qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,335,698 | <p>For this problem do I use the distance formula that I would use between two regular points? </p>
<p>$d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}$</p>
<p>The distance between points $u$ and $v$ on the $x$-axis is given by $|u-v|$. Solve $|x-5|+|x-6|=1$ (think geometrically).</p>
| Steven Alexis Gregory | 75,410 | <p>Write it as $|5-x|+|x-6|=|5-6|$</p>
<p>If $d(x,y)$ means the distance between points $x$ and $y$ on the number line, then this can be interpreted as $d(5,x) + d(x,6) = d(5,6)$. Which means that x is between $5$ and $6$. That is $5 \le x \le 6$.</p>
<p><img src="https://i.stack.imgur.com/bcGpj.jpg" alt="betweenness... |
1,937,826 | <p>Ok, this seems obvious to me, but how would one prove it?</p>
<p>Let $<f(t),g(t)>$ and $<h(t),p(t)>$ be parametrized arcs in the cartesian plot. If $f,g,h,p$ are all continous and the arcs don't intersect, then there will be a line between the two that will be the shortest distance. Prove this line is n... | dxiv | 291,201 | <p>(The following is more of a comment on the context and intuition behind the result, rather than a formal answer - which has been largely provided by @cjackal.)</p>
<p>Starting with a restatement of the problem: let $C_1, C_2$ be two smooth curves in the $2D$ plane. Assume that, among all segments $AB$ with $A \in C... |
1,392,257 | <p><strong>The definition of a conjugate element</strong> </p>
<p>We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$</p>
<p>Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = \big<a,b: a^4 =1,b^2 = a^2, b^{-1}ab = a^{-1} \big>$$</p>
<p>Now the elements of $Q_8$ a... | Nicky Hekster | 9,605 | <p>There is a connection to character theory of finite groups: a group is said to be <em>ambivalent</em> if every element is conjugate to its inverse. For a finite group, this is equivalent to every character of the group over complex numbers, being <em>real-valued</em>. It is easily seen that all symmetric groups $S_n... |
2,883,370 | <p>If I want to determine whether a sequence, ${a_n}$, is bounded above $\forall n \in \Bbb{N} $, is it enough to find a sequence that is larger than $a_n$, and show that it converges and is therefore bounded? For example:</p>
<p>$\forall n \in \Bbb{N}, let,$</p>
<p>$$
a_n = \frac{1}{n+1} + \frac{1}{n+2} + ...+\frac{... | Babelfish | 485,123 | <p>Yes, this is enough. If $(b_n)$ dominates $(a_n)$ and $b_n \rightarrow b$, then there is $S\in \mathbb R$ with $b_n<S$ for all $n \in \mathbb N$. Since $(b_n)$ dominates, we get $S>b_n \geq a_n$, so $(a_n)$ is bounded as well.</p>
|
61,106 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be Poisson random variables with means <span class="math-container">$\lambda$</span> and <span class="math-container">$1$</span>, respectively. The difference of <span class="math-container">$X$</span> and <span class="math-cont... | Ori Gurel-Gurevich | 1,061 | <p>Regarding the asymptotic behavior when $\lambda \to \infty$:</p>
<p>To get an estimate one simply finds that the dominating element among
$$A_k= \mathbb{P}(Y=k+1)\mathbb{P}(X=k) = e^{-1-\lambda} \frac{\lambda^k}{k!(k+1)!}$$
is $A_{\sqrt{\lambda}}$ which gives roughly $e^{2\sqrt{\lambda}-\lambda}$. Probably there's ... |
3,896,817 | <blockquote>
<p>Solve <span class="math-container">$x^2 \equiv 12 \pmod {13}$</span></p>
</blockquote>
<p>By guessing I can say that the solutions are <span class="math-container">$5$</span> and <span class="math-container">$8$</span>, but is there another way to find the solution besides guessing?</p>
| CopyPasteIt | 432,081 | <p>In the field <span class="math-container">${\displaystyle \mathbb {Z} /13\mathbb {Z}}$</span>, <span class="math-container">$\,[1] + [1] \ne [0]$</span>, and therefore there are either zero or two distinct <span class="math-container">$\text{modulo-}13$</span> solutions for,</p>
<p><span class="math-container">$\tag... |
2,732,562 | <p>I'm trying to figure out how to find the number of ternary strings of length $n$ that have 3 or more consecutive 2's.
So far I've been able to establish that there is $n(2^{n-1})$ with a single 2.
And I think (but am not certain) that this can be extrapolated to the number of strings with a single group of 2's of le... | G Cab | 317,234 | <p>Following another approach,
consider a binary word, where the one stands for the ternary $2$ and the zero stands for $0,1$.</p>
<p>Take a binary word of length $n$, having $s$ ones and $n-s$ zeros in total, with <em>no more than</em> $r$ consecutive ones.</p>
<p>Now the number of such binary words is given by<b... |
789,458 | <p>If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire <em>truly random</em> number generator?</p>
| Asimov | 137,446 | <p>Not random. Never random. Evenly distributed, and pseudo-random yes. Pi is defined before you calculate it, so its not random, just unknowable before you calculate each digit. If you pick digits that have not been calculated or formulated (or just not know to you) then it is for all practical purposes it is random. ... |
1,712,289 | <p>If from twice the greater of two numbers 17 is subtracted, the result is half the other number. If from half the greater number 1 is subtracted, the result is two-thirds of the smaller number.</p>
<p>$$2x - 17 = \frac{ 1 }{2}y$$</p>
<p>$$\frac{ x }{2} - 1 = \frac{ 2 }{3}y$$</p>
<p>$$-17 - 4 = \frac{ 1 }{2}y - \fr... | Brian Tung | 224,454 | <p><strong>Basic approach.</strong> Your equations look right to me, except that it should be $x/2$ in the second equation, not $x$.</p>
<p>Now, if you multiply both sides of the upper equation by $2$, and both sides of the lower equation by $3$, you will get two expressions for $y$ in terms of $x$ that can be equated... |
2,653,708 | <p>In this question I am using the euclidean metric to determine the distance between two points.</p>
<p>I want to make a function $f(x)=$ the minimum distance between $y=x$ and $y=e^x$ at each given point x, is there an efficient way of doing this?</p>
<p>Second related question if i knew $e^x$ was the shortest dist... | BoolHool | 532,226 | <p>Okay so, for your first question, all the points in the form of the line $y=x$ is in the form $(a,a)$ (So this is our best alternative, keep in mind however that the $x$'s in $y=x$ is not the same as the $x$'s in $y=e^x$, as discussed in the comments). As you mentioned, you are using the Euclidean Metric. So we need... |
3,016,169 | <p>I want to prove or disprove that <span class="math-container">$C^1([a,b], \mathbb{R}^n)$</span> equipped with the norm <span class="math-container">$||x||=\underset{t\in[a,b]}{\sup}|x(t)|_{\mathbb{R}^n}+\underset{t\in[a,b]}{\sup}|\dot{x}(t)|_{\mathbb{R}^n}$</span> is a reflexive Banach space. </p>
<p>I figured out ... | Nate Eldredge | 822 | <p>There's a standard theorem that for any Banach space <span class="math-container">$X$</span>, if <span class="math-container">$X'$</span> is separable then so is <span class="math-container">$X$</span>. Hence, if <span class="math-container">$X''$</span> is separable then so is <span class="math-container">$X'$</sp... |
222,555 | <p>I would like to find a simple equivalent of:</p>
<p>$$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $$</p>
<p>We have:</p>
<p>$$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$$</p>
<p>So $$ u_{n} \rightarrow 0$$</p>
<p>Clearly:</p>
<p>$$ u_{n} \sim \frac{1}{n!} \int_{\sin(1)}^1 (... | Did | 6,179 | <p>The change of variable $x=\cos\left(\frac{\pi s}{2n}\right)$ yields
$$
u_n=\frac1{n!}\left(\frac\pi2\right)^{n+2}\frac1{n^2}v_n,
$$
with
$$
v_n=\int_0^n\left(1-\frac{s}n\right)^n\,\frac{2n}\pi \sin\left(\frac{\pi s}{2n}\right)\,\mathrm ds.
