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 |
|---|---|---|---|---|
4,174 | <p>I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together a collection of suggested readings from the math education literature. I would be interested to hear your suggestio... | MathTeacher | 3,528 | <p>I'd suggest literature on students' understanding of the equals sign, e.g.,
1. "Concepts Associated with the Equality Symbol" by Kieran. <a href="http://link.springer.com/article/10.1007/BF00311062" rel="nofollow">http://link.springer.com/article/10.1007/BF00311062</a>
2.A Longitudinal Examination of Middle School ... |
3,333,928 | <p>I am reading an example of root test for a serie: <span class="math-container">$$\sum_{n=1}^\infty\frac{3n}{2^n}.$$</span> So applying the root test, we get <span class="math-container">$$\lim_{n \to \infty}\sqrt[n]{\frac{3n}{2^n}}=\lim_{n \to \infty}\frac{\sqrt[n]{3n}}{2}=\frac{1}{2}\lim_{n\to\infty}\exp(\frac{1}{n... | trula | 697,983 | <p>a transcendental function f(x) gives transcendental results for most rational x
example: e^x, sin(x) etc.
the simple seaming equation e^x=x or cos(x)=x have no formula for x as result, but must be calculated numerically.
also you cn not rewrite e^x as a polynomial or a fraction of polynoms
trula</p>
|
96,289 | <p>In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, Diamond, and Taylor.</p>
<p>I'm asking this as a matter of interest. Are there any other fields over which elliptic curv... | David Roberts | 4,177 | <p>In the <a href="http://www.ams.org/notices/199911/comm-darmon.pdf" rel="nofollow">article in the Notices of the AMS</a> which came out when the BCDT proof was announced, it says</p>
<blockquote>
<p><em>Generalizations to other number fields.</em> A number
of ingredients in Wiles’s method have been significantly... |
2,485,261 | <blockquote>
<p>$\displaystyle \sum_{k=0}^n k {n \choose k} p^k (1-p)^{n-k}$ with $0<p<1$</p>
</blockquote>
<p>I know of one way to evaluate it (from statistics) but I was wondering if there are any other ways. </p>
<p>This is the way I know:</p>
<p>Let </p>
<p>$$M(t)=\displaystyle \sum_{k=0}^n e^{kt} {n \c... | A.G. | 115,996 | <p>You can also notice that this is the expectation $E(X)$ where $X$ is a binomial random variable with parameters $n$ and $p$, and $E(X)=n\,p$.</p>
|
2,764,818 | <blockquote>
<p>Let $f(x)=ax^3+bx^2+cx+d$, be a polynomial function, find relation between $a,b,c,d$ such that it's roots are in an arithmetic/geometric progression. (separate relations)</p>
</blockquote>
<p>So for the arithmetic progression I took let $\alpha = x_2$ and $r$ be the ratio of the arithmetic progressio... | orangeskid | 168,051 | <p>To sum up, a cubic has its roots in arithmetic progression if and only if the arithmetic mean of the roots is a root of the cubic ( so equals one of the roots).</p>
<p>For the geometric progression, we could use the same trick, and say that the geometric mean of $x_1$, $x_2$, $x_3$ is a root of $P$. Alternatively, ... |
188,938 | <p>Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)</p>
<p>Why is tha... | Community | -1 | <p>I can think of two reasons.</p>
<ol>
<li><p>When we do geometry, we usually work in Euclidean space, where the intrinsic property of a line segment between two points is its length, given (in two dimensions) by $\ell^2 = \Delta x^2 + \Delta y^2$. We are allowed to change our reference frame as long as we preserve l... |
188,938 | <p>Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)</p>
<p>Why is tha... | Argon | 27,624 | <p><span class="math-container">$\sinh$</span> and <span class="math-container">$\cosh$</span> seem to appear less then their circular counterparts in real analysis as is explained well in some other answers. However, hyperbolic functions appear quite commonly in complex analysis. From <a href="http://en.wikipedia.or... |
188,938 | <p>Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)</p>
<p>Why is tha... | Tunococ | 12,594 | <p>I'd say because hyperbolic functions can be written pretty easily in terms of exponential functions. By that I mean you don't need $i$ like when you express $\sin$ and $\cos$ using $\exp$. That means in the "real" world, it's not necessary to use $\sinh$ and $\cosh$ because you can always resort to $\exp$, but the s... |
2,008,263 | <p>Solve $$(1+y^2\sin2x) \;dx - 2y\cos^2x \;dy = 0$$</p>
<p>Well, first of all I've written $M = 1+y^2\sin2x$ , $N = 2y\cos^2x$.</p>
<p>Then, I noticed that $M'_y$ <strong>does not</strong> equal to $N'_x$.</p>
<p>I'm trying to find something to multiply the equation with, but my math skills sucks. So I'm going for ... | Max Payne | 232,145 | <p>I think the equation is exact, as</p>
<p>$$2y \sin 2x = 4y\cos x\sin x$$</p>
|
918,689 | <p>of 5 be selected that contain /at least/ 1 of the broken bulbs?</p>
<p>So far, I have tried only 1 method, as it's the only one I've been taught, but I don't know if I am doing it right.
I tried doing C(100,1)/C(100,5) but it just doesn't seem right. Is it? If it isn't, what am I doing wrong?</p>
| voldemort | 118,052 | <p>Hints:</p>
<p>1) Total number of ways to choose a sample of $5$ bulbs from $100$= $100 \choose 5$.</p>
<p>2) Total number of ways to choose $5$ non defective bulbs= $98 \choose 5$- as there are $98$ non defective bulbs.</p>
<p>Now subtract $(2)$ from $1$ to get your answer.</p>
|
954,933 | <p>Let $\phi\in\ell^\infty$. For $p\in[1,\infty]$, define $M_\phi:\ell^p\to\ell^p$ by</p>
<p>$$M_\phi(f)=\phi f.$$</p>
<p>Show that $\Vert M_\phi\Vert=\Vert\phi\Vert_\infty$, and $M_\phi$ is compact if and only if $\phi\in c_0$, i.e. $\phi$ is a sequence that converges to $0$.</p>
<p>I only have problem with the par... | Jordan | 116,955 | <p>First I would point out that the $t$ you have defined may in fact be zero, even if the sequence $M_\phi (f_n)$ has no Cauchy subsequence - all you would need is for <em>some</em> pair $f_n,f_m$ to map to the same sequence under $M_\phi$.</p>
<p>Now, to prove the claim, one relatively easy way is to express $M_\phi$... |
2,114,446 | <p>But, just to get across the idea of a generating function, here is how a generatingfunctionologist might answer the question: the nth Fibonacci number, $F_{n}$, is the coefficient of $x^{n}$ in the expansion of the function $\frac{x}{(1 − x − x^2)}$ as a power series about the origin.</p>
<p>I am reading a book abo... | Peter Taylor | 5,676 | <p>Your comments on angryavian's answer suggest that it would be worth showing some more intermediate steps. Given $f(x) = F_0 + F_1 x + F_2 x^2 + F_3 x^3 + \ldots$ we have $$\begin{eqnarray}
f(x) = & F_0 + & F_1 x + & F_2 x^2 + F_3 x^3 + F_4 x^4 + F_5 x^5 + \ldots \\
xf(x) = & & F_0 x + & F... |
2,349,124 | <p>I keep on hitting a road block in trying to solve this, especially when trying to prove it going from the right hand side to the left hand side. </p>
| Bram28 | 256,001 | <p>From right to left:</p>
<p>$$X=$$</p>
<p>$$(X \cap Y) \cup (X \cap Y^C)=$$</p>
<p>$$(X \cap ((X \cap Y^C) \cup (X^C \cap Y))) \cup (X \cap Y^C)=$$</p>
<p>$$((X \cap X \cap Y^C) \cup (X \cap X^C \cap Y)) \cup (X \cap Y^C)=$$</p>
<p>$$((X \cap Y^C) \cup \emptyset) \cup (X \cap Y^C)=$$</p>
<p>$$(X \cap Y^C) \cup ... |
919,572 | <p>Do you know any nice way of expressing </p>
<p>$$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$
?</p>
<p>Some simple manipulations involving the integrals lead to an expression that also uses<br>
the hypergeometric series. Is there any way of getting a form that doesn't use the HG function?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
919,572 | <p>Do you know any nice way of expressing </p>
<p>$$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$
?</p>
<p>Some simple manipulations involving the integrals lead to an expression that also uses<br>
the hypergeometric series. Is there any way of getting a form that doesn't use the HG function?</p>
| Pedro | 23,350 | <p>Let $f(x)=\displaystyle\sum\limits_{n\geqslant 0}\frac{1}{n+1}x^n=-x^{-1}\log(1-x)$</p>
<p>Then $\displaystyle g(x)=\frac{1}{1-x}f(x)=\sum_{n\geqslant 0}\sum_{k=0}^n \frac{1}{k+1}x^n=\sum_{n\geqslant 0}H_{n+1} x^n$ and you want the coefficient of $x^n$ (i.e. $f^{(n)}(0)/n!$) in $$f(x)g(x)=x^{-2}\log^2(1-x)/(1-x)$$... |
1,102,758 | <p><strong>Problem</strong></p>
<p>Given a pre-Hilbert space $\mathcal{H}$.</p>
<p>Consider unbounded operators:
$$S,T:\mathcal{H}\to\mathcal{H}$$</p>
<p>Suppose they're formal adjoints:
$$\langle S\varphi,\psi\rangle=\langle\varphi,T\psi\rangle$$</p>
<p>Regard the completion $\hat{\mathcal{H}}$.</p>
<p>Here they'... | Disintegrating By Parts | 112,478 | <p>Let $S=\frac{d}{dx}$ and $T=-\frac{d}{dx}$ on the linear subspace $\mathcal{H}=\mathcal{C}_{0}^{\infty}(0,2\pi)\subset \hat{\mathcal{H}}=L^{2}[0,2\pi]$ consisting of infinitely differentiable functions on $[0,2\pi]$ which vanish outside some compact subset of $(0,2\pi)$. Then
$$
(Sf,g) = (f,Tg),\;\;\; f... |
2,127,494 | <p>Given two $3$D vectors $\mathbf{u}$ and $\mathbf{v}$ their cross-product $\mathbf{u} \times \mathbf{v}$ can be defined by the property that, for any vector $\mathbf{x}$ one has $\langle \mathbf{x} ; \mathbf{u} \times \mathbf{v} \rangle = {\rm det}(\mathbf{x}, \mathbf{u},\mathbf{v})$.
