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 |
|---|---|---|---|---|
4,452,636 | <p>Given two points, A and B;
Given two circles, having 2 points in common, I1 and I2:</p>
<ul>
<li>one circle at center C1, with radius r1, with the point A on to it</li>
<li>and another circle at center C2, with radius r2, with the point B on it.</li>
</ul>
<p><a href="https://i.stack.imgur.com/0rWdF.jpg" rel="nofoll... | Community | -1 | <p>Sum will be <span class="math-container">$\pi/2$</span>.</p>
<p>If you add, <span class="math-container">$$\lim_{n\to \infty}T_0 + T_1 + T_2 + ... T_n$$</span></p>
<p>you will get:
<span class="math-container">$$\lim_{n\to \infty}\underbrace{ \tan^{-1}(1) - \tan^{-1}(0)}_{ T_0}+ ... + \underbrace{\tan^{-1}(n) - \tan... |
2,653,084 | <p>How many different combinations $m$ are there to draw numbers $n_i$ with $n=\sum^{N}_i n_i$ from the set $\{1,\dots,n\}$ without regarding order and without replacement, and for a given number of draws $N$. </p>
<p>I am studying <a href="https://en.wikipedia.org/wiki/Pentagonal_number_theorem" rel="nofollow norefer... | Barry Cipra | 86,747 | <p>For what it's worth, here is an abstraction of the OP's spiral, extended to the prime $419$ in the lower right hand corner (and $313$ in the upper left), indicating only the locations of the twin primes congruent to $1$ mod $4$, with an $L$ if it's the "Lower" twin prime and a $U$ if it's the "Upper" (e.g., a $U$ fo... |
1,726,026 | <p>I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to factor this and also introduce me to the procedure that i need to follow to factorize expressions with degree higher t... | Siddd | 274,503 | <p>Can be done by a simple completion of squares. The presence of $x^4$ and 16 hints towards a possibility of a $(x^2+4)^2$ so just calculate that and compare it with your question function. $(x^2+4)^2 = x^4+8x^2+16$ which leaves us with a difference only in the $x^2$ term. This difference turns out to be $9x^2$. So ou... |
3,382,651 | <p>I'm trying to understand one of the steps taken during the process of getting a cnf in Boolean algebra but I just cant understand what is happening here. </p>
<p><span class="math-container">$$\bar A \bar B C + \bar A \bar C \bar D + A \bar C D + \bar A B \bar C$$</span>
<span class="math-container">$$\bar A \b... | rewritten | 43,219 | <p><span class="math-container">$$
\bar{A}B\bar{C} = \bar{A}B\bar{C}D + \bar{A}B\bar{C}\bar{D}
$$</span></p>
<p>the second piece <span class="math-container">$\bar{A}B\bar{C}\bar{D}$</span> is already a subset of <span class="math-container">$\bar{A}\bar{C}\bar{D}$</span> so you get</p>
<p><span class="math-container">... |
282,590 | <p>"Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$."
Exercise 27.5 from "Groups and Symmetry" M.A.Armstrong.</p>
<p>This should be an easy exercise but I'm completely unable to answer it. I know the group would be equal to the quotient group $F(X)/N$... | Ted | 15,012 | <p>In the definition of a presentation of a group by generators and relations, we take the quotient of the free group on the generators by the smallest <em>normal</em> subgroup generated by the relations. This has 2 consequences: (a) normality of the subgroup we quotient by is not a problem; (b) the subgroup we quotie... |
11,743 | <p>As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antipr... | Joseph Malkevitch | 1,618 | <p>There are many web sites where the information you want is in essence available.</p>
<p>One place in particular is:</p>
<p><a href="http://mathworld.wolfram.com/JohnsonSolid.html" rel="nofollow">http://mathworld.wolfram.com/JohnsonSolid.html</a></p>
<p>This site has nets of each of the Johnson Solids and it has a... |
11,743 | <p>As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antipr... | Gordon Williams | 4,490 | <p>If you are interested in abstract polyhedra you can consult the atlas of small regular polytopes put together by Michael Hartley, however some of the links seem to be broken right now (I'll follow up with him, so you don't have to).</p>
<p><a href="http://www.abstract-polytopes.com/" rel="nofollow">http://www.abstr... |
260,373 | <p>Given a vector $\vec{v} = (x, y, z)$, how do I find two vectors that make up an axis with $\vec{v}$? In other words, one of them is perpendicular and lies in the same plane and the other is normal to those two vectors.</p>
| Community | -1 | <p>You can use for example $(x,y,z)$,$(y,-x,0)$ and $(xz,yz,-x^2-y^2)$. In general, to find the second vector, use one which has dot product zero when taken with the first vector, and then take the cross product of the first two vectors to find the third.</p>
|
1,033,235 | <blockquote>
<p>How to show that $\lim \limits_{(x, y) \to (0,0)} f(x, y)$ does not
exist where, </p>
<p>$$f(x, y) = \begin{cases} \dfrac{x^3 + y^3}{x - y} \; ; & x \neq y \\
0 \; \;\;\;\;\;\;\;\;\;\;\; ; & x = y \end{cases} $$</p>
</blockquote>
<p>I tried bounding the value of the function as $... | heropup | 118,193 | <p>Hint: What happens on the curves $$(x(t), y(t)) = (1/t \pm e^{-t}, 1/t), \quad t > 0?$$</p>
<p>Here is a little animation of the surface in question. It was surprisingly challenging to obtain a smooth plot in the neighborhood of $(0,0)$, but I managed to find a nice parametrization. The red and green curves a... |
883,118 | <p>I think it's true, I just did this demo, please can you help me if I'm missing something or doing it wrong. Thanks.</p>
<p>Let $T\colon V \to W$ a linear transformation. </p>
<p>If $\dim V > \dim W$, then $T$ is not injective. </p>
<p>The contrapositive is: If $T$ is injective, then $\dim V \le \dim W$.</p>
... | hunter | 108,129 | <p>You're right. Remember that
$$
\text{rank } T + \text{nullity } T = \text{dim } V.
$$</p>
<p>What is the maximum possible rank of $T$? (It may help to think of $T$ as an $m \times n$ matrix.)</p>
|
95,236 | <p>I have been getting some ideas by reading other related questions in the forum, but the integral I have to do is not converging in many cases. The integrand is of the form:</p>
<pre><code>1/(a (qx^2+qy^2) + c qz^2)^2 * 1/( b(qx^2+qy^2+qz^2) + w) *
Abs[U[qx,qy,qz](qx Z1 + qx Z2 + qx Z3)]^2
</code></pre>
<p>and it's... | wikiwert | 34,198 | <p>So as I posted in the Edit, the initial problem was that I was trying to integrate a function of the form <code>1/x</code> around <code>0</code>. In this case, as I will describe later, it corresponds to integrating around the singularity given by <code>b(qx^2+qy^2+qz^2) + w ==0</code>. The integral diverges on each... |
1,719,772 | <p>I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both sides by A2 or transitivity. But I'm sure that to say $(-(-x)(-y))=-(xy)$ is outside the field axioms. So I'm stuck and w... | jimalton | 172,737 | <p>We are so used to the operators in arithmetic that we don't realise when they are being used in a way that has yet to be defined.</p>
<p>In the field axioms for the additive operator there is an identification of -x with the inverse of x, and the minus sign is blithely tossed around and assigned elsewhere as we kno... |
502,045 | <p>Let $E$ and $F$ be two Banach spaces. We know that, if $f:E\rightarrow F$ is a nonlinear continuous operator, then $f$ may fail to send weakly convergent sequences to weakly convergent sequences, i.e., $u_n \rightharpoonup u$ weakly in $E$ does not necessarily imply $f(u_n)\rightharpoonup f(u)$ weakly in $F$. Even ... | user37238 | 87,392 | <p>Here is an example (from Brézis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations exercise 4.20) that shows that it is much more restrictive than you think :</p>
<p>Take $\Omega = (0,1)$ and $a : \mathbb{R}\to \mathbb{R}$ a continous function such that</p>
<p>$$ |a(t)|\le a|t|+b$$</p>
<... |