$$
When $n\to\infty$, $\left(1-\frac{s}n\right)^n\mathbf 1_{0\leqslant s\leqs... |
2,698,121 | <p>Came across a question about CDF and PDF in my homework:</p>
<p><a href="https://i.stack.imgur.com/bYqvr.jpg" rel="nofollow noreferrer">Click here for the question</a></p>
<p><a href="https://i.stack.imgur.com/bYqvr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bYqvr.jpg" alt="enter image desc... | Siong Thye Goh | 306,553 | <p>a) </p>
<p>if $x \in [1,3]$,</p>
<p>\begin{align}Pr(X \le x) &= \int_{-\infty}^xf(t) \, dt \\
&=\int_0^1f(t) \, dt + \int_1^x f(t) \, dt\\
&= \int_0^1 \frac34 \, dt + \int_1^x 0 \, dt \\
&= \frac34\end{align}</p>
<p>Geometrically just find the area under the graph of the density when $X \le x$.</p... |
1,089,635 | <blockquote>
<p>Is it possible to display a result in Mathcad as a function of $\pi$? </p>
</blockquote>
<p>I'm studying physics and I need to show exact results at the exams. I know I can set Mathcad to give me the result in decimals or in fractions, but none of them are good enough. </p>
<p><strong>Example: calcu... | Community | -1 | <p>Solution (found by the OP): one can use "CTRL + ." instead of " = " in Mathcad. That gives the desired form of result.</p>
|
4,164,553 | <p>Can anybody enlighten me about the applications of <a href="https://en.wikipedia.org/wiki/Intuitionistic_logic" rel="noreferrer">intuitionistic logic</a>? I am familiar with this system only by <a href="https://www.maa.org/publications/maa-reviews/proof-theory" rel="noreferrer">G.Takeuti's book</a>, where it is desc... | HallaSurvivor | 655,547 | <p>One fun answer is "programming languages". It turns out a good way to study programming languages is by viewing them as proof systems, and the programming languages you get in this way tend to be intuitionistic proof systems. You <em>can</em> build programming languages which correspond to classical logic ... |
71,608 | <p>Consider the following question:</p>
<p>Is there a family $\mathcal{F}$ of subsets of $\aleph_\omega$ that satisfies the following properties?</p>
<p>(1) $|\mathcal{F}|=\aleph_\omega$</p>
<p>(2) For all $A\in \mathcal{F}$, $|A|<\aleph_\omega$</p>
<p>(3) For all $B\subset \aleph_\omega$, if $|B|<\aleph_\om... | Andy Voellmer | 14,794 | <p>I think the following diagonalization will show that there is no such set $\mathcal{F}$.</p>
<p>Suppose there were such an $\mathcal{F}$. Then we could split it up into $\omega$ many chunks $( \mathcal{F}_i ) _{i \in \omega} $ such that each $\mathcal{F} _i$ had exactly the sets of size $\aleph_i $ or smaller that... |
1,465,229 | <p>At p. 388 of Calculus, Spivak gives a formula:</p>
<p>$$\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$</p>
<p>Which can be integrated to find $\arctan(x)$.</p>
<p>I don't understand where this formula comes from, but I found it up to $(-1)^nt^{2n}$ by considering the geo... | skyking | 265,767 | <p>The term $(-x^2)^{n+1}/(1+x^2)$ is just the rest term:</p>
<p>$${1\over 1 - (-x^2)} = \sum_0^\infty (-x^2)^k = \sum_0^n (-x^2)^k + \sum_{n+1}^\infty (-x^2)^k$$</p>
<p>where</p>
<p>$$\sum_{n+1}^\infty (-x^2)^k = (-x^2)^{n+1} \sum_0^\infty (-x^2)^k = {(-x^2)^k\over1 - (-x^2)}$$</p>
<p>Alternately you can do it the... |
1,465,229 | <p>At p. 388 of Calculus, Spivak gives a formula:</p>
<p>$$\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$</p>
<p>Which can be integrated to find $\arctan(x)$.</p>
<p>I don't understand where this formula comes from, but I found it up to $(-1)^nt^{2n}$ by considering the geo... | egreg | 62,967 | <p>Consider
$$
1+x+x^2+\dots+x^n=\frac{1-x^{n+1}}{1-x}=\frac{1}{1-x}-\frac{x^{n+1}}{1-x}
$$
that can also be written
$$
\frac{1}{1-x}=1+x+x^2+\dots+x^n+\frac{x^{n+1}}{1-x}
$$
Now substitute $x=-t^2$, that gives
$$
\frac{1}{1+t^2}=1+(-t^2)+(-t^2)^2+\dots+(-t^2)^n+\frac{(-t^2)^{n+1}}{1+t^2}
$$
and not it's just a matter ... |
3,055,272 | <p><strong>Background</strong></p>
<p>A connected graph has an <em>Eulerian circuit</em> if every vertex has even degree. </p>
<p>I am thinking about a certain classification of connected graphs where, for a connected graph <span class="math-container">$G$</span>, every <a href="https://en.wikipedia.org/wiki/Cut_(gra... | Peter Taylor | 5,676 | <p>Given: <span class="math-container">$\forall v \in V: \textrm{deg}(v) \equiv 0 \pmod 2$</span>. Partition <span class="math-container">$V$</span> into <span class="math-container">$S$</span> and <span class="math-container">$T$</span>. We desire to show that the number of edges split by the cut is even.</p>
<p>By i... |
1,150,805 | <p>An unfair 3-sided die is rolled twice. The probability of rolling a 3 is $0.5$, the probability of rolling a 1 is $0.25$, and the probability of rolling a 2 is $0.25$. Let $X$ be the outcome of the first roll and $Y$ the outcome of the second.</p>
<ul>
<li><p>Find the Joint Distribution of $X$ and $Y$ in a Table.</... | Asaf Karagila | 622 | <p>For induction you have to define some explicit base case, what is the smallest finite set? The empty set. Define its power set explicitly.</p>
<p>Now suppose that you defined the power set of a set with $n$ elements, and $A=\{a_0,\ldots,a_n,a_{n+1}\}$, find a way to write $A$ as the union of $A'$ and a singleton, w... |
4,036,049 | <p>In order for <span class="math-container">$n^5 - n$</span> divisible by <span class="math-container">$5$</span>, <span class="math-container">$n^5 - n = 5 x + 0$</span> (for some <span class="math-container">$x$</span>, <span class="math-container">$x$</span> is a natural number)
I simplified the <span class="math-c... | Deepak | 151,732 | <p>You're solving a cubic here. While there is a general solution, it's no picnic. For most "elementary" (school-style) problems, you won't be expected to apply the full cubic solution. Instead you'll be expected to apply some combination of Rational Root Theorem, Factor Theorem and Polynomial Long Division (... |
4,036,049 | <p>In order for <span class="math-container">$n^5 - n$</span> divisible by <span class="math-container">$5$</span>, <span class="math-container">$n^5 - n = 5 x + 0$</span> (for some <span class="math-container">$x$</span>, <span class="math-container">$x$</span> is a natural number)
I simplified the <span class="math-c... | Barry Cipra | 86,747 | <p>This is mainly an extended comment on the answers that have already been posted, but it may be of help in approaching problems of this type, where you can assume the problem poser has carefully constructed things to have a nice answer.</p>
<p>As the other answers observe, the cubic simplifies initially to</p>
<p><sp... |
1,627,713 | <p>This is maybe math $101$ question:</p>
<p>Let $z_1=1+i$.</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$</p>
<p>But now if I take a look at</p>
<p>$z_2=-1-i$,</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(-1/-1)=\pi/4$ so $$z_1=\color{blue}{\sqrt ... | Neal | 20,569 | <p>That's actually correct. For any $a,b\in\Bbb{R}^2 - \Bbb{E}$, you need to find such a path. My hint to finding such a path is, try to think about what the space $\Bbb{R}^2 - \Bbb{E}$, the points with at least one irrational coordinate, might look like. Then see if you can visualize how you might go from one point wi... |
827,154 | <p>I need help with the definition of "within 1":</p>
<ul>
<li><p>If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. </p></li>
<li><p>If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$.</p></li>
<li><p>If $x = 8$ and $y = 8$, is $x$ still "within 1" of $y$?</p></li>
</ul>
<p>It's my understanding that this wou... | Earl Grey | 193,393 | <p>I have come across a similar query around the definition of "within". The dictionary definition (from www.oed.com) is "That which is within or inside".<br>
Given the "or" in the definition the "within 1 of" includes the value 1.</p>
|
878,373 | <p>Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics.</p>
<p>I have a good preparation in Algebra and Representation Theory (in particular about Representations of Lie Algebras), and I'm fascinated with Physics. My idea is t... | Salix Liu | 834,147 | <p>If your <span class="math-container">$f$</span> is continuous differentiable then this ito integral is a pathwise RS integral then of course you can find a bound</p>
|
118,298 | <p>I'm trying to work through "Elements of Functional Languages" by Martin Henson. On p. 17 he says:</p>
<blockquote>
<p>$v$ occurs free in $v$, $(\lambda v.v)v$, $vw$ and $(\lambda w.v)$ but not in $\lambda v.v$ or in $\lambda v.w$. And $v$ occurs bound in $\lambda v.v$ and in $(\lambda v.v)v$ but not in $v$, $vw$ ... | Community | -1 | <p>Since you implied you're comfortable on the computer side of the house... it's talking about the scope of a variable. "$\lambda x.$" introduces a new scope which lasts for the length of the lambda expression, and $x$ is a local variable in that scope.</p>
<p>A free variable is one not local to the expression. e.g. ... |
2,010,069 | <p>I am looking on the solution to this problem presented in the book <em>"Fifty Challenging Problems in Probability with Solutions"</em> by Mosteller (p.18-19).</p>
<blockquote>
<p>On average, how many times must a die be thrown until one gets a 6?</p>
</blockquote>
<p>There are many ways to solve this problem as... | user26872 | 26,872 | <p>We have
$x = x_i e_i = x'_i e'_i$
where $e_i$ and $e'_i$ are bases related by nonsingular linear transformation.