From this a number of properties... | Widawensen | 334,463 | <p>The attempt to prove</p>
<p>$|\mathbf{u} \times \mathbf{v}|^2 - |\mathbf{u}|^2 |\mathbf{v}|^2 + \langle \mathbf{u} ; \mathbf{v} \rangle ^2=0$,</p>
<p>with the use of formula </p>
<p>$\mathbf{u} \times \mathbf{v}= \mathbf {S(u)v}$ where $\mathbf {S(u)}$ is skew-symmetric matrix. </p>
<p>For simplification let... |
707,317 | <p>Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R</p>
<p>Fix $x$ in R. Show that there exists $M>0$ such that for all natural number n and all θ from 0 to 1 $$ |g^{(n)}(θx)|\leq M$$</p>
<p>Also, find the coefficients of the Taylor expa... | IV_ | 292,527 | <p><span class="math-container">$g''(x)-g(x)=0$</span> means <span class="math-container">$g''(x)=g(x)$</span>. And because <span class="math-container">$g$</span> is twice differentiable, <span class="math-container">$g''(x)$</span> is twice differentiable and so on. <span class="math-container">$g$</span> is infinite... |
773,880 | <p>What approach would be ideal in finding the integral $\int4^{-x}dx$?</p>
| user1337 | 62,839 | <p>Rewrite the integral as $$\int e^{(- \ln 4) x} \mathrm{d}x. $$</p>
|
1,342,570 | <p>So, this was my initial proof:</p>
<hr>
<p>Assume $R$ is a ring, and $a,b\in R$</p>
<p>Let $x_1$ and $x_2$ be solutions of $ax=b$</p>
<p>Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$</p>
<p>Thus, we have $x_1-x_2=0_R \Rightarrow x_1=x_2$, and only one solution exists.</p>
<hr>
<p>O... | egreg | 62,967 | <p>Other answers have been given, but I'll throw my 2 cents anyway.</p>
<p>For proving uniqueness you just need that $a$ is <em>left invertible</em>, that is, there exists $c\in R$ such that $ca=1$.</p>
<p>Indeed, if $ax_1=b=ax_2$, you get $a(x_1-x_2)=0$. Thus $ca(x_1-x_2)=c0=0$ and therefore $x_1-x_2=0$, because $ca... |
69,272 | <p>By the way, does anyone know how to prove in an elementary way (i.e. expanding) that $\prod_1^n (1+a_i r)$ tends to $e^r=\sum \frac{r^k}{k!}$ as you let $\max|a_i|\to 0$ with $0\leq a_i \leq 1$ and $\sum a_i = 1$? An easy solution goes by writing the product with the exponential function so that you get the exponent... | Anthony Quas | 11,054 | <p>$\prod_{i=1}^n (1+a_ir)=1+\sum_{k=1}^n r^k\sum_{i_1 < \ldots < i_k}a_{i_1}\ldots a_{i_k}$.</p>
<p>Notice that $1^k=\left(\sum_{i=1}^n a_i\right)^k =k!\sum_{i_1 < \ldots < i_k}a_{i_1}\ldots a_{i_k}+\text{other terms}$, where the other terms are (positive) terms with a repeated $a_i$. It follows that the... |
3,250,061 | <blockquote>
<p>Prove that if <span class="math-container">$p\equiv 5\pmod{8}$</span>, <span class="math-container">$p>5$</span> then <span class="math-container">$\zeta_p$</span> not constructible </p>
</blockquote>
<p>How to do this? There is a theorem in my book that says that the regular <span class="math-con... | Dzoooks | 403,583 | <p>Suppose that <span class="math-container">$p > 5$</span> is constructible and <span class="math-container">$p \equiv 5 \pmod{8}$</span>. Since <span class="math-container">$p$</span> must be a Fermat prime, we have <span class="math-container">$p=2^{2^n}+1$</span> for some <span class="math-container">$n \geq 2$... |
1,409,545 | <p>I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes:</p>
<p>I basically have three vertices of a rigid triangle with known 3D coordinates. The vertices are projected onto a 2D plane (by projection, I mean that each ... | k170 | 161,538 | <p>First note that
$$\frac{d}{dx}a^{f(x)}=a^{f(x)}(\ln a)\frac{d}{dx}f(x)$$
So now we have
$$\frac{d}{dx}\left[7+5^{x^2+2x-1}\right]$$
$$=\frac{d}{dx}[7]+\frac{d}{dx}\left[5^{x^2+2x-1}\right]$$
$$=0+5^{x^2+2x-1}(\ln 5)\frac{d}{dx}\left[x^2+2x-1\right]$$
$$=5^{x^2+2x-1}(\ln 5)\left(2x+2\right)$$
$$=2\cdot 5^{x^2+2x-1}(x... |
2,091,766 | <p>Suppose $h:R \longrightarrow R$ is differentiable everywhere and $h'$ is continuous on $[0,1]$, $h(0) = -2$ and $h(1) = 1$. Show that:
<p> $|h(x)|\leq max(|h'(t)| , t\in[0,1])$ for all $x\in[0,1]$</p>
<p>I attempted the problem the following way:
Since $h(x)$ is differentiable everywhere then it is also continuous ... | 5xum | 112,884 | <p>You ask "How can I disprove it", but you didn't really define a strict mathematical statement. Your statement</p>
<blockquote>
<p>As $d\to\infty$, $S=[a,b]$</p>
</blockquote>
<p>lacks definitions. You seem to imply that for a sequence of sets $A_1,A_2,\dots $, there exists a limit $$\lim_{n\to\infty} A_n$$
but l... |
2,018,239 | <p>I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.</p>
<p>For example, here is one of the things I tried:</p>
<p>Assuming that $21 |2^{4^k}+5$, we have to show that $21... | Community | -1 | <p>Note that $2^3 = 1\mod 7$ and hence $2^{3n} = 1 \mod 7$. Now, $4^k -1 = 0\mod 3$ and it follows that $2^{4^k-1} = 1\mod 7$ and $2^{4^k} = 2\mod 7$, Thus
$$2^{4^k}+ 5 = 0 \mod 7 $$</p>
|