77,053 | <p>In the proof of Chevalley's theorem on invariants in the polynomials on a semisimple Lie algebra, one uses the following general fact: the polynomials $(\sum m_ix_i)^n$ with $m_i$ positive integers span the homogeneous polynomials of degree $n$ in the variables $x_i$ with coefficients in a field of characteristic ze... | hmakholm left over Monica | 14,366 | <p>For brevity, define a <em>basal polynomial</em> to mean a positive integer combination of variables, $\sum_i m_ix_i$. Then the claim is equivalent to asserting that every monomial
$x_1^{i_1}x_2^{i_2}\cdots x_k^{i_k}$
is a linear combination $a_1p_1^n+\cdots+a_mp_m^n$ where $n=i_1+\cdots +i_k$ and each $p_j$ is basal... |
3,656,832 | <p>Let <span class="math-container">$ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $</span> . Find the maximum value of
<span class="math-container">$$I= \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) $$</span></p>
<p>I try: since <span class="math-container">$(a-b)^3=a^3-3a^2b+3ab^2-b^3$</span>,and <span class="ma... | epi163sqrt | 132,007 | <p>Here are some heuristic thoughts.</p>
<p><strong>A generalisation:</strong></p>
<p>We set <span class="math-container">$N=10$</span> and consider the generalisation
<span class="math-container">\begin{align*}
\color{blue}{\sum_{n=1}^N\left(na_n^2-n^2a_n\right)\qquad\to\qquad \max}\tag{1}
\end{align*}</span></p>
<p>W... |
3,498,919 | <p>I have been having some trouble showing conditions are met before applying green's theorem. For example, showing a set is a regular closed region is pretty hard. Showing that a set is compact is easy enough, but I am not sure how to tackle showing that the closure of it's interior is the set. </p>
<p>For example, h... | Michael Rozenberg | 190,319 | <p>2) If <span class="math-container">$\tan3x=0$</span>, so we can check it easily. </p>
<p>Let <span class="math-container">$\tan3x\neq0.$</span> </p>
<p>Thus, we need to solve:
<span class="math-container">$$\tan{x}\cot3x+1=\tan2x\tan4x+1$$</span> or
<span class="math-container">$$\frac{\sin4x}{\cos{x}\sin3x}=\fr... |
301,028 | <p>This limit</p>
<p>$$\lim_{\varepsilon\to0}\left(A+\frac{B}{\varepsilon}\right)^{-1}$$</p>
<p>for an invertible matrix $B$ is a null matrix (since it breaks down to $\varepsilon\cdot B^{-1}\to\bf{0}$)</p>
<blockquote>
<p>The real question is - what this limit in terms of $A$ and $B$ is when $B$ is singular?</p>
... | Mhenni Benghorbal | 35,472 | <p>You can use <a href="http://en.wikipedia.org/wiki/Variation_of_parameters" rel="nofollow">variational of parameters method</a>, since the Wronskian equals $1$ which makes the calculations easy. Here is the final result</p>
<p>$$ y \left( \tau \right) = A\cos \left( \tau \right)+B\sin\left( \tau \right)
+\frac{b\tau... |
3,512,971 | <p>In a group the ratio of men to women is 5:3
In the same group, the ratio of children to adults is 1:2</p>
<p>What is the ratio of men:women:children?</p>
<p>Through simple trial/error/obviousness, one can see that it is 5:3:4</p>
<p>I can’t wrap my head around a sound algebraic way to solve this.</p>
<p>Please h... | Bernard | 202,857 | <p>Let's use middle-school mathematics:</p>
<p>Denote <span class="math-container">$A, C, M,W$</span> the number of adults, children, men,women respectively.</p>
<p>We know that <span class="math-container">$\quad \dfrac M5=\dfrac W3$</span> and that <span class="math-container">$\enspace \dfrac A2 =C$</span>. By th... |
2,802,195 | <blockquote>
<p>For any two functions $f_1 : [0,1] →\mathbb R$ and $f_2 : [0,1]
→\mathbb R$, define the function $g : [0,1] →\mathbb R$ as $g(x) =
\max(f_1(x),f_2(x))$ for all $x ∈ [0,1]$. </p>
<p>A. If $f_1$ and $f_2$ are linear, then $g$ is linear</p>
<p>B. If $f_1$ and $f_2$ are differentiable, then g is... | José Carlos Santos | 446,262 | <p><strong>B.</strong> is false. Just take $f(x)=x$ and $g(x)=1-x$.</p>
<p><strong>C.</strong> is true. It follows from the definition of convex function.</p>
|
2,802,195 | <blockquote>
<p>For any two functions $f_1 : [0,1] →\mathbb R$ and $f_2 : [0,1]
→\mathbb R$, define the function $g : [0,1] →\mathbb R$ as $g(x) =
\max(f_1(x),f_2(x))$ for all $x ∈ [0,1]$. </p>
<p>A. If $f_1$ and $f_2$ are linear, then $g$ is linear</p>
<p>B. If $f_1$ and $f_2$ are differentiable, then g is... | Mike | 544,150 | <p>A. is not true. $f_1(x) = x$ and $f_2(x) = 1-x$.</p>
<p>B. is not true, see above, not differentiable at $x =\frac{1}{2}$.</p>
<p>C. is true though. Plug in definition of a convex function.</p>
|
4,518,177 | <p>I have just started learning Linear Algebra and in Vectors there is this concept of standard basis vectors , î and j, and all the vectors can be expressed as the sum of these two basis vectors. I want to know if any two random vectors can also serve as basis vectors ? What is the intuition behind this ?</p>
| JKL | 874,247 | <p>Continue to use orthonormality of the orthornomal basis (i.e. Pythagoras theorem) from what you have so far to show that
<span class="math-container">$$
\Vert Tv - \lambda v \Vert^2 = \sum_i |a_i|^2 \lambda_i^2 - 2 \lambda \sum_i |a_i|^2 \lambda_i + \lambda^2 = \sum_i |a_i|^2 (\lambda_i - \lambda)^2,
$$</span>
where... |
308,909 | <p>Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$
using Taylor's Theorem?</p>
<p>I am thinking of expanding it about $x=0$ but I got something like
$$f(x) = -x^2 + \frac{x^4}{2} - \dots$$</p>
<p>Is my approach correct? Could you give me some hints/guides here?</p>
<p>Thanks.</p>
| André Nicolas | 6,312 | <p>Yes, but it is a nuisance to use the Taylor expansion of $\log(1+x^2)$. Instead, let $t=x^2$, and look at what the Taylor expansion of $\log(1+t)$ tells you for non-negative $t$.</p>
<p>Use the Lagrange form of the remainder. The error when you truncate the expansion of $f(t)$ at the linear term is equal to $\frac... |
1,928,515 | <p>This subject is very foreign to me and has me kind of confused. This problem seems very easy, but because I'm new to this, its not to me. I'm not sure of the name of the law I'm using, or if its correct at all. I'm supposed to prove that:</p>
<pre><code>A \ (A ∩ B) = A \ B
x ∈ A\(A∩B) ⇔ (x ∈ A\A) ∩ (x ∈ A\B)) ... | Community | -1 | <p>This is the recurrence relation of the Heron method for computing the square root of $1$, also seen as the application of Newton's method to $a^2=1$.</p>
<p>As is shown by the relation</p>
<p>$$\frac{a_n-1}{a_n+1}=\frac1{2^{2^{n}}},$$</p>
<p>it has quadratic convergence and virtually equals $1$ for all $n$.</p>
... |
3,398,566 | <p>The number of integers in the range of 'c' such that there exists a line which intersects the curve <span class="math-container">$ y = x^4 – 6x^3 + 12x^2 + cx + 1$</span> at four distinct points. </p>
<p>My approach we need to intersect with line <span class="math-container">$y=mx+C$</span></p>
<p>Substituting w... | Oleg567 | 47,993 | <p>Consider equation
<span class="math-container">$$
x^4 – 6x^3 + 12x^2 - 9x + 2 = 0.\tag{1}
$$</span>
It is equivalent to
<span class="math-container">$$
(x - 2) (x - 1) (x^2 - 3 x + 1) = 0,\tag{2}
$$</span>
and has <span class="math-container">$4$</span> real solutions:
<span class="math-container">$$
x_{1} = 1,\\
x... |
3,398,566 | <p>The number of integers in the range of 'c' such that there exists a line which intersects the curve <span class="math-container">$ y = x^4 – 6x^3 + 12x^2 + cx + 1$</span> at four distinct points. </p>
<p>My approach we need to intersect with line <span class="math-container">$y=mx+C$</span></p>
<p>Substituting w... | Certainly not a dog | 691,550 | <p>Note that in order for a line to be able to meet this curve at four points, the curve must "change direction" thrice, which is to say that the derivative must have three roots.</p>