Note that
${e'}_i^T e'_j = g_{ij}$,
where $g$ is invertible.
Thus,
${e'}_i^T e_j x_j = {e'}_i^T e'_j x'_j = g_{ij}x'_j$
or
$$x'_i = (g^{-1})_{ij}{e'}_j^T e_k x_k
= (g^{-1})_{ij}{e'}_j^T x.$$
This give... |
2,010,069 | <p>I am looking on the solution to this problem presented in the book <em>"Fifty Challenging Problems in Probability with Solutions"</em> by Mosteller (p.18-19).</p>
<blockquote>
<p>On average, how many times must a die be thrown until one gets a 6?</p>
</blockquote>
<p>There are many ways to solve this problem as... | David Reed | 444,890 | <p>After reading your question further, I believe I misunderstood initially what you were asking.</p>
<p>In terms of orthogonality, what it tells you is that the row space is the orthogonal complement to the Nul space. Thus every vector can be written uniquely as the sum of a vector in the row space of A and in the Nu... |
119,481 | <p>I need to prove the following trigonometric identity:
$$ \frac{\sin^2(\frac{5\pi}{6} - \alpha )}{\cos^2(\alpha - 4\pi)} - \cot^2(\alpha - 11\pi)\sin^2(-\alpha - \frac{13\pi}{2}) =\sin^2(\alpha)$$</p>
<p>I cannot express $\sin(\frac{5\pi}{6}-\alpha)$ as a function of $\alpha$. Could it be a textbook error?</p>
| Blue | 409 | <p>Since all the trig values are squared, it seems as though the exercise is simply playing with shifts by odd or even multiples of $\pi/2$.</p>
<p>Loosely,</p>
<blockquote>
<ul>
<li>Shifting by "$\frac{\pi}{2} \cdot \text{odd}$" switches "sin" and "cos" (and possibly affects the sign)</li>
<li>Shifting by "$\f... |
2,028,703 | <p>I'm having this example for a simple <a href="https://en.wikipedia.org/wiki/Binary_symmetric_channel" rel="nofollow noreferrer">binary symmetric channel</a> (BSC) to bound the mutual information of $X$ and $Y$ as</p>
<p>\begin{align*}
I(X;Y) &= H(Y) - H(Y|X)\\
&= H(Y) - \sum p(x) H(Y \mid X = x) \\
&= H... | disaster | 280,592 | <p>It seems to me that you are confusing $H(X) = - \sum_i Pr(x_i) \log ( Pr(x_i) )$ (the entropy) and $H(p) = -p \log_2(p) - (1-p) \log_2(1-p)$, which is called the binary entropy function.</p>
<p>The difference is whether the input is a random variable or a probability.</p>
<p>Note that $H(\frac 1 3) = H(\frac 2 3)$... |
163,873 | <p>With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:</p>
<ol>
<li><p>Those that guarantee the existence of more complicated sets, given that simpler sets are already around (e.g. separation, replacement schema). Also, uniqueness of these entitie... | Andreas Blass | 6,794 | <p>I like to use the following very mild extension of ZFC. Add a predicate "Sat"; add axioms saying that this predicate satisfies the expected inductive clauses to define "satisfaction of $\in$-formulas in the full universe of sets"; and add replacement (or separation and collection) axioms for formulas in which Sat oc... |
1,476,313 | <p>I want to simplify this fraction</p>
<p>$$ \frac{\sqrt{6} + \sqrt{10} + \sqrt{15} + 2}{\sqrt{6} - \sqrt{10} + \sqrt{15} - 2} $$</p>
<p>I've tried to group up the denominator members like $ (\sqrt{6} + \sqrt{15}) - (\sqrt{10} + 2) $ and then amplify with $ (\sqrt{6} + \sqrt{15}) + (\sqrt{10} + 2) $ </p>
| mathlove | 78,967 | <p>HINT : </p>
<p>$$\sqrt 6\pm\sqrt{10}+\sqrt{15}\pm 2=\sqrt 3(\sqrt 5+\sqrt 2)\pm\sqrt{2}(\sqrt 5+\sqrt 2)$$
$$=(\sqrt 5+\sqrt 2)(\sqrt 3\pm\sqrt 2)$$</p>
|
1,346,286 | <p>Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? </p>
<p>At the very best, I know that the area $\sin x$ covers between $0$ to $\pi\over 2$ has the same magnitude bet... | Mark Viola | 218,419 | <p>The integral of interest $\int_0^{\pi}\frac{dx}{1-\sin x}$ does not converge. Rather, it diverges since </p>
<p>$$1-\sin x=-\frac12 (x-\pi/2)^2+O\left((x-\pi/2)^4\right)$$</p>
<p>Even if we interpret the integral in the sense of a Cauchy Principal Value, then we have</p>
<p>$$\begin{align}
\text{P.V.}\int_0^{\pi... |
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Community | -1 | <p>Anyone will understand</p>
<p>$$n\in\{1,2,\dots50\}$$
or even</p>
<p>$$n\in\{1,\dots50\}$$
without toil.</p>
<p>If it is clear from context that $n$ is an integer,</p>
<p>$$n\in[1,50]$$ is good enough (and is very compact from the standpoint of LaTeX formatting :) ).</p>
<p>Following @EspeciallyLime, $[50]$ i... |
2,567,607 | <p>$$\arctan 2x +\arctan 3x = \left(\frac{\pi}{4}\right)$$
$$\arctan \left(\frac{2x+3x}{1-2x*3x}\right)=\frac {\pi}{4}$$
$$\frac {5x}{1-6x^2}=\tan \frac{\pi}{4}=1$$
$$6x^2 + 5x -1 = 0$$
$$(6x-1)(x+1)=0$$
$$x=-1, \frac{1}{6}$$</p>
<p>The answer however rejects the solution $x=-1$ saying that it makes the L.H.S of the e... | Botond | 281,471 | <p>$\tan(x)$ is negative in the interval $\left(-\frac{\pi}{2},0\right)$, so it's inverse on the interval $(-\infty,0)$ will give a value from the interval $\left(-\frac{\pi}{2},0\right)$.</p>
|
1,307,269 | <p>In my textbook, before introducing the epsilon delta definition, they gave a working definition of what a limit is. The definition sounded something like this "$\lim \limits_{x \to a}f(x) = L$, if when $x$ gets closer to $a$, $f(x)$ gets closer to $L$" </p>
<hr>
<p>But is that always the case with limits? What if... | Michael Hardy | 11,667 | <blockquote>
<blockquote>
<p>when $x$ gets closer to $a$, $f(x)$ gets closer to $L$</p>
</blockquote>
</blockquote>
<p>That is wrong. Consider two examples:</p>
<ul>
<li><p>Let $f(x) = 6-(x-4)^2$. Clearly $f(x)$ never gets bigger than $6$, so the limit cannot be $7$, but $f(x)$ gets closer to $7$ (and to al... |
1,307,269 | <p>In my textbook, before introducing the epsilon delta definition, they gave a working definition of what a limit is. The definition sounded something like this "$\lim \limits_{x \to a}f(x) = L$, if when $x$ gets closer to $a$, $f(x)$ gets closer to $L$" </p>
<hr>
<p>But is that always the case with limits? What if... | CiaPan | 152,299 | <p>The 'working definition' is actually NOT working and you shoud NOT use it. To make it work replace it with </p>
<blockquote>
<p>$f$ gets <strong>arbitrarily</strong> close to $L$ if $x$ <em>sufficiently</em> approaches $a$</p>
</blockquote>
<p>where 'gets arbitrarily close' does not mean just $f$ <em>may</em> g... |
3,047,686 | <p>Prove that: <span class="math-container">$$(p - 1)! \equiv p - 1 \pmod{p(p - 1)}$$</span></p>
<p>In text it's not mentioned that <span class="math-container">$p$</span> is prime, but I checked and this doesn't hold for non-prime, so I guess <span class="math-container">$p$</span> is prime ..