1,933,744 | <p>I simulated the following situation on my pc. Two persons A and B are initially at opposite ends of a sphere of radius r. Both being drunk, can take exactly a step of 1 unit(you can define the unit, i kept it at 1m) either along a latitude at their current location, or a longitude. A and B are said to meet, if the a... | Bolton Bailey | 165,144 | <p>As stated in Daniel's answer, it is easier to think about the problem if we say only one of the two people is moving. Suppose person $B$ stays at the North pole and person $A$ starts at the South pole, and each time step, $A$ moves 1m either longitudinally or latitudinally. </p>
<p>If $A$ moves latitudinally, then ... |
248,733 | <p>Assume the following matrix
$$
C_p^{(a,b)}:=\left(
\begin{array}{cccccc}
a &a &0 &\cdots &\cdots &0 \\
0 &0 &a &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &0 \\
0 &a... | Robert Israel | 13,650 | <p>The characteristic polynomial of $C_p^{(a,b)}$ is $\lambda^p - (a+b) \lambda^{p-1}$. Therefore, for $m \ge p$ we have $$(C_p^{(a,b)})^m = (a+b)^{m-p} (C_p^{(a,b)})^{p-1}$$
It appears that $B = (C_p^{(a,b)})^{p-1}$ has entries
$$ \eqalign{b_{1j} &= a^{p-1}\cr
b_{ij} &= a^{p-i} b (a+b)^{i-2}\ \tex... |
723,633 | <p>My book asserts that for fixed $w$ where $w\neq 0$ that $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$</p>
<p>My book has a general corralary that $v\to P(v)$ is a bounded linear transformation and the fact that $P^2=P$ implies it is a projection. I'm not sure how they made the assertation. Any ideas?</p>
| Community | -1 | <p>We have</p>
<p>$$\require{cancel}P^2(v)=P(P(v))=P\left(\frac{\langle v,w\rangle }{||w||^2}w\right)=\frac{\langle v,w\rangle }{||w||^2}P\left(w\right)=\frac{\langle v,w\rangle }{||w||^2}\cancelto{=1}{\frac{\langle w,w\rangle }{||w||^2}}w=P(v)$$</p>
<p>Moreover, we have by the Cauchy-Schwarz inequality
$$||P(v)||=\l... |
3,290,095 | <p>Now first something that I already know;
<span class="math-container">\begin{eqnarray}
∞/ ∞ = undetermined ( ≠1 ) \\
∞- ∞ = undetermined (≠0)\\
\end{eqnarray}</span></p>
<p>So basically one reason for this is that the <span class="math-container">$∞$</span> I assume is not as same as the <span class="math-contain... | Mohammad Riazi-Kermani | 514,496 | <p>Infinity is not a real number but you can do some algebra with infinity. </p>
<p>For example we define
<span class="math-container">$$ \lambda +\infty =\infty$$</span>for any real <span class="math-container">$\lambda$</span></p>
<p><span class="math-container">$$ \infty +\infty=\infty$$</span>
<span class="math... |
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | AlgorithmsX | 355,874 | <p>$$\begin{align}\sin^2x&=1-\cos^2x&\text{Pythagorean Identity}\\
\sin2x&=2\sin x\cos x&\text{Double Angle}\\
\cos2x&=2\cos^2x-1&\text{Double Angle}\\
\tan x&=\frac{\sin x}{\cos x}&\text{Def. of $\tan x$}
\end{align}$$
You should be able to take it from there.</p>
|
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | Jam | 161,490 | <p>Think back to the definition of $\cos(x)$ and try to draw a right angled triangle. Since $x$ is in the fourth quadrant, also know that it's in the bottom right of the plane. We also know that $\cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}}$ so we have some information about our triangle. From the question, we can... |
1,958,491 | <p>Let $t^k$ act as the $k$-th derivative operator on the set of polynomials. So</p>
<p>$$t^k(x^n)=t^k x^n=(n)_kx^{n-k}$$</p>
<p>where $(n)_k=n(n-1)(n-2)...(n-k+1)$ is the falling factorial. Then with a formal power series, $f(t)=\sum_{k\ge 0}a_k\frac{t^k}{k!}$, the linear operator $f(t)$ acts as such that</p>
<p>... | epi163sqrt | 132,007 | <p><em>Note:</em> OPs calculations are quite ok and it shows the operators are closely related, but different. I don't think there is a necessity to <em>fix</em> anything.</p>
<p>I skimmed through the classic <em><a href="http://rads.stackoverflow.com/amzn/click/0486441393" rel="nofollow">The Umbral Calculus</a></em> ... |
748,815 | <blockquote>
<p>$\displaystyle\sum\limits_{k=1}^nk^2(k-1){n\choose k}^2 = n^2(n-1)
{2n-3\choose n-2}$ considering $n\ge2$</p>
</blockquote>
<p>Can somebody help with this combinatorial proof?
I'm struggling a lot.
Thanks.</p>
<p><strong>EDIT:</strong> Ok. I could figure it out, if we had $\displaystyle\sum\limits_{... | robjohn | 13,854 | <p><strong>Hint:</strong> Note that because choosing $k$ elements from a set of $n$ is the same as choosing the complement of the $k$ elements, we have
$$
\binom{n}{k}=\binom{n}{n-k}\tag{1}
$$
and since choosing a team of $k$ people and then a leader from those chosen is the same as choosing a leader and then choosing ... |
189,650 | <p>let $S=\{s_1, s_2, s_3 \}$, if $s_1$ can be represented as a linear combination of $s_2$ and $s_3$, $s_2$ can be represented as a linear combination of $s_1$ and $s_3$ but $s_3$ can not be represented as a linear combination of $s_1$ or $s_2$ or $s_1$ and $s_2$, can we call $S$ a linearly dependent set? </p>
| Hagen von Eitzen | 39,174 | <p><strong>Warning!</strong>
By strictly adhering to notation, there is a special case where your $S=\{s_1, s_2, s_3\}$ with the given conditions is linearly independent, namely if $s_1=s_2$ and $s_1, s_3$ are linearly independent.