<p>The derivative in question is visibly <span class="math-container">$$g'(x) = 4x^3 -18x^2+24x+c$$</span>
For it to have three roots,... |
1,243,548 | <p>Suppose we have a set of $N$ elements, each of which is considered distintic from all others. If we ask ourselves the number of ways to order those $N$ elements the reasoning is based on this: for the first element we have $N$ choices, since they are all equivalent after removing the first, there are $N-1$ choices f... | Andrew D. Hwang | 86,418 | <p>"Yes", the formula in your original question appears to be correct.</p>
<p>Let $S$ be a sphere, $\ell$ a line through the center of $S$, and $C$ the circumscribed cylinder with axis $\ell$. A remarkable theorem of Archimedes asserts that <em>axial projection away from $\ell$ is area preserving</em>.</p>
<p><img sr... |
42,879 | <p>Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ by any element $\sigma$ of the Galois group. What's the shortest way to prove that $\sigma(\chi)$ is also a character ... | Jack Schmidt | 583 | <p>This is just a slightly more quantitative version of Matt E's answer. I agree also that this is the obvious and implicit argument.</p>
<p><strong>C</strong><em>G</em> is a direct product of matrix rings of size χ(1). The primitive central idempotents lie in <em>KG</em>, so <em>KG</em> is a direct product of the s... |
1,641,012 | <blockquote>
<p>If $S$ and $T$ are subrings of $R$, is $S+T=\{s+t\mid s\in S, t\in T\}$ a subring of $R$?</p>
</blockquote>
<p>So I think that $S+T$ is a subring, but I am getting stuck trying to prove it. </p>
<p>Clearly since $S$ and $T$ are rings, $0,1\in S$ and $0,1\in T$. So, $0+0=0\in S+T$ and $0+1=1\in S+T$.... | Jendrik Stelzner | 300,783 | <p>Take $R = \mathbb{C}$ and consider the subfields $S = \mathbb{Q}[\sqrt{-1}] = \{a+bi \mid a,b \in \mathbb{Q}\}$ and $T = \mathbb{R}$. Then $S + T = \{a + bi \mid a \in \mathbb{R}, b \in \mathbb{Q}\}$. This contains $i$ and $\sqrt{2}$, but not $i\sqrt{2}$.</p>
|
4,309,047 | <p>Toss a biased coin <span class="math-container">$n$</span> times with a probability <span class="math-container">$p$</span> to land tails. What is the expected value for occurrences of <span class="math-container">$m$</span> consecutive heads that don't overlap with each other? We call the amount of those occurrence... | Thomas Andrews | 7,933 | <p>Full answer, but a little sloppy, using Markov chains.</p>
<p>Consider the Markov chain with states <span class="math-container">$0,1,\dots,m$</span> with <span class="math-container">$0$</span> starting state, and transition matrix:</p>
<p><span class="math-container">$$T=\begin{pmatrix}p&(1-p)&0&\cdots... |
3,172,379 | <p>how he found the two inequalities (56 and the last)?
<a href="https://i.stack.imgur.com/HajrS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HajrS.png" alt="enter image description here"></a></p>
<p>Thanks!</p>
| angryavian | 43,949 | <p>For (56), apply the triangle inequality to
<span class="math-container">$$z^n = P(z) - a_0 - a_1 z - \cdots - a_{n-1} z^{n-1}$$</span>
to obtain
<span class="math-container">$$R^n \le |P(z)| + |a_0| + |a_1| R + \cdots + |a_{n-1}| R^{n-1}.$$</span></p>
<p>I believe the same idea works for the last inequality in the ... |
3,172,379 | <p>how he found the two inequalities (56 and the last)?
<a href="https://i.stack.imgur.com/HajrS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HajrS.png" alt="enter image description here"></a></p>
<p>Thanks!</p>
| Myo Nyunt | 828,003 | <p>The last inequality is achieved as follows.</p>
<p><span class="math-container">$ |Q(re^{i\theta})|$</span><br />
<span class="math-container">$=|1+b_k r^k e^{ik\theta}+b_{k+1}r^{k+1}e^{i(k+1)\theta}+ \dots +b_{n}r^{n}e^{in\theta}|$</span><br />
<span class="math-container">$\le|1+b_k r^k e^{ik\theta}|+|b_{k+1}r^{k+... |
629,460 | <p>How do I find the Galois group of $x^4-5$ over $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt5)$ and $\mathbb{Q}(\sqrt{-5})$?
I've managed to do so over $\mathbb{Q}$ but I don't know how to find the others.</p>
<p>I'd appreciate any help.
Thanks!</p>
| ccorn | 75,794 | <p>You have already found that the splitting field is
$E=\mathbb{Q}(\mathrm{i},\sqrt[4]{5})$
and that $\operatorname{Gal}(E/\mathbb{Q})\cong D_8$ (I use the algebraic
convention where the index denotes the group order, not the vertex count).
Therefore the other fields you cited are intermediate fields $K$
between $E$ a... |
1,502,610 | <p>My exercise book and Wolfram Alpha give:</p>
<p>$$\lim\limits_{x\to\infty}(\sqrt{9x^2+x}-3x)=\frac{1}{6}$$</p>
<p>When I work it out I get 0:</p>
<p>$$(\lim\limits_{x\to\infty}x\sqrt{9\frac{x^2}{x^2}+\frac{x}{x^2}}-\lim\limits_{x\to\infty}3x)$$</p>
<p>$$(\lim\limits_{x\to\infty}x*\sqrt{\lim\limits_{x\to\infty}9+... | Community | -1 | <p>$$\lim\limits_{x\to\infty}(\sqrt{9x^2+x}-3x)\frac{\sqrt{9x^2+x}+3x}{\sqrt{9x^2+x}+3x} = \lim\limits_{x\to\infty} \frac{(9x^2+x)-9x^2}{\sqrt{9x^2+x}+3x} = \lim\limits_{x\to\infty} \frac{x}{\sqrt{9x^2+x}+3x}= \lim\limits_{x\to\infty}\frac{1}{\sqrt{9+1/x}+3} = \frac{1}{6}$$</p>
|
3,483,524 | <p>I have a 4-dimensional second-order PDE, which needs to be reduced to the canonical form.</p>
<p><span class="math-container">$u_{xx}+2u_{xy}-2u_{xz}-4u_{yz}+2u_{ty}+u_{zz}=0$</span></p>
<p>I know and understand the 2-dimensional (xy space) theory, but how does one approach this case of 4 dimensions? Should one ex... | Harjeq | 732,916 | <p>I've gone through pages <span class="math-container">$39$</span> - <span class="math-container">$41$</span> of your text. I'm going to restate some things in a hopefully clearer manner. </p>
<p>As with the usual <span class="math-container">$2$</span>-D case, we seek a transformation which transforms the equation t... |
2,301,483 | <p>The integral:</p>
<p>$$\int_0^{+\infty}\frac{\arctan x}{(1+x^2)^{\frac{3}{2}}}\;\mathrm{d}x$$</p>
<p>so there's the function and its derivative, I was thinking of substitution, but not sure which one...</p>
| Jan Eerland | 226,665 | <p>Well, when you use integration by parts:</p>
<p>$$\mathscr{I}:=\int_0^\infty\frac{\arctan\left(x\right)}{\left(1+x^2\right)^\frac{3}{2}}\space\text{d}x=\lim_{\text{n}\to\infty}\space\left[\frac{x\cdot\arctan\left(x\right)}{\sqrt{1+x^2}}\right]_0^\text{n}-\int_0^\infty\frac{x}{\left(1+x^2\right)^\frac{3}{2}}\space\t... |
2,301,483 | <p>The integral:</p>
<p>$$\int_0^{+\infty}\frac{\arctan x}{(1+x^2)^{\frac{3}{2}}}\;\mathrm{d}x$$</p>
<p>so there's the function and its derivative, I was thinking of substitution, but not sure which one...</p>
| Ghartal | 83,884 | <p>Letting $x=\tan t$ gives$$I=\int_{0}^{\pi/2} t \ \cos t \ \text{d}t$$You can use integration by parts now.</p>
<p>Another way to continue: Let $$I(a)=\int_{0}^{\pi/2} \sin (at) \ \text{d}t=\frac{1-\cos \left(\frac{\pi a}{2}\right)}{a}.$$Then</p>
<p>$$I'(a)=\int_{0}^{\pi/2} t \cos (at) \ \text{d}t=\frac{\text{d}}{\... |
1,092,152 | <p>It's said (proven in some reduction to the Gödel–Rosser theorem?) that second order logic and higher fails to be complete for full semantics; that is there isn't any semi-algorithm for determining if a sentence is valid in full semantics. Yet full semantics I understand is reducible to set theory, and set theory is ... | Hanno | 81,567 | <blockquote>
<p><em>Why can't we simply translate a higher order logic sentence into first order set theory to test it for validity?</em></p>
</blockquote>
<p>I want to focus on this (interesting!) confusion as to why the sentences of second-order arithmetic which are valid w.r.t full semantics need not be recursive... |
3,755,509 | <p>Let <span class="math-container">$A \in \mathbb R^{n\times n}$</span> be an invertible block anti-diagonal matrix (with <span class="math-container">$d$</span> blocks), i.e.