I know that <span class... | user10354138 | 592,552 | <p>Apply <a href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem" rel="nofollow noreferrer">Chinese remainder theorem</a> to the coprime moduli <span class="math-container">$p$</span> and <span class="math-container">$p-1$</span>.</p>
<p>However, if you can't use CRT, you can simply observe <span class="math-... |
3,929,089 | <p>In the construction of <span class="math-container">$\operatorname{Frac}(R)$</span>, where <span class="math-container">$R$</span> is a domain, we define a partition on <span class="math-container">$R \times R^\times$</span> where <span class="math-container">$R^\times:= R \setminus \{0\}$</span>. which in turn beco... | Vercassivelaunos | 803,179 | <p>Your understanding has a gap starting at the ring of polynomials <span class="math-container">$\mathbb Q[X]$</span>, not at the field of fractions. So I'm going to focus on the former, not the latter.</p>
<p>To start with, polynomials are <em>not</em> functions. Memorize this sentence and never forget it: Polynomial... |
422,761 | <blockquote>
<p>Prove that there is no Integer such that $x≡2 \pmod 6$ and $x≡3 \pmod 9$ are both true.</p>
</blockquote>
<p>How should I approach this question?<br>
I attempted using contra-positive proof, so $x=6p+2$ and $x=9q+3$ where $p,q$ are integers.<br>
Then $6p+2=9q+3$. </p>
| rurouniwallace | 35,878 | <p>I'll go ahead and answer what I think you were <em>trying</em> to ask. You asked if the series converges. The series you typed is a finite series, but if it were an infinite series it would diverge because:</p>
<p>$$\lim_{n\to\infty}\sum_{k=0}^n{\ln^n(3)}=\lim_{n\to\infty}n\ln^n(3)$$</p>
<p>Also, if the series wer... |
2,349,982 | <p>According to the <a href="http://www.fftw.org/fftw3_doc/1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" rel="nofollow noreferrer">FFTW Website</a>, the Fourier Sine Transform (FST) returns:</p>
<p>$$Y_k = 2 \sum_{j=0}^{N-1} X_i \sin [\pi (j+1)(k+1)/(N+1)]$$</p>
<p><a href="http://r... | Michael Hardy | 11,667 | <p>$\newcommand{\E}{\operatorname{E}}\newcommand{\v}{\operatorname{var}}\newcommand{\c}{\operatorname{cov}}$First, note that $h\sum_{i=1}^n x_i^2+k\left(\sum_{i=1}^nx_i\right)^2$ is merely a number, so it cannot be biased or unbiased, but $h\sum_{i=1}^n X_i^2 + k \left( \sum_{i=1}^n X_i\right)^2$ is a random variable, ... |
1,740,458 | <p>I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ <strong>without</strong> using Weierstrass substitution, which is the usual technique. </p>
<p>When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem ... | Justin Benfield | 297,916 | <p>If $a=b$ then you can modify the technique for $a=b=1$ slightly to obtain:</p>
<p>$\int \frac{dx}{b+b\cos x}=\int\frac{b-b\cos x}{(b+b\cos x)(b-b\cos x)}dx$</p>
<p>$=\int\frac{b-b\cos x}{b^2-b^2\cos^2 x}dx=\int\frac{b-b\cos x}{b^2(1-\cos^2 x)}dx=\frac{1}{b}\int\frac{1-\cos x}{\sin^2 x}dx$</p>
<p>Splitting the num... |
1,740,458 | <p>I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ <strong>without</strong> using Weierstrass substitution, which is the usual technique. </p>
<p>When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem ... | Adhvaitha | 228,265 | <p>If the integral is a definite integral (typically from $0$ to $\pi/2$ or some other variants of this), then we can <a href="https://www.dropbox.com/s/imfywebvbabebj0/1_over_a_b_sin.pdf?dl=0" rel="nofollow">follow the technique here</a> to obtain the integral. The key ingredient is to write $\dfrac1{a+b\cos(x)}$ as a... |
19,325 | <p>I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.</p>
<p>I'm hoping some Algebraist could help me properly state the following:</p>
<blockquote>
<p>A number in and of itself has no true meaning. True in the sense that it relates ... | Rusty Core | 7,930 | <p>Forget New Math, it does now work with young kids, and it is not needed for learning arithmetics in elementary school.</p>
<p>Instead, first teach the concepts of "same as", "more than", "less than" by lining up real objects like apples or backpacks or horses one against another, making... |
19,325 | <p>I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.</p>
<p>I'm hoping some Algebraist could help me properly state the following:</p>
<blockquote>
<p>A number in and of itself has no true meaning. True in the sense that it relates ... | jfkoehler | 7,003 | <p>I would suggest taking it easy to start but maybe something like base numeration is a way in here. There is no reason we have to use base 10 and students use of different bases can be very important to understanding operations that require regroupings, and that weird alignment under the typical presentation of mult... |
4,257,962 | <p>By definition - A real number is algebraic if it is a root of a non-zero polynomial equation with rational coefficients. What does non-zero polynomial equation mean?</p>
<p>Well, an equation f(x) = x -5, becomes zero when x = 5, so this is a zero polynomial equation. Is the definition saying that the equation should... | Rounak Sarkar | 831,748 | <p>A zero polynomial is a polynomial which gives <span class="math-container">$0$</span> for all values of <span class="math-container">$x$</span>,</p>
<p>Basically <span class="math-container">$P(x)=0$</span> means that no matter what <span class="math-container">$x$</span> you put in there, the result will always be ... |
3,970,959 | <blockquote>
<p><span class="math-container">$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$</span>.</p>
</blockquote>
<blockquote>
<p>Prove that <span class="math-container">$\dfrac{29}{27}<S<\dfrac{7}{6}$</span>.<br></p>
</blockquote>
<p>My Attempt:<br>
<span class="math-container... | Claude Leibovici | 82,404 | <p>We can find an <em>approximation</em> of the result.</p>
<p>Consider
<span class="math-container">$$\sum_{i=1}^{2001}\frac1{1000+i}=\sum_{i=1}^{3001}\frac1{i}-\sum_{i=1}^{1000}\frac1{i}$$</span> and we shall use the fact that
<span class="math-container">$$\sum_{i=1}^{n}\frac1{i}=H_n$$</span></p>
<p>Now, the asympto... |
3,970,959 | <blockquote>
<p><span class="math-container">$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$</span>.</p>
</blockquote>
<blockquote>
<p>Prove that <span class="math-container">$\dfrac{29}{27}<S<\dfrac{7}{6}$</span>.<br></p>
</blockquote>
<p>My Attempt:<br>
<span class="math-container... | Erik Satie | 698,573 | <p>Well not an answer but an idea to inspire someone :</p>
<p>I remenber the Gauss's solution of the first case concerning the Faulhaber's formula :</p>
<p>We have :</p>
<p><span class="math-container">$$\frac{1}{1001}+\frac{1}{3001}>\frac{1}{1002}+\frac{1}{3000}>\cdots>\frac{1}{2002}+\frac{1}{2000}$$</span></... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| J. M. ain't a mathematician | 498 | <p>The required tetrahedral angle is $\arccos\left(-\frac13\right)\approx109.5^\circ$. You can use the law of cosines to show this... or more transparently, you can exploit the fact that a tetrahedron is easily embedded inside a cube:</p>
<p><img src="https://i.stack.imgur.com/stZDK.gif" alt="tetrahedron in a cube"></... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| Narasimham | 95,860 | <p>If the tetrahedron is regular, we can use statical equilibrium of four equal isotropic forces with angle $ \theta$ between any two of them. Referencing with reference to any one force,</p>