This happens when one uses sets instead of families to talk about linear dependence and... |
12,690 | <p>I understand Lie groups are defined by the structure constants associated with the lie brackets, which are treated as commutators in quantum mechanics, but i dont know of a math theory related to group theory to define or use an anti commutator. If Lie groups theory uses the commutator, what theory uses the anti com... | T.. | 467 | <p>If Lie algebras are (in light of the Poincare-Birkhoff-Witt theorem) a complete axiomatization of the antisymmetric multiplication $AB-BA$ in an associative algebra, Jordan algebras are an almost-complete axiomatization of the symmetric multiplication $(AB+BA)/2$. ( <a href="http://en.wikipedia.org/wiki/Jordan_algeb... |
1,180,437 | <p>I am trying to understand this proof. Rather an important part of the proof. I have already shown this is true for $n=2$ and am assuming the $a_n$ case is true.</p>
<p>$$(a_1^2+a_2^2+...+a_n^2) \le (a_1+a_2+...+a_n)^2$$
Want to show that
$$(a_1^2+a_2^2+...+a_n^2 + a_{n+1}^2) \le (a_1+a_2+...+a_n+a_{n+1})^2$$
$=$... | Joffan | 206,402 | <p>If you really need to use induction, here's what you need for the inductive step:</p>
<p>Assuming $(a_1^2+a_2^2+...+a_n^2 ) \le (a_1+a_2+...+a_n)^2$</p>
<p>then
$$\begin{align} (a_1+a_2+...+a_n+a_{n+1})^2 &= ((a_1+a_2+...a_n)+(a_{n+1}))^2 \\
&= (a_1+a_2+...+a_n)^2+2(a_1+a_2+...+a_n)a_{n+1} + a_{n+1}^2 \\
... |
3,298,412 | <blockquote>
<p>For an n-dimensional vector space <span class="math-container">$V$</span> and an ordered basis <span class="math-container">$B$</span> of <span class="math-container">$V$</span>
, the mapping <span class="math-container">$\Phi : \mathbb{R}^n → V , \Phi(e_i) = b_i, i = 1,...,n$</span>
is linear , w... | InsideOut | 235,392 | <p>Generally, vector spaces are defined in an absolutely abstract way. Namely, a vector space <span class="math-container">$V$</span> over a field <span class="math-container">$\Bbb K$</span> is just a set on which are defined two operations (inner operation and an external operation) that satisfy a precise list of axi... |
3,111,985 | <p><span class="math-container">$f_n(x)= \frac{x}{(1+x)^n}\quad f_n(0)=0$</span></p>
<p>pointwise convergence: <span class="math-container">$\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}=x \sum_{n=1}^{\infty} \frac{1}{(1+x)^n}$</span> and the series is a geometric series convergent if <span class="math-container">$|x+1|>1... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$S_N$</span> is the N-th partial sum then <span class="math-container">$S_N-1=\frac 1 {(1+x)^{N}}$</span> (by the formula for sum of a finite geometric sum). Hence the series converges uniformly on a set iff <span class="math-container">$|(1+x)|^{N} \to \infty$</span> uniformly on <sp... |
3,362,000 | <p>From listing the first few terms, I suspect that the sequence is increasing, so I wanted to use mathematical induction to verify my suspicion.</p>
<p>I have assumed that <span class="math-container">$a_k<a_{k+1}$</span>, I don't see how I can obtain <span class="math-container">$a_{k+1}<a_{k+2}$</span> becaus... | JezuzStardust | 213,886 | <p>Prove that <span class="math-container">$a_n > 0$</span> for all <span class="math-container">$n$</span>. </p>
<p>Then use that
<span class="math-container">$$
a_n = a_{n-1} + \frac{1}{a_{n-1}} > a_{n-1},
$$</span>
since <span class="math-container">$1 / a_{n-1} > 0$</span>. </p>
|
2,780,597 | <p>The <strong>definition</strong> of a <em>convex set</em> is geometrically intuitive. But the definition of <em>convex function</em> doesn't seem so intuitive: $S \subset \mathbb{R}^n$ is convex if given $x,y\in S$ the line segment joining $x,y$ is in $S$. </p>
<p>Let $f$ be a real valued function from an open inter... | DechiWords | 839,564 | <p>I'm refer Boris T. Polyak in his book 'INTRODUCTION TO OPTIMIZATION', page 8.</p>
<p><strong>Definition of convex function</strong></p>
<p>A scalar function <span class="math-container">$f(x)$</span> on <span class="math-container">$\mathbb R^n$</span> is said to be <em>convex</em> if</p>
<p><span class="math-contai... |
1,511,246 | <blockquote>
<p>What is the value of $0.7\overline{54}$ +$0.69\overline2$?</p>
<p>(a) $\frac{1813}{900}$ (b) $\frac{1783}{910}$ (c) $\frac{14323}{9900} (d) \frac{13243}{9900}$</p>
</blockquote>
<p>I get</p>
<p>@edit</p>
<p>$$754-7/990 + 692-69/900$$=$747$/$990$ + $623$/$900$=$1$/$90$($747$/$11$ + $623$/$10$)<... | Akash SSM 2 8 std | 286,933 | <p>0.7545454...(+)
0.6922222...[=]
1.44676767...
Now convert this into rational number.
I got the answer as 14323/9900.
Hope this helps.</p>
|
2,010,693 | <p>How can I prove that $x_{n+1}=c+\sqrt{x_n}$, $x_1=a>0$ and $c>0$ converges?
I know that the limit (if it exists) is $L={{2c+1+\sqrt{4c+1}}\over 2}$.
I have already prove that if $x_1<L$ then $x_n<L$ so its bounded from above but how can I prove that if $x_1<L$ then the sequence is increasing?
I would... | Zongxiang Yi | 388,565 | <p>So you have $x_{n+1}-x_n=c+\sqrt{x_n}-x_n$. Now consider the function:
$$f(x)=c+\sqrt{x}-x,x\in R.$$
It follows
$$f'(x)=\frac{1}{2\sqrt{x}}-1.$$
You can see that when $x< \frac{1}{4}$, it has
$$f'(x)>0.$$
This means
$$f(x)>f(\frac{1}{4})=1+\sqrt{\frac{1}{4}}-\frac{1}{4}=\frac{5}{4}>0.$$
That's
$$x_{n+1}-... |
869,337 | <p>"Abstract index" and "coordinate free notations" are often submitted as alternatives to Einstein Summation notation. Could you illustrate their use using an example?</p>
<p>Here's a sum written in Einstein's notation:</p>
<p>$a_{ij}b_{kj} = a_{i}b_{k}$</p>
<p>How would you rewrite it in a modern way? </p>
| reuns | 276,986 | <p><span class="math-container">$$R[y]/(f(y))\cong R[x]/(g(x))$$</span> as <span class="math-container">$R$</span>-algebras iff there exists <span class="math-container">$\phi,\varphi\in R[t]$</span> such that <span class="math-container">$$f(\phi(x))\in (g(x)),\quad g(\varphi(y))\in (f(y)),\quad \phi(\varphi(y))-y\in ... |
4,416,063 | <p>How to solve <span class="math-container">$\int\frac{\ln(x \ln(x))}{x} dx$</span>?</p>
<p>My work:<br />
Let <span class="math-container">$t = \ln(x) \implies x= e^t ; dt = \dfrac{dx}{x}$</span></p>
<p>So above integral changes to,
<span class="math-container">$$\int t ( e^t t) dt$$</span>
<span class="math-contain... | Dr. Sundar | 1,040,807 | <p>We make the substitution <span class="math-container">$t = \ln| x | $</span> or <span class="math-container">$x = e^t$</span>.</p>
<p>Then <span class="math-container">$dt = {1 \over x} dx$</span> or <span class="math-container">${dx \over x} = dt$</span>.</p>
<p>Thus, the given integral can be simplified as
<span c... |
3,415,378 | <p>I am looking for an estimation or an approximation of </p>
<p><span class="math-container">$\sum _{k=1}^{n}{\log(k)\binom {n}{k}}$</span></p>
<p>Any hints will be appreciated.
Thank you.</p>
| metamorphy | 543,769 | <p>I'm continuing the computations by Jack D'Aurizio (following <a href="https://math.stackexchange.com/a/3399858">myself</a>).
<span class="math-container">\begin{align}\sum_{k=1}^{n}\binom{n}{k}\ln k&=\sum_{k=1}^{n}\binom{n}{k}\int_0^1\frac{x^{k-1}-1}{\ln x}\,dx\\\color{gray}{[\text{note the sign}]}\quad&=\in... |
1,606,202 | <p>I'm having trouble figuring out why these two different ways to write this combination give different answers. Here is the scenario:</p>
<p>Q: Choose a group of 10 people from 17 men and 15 women, in how many ways are at most 2 women chosen?</p>
<p>Solution A: From 17 men choose 8, and from 15 women choose 2. Or f... | Brian M. Scott | 12,042 | <p>Suppose that the men are $M_1,\ldots,M_{17}$, and the women are $W_1,\ldots,W_{15}$. Consider the group</p>
<p>$$\{M_1,M_2,\ldots,M_9,W_1\}\;.$$</p>
<p>Your second approach counts this $9$ times: once as $\{M_1,\ldots,M_8\}$ for the $8$ men and $\{M_9,W_1\}$ for the last two, once as $\{M_1,M_2,M_3,M_4,M_5,M_6,M_7... |
2,032,387 | <p>I know this is somewhat of an odd question, but I am having trouble with my TI-84 calculator and I don't know why.</p>
<p>I'm trying to find the RREF of the transpose of a <span class="math-container">$4\times6$</span> matrix; for some reason my graphing calculator gives me an error. Something to do with the dimensi... | perplexed | 179,093 | <p><a href="http://tibasicdev.wikidot.com/rref" rel="nofollow noreferrer">The TI-84's rref function throws an error if there are more rows than columns</a>, and the transpose has more rows than columns.</p>
|
2,032,387 | <p>I know this is somewhat of an odd question, but I am having trouble with my TI-84 calculator and I don't know why.</p>
<p>I'm trying to find the RREF of the transpose of a <span class="math-container">$4\times6$</span> matrix; for some reason my graphing calculator gives me an error. Something to do with the dimensi... | user399923 | 399,923 | <p>Change your matrix from 6x4 to 6x6 by adding two columns of zeros. Then you can use the rref or ref functions. Then just ignore the added columns.</p>
|
3,130,939 | <p>Suppose the following function with pi notation, with the pi denoting the iterated product, multiplying from <span class="math-container">$i = 0$</span> to <span class="math-container">$i = n$</span>:</p>
<p><span class="math-container">$$\prod_{i=0}^n \ln(y_i^{x - 1})$$</span></p>
<p>That is, the natural logarith... | clathratus | 583,016 | <p>Let
<span class="math-container">$$f(x)=\prod_{i=0}^{n}f_i(x)$$</span>
and let <span class="math-container">$$g^{(m)}(x)=\left(\frac{d}{dx}\right)^mg(x),\qquad m=0,1,2,...$$</span>
as well as <span class="math-container">$\delta_{ij}$</span> denote the Kronecker Delta.</p>
<p>We have that
<span class="math-contain... |
634,127 | <p>How to prove this (true or not)?</p>
<blockquote>
<p>$f(a,b) = f(a,c)$ must hold if $b = c$</p>
</blockquote>
<p><b>Note:</b> <i><b>f(a,b)</b> is a function with <b>a</b> & <b>b</b></i> parameters</p>
<p>thanks</p>
| arkadeep | 120,499 | <p>See it is a bi variate function that you have given here.