<span class="math-container">$$
A = \begin{pmatrix} & & & A_1 \\ & & A_2 & \\ & \cdot^{\textstyle \cdot^{\textstyle... | Claude Leibovici | 82,404 | <p><span class="math-container">$$f(x, y)=4 x^{2}+9 y^{2}-12 x-12 y+14$$</span>
<span class="math-container">$$\frac{\partial f(x,y)}{\partial x}=8x-12$$</span>
<span class="math-container">$$\frac{\partial f(x,y)}{\partial y}=18y-12$$</span>
Set the partial derivatives equal to <span class="math-container">$0$</span>;... |
2,572,000 | <p>Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:<br>
$2x+3 \neq 0$<br>
$2x \neq -3$<br>
$x \neq - \frac{3}{2}$<br>
So, the domain of $x$ (for fraction to be valid) is $x \in \left(- \infty, - \frac{3}{2}\right) \cup \left(- \fr... | alans | 80,264 | <p>($1-2x\geq 0\text{ and }2x+3>0$) or ($1-2x\leq0\text{ and }2x+3<0$). From the first condition we get $x\in (\frac{-3}{2},\frac{1}{2}]$. Second condition isn't possible.</p>
|
2,572,000 | <p>Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:<br>
$2x+3 \neq 0$<br>
$2x \neq -3$<br>
$x \neq - \frac{3}{2}$<br>
So, the domain of $x$ (for fraction to be valid) is $x \in \left(- \infty, - \frac{3}{2}\right) \cup \left(- \fr... | Taamer | 507,094 | <blockquote>
<p>Find the domain of $y=\sqrt{\frac{1-2x}{2x+3}}$</p>
</blockquote>
<p>Draw a simple table:</p>
<p>$$
\begin{array}{c|ccccc}
x & \text{under $-3/2$} & \text{$-3/2$} & \text{$]-3/2;1/2[$} & \text{$1/2$} & \text{over $1/2$} \\
\hline
1-2x & + & + & + & 0 & - \\
2x... |
9,648 | <p>To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution.</p>
<p>One can show this as follows: Consider the following polynomial in ${\mathbb Z}_p[x]$: $x^{4k}-1$, where $p=4k+1$. The roots of this polyno... | T.. | 467 | <p>If I remember correctly, this approach with a Gauss sum (associated to an 8th root of 1 in a finite field) is used in the first page or two of Serre's <em>Cours d'Arithmetique</em> to determine the "supplementary law" for $\left( \frac{2}{p} \right)$. The magical algebraic identity in this case is that if $a^4 = -1... |
4,086,894 | <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$V,W$</span> be vector spaces. Let <span class="math-container">$\rho: G \to GL(V)$</span> and <span class="math-container">$\tilde{\rho}: G \to GL(W)$</span> be two representations of <span class="math-container">$G$</spa... | Ruy | 728,080 | <p>It all boils down to how much importance one gives to the action of some algebraic structure on another.</p>
<p>The scalar multiplication in a vector space is perhaps the example we least pay attention to, so we just write
"<span class="math-container">$\lambda v$</span>",
a notation that does not ev... |
2,482,449 | <p>In the process of constructing a highway across a certain region in which there are many hills and valleys. the engineer will be certain that</p>
<p>There is some level in between the elevations of the highest hill and the
lowest valley at which the surface of the highway can be laid using the
tops of the hills as ... | Sunyam | 463,614 | <p>Multiply whole equation by $M$ and apply $U^{-1}$ and $U$ (matrix to diagonalize $M$)to the resulting equation. Then you know what to do.</p>
|
2,482,449 | <p>In the process of constructing a highway across a certain region in which there are many hills and valleys. the engineer will be certain that</p>
<p>There is some level in between the elevations of the highest hill and the
lowest valley at which the surface of the highway can be laid using the
tops of the hills as ... | user438666 | 438,666 | <p>$$I=M^{-1}M=\frac{M^3}{\alpha}-M^2-\frac{11}{a}M$$
thus
$$\frac{M^3}{\alpha}-M^2-\frac{11}{a}M-I=0$$
or
$$M^3-\alpha M^2-11M-\alpha I=0$$</p>
<p>By the Cayley Hamilton theorem, a matrix is a root of its characteristic polynomial, thus</p>
<p>$$M^3-\alpha M^2-11M-\alpha I=(M-3I)(M-2I)(M-I) $$</p>
<p>Now,</p>
<p>W... |
682,379 | <p>$$
f(x,y) =
\begin{cases} \frac{x^3}{x^2+y^2} & \text{ for } (x,y) \ne (0,0)\\
0 & \text{ for } (x,y) = (0,0)
\end{cases}$$</p>
<p>I know how to prove the function is continuous at $(x,y) \ne (0,0)$ but I don't know how to prove it is continuous at zero. I know it is possible with ... | Ant | 66,711 | <p>if you set $$x = \rho \cos\theta, y = \rho \sin\theta$$ the limit becomes</p>
<p>$$\lim_{\rho \to 0} \frac{\rho^3 \cos^3{\theta}}{\rho^2} = \lim_{\rho \to 0} \rho \cos^3{\theta} = 0 $$</p>
<p>Because $\cos\theta \le 1$, so $|\rho \cos^3{\theta}| \le |\rho| \to 0$</p>
|
102,068 | <p>I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve partial differential equations using a finite difference scheme, I'm generally more interested in the solution, its stab... | Willie Wong | 1,543 | <p>For your specific question, observe that
$$ \nabla^2 u = 0 $$
is a special case of the eigenvalue equation
$$ \nabla^2 u = \lambda u $$
with $\lambda = 0$. So understanding the eigenvalue problem certain helps understanding the homogeneous problem. </p>
<p>For the inhomogeneous problem, formally if one knows all ... |
65,944 | <p>How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$?</p>
<p>I have to prove it using lattice paths, it should be related to Catalan numbers</p>
<p>The $n$th Catalan number $C_n$ counts the number of monotonic paths along the edges of a grid with $n\times n$ square... | Marko Riedel | 44,883 | <p>This one can also be done using complex variables.
Suppose we seek to evaluate
$$\sum_{k=1}^n \frac{1}{k} {2k-2\choose k-1}
{2n-2k+1\choose n-k}.$$</p>
<p>Introduce the integral representation
$${2n-2k+1\choose n-k} =
\frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{(1+z)^{2n-2k+1}}{z^{n-k+1}} \; dz.$$
This has the prop... |
1,179,036 | <p>I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations:</p>
<p>$\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$</p>
<p>$\sqrt{(x-x_2)^2+(y-y_2)^2+(z-z_2)^2}=r_2$</p>
<p>$\sqrt{(x-x_3)^2+(y-y_3)^2+(z-z_3)^2}=r_3$</p>
<p>$(x,y,z)$ is ... | Amzoti | 38,839 | <p>The regular Newton-Raphson method is initialized with a starting point $x_0$ and then you iterate $x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}$. </p>
<p>In higher dimensions, there is an exact analog. We define:</p>
<p>$$F\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}f_1(x,y,z) \\ f_2(x,y,z) \\ f_3(x,y,z)\end{bma... |
302,162 | <p>How can we determine if any pair of the following graphs are isomorphic to each other? Is there an efficient way to know for sure? The obvious things to check for (number of edges, vertices, degrees) aren't fruitful because all three graphs have the same of each. Any suggestion appreciated.<img src="https://i.stack.... | Chris Eagle | 5,203 | <p>Two graphs are isomorphic if and only if their complements are isomorphic. The complement of $G_1$ is a $7$-cycle, while the complements of $G_2$ and $G_3$ are both the disjoint union of a $4$-cycle and a $3$-cycle. Thus $G_2$ and $G_3$ are isomorphic to each other but not to $G_1$.</p>
|
302,162 | <p>How can we determine if any pair of the following graphs are isomorphic to each other? Is there an efficient way to know for sure? The obvious things to check for (number of edges, vertices, degrees) aren't fruitful because all three graphs have the same of each. Any suggestion appreciated.<img src="https://i.stack.... | hmakholm left over Monica | 14,366 | <p>As for the general question: No efficient <em>general</em> procedure is known for determining whether two graphs are isomorphic.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Graph_isomorphism_problem" rel="nofollow">graph isomorphism problem</a> is somewhat famous for being one of the few problems in NP that ar... |
1,417,155 | <p>Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$.
Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the distribution of $X_2$ that would be helpful. I saw this in a problem and I can't really start trying it. Thanks for any he... | air | 181,046 | <p>You already know the distribution of $X_2$ conditionally on $X_1$, i.e.:</p>
<p>$$ X_2 | X_1 = x_1 \sim U[0,x_1]$$</p>
<p>This means that for $t \in [0,1]$:</p>
<p>$$ F_{X_2|X_1=x_1}(t) = \Pr[X_2 \leq t \vert X_1=x_1] = \min\left(\frac{t}{x_1},1\right)$$</p>
<p>Then the unconditional result follows from $X_1 \si... |
1,279,955 | <blockquote>
<p>Let A be real square matrix of order 7, then A has 6-dimensional invariant subspace.</p>
</blockquote>
<p>How to prove it?</p>
| Adelafif | 229,367 | <p>First, since the characteristic polynomial has degree 7, there is at least one real eigenvalue. Going to the Jordan form we see that either we have one block or more. This determines the invariant spaces. If we have one block then we have only one dimensional invariant subspace. </p>
|
778,118 | <p>I have a textbook question that asks to use L'Hopital to show which of the two functions: $e^{0.1 x}$ vs $x^{10}$ is dominant as $x \to \infty$. That is, using:
$$\lim_{x \to \infty} \left( \frac {x^ {10} } {e^{0.1 x}} \right) =
\lim_{x \to \infty} \left( \frac {10x^ 9 } {0.1e^{0.1 x}} \right) =
\lim_{x \to \inft... | Ellya | 135,305 | <p>Basically $e^{ax}=1+ax+(ax)^2/2+...+(ax)^{10}/((10)!)+...=\sum_{n=0}^\infty\frac{(ax)^n}{n!}$</p>
<p>so you will see that eventually $e^{ax}$ is dominant over any polynomial as it has "infinite" degree.</p>
<p>You are correct about the root as well, but the problem is that $e^{0.1x}=x^{10}$ is an implicit equatio... |
778,118 | <p>I have a textbook question that asks to use L'Hopital to show which of the two functions: $e^{0.1 x}$ vs $x^{10}$ is dominant as $x \to \infty$. That is, using:
$$\lim_{x \to \infty} \left( \frac {x^ {10} } {e^{0.1 x}} \right) =
\lim_{x \to \infty} \left( \frac {10x^ 9 } {0.1e^{0.1 x}} \right) =
\lim_{x \to \inft... | Community | -1 | <p>Consider the ratios of successive values $\frac{e^{0.1(x+1)}}{e^{0.1x}}$ vs. $\frac{(x+1)^{10}}{x^{10}}$, i.e. $e^{0.1}$ vs. $(1+\frac{1}{x})^{10}$.</p>
<p>In the first case, you multiply each time by a constant. In the second, by a term that goes decreasing. The first function grows faster when $e^{0.1}\gt(1+\frac... |
2,632,668 | <p>It is well known that gamma function is not defined at negative integers , but my question is to know how i take the value of $\binom{n}{p}$ for $p>n$ then is this make a sense or it is $0$ by convention ? </p>
| Albert | 19,331 | <p>Define $\binom{n}{p}$ as the number of subsets of $\{1, \ldots, n\}$ having exactly $p$ elements.</p>
<p>Then it makes mathematical sense to say that $\binom{n}{p}=0$ if $p>n$.
Of course, if you choose this definition then you have to prove that $\binom{n}{p} = \frac{n!}{p!(n-p)!}$ for all $p, n \in \mathbb N$ s... |
761,947 | <p>Suppose that we are given the following integral:</p>
<p>$$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$</p>
<p>(<a href="https://i.stack.imgur.com/SLzxw.png" rel="nofollow noreferrer">Original screenshot</a>)</p>
<p>And the answer is one of these :- </p>
<blockquote>
<ol>
<li>$4\sqrt3-4-\frac\... | Anastasiya-Romanova 秀 | 133,248 | <p>\begin{align}
\int_0^\pi\sqrt{4\sin^2\frac{x}{2}-4\sin\frac{x}{2}+1}\,dx&=\int_0^\pi\sqrt{\left(2\sin\frac{x}{2}-1\right)^2}\,dx\\
&=\int_0^\frac{\pi}{3}\left(1-2\sin\frac{x}{2}\right)\,dx+\int_\frac{\pi}{3}^\pi\left(2\sin\frac{x}{2}-1\right)\,dx\\
&=\left[x+4\cos\frac{x}{2}\right]_0^\frac{\pi}{3}+\left[... |
4,081,889 | <p>For which <span class="math-container">$\alpha\in\mathbb{R}$</span> the integral <span class="math-container">$I=\int_0^\infty \frac{t^{\frac{\alpha-n+1}{n}}}{e^t} dt$</span> converges?</p>
<p>I tried to write I as sum of two integrals, from 0 to 1 and from 1 to <span class="math-container">$\infty$</span>:</p>
<p>... | Olivier Oloa | 118,798 | <p>We assume <span class="math-container">$n>0$</span>.</p>
<p><strong>Hint</strong>. As <span class="math-container">$t\to 0^+$</span>,
<span class="math-container">$$
\frac{t^{\frac{\alpha-n+1}{n}}}{e^t}\sim t^{\frac{\alpha-n+1}{n}}
$$</span> giving that <span class="math-container">$\displaystyle\int_0^1 \frac{t^... |
3,156,557 | <p>I've having trouble understanding what the question is trying to ask. And I am not sure how to start to answer the question.</p>
<hr>
<p>The diagram below shows what happens for waves on the surface of a pond. If you
drop a stone in the point at the point <span class="math-container">$F_1$</span> at time <span cla... | Maxime Ramzi | 408,637 | <p>Your reasoning doesn't work per se : <span class="math-container">$k$</span> might have an uncountable cardinality that is <span class="math-container">$<|\mathbb{C}|$</span> (for instance if the continuum hypothesis fails). </p>
<p>But in this situation you can fix the issue. Here's a general situation where yo... |
3,150,437 | <p>I want to show that for a sequence <span class="math-container">$a=\{a_i\}^{\infty}_1 \in l^1$</span> I can write
<span class="math-container">$$a= \sum ^\infty _{i=1} a_i e_i $$</span>
For <span class="math-container">$\{e_i\}^\infty_1 \in l^1$</span> where the ith term is 1 and all else zero.</p>
<p>I know for <... | mfl | 148,513 | <p>In my opinion your answer is correct. Mimicking your computations I got the same result.</p>
<p>Defining <span class="math-container">$t=\dfrac{x^2+1}{x^2}$</span> we got that <span class="math-container">$$x^2=\dfrac{1}{t-1}.$$</span> So</p>
<p><span class="math-container">$$f(t)=f\left(\dfrac{x^2+1}{x^2}\right)=... |
1,025,175 | <p>I do not understand what sets like these are. I know what something like $\mathbb{Z}_7$ is. It is the ring of integers modulo 7 so it is equal to ${0,1,2,3,4,5,6}$. But what is $\mathbb{F}_7[X]$ equal to. I don't understand. I have spent ages searching on the internet but can't find anything on it. All I know is tha... | layman | 131,740 | <p>Well, you know what the ring $\mathbb{Z}_{n}$ is for any integer $n$. It turns out that if $n$ is a prime number (like $2$, $3$, $5$, $7$, etc.) (when $n$ is prime, it is usually replaced with the letter $p$) then $\mathbb{Z}_{p}$ is actually a field! It's not just a ring, but it also has a multiplicative identity... |
911,370 | <p>I need to prove the following:</p>
<blockquote>
<p>A set $X$ is infinite if and only if it is equipotent to a proper subset of itself</p>
</blockquote>
<p>Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or equivalently, there is no bijection $\mathbb{N}_n \rightarrow X$ for $n\in \math... | Asaf Karagila | 622 | <p>Let me address quickly the first part of the proof, $\Rightarrow$. You don't need to do it by contradiction. You can show that if $f\colon X\to A$ is a bijection, where $A$ is a proper subset of $X$, then there is an injection from $\Bbb N$ into $X$, and therefore $X$ is infinite.</p>
<p>To your second question, pe... |
1,259,383 | <p>I have a distribution with literally an infinite number of potential data points. I need the standard deviation. I generate about a hundred points and take the standard deviation of the points. This gives a hopefully good approximation of the true standard deviation, but it won't, of course, be exact. How do I e... | MichaelChirico | 205,203 | <p>If you're allowed to take that sample repeatedly, it's basically bootstrapping.</p>
<p>Procedure:</p>
<ol>
<li><p>Draw 100 points</p></li>
<li><p>Calculate standard deviation</p></li>
<li><p>Repeat Steps 1 & 2 a lot of times (empirically, I've found 5-10,000 to be enough), keeping track of the results of step ... |
1,781,293 | <p>The limit:</p>
<p>$\lim_{x\rightarrow\infty}
\left( {\frac {2\,x+a}{2\,x+a-1}} \right) ^{x}$</p>
<p>I make this:</p>
<p>$\left( {\frac {2\,x+a}{2\,x+a-1}} \right) ^{x}$=${{\rm e}^{{\it x\ln} \left( {\frac {2\,x+a}{2\,x+a-1}} \right) }}$</p>
<p>Then:</p>
<p>${{\rm e}^{{\it \lim_{x\rightarrow\infty}
x\ln} \left... | MoonKnight | 115,071 | <p>Hint:
$$
\frac{2x+a}{2x+a-1} = 1 + \frac{1}{2x+a-1}
$$</p>
<p>Does that look a little similar to the following now?