<p>$$ F \cos \theta + .. + .. + F = 0$$</p>
<p>$$ \cos \theta = -\frac13 $$</p>
<p>To generalize for all $i$ direction forces... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| tclayton2k | 428,796 | <p>Using the illustration of the tetrahedron embedded into the cube, you can lay out some dimensions. Assign a length of 2 to the sides of the cube, and then according to an old guy name Pythagoras, the hypotenuse of each side will be d 2*√2. Since the center of the tetrahedron is also the center of the cube, then you ... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| Roy | 894,041 | <p>Another possible solution:</p>
<p>Consider the equilateral triangle, edge length x, formed by any 3 tips of a regular tetrapod (a face on it's corresponding tetrahedron).</p>
<p>The 3 medians of this equilateral triangle section it into 6 congruent right triangles with hypotenuse length y, and angles of 60° between ... |
3,001,335 | <p>I know little formal math terminology and don't understand much of anything about complex analysis. Also, if this isn't a good starting point for complex integration feel free to say (I'm learning about it partly for Cauchy's residue theorem). </p>
<p>My first and intuitive idea of residue has to do with remainder,... | Masacroso | 173,262 | <p>Yes, it is. The residue of the Laurent series of a function in an annulus is the coefficient <span class="math-container">$c_{-1}$</span> of this series, that is what is left after integration of the function in a loop on the annulus (that is, after the integration of the Laurent series that represent the function i... |
234,945 | <p>Let $a \in \mathbb R$, what values of $t$ solve the equation $at + \sin(t) = 0$?</p>
| Berci | 41,488 | <p>First, $t=0$ is a solution. Otherwise, it can be written as
$$-a=\frac{\sin t}t.$$
One can find the bounds of $\frac{\sin t}t$ so that if $-a$ is out of that range, there is no solution. But, in general, it seems only numerical solutions could be found, see also <a href="http://www.wolframalpha.com/input/?i=%28sin%2... |
2,734,338 | <p>I would like to show that
$$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$$</p>
<p>I'm interested in more ways of proofing this.</p>
<p>My method :</p>
<p>suppose that $\sqrt{\frac{n}{n+1}}\in \mathbb{Q}$ then there exist $(p,q)\in\mathbb{Z}\times \mathbb{N}^*$ such that $\sqrt{\frac{n}... | fleablood | 280,126 | <p>Finishing your proof</p>
<p>$n = \frac {p^2}{q^2 - p^2} = \frac {p^2}{(q-p)(q+p)}$</p>
<p>All prime or unitary factors of $q-p$ and of $q+p$ must be prime or unitary factors of $p$ so then must be common factors of both $p$, and $q$. But $p$ and $q$ are presumed to be relatively prime. So the only factors of $q-... |
959,219 | <p>let $a,b,c>0$, and such
$$a^2+b^2+c^2<2ab+2bc+2ca$$</p>
<p>show that
$$a^4+b^4+c^4+6(a^2b^2+b^2c^2+a^2c^2)+4abc(a+b+c)<4(ab+bc+ac)(a^2+b^2+c^2)$$</p>
<p>I know this indentity:
$$a^2+b^2+c^2-2(ab+bc+ac)
=-(\sqrt{a}+\sqrt{b}+\sqrt{c})(-\sqrt{a}+\sqrt{b}+\sqrt{c})(\sqrt{a}-\sqrt{b}+\sqrt{c})(\sqrt{a}+\sqrt{b... | Macavity | 58,320 | <p>Let $p = a+b+c, q = ab+bc + ca, r = abc$ then the inequality we need to show is equivalent to:
$$p^4+16q^2 < 8p^2q+4pr$$ </p>
<p>Using the condition, we know $p^2 < 4q$ and it is well known that $3q \le p^2$. Thus we have $(4q-p^2)(p^2-3q) \ge 0 \implies 7p^2q \ge p^4+12q^2$.</p>
<p>Using this, it is enough... |
3,836,695 | <p>Is there a way to evaluate <span class="math-container">$\int\cos^2xdx$</span> without using <span class="math-container">$\cos^2x=\frac12(1+\cos(2x))$</span>?</p>
| rash | 650,763 | <p><span class="math-container">$$I=\int cos^2 xdx= x\cos^2 x+\int x(2\sin x\cos x)dx \text{ (Integration by Parts)} $$</span>
<span class="math-container">$$=x\cos^2 x+\int x\sin 2xdx$$</span>
Applying Integration by parts for <span class="math-container">$\int x\sin 2xdx$</span>,
<span class="math-container">$$\i... |
3,836,695 | <p>Is there a way to evaluate <span class="math-container">$\int\cos^2xdx$</span> without using <span class="math-container">$\cos^2x=\frac12(1+\cos(2x))$</span>?</p>
| Aadhaar Murty | 826,105 | <p>You could use the complex definitions of sine and cosine -</p>
<p><span class="math-container">$$\cos x = \frac {e^{ix}+ e^{-ix}}{2}, \sin x = \frac {e^{ix}- e^{-ix}}{2i}$$</span></p>
<p><span class="math-container">$$I = \int \frac {e^{2ix}+e^{-2ix} + 2}{4} dx = \frac {1}{4}\left(2x + \frac {e ^{2ix}}{2i} - \frac {... |
227,873 | <p>I am looking for robust generalizations of matrix rank.</p>
<p>Think of the the following problem: A big matrix of low rank is perturbed by random noise, such that it becomes a full-rank matrix. Is there a generalization of matrix rank that still 'sees' that the perturbed matrix is close to a low-rank matrix?</p>
| Carlo Beenakker | 11,260 | <p><A HREF="http://www.sciencedirect.com/science/article/pii/016501149090208N" rel="nofollow">The rank of a fuzzy matrix and its evaluation</A></p>
<blockquote>
<p>A new type of matrix rank, which is called margin rank in this
article, is introduced to a fuzzy matrix defined to be rectangular
array of fuzzy numb... |
4,305,538 | <p>I am trying to compute the Fourier transform of <span class="math-container">$(x_{1}+ix_{2})^{-1}$</span> in <span class="math-container">$S'(\mathbb{R}^{2})$</span>. i.e. as a tempered distribution.</p>
<p>It might be useful to note that for <span class="math-container">$\mu \in S'(\mathbb{R}^{2})$</span> and <span... | Svyatoslav | 869,237 | <p>There is a comprehensive solution by @Ninad Munshi. We can also try another approach.</p>
<p>Let's denote <span class="math-container">$x=\sqrt{x_1^2+x_2^2}$</span> and <span class="math-container">$\lambda=\sqrt{\lambda_1^2+\lambda_2^2}$</span>. We consider the case <span class="math-container">$\lambda\neq0$</span... |
61,047 | <p>I can add the value of a slider to the right of it using the Appearance-->Labelled option, but what if I want to add text after the automatic label. How can I do that?</p>
<p>Normally I want to do this to show the units of the value. For example, if the slider label is "4.7", I might want it to read "4.7 meters".</... | eldo | 14,254 | <pre><code>Framed[
Row[{
"Interval",
IntervalSlider[Dynamic[m], {0, 3, 0.1}, Appearance -> "Markers"],
Dynamic[m],
" meters"}],
Background -> GrayLevel@0.9,
FrameMargins -> 15]
</code></pre>
<p><img src="https://i.stack.imgur.com/dJpfT.jpg" alt="enter image description here"></p>
<p>Thanks to ... |
1,712,256 | <p>I'm given some equations.</p>
<p>The first one, $x^3+2x^2-8x+1$ wants me to find the tangent line at $x=2$.</p>
<p>The second, (x^1.5) - (x^1/2) wants me to find the tangent line at $x=4$.</p>
<p>How would I go about solving this algebraically? I have to be able to prove the answers are $y=12x-23$ and $y=2.75x-5$... | Randy | 310,503 | <p>The tangent line can be written as
$$
y = m x + b
$$
Here, $m$ would be equal to the slope, and at the tangent line, the slope is equal to the derivative of a function in consideration. Thus for your first function, the derivative at $x=2$ will have
$$
\frac{dx^3+2x^2-8x+1}{dx}|_{x=2} = 3x^2+4x-8|_{x=2}= 12.