Just think of a 3-D(2 dimentional system of co-ordinate).
Now if a function is a bivariate one then the functional value will have the value on axis which one is mutually perpendicular to the other two axis.
Now you have the equation as f(a,b)=f(a,c).Now we h... |
203,456 | <p>Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.</p>
| Madrit Zhaku | 34,867 | <p>Before we prove the given identity proof this idenity</p>
<p>$$\log_b a\log_c b=\log_c a$$</p>
<p>Proof: Implement the formula $\log_a b=\frac{\log_x b}{\log_x a}$</p>
<p>$$\frac{\log a}{\log b}\cdot\frac{\log b}{\log c}=\frac{\log a}{\log c}=\log_c a$$</p>
<p>Now proof the given identity.</p>
<p>$$\log_b a\cdo... |
500,632 | <p>Find all such lines that are tangent to the following curves:</p>
<p>$$y=x^2$$ and $$y=-x^2+2x-2$$</p>
<p>I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will... | Old John | 32,441 | <p>Here is a hint for a method which avoids calculus:</p>
<p>The line $y=ax+b$ is a tangent to a quadratic such as $y=x^2$ if and only if the quadratic equation you get by solving these equations simultaneously has a double root. This will give you an equation which must be satisfied by the unknowns $a$ and $b$.</p>
... |
500,632 | <p>Find all such lines that are tangent to the following curves:</p>
<p>$$y=x^2$$ and $$y=-x^2+2x-2$$</p>
<p>I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will... | Kaster | 49,333 | <p>Tangent line of first equation through some point $(x_1,f_1(x_1))$ is
$$
y = f_1(x_1) + f'_1(x_1)(x-x_1) = x_1^2 + 2x_1(x-x_1) = 2x_1x-x_1^2
$$
Tangent line of second equation through some point $(x_2, f_2(x_2))$ is
$$
y = f_2(x_2) + f'_2(x_2)(x-x_2) = -x_2^2+2x_2-2 + (-2x_2+2)(x-x_2)
= 2(1-x_2)x+x_2^2-2
$$
In orde... |
105,190 | <p>Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/conjectured about the next term?</p>
<p>On a related note, the BSD conjecture predicts the value of the first non-vanishi... | paul garrett | 15,629 | <p>The questions about Birch-SwinnertonDyer are much subtler than the first question, and I do not pretend to have anything to say about it.</p>
<p>Edit: and, indeed, the following bits of information are a "weak" answer, at the level of saying "yes, just as the Euler-Mascheroni constant (and a family of such constant... |
9,629 | <p>are people facing problem of not loading latex symbols in MSE? I have high speed internet connection but I am facing this problem from yesterday,any suggestion?It says "math processing error" if my connection is low speed but this is not the case, I am just watching all latex symbols instead of compiled complete pi... | Balbichi | 24,690 | <p>Uncaught TypeError: Cannot read property 'strings' of undefined TeX-AMS_HTML.js:43
a.CreateLocaleMenu TeX-AMS_HTML.js:43
a.showRenderer.a.cookie.showRenderer.p.showRenderer TeX-AMS_HTML.js:43
CALLBACK.execute MathJax.js:29
(anonymous function)
MathJax.Object.Subclass.Execute MathJax.js:29
QUEUE.Subclass.ExecuteHooks... |
1,380,508 | <p>Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?</p>
| mathcounterexamples.net | 187,663 | <p>Runge Phenomena doesn't occur for all functions. For a detailed analysis on polynomial interpolation at equidistant points you can have a look <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Epperson329-341.pdf" rel="nofollow">here</a></p>
|
4,046,356 | <p>Recently, some of the remarkable properties of second-order
Eulerian numbers <span class="math-container">$ \left\langle\!\!\left\langle n\atop k\right\rangle\!\!\right\rangle$</span> <a href="https://oeis.org/A340556" rel="nofollow noreferrer">A340556</a> have been proved on MSE [ <a href="https://math.stackexchan... | Marko Riedel | 44,883 | <p>In trying to verify the identity</p>
<p><span class="math-container">$$\sum_{j=0}^{k} {n-j \choose n-k}
\left\langle\!\! \left\langle n\atop j
\right\rangle\!\! \right\rangle
= \sum_{j=0}^k (-1)^{j+k} {n+k \choose n+j}
\left\{ n+j \atop j\right \}$$</span></p>
<p>we quote from <a href="https://math.stackexchan... |
237,708 | <p>Does the series </p>
<p>$$\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$$</p>
<p>converge?</p>
| WimC | 25,313 | <p>Let $n \geq 3$ then</p>
<p>$$
\log(n) - \log(n)^{n/(n+1)} = \frac{\log(n)^{n/(n+1)}}{n+1} (n+1) \left(\log(n)^{1/(n+1)} - 1 \right) \geq \frac{\log(\log(n))}{n+1}
$$</p>
<p>Since $\log(\log(n)) \to \infty$ and $\sum 1/(n+1)$ diverges, this series itself diverges.</p>
|
96,191 | <p>I am trying to calculate the following integral which contains a parameter.