$$
\lim_{n->\infty} \left(1+\frac{1}{n}\right)^n
$$</p>
|
1,692,987 | <p>General solutions of trigonometric equations are given by: </p>
<blockquote>
<p>If $\sin(x) = \sin(y)$, then $x = n \pi + (-1)^ny$<br>
If $\cos(x) = \cos(y)$, then $x = 2n \pi \pm y$ </p>
</blockquote>
<p>If we consider an example, </p>
<p>$$
\sin(x) = \sin(30^\circ) \\
\implies x = n \pi + (-1)^n30^\cir... | Steven Alexis Gregory | 75,410 | <p>Its best in situations like this to think in terms of the unit circle.</p>
<p><a href="https://i.stack.imgur.com/0TmYY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0TmYY.jpg" alt="enter image description here"></a></p>
|
172,924 | <p>Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?</p>
| Akhil Mathew | 344 | <p>Let $\alpha \in \pi_s(S^0)$ for $s > 0$ be an element of positive degree in the stable stems. Then $\alpha$ has positive-dimensional filtration in the Adams-Novikov spectral sequence: in other words, it is annihilated by complex bordism. </p>
<p>The $E_\infty$-page of the Adams-Novikov spectral sequence for the ... |
2,431,287 | <p>Suppose there is a coin toss game where quarters are thrown onto a checkerboard. Management keeps all of the quarters; however, if a quarter lands entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of the quarter, and that outcomes a... | quasi | 400,434 | <p>Hints:
<p>
Assume the radius of the quarter is $1$ unit.
<p>
The center of the tossed coin will land at a uniformly random point in some $4{\,\times\,}4$ square.
<p>
Where do the centers need to be in order for the player to win?
<p>
Graph the region of winning centers.
<p>
The area of the region of winning cente... |
1,735,087 | <p>I have the following question in my notes. </p>
<blockquote>
<p>Let $A \in H^*$ and let $F=A^{-1}({0})$ $F$ is a closed linear
subspace. Show that for any choice of $u,w \in H$ with $Au$ non zero
the vector is $w-\frac{Aw}{Au} u$ is an element of F.</p>
</blockquote>
<p>The solution is as below. </p>
<bloc... | Ian | 83,396 | <p>It seems more complicated than it has to be. The point is that:</p>
<p>$$A \left ( w - \frac{Aw}{Au} u \right ) = Aw - \frac{Aw}{Au} Au = Aw - Aw = 0$$</p>
<p>which is the defining property for a vector to be in $F$. Here all I have used is the fact that $A$ is a linear functional defined on all of $H$. No continu... |
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| Samuel | 533,774 | <p>Look into <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling's Approximation</a> and use asymptotics.</p>
|
4,476,229 | <p>In Theorem 5.5, Rudin proves the chain rule, but does so in a somewhat different fashion than expected. It seems we can prove the chain rule more easily.</p>
<p>Theorem:
Suppose <span class="math-container">$f:[a,b]\to\mathbb{R}$</span> is continuous on <span class="math-container">$[a,b]$</span> and <span class="m... | Ethan Bolker | 72,858 | <p>Suggested start too long for a comment.</p>
<p>The third and fourth equations have the form</p>
<p><span class="math-container">$$
x = r\cos \sigma + s \sin \tau
$$</span>
<span class="math-container">$$
y = r\sin \sigma + s \cos \tau
$$</span>
with <span class="math-container">$x$</span>, <span class="math-containe... |
2,190,883 | <p>My textbook asks me to prove that the number $$1\underbrace{000\ldots0}_{2012}5\underbrace{000\ldots0}_{2012}1$$
is not an exact cube of any natural number. And in the answers section it states that one shall use the properties of modulo 9, however I can't understand, why the number 9 is chosen for the task. Why do... | Misha Lavrov | 383,078 | <p>The first step of any problem in enumerative combinatorics is to compute the first few cases, then look the sequence up on OEIS. If we do that here, we find <a href="http://oeis.org/A002464" rel="noreferrer">A002464</a>:</p>
<blockquote>
<p>Hertzsprung's problem: ways to arrange n non-attacking kings on an n X n ... |
1,803,334 | <p>I am confusing how to determine the set is clopen, neither open or closed, open but not closed and closed but not open. I read an example from "Topology without Tears". </p>
<p>Let $X=\{a,b,c,d,e,f\}$ and $\tau=\{X,\emptyset,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$. $\tau$ is a topology on $X$. Then </p>
<ol>
<li>... | Daron | 53,993 | <p>The set $A$ is open if $A$ is listed as an element of $\tau$. To check just look at the list.</p>
<p>The set $A$ is closed if $X-A$ is listed as an element of $\tau$. To check first calculate $X-A$ then look at the list. </p>
<p>For example if $A = \{b,c\}$ then $X-A = \{a,d,e,f\}$.</p>
<p>The set $A$ is clopen i... |
167,981 | <p>My question is:</p>
<blockquote>
<p>Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$</p>
</blockquote>
<p>Any help to solve this question would be greatly appreciated.</p>
| Gigili | 181,853 | <p>$$
\begin{align}
& {}\quad (x^{11} + x^{10}) + (x^9 + x^8)+(x^7+x^6)+(x^5+x^4)+(x^3+x^2 )+( x + 1)\\[8pt]
& =x^{10}(x+1)+x^8(x+1)+x^6(x+1)+x^4(x+1)+x^2(x+1)+(x+1)\\[8pt]
& =(x+1)(x^{10}+x^8+x^6+x^4+x^2+1)\\[8pt]
& =(x+1)(x^8(x^2+1)+x^4(x^2+1)+x^2+1)\\[8pt]
& =(x+1)((x^2+1)(x^8+x^4+1))\\[8pt]
&... |
2,510,712 | <blockquote>
<p>If A is an integral domain, we have seen that in $A[x]$, $\text{deg }a(x)b(x)=\text{deg }a(x)+\text{deg }b(x)$. Show that if $A$ is not an integral doamin we can find polynomails $a(x), b(x)$ such that $$\text{deg}\ a(x)b(x) <\text{ deg}\ a(x)+ \text{deg}\ b(x).$$</p>
</blockquote>
<p>So if we don... | Alex Provost | 59,556 | <p>Yes, that's the idea. Suppose that $a \in A$ is a zero divisor, so that $ab = 0$ for some $b \neq 0$. Consider the constant polynomial $a(x) = a$ and the linear polynomial $b(x) = bx$. Then $$\deg a(x)b(x) = \deg abx = \deg 0 = -\infty < 1 = 0 + 1 = \deg a + \deg bx = \deg a(x) + \deg b(x)$$</p>
|
760,330 | <p>For every positive integer $n$, prove that
$$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$</p>
<p>Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ denotes the greatest integer not exceeding $x$. </p>
<p>This question was posed to me in class by my teacher.... | lab bhattacharjee | 33,337 | <p>Observe that $$\lfloor\sqrt{4n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$$</p>
<p>unless $4n+2$ is perfect square</p>
<p>But any square $\equiv0,1\pmod4$ </p>
<p>$$\implies\lfloor\sqrt{4n+1}\rfloor=\lfloor\sqrt n+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$$</p>
|
2,768,608 | <p>$$\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{\sqrt i}$$</p>
<p>This question was asked in an entrance test for an undergraduate program in India. I want to know how to approach such questions.</p>
<p>I was thinking of finding the partial sum till n and applying limit concepts to get an answer, but couldn't find such a... | D.B. | 530,972 | <p>The sum diverges. Its partial sums are larger than the harmonic series, which is know to diverge.</p>
|
2,737,911 | <p>We can find a countable base for $\mathbf{R}^{n}$ using fact that $\mathbf{Q}$ is dense in the reals (example of proof by someone else is here: <a href="https://math.stackexchange.com/questions/2483544/proof-that-mathbfrn-has-a-countable-base">Proof that $\mathbf{R}^n$ has a countable base.</a>).</p>
<p>We also hav... | Sally G | 156,064 | <p>If $d$ and $d’$ are two metrics on a set $X$ that induce the topologies $T$ and $T'$, then $T'$ is finer than T if for any x in X and any $\epsilon > 0$ , there is some $\delta > 0$ such that the $\delta-neighborhood$ of $x$ in the $d'$ metric is contained in the $\epsilon-neighborhood$ of the $d$ metric.</p>