$$
Thus... |
2,502,224 | <p>I don't quite understand <a href="https://math.stackexchange.com/a/1866099/185631">this example given by Mike Haskel</a>. I want to find an example about</p>
<p><span class="math-container">$$\operatorname{Hom}_R\left ( M ,\bigoplus_{i\in I} N_{i}\right )\not \cong\bigoplus_{i\in I} \operatorname{Hom}_R\left ( M ,N_... | Andreas Blass | 48,510 | <p>I think you're right that the vector spaces in this example will be isomorphic when $I$ is countably infinite. Probably what Mike Haskel had in mind is that the canonical map from the sum-of-Homs to the Hom-into-sum is not an isomorphism. </p>
<p>The example becomes correct if $I$ is chosen more carefully. What o... |
512,118 | <p>Suppose $X$ is a metric space, $z$ is in $X$ and $(x_n)$ is a sequence in $X$. </p>
<p>Then what does it mean to say that, $z$ is in the "<em>closure of every tail of $(x_n)$</em>."</p>
<p>What does "<em>closure</em>" of every tail, mean ?</p>
| Brian M. Scott | 12,042 | <p>First, if $m\in\Bbb N$, the $m$-tail of the sequence is the subsequence $\langle x_n:n\ge m\rangle$ consisting of all of the terms from $x_m$ on. Now instead of viewing this as a sequence, consider just the set of terms in the $m$-tail: that set is $T_m=\{x_n:n\ge m\}$. It’s just a subset of $X$, so it has a closure... |
3,445,768 | <p>I am trying to solve a nonlinear differential equation of the first order that comes from a geometric problem ; <span class="math-container">$$x(2x-1)y'^2-(2x-1)(2y-1)y'+y(2y-1)=0.$$</span></p>
<p>edit1 <strong><em>I am looking for human methods to solve the equation</em></strong> </p>
<p>edit2 the geometric pr... | user577215664 | 475,762 | <p>Not a complete solution, but you can try a polynomial solution approach:
<span class="math-container">$$y(x)=\sum_{n=0}^{m}a_nx^n$$</span>
Plug <span class="math-container">$y(x)=a_mx^m$</span> in the equation to find the degree of the polynomial:
<span class="math-container">$$2m^2a_m^2-4a_m^2m+2a_m^2=0$$</span>
<s... |
2,704,247 | <p>I was given this question to prepare for an exam:</p>
<p><em>Show that the set of all functions $f(x)$ such that $f''(x)$ = -3 on ($-\infty$, $\infty$) is uncountable.</em></p>
<p>I know that this gives me a set of parabolas $f(x) = -\frac{3}{2}x^{2} + ax + b$, but I'm unsure of how to show this set is uncountable... | grontim | 283,574 | <p>You have $f(x) = \frac{-3}{3}x^2+ax+b$. Note that every pair of $a,b$ gives you a different function $f$. Why? Because a polynomial function is uniquely determined by its coefficients i.e., if you have two polynomials $f$ and $g$ and they have same values every where,then they have the same coefficients. This is tru... |
3,746,402 | <p>For example, if we define <span class="math-container">$F(x)=\int^x_a f(t)dt$</span>, where <span class="math-container">$f$</span> is Riemann integrable, then <span class="math-container">$F(x)$</span> is a function. Or for a 2 variables real-valued integrable function <span class="math-container">$f(x, y)$</span>,... | Xander Henderson | 468,350 | <h3>The Riemann integral is a number.</h3>
<p>If <span class="math-container">$f : [a,b]\to\mathbb{R}$</span> is a Riemann integrable function then, by definition, the <em>Riemann integral</em> of <span class="math-container">$f$</span> is given by
<span class="math-container">$$ \int_{a}^{b} f(x)\,\mathrm{d}x
= \lim_{... |
1,362,220 | <p>My question is regarding the validity of the following statement:</p>
<p>$$ (\forall a (\phi \implies \psi)) \equiv (\phi \implies \forall a \psi ),$$</p>
<p>provided, of course, there are no free occurrences of $a$ in $\phi$.</p>
<p>I am using the axiom system from <a href="http://rads.stackoverflow.com/amzn/cli... | Trung Ta | 214,606 | <p>My sequences of transformation is as follow:</p>
<p>$$
\forall a (\phi \Rightarrow \psi)
$$</p>
<p>$$
\equiv~
\forall a (\neg\phi \vee \psi)
$$
$$
\equiv~
\neg\phi \vee \forall a (\psi)
$$
$$
\equiv~
\phi \Rightarrow \forall a (\psi)
$$</p>
<p>Note that from step 2 to 3, since $a$ does not occur fre... |
2,919,561 | <p>What is the following limit ? How do I solve it ?
$$\lim \limits_{x\to0}\frac{6\sin x}{x-3\tan x}$$</p>
| Cesareo | 397,348 | <p>$$
\frac{6\sin x}{x-3\tan x} = \frac{6\sin x\cos x}{x(\cos x-3\frac{\sin x}{x})} = 6\cos x\left(\frac{\sin x}{x}\right)\frac{1}{\cos x-3\left(\frac{\sin x}{x}\right)}
$$</p>
<p>hence</p>
<p>$$
\lim_{x\to 0}\frac{6\sin x}{x-3\tan x} = \lim_{x\to 0}6\cos x\left(\frac{\sin x}{x}\right)\frac{1}{\cos x-3\left(\frac{\si... |
2,683,032 | <blockquote>
<p>Show the sum of the first $n$ positive even integers is $n^2 + n$ using strong induction.</p>
</blockquote>
<p>I can't solve the above problem using strong induction. It will be very helpful if I can get the solution. Thanks in advance.</p>
| Graham Kemp | 135,106 | <p>In your Attempt A, the "bigram" you discuss is not $\mathsf P(w_i\mid w_{1:i-1})$ but rather it is $\mathsf P(w_i\cap w_{1:i-1})$ - the <em>joint probability</em> for the $i$th word choice and the preceeding sequence of $i-1$ word choices. Which is of course, $\mathsf P(w_{1:i})$.</p>
<p>What is $\mathsf P(w_i\mid ... |
3,896,314 | <p>I am trying to show that <span class="math-container">$\log 15$</span> and <span class="math-container">$\log 3$</span> + <span class="math-container">$\log 5$</span> is irrational.</p>
<p>For <span class="math-container">$\log 15$</span> I feel like I have no issues showing this is irrational.</p>
<p>By contradicti... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\log_{10}3+\log_{10}5=\log_{10}(3\times5)=\log15$</span> and since <span class="math-container">$\log_{10}15$</span> is irractional…</p>
<p>On the other hand, the problem with the equality <span class="math-container">$3^n5^n=5^m2^m$</span> lies in the fact that <span class="math-... |
3,896,314 | <p>I am trying to show that <span class="math-container">$\log 15$</span> and <span class="math-container">$\log 3$</span> + <span class="math-container">$\log 5$</span> is irrational.</p>
<p>For <span class="math-container">$\log 15$</span> I feel like I have no issues showing this is irrational.</p>
<p>By contradicti... | peterwhy | 89,922 | <p>Where does the restriction end? Without using the log identity directly, I can still assume <span class="math-container">$\log 3 + \log 5 = \frac mn$</span> and then</p>
<p><span class="math-container">$$10^{\log 3+\log 5} = 10^{m/n} \implies 10^{\log 3}\cdot 10^{\log 5} = 10^{m/n} \implies 3\cdot 5 = 10^{m/n}$$</sp... |
1,311,466 | <p>My concept of real no. Is not very clear. Please also tell the logic behind the question.
The expression is true for 19, is it true for all the multiples? </p>
| Asinomás | 33,907 | <p>You want to prove $\frac{(3n)!}{6^n}$ is an integer. Just use $\frac{3n!}{6^n}=\frac{1\cdot2\cdot3}{6}\frac{4\cdot 5\cdot 6}{6}\frac{7\cdot8\cdot 9}{6}\dots \frac{(3n-2)(3n-1)(3n)}{6}$ and each fraction is an integer since $k(k+1)(k+2)$ is always a multiple of $2$ and of $3$ since three consecutive integers always c... |
1,311,466 | <p>My concept of real no. Is not very clear. Please also tell the logic behind the question.
The expression is true for 19, is it true for all the multiples? </p>
| Shailesh | 241,153 | <p>Very simply the expression is the number of ways of arranging $ 3n $ objects, of which there are $n$ distinct group of $3$ alike objects. eg. Number of ways of arranging 12 balls of which 3 are blue, 3 are red, 3 are green and 3 are yellow. Since this expression is the number of ways, so it is an integer alright. ... |
1,311,466 | <p>My concept of real no. Is not very clear. Please also tell the logic behind the question.