<a href="https://i.stack.imgur.com/qUJ9f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qUJ9f.jpg" alt="enter image description here"></a></p>
<p>I have used the Integrate and FullSimplify using assumptions but Mathemati... | Jason B. | 9,490 | <p>It really depends on the level at which you want to estimate this function. Do you want to end up with a nice closed expression? Do you simply need <strong>an expression</strong> to model the data? You can't be sure that an analytic solution exists. I tried the Rubi package (apmaths.uwo.ca/~arich) and it didn't giv... |
86,762 | <p>The other day, my teacher was talking infinite-dimensional vector spaces and complications that arise when trying to find a basis for those. He mentioned that it's been proven that some (or all, do not quite remember) infinite-dimensional vector spaces have a basis (the result uses an Axiom of Choice, if I remember ... | Qiaochu Yuan | 232 | <p>It's known that the statement that every vector space has a basis is equivalent to the <a href="http://en.wikipedia.org/wiki/Axiom_of_choice">axiom of choice</a>, which is independent of the <a href="http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">other axioms of set theory</a>. This is generally t... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Gautam Shenoy | 35,983 | <p>I'm sure everyone has answered the question appropriately. But here's my 2 cents:</p>
<p>From the Argand plane perspective, multiplying a complex number by $i$ is equivalent to rotating it about a circle (with radius = modulus of complex number) counterclockwise by 90 degrees. So ask yourself where you end up when ... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Lucian | 93,448 | <p>$\sqrt[n]z$ does not return a single value, but <em>n</em> complex values. Hence your confusion, since both <em>i</em> and $-i$ are among the square roots of $-1$.</p>
|
455,230 | <p>I found this proposition and don't see exactly as to why it is true and even more so, why the converse is false:</p>
<p>Proposition 1. The equivalence between the proposition $z \in D$ and the proposition $(\exists x \in D)x = z$ is provable from the definitory equations of the existential quantifier and of the equ... | Kevin Ventullo | 546 | <p>Consider the algebra $K[T]$. The simple $K[T]$-modules are of the form $K[T]/P(T)$ for some irreducible polynomial $P$. These do not occur as submodules of $K[T]$, since every such submodule contains a free module, and hence is infinite dimensional over $K$. </p>
|
2,734,374 | <p>I don't think it is possible because that entails that only the <span class="math-container">$\mathbf 0$</span>-vector is in the eigenspace, but <span class="math-container">$\mathbf 0$</span> is not an eigenvector by definition. </p>
<p>However, my textbook says:</p>
<blockquote>
<p>For an <span class="math-con... | the_candyman | 51,370 | <p>Following the definition, <span class="math-container">$\lambda$</span> is an eigenvalue of the matrix <span class="math-container">$A$</span> if there exists a non-zero vector <span class="math-container">$v$</span> such that:</p>
<p><span class="math-container">$$Av = \lambda v.$$</span></p>
<p>The definition it... |
2,734,374 | <p>I don't think it is possible because that entails that only the <span class="math-container">$\mathbf 0$</span>-vector is in the eigenspace, but <span class="math-container">$\mathbf 0$</span> is not an eigenvector by definition. </p>
<p>However, my textbook says:</p>
<blockquote>
<p>For an <span class="math-con... | Gatgat | 550,782 | <p>It doesn't imply that dimension 0 is possible. You know by definition that the dimension of an eigenspace is at least 1. So if the dimension is also at most 1 it means the dimension is exactly 1. It's a classic way to show that something is equal to exactly some number. First you show that it is at least that number... |
2,734,374 | <p>I don't think it is possible because that entails that only the <span class="math-container">$\mathbf 0$</span>-vector is in the eigenspace, but <span class="math-container">$\mathbf 0$</span> is not an eigenvector by definition. </p>
<p>However, my textbook says:</p>
<blockquote>
<p>For an <span class="math-con... | Marc van Leeuwen | 18,880 | <p>It is a matter of convention. What everybody should agree on is that $\lambda$ being an eigenvalue of$~A$ means that $\dim(\ker(A-\lambda I))>0$, so the dimension of the eigenspace associated to an eigenvalue is never$~0$. However if the dimension in that formula is$~0$, so if $\lambda$ is <em>not</em> an eigenva... |
1,587,498 | <p>I need some help with this (seemingly) simple problem. As before, it comes from Apostol "Calculus", Volume 1, Section 8.28, Question 23 and it states:</p>
<p>Solve the differential equation $(1+y^2e^{2x})y^{'} + y = 0$ by introducing a change of variable of the form $y = ue^{mx}$, where $m$ is constant and $u$ is a... | André Nicolas | 6,312 | <p>Hint: We can choose $m$ freely. Let $m=-1$.</p>
|
535,080 | <p>For the following example: </p>
<blockquote>
<p>Let the topological space $X$ be the real line $\mathbb{R}$. An open set is any set whose complement is finite. Let $S=[0,1]$. Find the closure, the interior, and the boundary of $S$. </p>
</blockquote>
<p>What is meant by let the topological space $X$ be the real... | D Left Adjoint to U | 26,327 | <p>What it should say is let $X$ be a topological space on $\Bbb{R}$ whose open sets consist of all subsets of $\Bbb{R}$ that have a finite complement in $\Bbb{R}$. </p>
|
2,872,492 | <p>My work starts with a supposition of $N$, so that for $n > N$ we have $\vert b \vert ^n < \epsilon$.</p>
<p>Since $0 < \vert b \vert < 1$, we see the logarithm with base $\vert b \vert$ is a decrescent function meaning it will invert the inequality once taken.
$$\vert b \vert ^n < \epsilon $$
$$n >... | Lev Bahn | 523,306 | <p>My answer can be over killing, but it has different view point.</p>
<p>If $0<|b|<1$, note that</p>
<p>$\sum_{n=0}^{\infty}|b|^n = \frac{1}{1-|b|} <\infty$.</p>
<p>Thus, $\lim_{n\rightarrow \infty}|b|^n=0$ so</p>
<p>$|\lim_{n\rightarrow \infty} b^n|\leq |\lim_{n\rightarrow \infty}|b|^n| =0$ $\implies \li... |
1,621,347 | <p>Is there a closed-form expression for the following definite integral?
\begin{equation}
\mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx,
\end{equation}
where $A$, $B$, $\delta_1$, and $\delta_2$ are positive constant. $L$ is a positive integer.</p>
<p>I am facing problem due to finite... | Robert Israel | 8,508 | <p>Since your endpoints are arbitrary, what you need is an antiderivative.<br>
For convenience, apply scaling so that $A=1$. Let</p>
<p>$$ f_L(x) = \int \left( \dfrac{x}{1+x}\right)^L e^{-Bx}\; dx $$</p>
<p>The generating function is</p>
<p>$$ \eqalign{g(t,x) &= \sum_{L=0}^\infty f_L(x) t^L\cr &= \int \sum_{... |
452,653 | <p>If $f:X\rightarrow Y$ is initial in category <strong>Top</strong> then
it is easy to proof that </p>
<blockquote>
<p>(!) the topology on $X$ is the set of preimages of open sets in $Y$. </p>
</blockquote>
<p>Just construct topology $Z$ having
the same underlying subset as $X$ and let the set of these preimages
s... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Use <a href="http://en.wikipedia.org/wiki/Alternating_series_test" rel="nofollow">alternating series test</a>.</p>
|
452,653 | <p>If $f:X\rightarrow Y$ is initial in category <strong>Top</strong> then
it is easy to proof that </p>
<blockquote>
<p>(!) the topology on $X$ is the set of preimages of open sets in $Y$. </p>
</blockquote>
<p>Just construct topology $Z$ having
the same underlying subset as $X$ and let the set of these preimages
s... | Clement C. | 75,808 | <p><strong>Hint:</strong>
For $a_n=\frac{1}{n^\alpha \ln^\beta n}$ ($n\geq 2$), the positive series $\sum a_n$ converges if</p>
<ul>
<li>$\alpha > 1$; or</li>
<li>$\alpha = 1$ and $\beta > 1$</li>
</ul>
<p>(and diverges otherwise.)</p>
<p>This'll allow you to see if your series converges absolutely.</p>
|
3,153,306 | <p>In other words, say I am looking for multiple X</p>
<p>let: </p>
<p>X < 1000005</p>
<p>let the fist 18 divisors of X be:
1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 25 | 32 | 40 | 50 | 64 | 80 | 100 | 125 | 160 | 200 </p>
<p>finally, I also know: X has exactly 49 divisors. </p>
<p>I will tell you what the answer is.... | Henry Lee | 541,220 | <p><span class="math-container">$$f(p)=\frac{1}{p^2-1}\sum_{q=3}^p\frac{q^2-3}{q}$$</span>
If we use the fact that:
<span class="math-container">$$\sum_{q=3}^p\frac{q^2-3}{q}=\sum_{q=3}^pq-3\sum_{q=3}^p\frac1q$$</span>
Now we know that:
<span class="math-container">$$\sum_{q=3}^pq=\sum_{q=1}^pq-\sum_{q=1}^2q=\frac{p(p+... |
2,215,087 | <p>I'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$</p>
<p>By multiplicativity, it suffices to show that $\forall x \in \mathbb{Q}(\sqrt{11}) \exists n \in \mathbb{Z}(\sqrt{11}):|N(n-x)| < 1$</p>
<p>For the analogous s... | Chan Tai Man | 876,234 | <p>I am a novice and learning to write simple proofs. I welcome corrections and suggestions. Below is a detailed workout. Most maths students will find it unnecessarily verbose. Oppenheim (1934) proved, among other proofs, that <span class="math-container">$\mathbb{Z}[\sqrt{11}]$</span> has a division algorithm and hen... |
4,331,081 | <p>Suppose <span class="math-container">$a, b >0$</span>. I'm looking for closed expressions for the following integral:
<span class="math-container">$$\int_{-\pi}^{\pi}\sqrt{a^{2}-2ab\cos(x)+b^{2}}dx $$</span>
I tried to solve this by myself and got nowhere and even wolfram alpha couldn't get me an answer so maybe ... | projectilemotion | 323,432 | <p>This does not admit a closed form in terms of elementary functions for general <span class="math-container">$a,b>0$</span>. However, using some elementary trigonometric identities and symmetry properties, one can get a solution in terms of elliptic integrals.</p>
<hr />
<p>Firstly, by the substitution <span class... |
30,292 | <p>One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a natural continuous analog of discrete self-avoiding walks?