... |
2,737,911 | <p>We can find a countable base for $\mathbf{R}^{n}$ using fact that $\mathbf{Q}$ is dense in the reals (example of proof by someone else is here: <a href="https://math.stackexchange.com/questions/2483544/proof-that-mathbfrn-has-a-countable-base">Proof that $\mathbf{R}^n$ has a countable base.</a>).</p>
<p>We also hav... | WKhan | 551,413 | <p>Let $d$, $d_1$, and $d_2$ be metric spaces on $X$ inducing topologies $\it T_d$, $\it T_{d_1}$, and $\it T_{d_2}$.</p>
<p>I want to show that $\it T_d$ is finer than both $\it T_{d_1}$, and $\it T_{d_2}$.</p>
<p>$\it T_d$ is finer than $\it T_{d_1}$ if and only if for all $x\epsilon X$ and $\epsilon>0$, there i... |
1,095,416 | <p>$e$ and $\pi$ are rather peculiar numbers. It turns out that, in
addition to being irrational numbers, they are also transcendental
numbers. Basically, a number is transcendental if there are no
polynomials with rational coefficients that have that number as a
root.</p>
<p>Clearly, $p(x) = (x-e)(x-\pi)$ is a po... | Math137 | 60,099 | <p>I would like to post an alternative proof:</p>
<p>By <strong>Proposition 2.12</strong> of the ops answer (in other words by the universal property of tensor product) the following bilinear map</p>
<p>\begin{eqnarray*}
\psi:A/\mathfrak{a}\times M & \to & M/\mathfrak{a}M \\
(x+\mathfrak{a},M) &a... |
1,095,416 | <p>$e$ and $\pi$ are rather peculiar numbers. It turns out that, in
addition to being irrational numbers, they are also transcendental
numbers. Basically, a number is transcendental if there are no
polynomials with rational coefficients that have that number as a
root.</p>
<p>Clearly, $p(x) = (x-e)(x-\pi)$ is a po... | egreg | 62,967 | <p>This is mostly similar to your argument, but much easier.</p>
<hr>
<p>From the exact sequence $0\to\mathfrak{a}\to A\to A/\mathfrak{a}\to 0$ we get the commutative diagram with exact rows
$$\require{AMScd}\def\ma{\mathfrak{a}}
\begin{CD}
{} @. \ma\otimes_AM @>>> A\otimes_AM @>>> A/\ma\otimes_AM @... |
107,443 | <p>Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure.
Is it true that
$$
\int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx)
$$
where $f(x)$ is some continuous function?</p>
| Norbert | 19,538 | <p>In general this formula doesn't hold.</p>
<p>Consider case $n=3$, and $M=\{(x,y,z):0\leq x\leq 1,\quad 0\leq y\leq 1\quad z=0\}$ is a square on $xy$-plane. Define linear transformation $T$ by matrix
$$
T_k=\begin{vmatrix}1 && 0 && 0\\0 && 1 && 0\\0 && 0 && k \end{vmat... |
666,229 | <p>What is ML inequality property in complex integral which says $|\int_{c}f(z)dz| \leq ML$. I can't understand a thing from this expression.
I want to understand it conceptually(if that helps).<br> How can we find the upper bound of a complex integral</p>
| Ulrik | 53,012 | <p>$L$ is the arc length of $c$, $M$ is an upper bound for the absolute value of $f$ on $c$.</p>
<p>Let's compare the result to real integrals: Let $f$ be defined on $[a,b]$ and $|f|$ bounded by $M$. Then:</p>
<p>$\left| \int_a^b f(x) dx \right| \leq \int_a^b |f(x)| dx \leq \int_a^b M dx = M \int_a^b dx = M(b-a)$.</p... |
1,426,383 | <p>Is there any equation which describes or estimates the number of singular values of a Matrix $X$ ?<br>
I found out that the number is equal to the number of eigenvalues of the Matrix $X^{*} X$, which are calculates as: $det( \lambda *I- X^{*} X)=0$. From these Eigenvalues I have to take the square roots and will get... | mickep | 97,236 | <p>The integral in the numerator is zero, since the function you integrate is odd and the interval you integrate over is symmetric with respect to zero.</p>
|
2,658,126 | <p>I'm reading this textbook question and I'm confused as to why this answer is in radians and not degrees. How did that happen?</p>
<p><a href="https://i.stack.imgur.com/UvmeB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UvmeB.png" alt="enter image description here"></a></p>
| welshman500 | 393,140 | <p>If the question is about "why we use radians instead of degrees in calculus", look at following thread: <a href="https://math.stackexchange.com/questions/720924/why-do-we-require-radians-in-calculus">Why do we require radians in calculus?</a></p>
|
2,658,126 | <p>I'm reading this textbook question and I'm confused as to why this answer is in radians and not degrees. How did that happen?</p>
<p><a href="https://i.stack.imgur.com/UvmeB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UvmeB.png" alt="enter image description here"></a></p>
| Peter Green | 278,485 | <p>The set of trig differentiation rules pretty much everyone uses is for the radians version of the trig functions. We use radians because they make the differentiation and integration of trig functions come out without ugly constants.</p>
<p>You <em>could</em> write a set of differentiation rules for "trig function ... |
1,507,187 | <p>I am reading "Topology without tears " book, and got confused about proposition 2.3.4 page 60:</p>
<p>$\tau_1 =\tau_2$ iif</p>
<p>1) for each $B \in \mathscr B_1 $ and $ \forall x \in B , \exists B' \in \mathscr B_2 $ such that $ x\in B' \subseteq B $</p>
<p>2) for each $B \in \mathscr B_2 $ and $ \forall x \in... | Dylan | 135,643 | <p>Don't integrate by parts right away. Note that $d(x^2)=2x\,dx$. Let $u = x^2$</p>
<p>$$ \int x\arctan x^2 dx = \frac{1}{2} \int\arctan u\,du$$</p>
<p>You can take it from here.</p>
|
3,811,012 | <p>I wish to find <span class="math-container">$\displaystyle \lim_{n \rightarrow \infty}\frac{n+1}{\sqrt{n}}$</span>.</p>
<p>Here is what I did:</p>
<p><span class="math-container">$1.$</span> Rewrite <span class="math-container">$\frac{n+1}{\sqrt{n}}$</span> to <span class="math-container">$(n+1) \cdot \frac{1}{\sqrt... | Community | -1 | <p>There are two problems in your attempt:</p>
<ol>
<li><p><span class="math-container">$\lim a\cdot b=\lim a\cdot\lim b$</span> can only be used if the limits exist, and this is not the case;</p>
</li>
<li><p>the expression <span class="math-container">$\infty\cdot0$</span> has no meaning as a function from <span clas... |
3,811,012 | <p>I wish to find <span class="math-container">$\displaystyle \lim_{n \rightarrow \infty}\frac{n+1}{\sqrt{n}}$</span>.</p>
<p>Here is what I did:</p>
<p><span class="math-container">$1.$</span> Rewrite <span class="math-container">$\frac{n+1}{\sqrt{n}}$</span> to <span class="math-container">$(n+1) \cdot \frac{1}{\sqrt... | Anas anas | 829,025 | <p>Answer :</p>
<p><span class="math-container">$\lim_{n \to +\infty } \frac{n+1}{\sqrt{n}}$</span>= <span class="math-container">$\frac{\sqrt{n}(\sqrt{n} +\frac{1}{\sqrt{n}})}{\sqrt{n}} $</span>= <span class="math-container">$\lim_{n \to +\infty } \sqrt{n} +\frac{1}{\sqrt{n}}$</span> = <span class="math-container">$+\... |
2,436,634 | <p>The limit is $$ \lim_{(x,y)\to(0,0)} \frac{x\sin(y)-y\sin(x)}{x^2 + y^2}$$</p>
<p>My calculations: I substitute $y=mx$</p>
<p>\begin{align}\lim_{x\to 0} \frac{x\sin(mx)-mx\sin(x)}{x^2 + (mx)^2} &= \lim_{x\to 0} \frac{x(\sin(mx)-m\sin(x)}{x^2(1 + m^2)}\\ &= \lim_{x\to 0} \frac{1}{1+m^2}\bigg[\frac{\sin(mx)}... | José Carlos Santos | 446,262 | <p>No, you cannot. The limit$$\lim_{x\to0}\frac{\sin(mx)}x$$<em>does</em> exist. It is equal to $m$.</p>