The expression is true for 19, is it true for all the multiples? </p>
| miniparser | 99,402 | <p>prove $(3n)!\over{6^n}$ is integer. no need for induction, etc. $6^n=2^n3^n$</p>
<p>$(3n)!=(3n)(3n-1)(3n-2)...(3)(2)(1)$ with $3n$ total factors. half of those factors must be even, which takes care of $2^n$. for $3^n$ every $3rd$ factor going back from $3n$ must be divisible by $3$:</p>
<p>$\{3n,3n-3,3n-6,...... |
3,484,136 | <blockquote>
<p>Show that <span class="math-container">$n^2+n$</span> is even for all <span class="math-container">$n\in\mathbb{N}$</span> by contradiction.</p>
</blockquote>
<p>My attempt: assume that <span class="math-container">$n^2+n$</span> is odd, then <span class="math-container">$n^2+n=2k+1$</span> for all <... | Robert Lewis | 67,071 | <p>I hate to be the bearer of bad tidings, but the presented proof attempt is flawed. There is no guarantee that <em>any</em> <span class="math-container">$k$</span> corresponds to a given <span class="math-container">$n^2 + n$</span>, so the assumption that</p>
<p><span class="math-container">$\forall n \in \Bbb N, ... |
55,737 | <p>Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$.</p>
<blockquote>
<p><strong>Question:</strong> How many of the $4\cdot 3^{n}$ words of length $n+1$ in the generators and their inverses c... | David Ben-Zvi | 582 | <p>Since there's no response to this question so far I can break out my broken record and point out such theorems (sheaves on the product are tensor product of categories of sheaves on the factors) are true, and in fact extremely easy, in the derived setting (modulo the appropriate $\infty$-categorical technology). Fir... |
55,737 | <p>Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$.</p>
<blockquote>
<p><strong>Question:</strong> How many of the $4\cdot 3^{n}$ words of length $n+1$ in the generators and their inverses c... | Daniel Schäppi | 1,649 | <p>I have recently proved a result that gives a partial answer to this question, see <a href="http://arxiv.org/abs/1211.3678" rel="nofollow">here</a> (I need to make some assumptions on $X$ and $Y$). Let $X$ and $Y$ be noetherian schemes with the resolution property (that is, every coherent sheaf is a quotient of a loc... |
55,737 | <p>Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$.</p>
<blockquote>
<p><strong>Question:</strong> How many of the $4\cdot 3^{n}$ words of length $n+1$ in the generators and their inverses c... | Martin Brandenburg | 2,841 | <p>More generally, I have proven that for quasi-compact and quasi-separated schemes <span class="math-container">$\mathrm{Qcoh}(X \times_S Y)$</span> is the bicategorical pushout of <span class="math-container">$\mathrm{Qcoh}(X)$</span> and <span class="math-container">$\mathrm{Qcoh}(Y)$</span> over <span class="math-c... |
4,090,259 | <p>I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (<a href="https://youtube.com/play... | user20672 | 512,659 | <p>Consider the system of equations</p>
<p><span class="math-container">$$
Ax = b
$$</span></p>
<p>In this system, <span class="math-container">$A$</span> is a linear transformation. When that transformation is applied to <span class="math-container">$x$</span> it equals <span class="math-container">$b$</span>. In that... |
4,090,259 | <p>I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (<a href="https://youtube.com/play... | Ethan Bolker | 72,858 | <p>You ask</p>
<blockquote>
<p>Since matrices are used to represent all sorts of things (linear
transformations, systems of equations, data, etc.), how come
operations that are seemingly defined for use with linear maps the
same across all these different contexts?</p>
</blockquote>
<p>Other answers and comments addres... |
2,417,356 | <p>I found the following very nice post yesterday which presented the conditional expectation in a way which I found intuitive;</p>
<p><a href="https://math.stackexchange.com/questions/1492306/conditional-expectation-with-respect-to-a-sigma-algebra?noredirect=1&lq=1">Conditional expectation with respect to a $\sig... | H. H. Rugh | 355,946 | <p>Your intuition and formula makes sense when each $E_i$ is an elementary event in ${\cal F} $ which can not be decomposed into 2 disjoint events, both of positive probability. If it can be decomposed then there is a mis-match as in general $E(X|{\cal F})(\omega)$ is not constant over $\omega\in E_i$ while your formul... |
1,036,636 | <p>The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.)</p>
<blockquote>
<p>If $f$ is continuous on an interval $I$ and $x_0$ is the <strong>only</strong> relative (local) extremum, then $x_0$ is actua... | Jeffrey Rolland | 329,781 | <p>Suppose $x_0$ is a local maximum. Assume $x_0$ is not an absolute maximum. Then $\exists x_1 \in I$ with $f(x_1) > f(x_1)$. Let $I'$ be the closed interval from $x_0$ to $x_1$. Let $x'$ be the global minimum of $f$ on $I'$.</p>
<p>Then we can choose $x' \ne x_0$, as $x_0$ is a local maximum for $f$, so there is ... |
2,419,753 | <p>How would you approach to solve questions like:</p>
<p>$\sin(z)=\sin(2)$, $z$ is an arbitrary complex number.</p>
| user577215664 | 475,762 | <p>When exponents are at the same level you only deal with addition and multiplication... So $x^{10n} \left( x^5 \left(x^{-5}\right)^n \right) = x^{-10}$
Multiply n and 5</p>
<p>$x^{10n} x^5 x^{-5n} = x^{-10}$</p>
<p>Adding 5 and 5n</p>
<p>$x^{10n} x^{5-5n} = x^{-10}$</p>
<p>$x^{10n+5-5n} = x^{-10}$</p>
<p>$5+5n ... |
4,050,336 | <p>I am familiar with the Negative Binomial distribution <span class="math-container">$NB(p, k)$</span>, which gives the number of failures before <span class="math-container">$k$</span> successes occur in a Bernoulli process with parameter <span class="math-container">$p$</span>. I am wondering, however, if there the ... | Wuestenfux | 417,848 | <p>Hint: <span class="math-container">$x^2+1$</span> has a unique factorization in an appropriate extension field, say <span class="math-container">$\Bbb Q(i)$</span>. Then <span class="math-container">$x^2+1 = (x+i)(x-i)$</span>.</p>
<p>You have to show that <span class="math-container">$i\not\in \Bbb Q(\sqrt 2 i)$</s... |
4,050,307 | <p>You have a black box function to which you can give real number inputs and from which you can receive real number outputs. <strong>How would you test whether it is likely to be a polynomial?</strong></p>
<p>One expensive idea is to use finite differences:</p>
<ol>
<li>Choose a maximum degree <em>n</em> of the "... | preferred_anon | 27,150 | <p>I don't know much about "likely". But <span class="math-container">$N+1$</span> distinct points uniquely determine a degree <span class="math-container">$\le N$</span> polynomial by <a href="https://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_theorem" rel="nofollow noreferrer">Lagrange int... |
2,359,455 | <blockquote>
<p>Let <span class="math-container">$M$</span> be the largest subset of <span class="math-container">$\{1,\dots,n\}$</span> such that for each <span class="math-container">$x\in M$</span>, <span class="math-container">$x$</span> divides at most one other element in <span class="math-container">$M$</span>. ... | Sarvesh Ravichandran Iyer | 316,409 | <p>You are right. We first divide $M$ into two subsets , $M_1$ and $M_2$, where the former contains all $x$ such that $x$ divides exactly one other element, and $M_2$ contains all the others.</p>
<p>It is easy to see that $M_1$ is an antichain with respect to two elements being linked if one divides the other. This i... |
1,700,608 | <p>In <a href="https://math.stackexchange.com/a/1700505/132192">this</a> answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.</p>
<p>In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumfer... | Mark Viola | 218,419 | <p>The bound <span class="math-container">$-1\le \cos(x)\le 1$</span> is correct, but rather a crude one.</p>
<p>We can easily obtain tighter bounds using the inequality from elementary geometry</p>
<p><span class="math-container">$$|\sin(\theta)|\le |\theta| \tag 1$$</span></p>
<p>for all <span class="math-container">... |
1,700,608 | <p>In <a href="https://math.stackexchange.com/a/1700505/132192">this</a> answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.</p>
<p>In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumfer... | ajk68 | 558,962 | <p>ncmathsadist's technique can be extended for a tighter bound as described below. It also extends the region over which the bound is better than <span class="math-container">$1 - \cos (x) \leq 2$</span> per Mark Viola's answer.</p>
<p>Cutting off the Taylor series for <span class="math-container">$\sin(t)$</span> bef... |
2,646,251 | <p>Given the linear system in $\mathbb{Z_3}$:</p>
<p>$$
\left\{
\begin{array}{c}
a+b+c+d=1 \\
b+c+e=2 \\
a+2e=0
\end{array}
\right.