I am particularly interested in self-avoiding polygons,
i.... | Yuri Bakhtin | 2,968 | <p>In 2D the scaling limit is believed to be SLE with parameter 8/3. This was conjectured by Lawler, Schramm and Werner and, to the best of my knowledge, still remains open.</p>
|
8,107 | <p>Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on resear... | Hahn | 93,001 | <p>The other answers are better, but knowing something about the marketplace for your widgets I will add a few more considerations. </p>
<p>If (1 & A) you are on an island with consumers (or manufacturing bases) in distant lands that are susceptible to hostile takeovers that will dramatically decrease your monthly... |
24,230 | <p>$f$ is continuous between $[0,1]$, and $f(0)=f(1)$.</p>
<p>I want to prove that there is an $a \in [0,0.5]$ such that $f(a+0.5)=f(a)$.</p>
<p>ok, so Rolle's theorem can be useful here, but I can't see the connection to the derivative,</p>
<p>(Weierstrass, Uniform continuity?) I'll be glad to instructions.</p>
<p... | Eelvex | 7,476 | <p><em>Or</em> </p>
<p>consider if there is a $b\in [0,1]$ such that $f(b) = f(0) = f(1)$.</p>
<p>What if there is no such $b$?</p>
<p>What if $b = 0.5$?</p>
<p>[ You don't need Rolle's theorem this way ]</p>
|
721,449 | <p>I need to determine all the positive divisors of 7!. I got 360 as the total number of positive divisors for 7!. Can someone confirm, or give the real answer?</p>
| copper.hat | 27,978 | <p>360 is incorrect.</p>
<p>$7! = 2^4 3^2 5^1 7^1$. Now start counting...</p>
<p><strong>Note</strong>: Count $\{0,1,2,3,4\} \times \{0,1,2\} \times \{0,1\} \times \{0,1\}$.</p>
|
1,963,295 | <p>I have the following equation for a decision boundary line: $-w_0 = w_1x_1 + w_2x_2$ and I want to prove that the distance from the decision boundary to the origin is $l = \frac{w^Tx}{||w||}$. I am having trouble wrapping my mind around how I can just get the distance from a line to a point. Am I supposed to be aver... | qwr | 122,489 | <p>Mean and variance derivations are given by Maxime Beauchamp, "On numerical computation for the distribution of the convolution of N independent rectified Gaussian variables" at <a href="http://journal-sfds.fr/article/view/669" rel="nofollow noreferrer">http://journal-sfds.fr/article/view/669</a></p>
<p><sp... |
157,587 | <p>I know the following is a well-known result.</p>
<p>Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is equality if and only if $f$ is linear.</p>
<p>I need some reference about the second part, i.e. there is equal... | Malik Younsi | 1,162 | <p>This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the <em>paper Area, capacity and diameter versions of Schwarz's lemma</em> by Burckel, Marshall, Minda, Poggi-Corradini and Ransford.</p>
<p>See Theorem 1.3 <a href="http://arxiv.org/pdf/0801.3629v1.pdf" rel... |
2,283,123 | <p>Let $(\mathbb{R}, +, 0)$ be the additive group of reals. Is this structure $\aleph_0$-saturated? </p>
<p>I don't really see how to go about showing this. To show it is not saturated, it is enough to exhibit a type omitted in $(\mathbb{R}, +, 0)$. The interesting statements we can make about groups are usually to do... | zarathustra | 73,997 | <p>Every (nontrivial) torsion-free divisible group has a structure of $\Bbb Q$-vector space. This shows that the theory $T$ of torsion-free divisible group is uncountably categorical (because a vector space is identified up to isomorphism by its dimension, and all the uncountable $\Bbb Q$-vector spaces have the same di... |
3,890,382 | <blockquote>
<p>Find the locus of <span class="math-container">$z$</span> such that <span class="math-container">$\arg \frac{z-z_1}{z-z_2} = \alpha$</span>.
Use and draw <span class="math-container">$w = \frac{z-z_1}{z-z_2}$</span>.</p>
</blockquote>
<p>This exercise was discussed many times -- <a href="https://math.st... | user2661923 | 464,411 | <p>You know that (within a modulus of <span class="math-container">$2\pi$</span>), <br>
<span class="math-container">$\arg \left(w_1 \times w_2\right) ~=~ \arg(w_1) + \arg(w_2).$</span></p>
<p>Therefore, <span class="math-container">$\arg \left(\frac{w_1}{w_2}\right) ~=~ \arg(w_1) - \arg(w_2).$</span></p>
<blockquote>
... |
560,929 | <p>Consider a circle with two perpendicular chords, dividing the circle into four regions $X, Y, Z, W$(labeled):</p>
<p><img src="https://i.stack.imgur.com/2TDK5.png" alt="enter image description here"></p>
<p>What is the maximum and minimum possible value of </p>
<p>$$\frac{A(X) + A(Z)}{A(W) + A(Y)}$$</p>
<p>where... | Christian Blatter | 1,303 | <p>When the considered quotient is not constant it has a maximum value $\mu>1$, and the minimum value is then ${1\over \mu}$. I claim that
$$\mu={\pi+2\over\pi-2}\doteq4.504\ ,\tag{1}$$
as conjectured by MvG.</p>
<p><img src="https://i.stack.imgur.com/s8n29.jpg" alt="enter image description here"></p>
<p><em>Proof... |
3,206,730 | <blockquote>
<p>Let <span class="math-container">$f : (-1,1)\to (-\pi/2,\pi/2)$</span> be the function defined by <span class="math-container">$f(x)= \tan^{-1}\left(\frac{2x}{1-x^2}\right)$</span> the verify that <span class="math-container">$f$</span> is bijective</p>
</blockquote>
<p>To check objectivity I assumed... | Community | -1 | <p><span class="math-container">$$\forall x\in(-1,1):\left(\frac x{1-x^2}\right)'=\frac{1+x^2}{(1-x^2)^2}>1$$</span></p>
<p>and</p>
<p><span class="math-container">$$\forall t:(\arctan(t))'=\frac1{1+t^2}>0.$$</span></p>
<p>Both functions are monotonous and continous.</p>
|
80,899 | <p>This is related to a previous post of mine (<a href="https://math.stackexchange.com/questions/78669/limit-superior-of-a-sequence-showing-an-alternate-definition">link</a>) regarding how to show that for any sequence $\{x_{n}\}$, the limit superior of the sequence, which is defined as $\text{inf}_{n\geq 1}\text{sup }... | Florian | 1,609 | <p>The sequence does not converge to anything, but a subsequence might converge. In fact there exists a subsequence whose limit is 1.</p>
|
892,986 | <p>Can someone help me to compute:
$$\int \frac{x}{(x^2-4x+8)^2}\mathrm dx$$
And, in general, the type:</p>
<p>$$\int \frac{N(x)}{(x^2+px+q)^n}\mathrm dx$$
with the order of polynomial $N(x)<n$ and $n$ natural greater than 1?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $\displaystyle x^2-4x+8=(x-2)^2+2^2,$</p>
<p>use <a href="http://en.wikipedia.org/wiki/Trigonometric_substitution" rel="nofollow">Trigonometric substitution</a> as $x-2=2\tan\theta$</p>
|
892,986 | <p>Can someone help me to compute:
$$\int \frac{x}{(x^2-4x+8)^2}\mathrm dx$$
And, in general, the type:</p>
<p>$$\int \frac{N(x)}{(x^2+px+q)^n}\mathrm dx$$
with the order of polynomial $N(x)<n$ and $n$ natural greater than 1?</p>
| amWhy | 9,003 | <p>$$\int \frac{x}{(x^2-4x+8)^2} dx = \int\frac{x - 2 + 2}{(x^2 - 4x + 8)^2}\,dx $$
$$= \frac 12\int \frac{2x - 4}{(x^2 - 4x + 8)^2}\,dx + \int \frac 2{((x-2)^2 + 2^2)^2}\,dx$$</p>
<p>For the first integral, use $u = x^2 - 4x + 8 \implies du = (2x-4)\,dx$.</p>
<p>For the second integral, put $2\tan \theta = (x-2)\imp... |
523,529 | <p>I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:<br>
$V(1): 1≤1 \text{ true}$ <br>
$V(n): n!≤((n+1)/2)^n$ <br>
$V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$<br><br></p>
<p>and I've got : <br>$(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ <br>$((n+1)^n)n(n+1)... | Jeyekomon | 29,060 | <p>An induction proof:</p>
