{2,1,3,4,8,10\}=1$$</p>
<p>In case of functions, you can represent its minimum ... |
40,463 | <p>If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it ... | Jack Lee | 6,751 | <p>No rhombus on the round sphere is rigid, if we define a rhombus $ABCD$ to be a union of four geodesic segments $\overline{AB}\cup \overline{BC}\cup \overline{CD}\cup \overline{DA}$, such that $A$, $B$, $C$, and $D$ are all distinct points, no three of which are collinear, and such that all four segments have the sam... |
3,690,076 | <p>Is there a rigorous proof that <span class="math-container">$|G|=|\text{Ker}(f)||\text{Im}(f)|$</span>, for some homomorphism <span class="math-container">$f\,:\,G\rightarrow G'$</span>? Can anyone provide such a proof with explanations?</p>
| Maryam | 626,408 | <p>If <span class="math-container">$f:G\to H$</span> is a group homomorphism, then the subgroup <span class="math-container">$\operatorname{Im}(f)$</span> of <span class="math-container">$G'$</span> is isomorphic to the factor group <span class="math-container">$G/\operatorname{Ker}(f)$</span>. Therefore, the order of ... |
4,486,956 | <p>If <span class="math-container">$f$</span> is a convex increasing function, then I need to show that
<span class="math-container">$$ \lim_{x\rightarrow -\infty} \frac{1}{x}\int_x^{x_0}f(y)dy=-\lim_{x\rightarrow -\infty}f(x).$$</span></p>
<p>I can prove this equality using L'Hopital's rule and the Fundamental Theorem... | Cauchy's Sequence | 744,327 | <p>I think the limit on the right should have a negative sign in front of it. I didn't need the convexity assumption for this proof. Let <span class="math-container">$L = \lim_{x \to -\infty}f(x)$</span> and <span class="math-container">$g(x) = f(x) - L$</span>. Suppose <span class="math-container">$L$</span> is finite... |
694,090 | <p>Let $V$ and $W$ be vector spaces over $\Bbb{F}$ and $T:V \to W$ a linear map. If $U \subset V$ is a subspaec we can consider the map $T$ for elements of $U$ and call this the restriction of $T$ to $U$, $T|_{U}: U \to W$ which is a map from $U$ to $W$. Show that</p>
<p>$$\ker T|_{U} = \ker T\cap U.$$</p>
<p>I know ... | hmmmm | 18,301 | <p>We know that $kerT=\{v\in V| f(v)=0\}$ </p>
<p>So that $kerT|_U=\{v\in U|f(v)=0\}$</p>
<p>So if $v\in ker T|_U$ then $v\in U$ and $v\in ker T$ which shows that $kerT|_U \subset ker T \cap U$</p>
<p>Now if we take $v\in ker T \cap U$ then $v\in U$ and $v\in ker T$ and so $v\in ket T|_U$ which gives $KerT\cap U\sub... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | Carlo Beenakker | 11,260 | <p>If you have numerical evidence in support of the conjecture, the journal of <A HREF="https://en.wikipedia.org/wiki/Experimental_Mathematics_(journal)" rel="noreferrer">Experimental Mathematics</A> seems to fit the bill:</p>
<blockquote>
<p>Experimental Mathematics publishes original papers featuring formal
resu... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | bof | 43,266 | <p>The editorial policy of the <a href="http://ulamquarterly.dyndns.org/editor.html" rel="noreferrer"><em>Ulam Quarterly Journal</em></a> claims that the journal is "devoted to the publication of original research <strong>and open problems</strong> in all areas of mathematics."</p>
|
325,519 | <p>I'm reading a book about inverse analysis and trying to figure out how the authors do the inversion.</p>
<p>Assume that matrix $C$ is
$$
C
~=~
\begin{bmatrix}
88.53 & -33.60 & -5.33 \\
-33.60 & 15.44 & 2.67 \\
-5.33 & 2.67 & 0.48
\end{bmatrix}
$$
and at some point authors d... | Amzoti | 38,839 | <p>Have you heard of <a href="http://en.wikipedia.org/wiki/Jordan_normal_form" rel="nofollow"><em>Jordan Normal Form</em></a></p>
<p>For your matrix:</p>
<p>$$C=\left(\begin{matrix}
88.53 & -33.60 &-5.33\\
-33.60 & 15.44 & 2.67\\
-5.33 & 2.67 & 0.48
\end{matrix}\right)$$</p>
<p>We would find ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.