<p>But, and that's more importante, the limit$$\lim_{x\to 0} \frac{1}{1+m^2}\bigg[\frac{\sin(mx)}{x}- \frac{m\sin(x)}{x}\bigg]$$is equal to $0$ for every $m$.</p>
|
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Tanner Swett | 13,524 | <p>We can do anything we want!</p>
<p>Specifically, we can <em>define</em> anything we want (as long as our definitions don't contradict each other). So if we want to allow ourselves to use imaginary numbers, all we have to do is write something like the following:</p>
<blockquote>
<p>Define a <strong>complex numbe... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Red Banana | 25,805 | <ul>
<li><p>What allow us to use anything in mathematics are the axioms. Real numbers form a complete ordered field, if you take the set of real numbers and "add" <span class="math-container">$i$</span> to it (considering its usual behavior), you lose this order (that is: They won't be a complete ordered field anymore)... |
3,534,075 | <p>Let </p>
<p><span class="math-container">$$
Y = \{ x = (x_n) \in l^2: x_{2n} = 0, \, n \in \mathbb{N}\}
$$</span></p>
<blockquote>
<p>If <span class="math-container">$x = (x_n) \in Y^{\perp}$</span> then <span class="math-container">$$ x_{2n + 1} = \langle x, e_{2n +
1} \rangle = 0, \, \forall n \in \mathbb{N}... | Mark Bennet | 2,906 | <p>Since an answer has been given, here is a hint of another way to proceed.</p>
<p>Let <span class="math-container">$B=A^2+A+I$</span> and for an arbitrary vector <span class="math-container">$x$</span> consider the vector <span class="math-container">$y=Bx$</span>.</p>
|
4,215,924 | <p>The Weak Factorization Theorem tells us that birational map of varieties over field of perfect characteristic which has resolution of singularities can be factored into blow-ups and blow-downs. My question is what happens when we restrict ourself to birational morphisms instead of birational maps? Can we then assume... | Evans Gambit | 951,422 | <p>This is true for surfaces for example. See [1] for general result. But in dimension >2 I am quite sure there will be an example of a birational morphism <span class="math-container">$f:X\to Y$</span> which cannot be factored as a composition of blow-up.</p>
<p>[1] Stacks Project : <a href="https://stacks.math.col... |
1,714,965 | <p>I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic?</p>
<p>My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have the quality that $p_1 \times a = p_2 \times b$ where $a \space\text{and}\space b$ are integers, then the enti... | hmakholm left over Monica | 14,366 | <p>If $f(x)=\sin(x)\sin(\pi x)$ were periodic, then in particular its set of zeroes would be periodic. But the zero set is $\mathbb Z\cup\pi\mathbb Z$, and $a\mathbb Z\cup b\mathbb Z$ (for nonzero $a$, $b$) is not periodic unless $b/a$ is rational.</p>
<p>(Suppose $a\mathbb Z\cup b\mathbb Z$ is periodic with period $P... |
19,797 | <p>I'm trying to compute a multidimensional integral with a variable number of dimensions.</p>
<p>The integral is as follows:
$$
\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.
$$</p>
<p>I have tried this</p>
<pre><code>Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]
</code></pre>
<p>but it's not wor... | Carl Woll | 45,431 | <p>For your particular example, you could use <code>FullRegion[n]</code> as the integration domain:</p>
<pre><code>Integrate[Exp[-(β/(2m)) p.p], p ∈ FullRegion[6], Assumptions->β>0 && m>0]
</code></pre>
<blockquote>
<p>(8 m^3 π^3)/β^3</p>
</blockquote>
<p>which is the same as whuber's answer.</p>
|
4,610,892 | <p>Find all <strong>positive integers</strong> <span class="math-container">$x, y, z$</span> such that <span class="math-container">$x^2y+y^2z+z^2x = 3xyz$</span>.</p>
<blockquote>
<p>I first tried to solve it by spilting <span class="math-container">$xyz$</span> to every expression and factor it. But it fails.</p>
</b... | Raul Fernandes Horta | 1,125,344 | <p>If you assume one of the variables is <span class="math-container">$0$</span> (say <span class="math-container">$x=0$</span>), then we must have <span class="math-container">$y^{2}z = 0$</span>, so either <span class="math-container">$y=0$</span>, <span class="math-container">$z=0$</span> or both, hence the triples ... |
49,787 | <p>From the link in wikipedia </p>
<p><a href="http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii" rel="nofollow noreferrer">http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii</a></p>
<p><strong>OPEN QUESTION:</strong></p>
<p><strong>What is the e... | Jason | 202,602 | <p>In the special case of all 3 circles of the same radius $r$, the area is given by the equation $k r^2$ where $k=0.161254481$.</p>
|
2,574,922 | <p>I'm sorry this is really basic. I'm terrible with math and I'm struggling to help my son with homework:</p>
<p>Find an expression in terms of n for the nth term in this sequence:
$0 , 9 , 22 , 39, 60, \ldots$</p>
<p>We can get the $2n^{2}$ part of the answer but just can't get the rest. Any help gratefully appreci... | Franklin Pezzuti Dyer | 438,055 | <p>Given that the sequence is quadratic, start by assuming that
$$a_n=An^2+Bn+C$$
for some $A,B,C$. Since $a_0=0$, it follows immediately that $C=0$ and
$$a_n=An^2+Bn$$
Furthermore, since $a_1=9$ and $a_2=22$, you have
$$a_1=A+B=9$$
$$a_2=4A+2B=22$$
and you have the system of equations
$$A+B=9$$
$$4A+2B=22$$
When solve... |
1,817,609 | <p>Here is a list of other systems:</p>
<ul>
<li>Babylonian numerals</li>
<li>Egyptian numerals</li>
<li>Aegean numerals</li>
<li>May numerals</li>
<li>Chinese numerals</li>
</ul>
<p>These system are far older than the current system. How did it get to be known and used internationally by nearly every cultures these ... | Bill Thomas | 165,262 | <p>First of all, the Egyptian and Aegean systems are "agglutinative" rather than positional. I'm looking at the Unicode Aegean numbers block and I see that they have a specific symbol for ninety thousand. What if you need to represent one hundred thousand? Do you just invent a new symbol for it?</p>
<p>Scientists esti... |
390,438 | <p>Suppose X,Y are sets with at least 2 elements. Show that
$X\cup Y\le X\times Y$</p>
<p>So my first thought was that cardinality $|X|\ge 2$ and the same for $|Y|\ge 2$ but by the inclusion-exclusion principle we have $|X\cup Y|=|X|+|Y|-|X\cap Y|$ but the problem does not say if they are disjoint or not.
If we assum... | Edoardo Lanari | 77,181 | <p>We can do far better: suppose $(\kappa_{i})_{i \in I},(\lambda_{i})_{i \in I}$ are two families of cardinals with $ \kappa_{i} <\lambda_{i} \forall i \in I$, then $ \coprod_{i\in I} k_i < \prod_{i\in I}\lambda_i$ (the equality case is easily treated separatedly).</p>
|
390,438 | <p>Suppose X,Y are sets with at least 2 elements. Show that
$X\cup Y\le X\times Y$</p>
<p>So my first thought was that cardinality $|X|\ge 2$ and the same for $|Y|\ge 2$ but by the inclusion-exclusion principle we have $|X\cup Y|=|X|+|Y|-|X\cap Y|$ but the problem does not say if they are disjoint or not.
If we assum... | AndreasT | 53,739 | <p>For any sets $A,B$ (either disjoint or not) the cardinality of the Cartesian product is the product of their cardinality, while the cardinality of their union is less or equal the the sum of their cardinality. Let $a:=|A|$ and $b:=|B|$, then
$$
|A\times B| = ab \quad\text{and}\quad |A\cup B|=|A|+|B|-|A\cup B|\leq a+... |
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