<p>First, let's make it a little bit more eye-candy:</p>
<p>$$
n! \cdot 2^{n} \leq (n+1)^n
$$</p>
<p>Now, for $n=1$ the inequality holds. For $n=k\in\mathbb{N}$ we know that:</p>
<p>$$
k! \cdot 2^{k} \leq (k+1)^k
$$</p>
<p>holds and we need to prove:</p>
<p>$$
(k+1)! \cdot 2^{k+1} \leq ... |
479,551 | <p>A store carries three types of donuts: Strawberry, Chocolate and Glazed</p>
<p>Suppose you bought $4$ of each kind and in addition, you have the option to apply sprinkles on your donuts. How many ways are there to eat the donuts if you never eat two donuts in a row that both have sprinkles? </p>
<p>The idea I have... | Brian M. Scott | 12,042 | <p>ShreevatsaR has already broken the problem into its component parts, and you’ve correctly solved the first part.</p>
<p>For the recurrence, let $u_n$ be the number of ways to apply sprinkles to a row of $n$ doughnuts so that no two adjacent doughnuts have sprinkles. This is the same as the number of $n$-bit binary ... |
1,027,807 | <p>So I have this question that looks like</p>
<p>$$ \frac{x^3 + 3x^2 - x - 8}{x^2 + x - 6} $$</p>
<p>and first I got the partial fraction so getting </p>
<p>$$ x + 2 + \frac{3x + 4}{x^2 + x -6} $$</p>
<p>but now I'm trying to integrate it and I cannot remember for the life of me how I should integrate the fraction... | Khosrotash | 104,171 | <p>$$3x+2+\frac{3x+4}{x^2+x−6} =\\3x+2+\frac{3x+4}{(x-2)(x+3)} =\\
3x+2+\frac{a}{(x-2)}+\frac{b}{(x-2)} =\\
$$now find a,b
$$\frac{a}{(x-2)}+\frac{b}{(x-2)} =\frac{a(x+3)+b(x-2)}{(x-2)(x+3)}=\frac{3x
+4}{(x-2)(x+3)}\\\rightarrow \\(a+b)x=3x\\3a-2b=4\\a=2,b=1\\
$$</p>
|
4,601,113 | <p>I have a plane, defined by ax1+bx2+cx3=d, and a point which I know is on said plane. How could I convert the coordinates of the point to coordinates relative to the plane? I have attempted to find a solution online, but so far have been met with confusing answers such as</p>
<blockquote>
<p>Find the dot products <... | geetha290krm | 1,064,504 | <p>This follows immediately from SLLN's since <span class="math-container">$(f(X_i))$</span> is also i.i.d. Measurability and boundedness is enough for this; continuity is not needed. [Convergence holds in the almost sure sense].</p>
|
4,601,113 | <p>I have a plane, defined by ax1+bx2+cx3=d, and a point which I know is on said plane. How could I convert the coordinates of the point to coordinates relative to the plane? I have attempted to find a solution online, but so far have been met with confusing answers such as</p>
<blockquote>
<p>Find the dot products <... | Balaji sb | 213,498 | <p>By chebyshev inequality,
<span class="math-container">$P(|\frac{1}{N} \sum_i f(X_i)-E(f(X))|^2 \geq \frac{1}{N}) \leq \frac{Var(\frac{1}{N} \sum_i f(x_i))}{\frac{1}{N}} = \frac{Var(f(X))}{N} \rightarrow 0 \ as \ N \rightarrow \infty.$</span></p>
<p>where we used the fact that <span class="math-container">$Var(f(X))$... |
3,669,080 | <p>I would love to get some insight on how to solve <span class="math-container">$\int_0^{\frac\pi4}\log(1+\tan x)\,\mathrm dx$</span> using Leibniz rule of integration. I know it can be solved using the property<span class="math-container">$$\int_a^bf(x)\,\mathrm dx=\int_a^bf((a+b)-x)\,\mathrm dx,$$</span>but I find t... | heropup | 118,193 | <p>Your choice of where to put the parameter results in significant problems, which I will illustrate with an example. Suppose we are interested in computing
<span class="math-container">$$\int_{x=0}^{\pi/4} \log (1+x) \, dx.$$</span>
If we introduce a parameter <span class="math-container">$t$</span> as you did, we h... |
487,084 | <p>I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?</p>
| Alexander Gruber | 12,952 | <p>Not the group of order $p^0=1$!</p>
<p>Other than that, first prove that if the order of a group element $x$ is $mn$, then the order of $x^m$ is $n$. Then you can either show directly that if $x\in G$ and $|G|$ is finite, $x^{|G|}$ is the identity, or apply Lagrange's theorem to $\langle x \rangle$.</p>
|
2,292,511 | <p>I need some help to solve this problem:</p>
<blockquote>
<p>Evaluate A such that the exponential distribution with parameter $\alpha, P(X = x) = Ae^{−\alpha x}$, is normalized.
Here, $\alpha > 0$ and $\Omega = \mathbb{R}_{+}$.</p>
</blockquote>
<p>I've been trying to evaluate the following Integral </p>
<p... | Darío A. Gutiérrez | 353,218 | <p>$$\int_0^{\frac{\pi}{2}}(\sin^2\theta+a)^{-\frac{3}{2}}\,d\theta$$</p>
<p>$$\int_0^{\frac{\pi}{2}} \frac{1}{(\sin^2\theta+a)\sqrt{\sin^2\theta+a}} \,d\theta$$</p>
<p>$$\int_0^{\frac{\pi}{2}} \frac{1}{a(\frac{1}{a}\sin^2\theta+1)\sqrt{a\left( \frac{1}{a}\sin^2\theta+1\right) }} \,d\theta$$</p>
<p>$$\frac{1}{a\sqrt... |
4,185,658 | <p>In the proof of Proposition. 1.3. page 100 Functional Analysis book of Conway the following claim (<span class="math-container">$X$</span> is a TVS and <span class="math-container">$p$</span> is a seminorm.)</p>
<p>If <span class="math-container">$0 \in \operatorname{Int}{\{x \in X : p(x) \le 1}\}$</span> then <span... | Martin Argerami | 22,857 | <p>Fix <span class="math-container">$\epsilon>0$</span>. If <span class="math-container">$$0 \not\in \operatorname{Int}{\{x \in X : p(x) \le \epsilon}\},$$</span> there exists a net <span class="math-container">$\{x_j\}$</span> such that <span class="math-container">$p(x_j)>\epsilon$</span> and <span class="math-... |
2,777,631 | <p>Angle bisectors of traingle $ABC$ meet its circum-circle ( after passing through in-center) at opposite points $P, Q$, and $R$ respectively on the circumcircle. </p>
<p>Find $\angle RQP.$ </p>
<p>Is there any way of getting the answer through its in-center properties?</p>
<p>Ans = $90-\frac{B}{2}$ </p>
| Arthur | 15,500 | <p>It's not a contradiction yet, because $c$ could be $0$. Specifically pick $c\geq0$ such that $c\neq0$ (if you can guarantee that such a thing exists), and you will have your contradiction.</p>
|
4,166,894 | <p>Let <span class="math-container">$N\triangleleft G$</span> and <span class="math-container">$K\leqslant G$</span>. Consider <span class="math-container">$\phi :G \mapsto G/N$</span> onto group homomorphism. Show that <span class="math-container">$\phi(K)=KN/N$</span>.</p>
<p>I thought using the equality <span class=... | Steven | 849,372 | <p>Intuitively this is easy to see: with a fixed number of hyperedges, you will maximize the number of complete subgraphs when the hyperedges share as many vertices as possible.
So in your case: consider the complete 3-uniform hypergraph on <span class="math-container">$x$</span> vertices. It has <span class="math-cont... |
1,081,717 | <p>I have a vector valued mapping $F:\mathbb{R}^2\rightarrow\mathbb{R^2}$, I'm wondering whether there's a sufficient condition for it to be a contraction mapping. </p>
<p>For example, if $F$ is $:\mathbb{R}\rightarrow\mathbb{R}$, and $F\in C^1$, then a sufficient condition is $F'(\cdot)<1$ in all its domain. So fo... | Community | -1 | <p>Disclaimer: these are my musings about what's going on, without actually having seen anything that properly explains things.</p>
<hr>
<p>First the stuff I do know. Let $V^*$ denote the space of all linear functionals on a vector space $V$.</p>
<p>An important part of <em>multilinear algebra</em> is the tensor pro... |
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