$$</p>
<p>I used the row reduction with matrices and I got:</p>
<p>$$
\left\{
\begin{array}{c}
a+b+c+d=1 \\
b+c+e=2 \\
d=2
\end{array}
\right.
$$</p>
<p>But now I don't know ho... | Martin Argerami | 22,857 | <p>What you probably did is add the last two equations to get
$$
2=a+b+c,
$$
since $3e=0$. Now the first equation gives you $d=1-2=2$. Going back to the system (and again using $2=-1$),
$$
\left\{
\begin{array}{c}
a+b+c=2 \\
b+c+e=2 \\
a-e=0
\end{array}
\right.
$$
and we already know $d=2$, $a=e$. Now theh first t... |
1,265,026 | <p>Suppose I have some vector field
\begin{align}
\vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1}
\end{align}
Would it be correct for me to say
\begin{align}
\mathbb{R}^3\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3\;?\tag{2}
\end{align}</p>
| Kyle Miller | 172,988 | <p>It looks like your vector field is also parameterized by time, so writing $\vec{F}_t:\mathbb{R}^3\to\mathbb{R}^3$ might be better.</p>
<p>For more notation: each of the component for the vector field are functions $G_t,H_t,T_t:\mathbb{R}^3\to\mathbb{R}$.</p>
|
450,785 | <p>I want to obtain the formula for binomial coefficients in the following way: elementary ring theory shows that $(X+1)^n\in\mathbb Z[X]$ is a degree $n$ polynomial, for all $n\geq0$, so we can write</p>
<p>$$(X+1)^n=\sum_{k=0}^na_{n,k}X^k\,,\ \style{font-family:inherit;}{\text{with}}\ \ a_{n,k}\in\mathbb Z\,.$$</p>
... | user136023 | 136,023 | <p>I think Mathematica command for solving recurrence relation can be used:</p>
<p>RSolve[{a[n,k]==a[n-1,k]+a[n-1,k-1],a[n,0]==1,a[n,n]==1},a[n,k],{n,k}]</p>
|
1,135,045 | <p>I need to compute
\begin{align}
S = \sum_{k=-\infty}^j \sum_{m=-1}^2 w_{k,m} f_{k+m-1}
\end{align}
but I only want to access the elements of $f$ once, so I would prefer something like
\begin{align}
\sum_k f_k \sum_m ...
\end{align}
Here is what I did: substitute $l=m-1+k$ to get
\begin{align}
S &= \sum_{k=-\inf... | Mark Fischler | 150,362 | <p>Step 1: break up the inner sum into 4 parts corresponding to four values of $m$. If the range were larger you could do this for arbitrary upper limit, but I will illustrate this for the specific case of upper limit $u=2$:
$$
S(j,2) = \sum_{k=-\infty}^j \left[ w_{k,-1}f_{k-2}+ w_{k,0}f_{k-1} + w_{k,1}f_l+w_{k,2}f_{k... |
1,271,942 | <p>I am a little bit confused with the definition of finitely presented modules. In Lang's <em>Algebra</em> he defines a module <span class="math-container">$M$</span> to be finitely presented if and only if there is a exact sequence <span class="math-container">$F'\to F\to M \to 0$</span> such that both <span class="m... | Bernard | 202,857 | <p>In Lang's definition, both $F$ and $F'$ must be finitely generated (and free). This definition is equivalent to: ‘There exists a short exact sequence:
$$0\to K\to F\to M\to 0$$
such the $T$ is a finitely generated free module, and the module of relations $K$ is finitely generated. It is equivalent because there is ... |
2,634,791 | <blockquote>
<p>How can I show that the map $f: GL_n(\mathbb R)\to GL_n(\mathbb R)$ defined by $f(A)=A^{-1}$ is continuous?</p>
</blockquote>
<p>The space $GL_n(\mathbb R)$ is given the operator norm and so I want to show for all $\epsilon$ there exists $\delta$ such that $\|A-B\|<\delta \implies \|A^{-1}-B^{-1}\... | Giuseppe Negro | 8,157 | <p>C.Falcon gave the simplest and most elegant answer with the algebraic formula involving the determinant. That formula actually implies that $A^{-1}$ is an <strong>analytic</strong> function of $A$. We can prove this fact with a direct, determinant-free proof that extends to the infinite dimensional case (see also th... |
2,634,791 | <blockquote>
<p>How can I show that the map $f: GL_n(\mathbb R)\to GL_n(\mathbb R)$ defined by $f(A)=A^{-1}$ is continuous?</p>
</blockquote>
<p>The space $GL_n(\mathbb R)$ is given the operator norm and so I want to show for all $\epsilon$ there exists $\delta$ such that $\|A-B\|<\delta \implies \|A^{-1}-B^{-1}\... | Community | -1 | <p>The crucial inequality is this one:</p>
<p>$$\|A^{-1}-B^{-1}\|=\|-A^{-1}(A-B)B^{-1}\|\le\|A^{-1}\|\|A-B\|\|B^{-1}\|\le \|A^{-1}\|\|A-B\|(\|A^{-1}\|+\|A^{-1}-B^{-1}\|)$$</p>
<p>The rest is just finding how small to make $\|A-B\|$ to make $\|A^{-1}-B^{-1}\|\lt\varepsilon$. I've done this calculation, and it turns ou... |
1,288,812 | <p><a href="https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#Exotic_map">This section</a> says:</p>
<blockquote>
<blockquote>
<p>There is a subgroup (indeed, $6$ conjugate subgroups) of $S_6$ which are abstractly isomorphic to $S_5$,</p>
</blockquote>
</blockquote>
<p>At thi... | Josh B. | 4,308 | <p>Note that $S_5$ contains a subgroup of order $20$ (generated by, say, $(1,2,3,4,5)$ and $(1,3,4,2)$ ). The action of $S_5$ on the $6$ cosets of a subgroup of order $20$ provides a permutation representation of $S_5$ on $6$ points. And yes, an outer automorphism maps this sort of $S_5$ to the first type you were thin... |
1,288,812 | <p><a href="https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#Exotic_map">This section</a> says:</p>
<blockquote>
<blockquote>
<p>There is a subgroup (indeed, $6$ conjugate subgroups) of $S_6$ which are abstractly isomorphic to $S_5$,</p>
</blockquote>
</blockquote>
<p>At thi... | David Wheeler | 23,285 | <p>I will show you in principle how this is done, but the actual mechanics are tedious. Let:</p>
<p>$$P_1 = \{e,(1\ 2\ 3\ 4\ 5),(1\ 3\ 5\ 2\ 4), (1\ 4\ 2\ 5\ 3),(1\ 5\ 4\ 3\ 2)\}\\
P_2 = \{e, (1\ 2\ 3\ 5\ 4),(1\ 3\ 4\ 2\ 5),(1\ 5\ 2\ 4\ 3),(1\ 4\ 5\ 3\ 2)\}\\
P_3 = \{e,(1\ 2\ 4\ 3\ 5),(1\ 4\ 5\ 2\ 3), (1\ 3\ 2\ 5\ 4),... |
2,232,095 | <blockquote>
<p>Let $a, b, c, p, q$ be real numbers. Suppose $\{α, β\}$ are the roots
of the equation $x^2 + 2px+ q = 0$ and $\{α,\frac{1}{β}\}$ are the
roots of the equation $ax^2 + 2bx+ c = 0$, where $β \notin \{−1, 0, 1\}$.</p>
</blockquote>
<p>STATEMENT-1 : $(p^2 − q)(b^2 − ac) ≥ 0$</p>
<p>STATEMENT 2: $b \... | Twenty-six colours | 424,197 | <p>This is a reply to the comment </p>
<blockquote>
<p>"Does statement 2 imply statement 1?" </p>
</blockquote>
<p>I don't have enough rep to comment.<br>
The answer is yes.<br>
Statement 1 was derived from the condition that $\beta \notin \{-1,0,1\}.$<br>
Since statement 2 also requires that condition on $\beta$... |
3,695,868 | <p>In right triangle <span class="math-container">$ABC,$</span> <span class="math-container">$\angle C = 90^\circ.$</span> Let <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> be points on <span class="math-container">$\overline{AC}$</span> so that <span class="math-container">$AP = P... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/RIWYl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RIWYl.png" alt="enter image description here"></a></p>
<p>Let <span class="math-container">$|BC|=a$</span>,
<span class="math-container">$|AB|=c$</span>,
<span class="math-container">$|AC|=b$</span>,
<span... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.