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 |
|---|---|---|---|---|
4,242,133 | <p>I was having trouble with the question:</p>
<blockquote>
<p>Prove that
<span class="math-container">$$I:=\int_0^{\infty}\frac{\ln(1+x^2)}{x^2(1+x^2)}dx=\pi\ln\big(\frac e 2\big)$$</span></p>
</blockquote>
<p><strong>My Attempt</strong></p>
<p>Perform partial fractions
<span class="math-container">$$I=\int_0^{\infty}... | Ninad Munshi | 698,724 | <p>Define</p>
<p><span class="math-container">$$J[a] = \int_0^\infty \frac{\ln(a^2+x^2)}{1+x^2}\:dx \implies J'[a] = \frac{2a}{a^2-1}\int_0^\infty \frac{1}{x^2+1}-\frac{1}{x^2+a^2}\:dx$$</span></p>
<p><span class="math-container">$$J'[a] = \frac{\pi}{1+a} \implies J[a] = \pi\ln(1+a) + C$$</span></p>
<p>We can see that<... |
221,667 | <p>I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of duality?</p>
| Hew Wolff | 34,355 | <p>Duality is a simple way to make new vector spaces. These dual spaces are useful in functional analysis, for example when you want to define the integral of a function, or you want to analyze a probability distribution. In this case there's a vector space of functions and a linear way to map those functions to numb... |
221,667 | <p>I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of duality?</p>
| hmakholm left over Monica | 14,366 | <p>For what it's worth, you're not the only one having trouble seeing the immediate relevance of dual spaces. In the preface to Michael Artin's algebra textbook, he says:</p>
<blockquote>
<p>(2) The book is not intended for a "service course," so technical points should be presented only if they are needled in the b... |
1,515,667 | <blockquote>
<p>Show that the limit $\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$
does not exist</p>
</blockquote>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$$</p>
<p>Divide by $y^2$:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x}{\frac{x^2}{y^2}+1}$$</p>
<p>$$=\frac{2(1)}{\frac{0}{0}+1}$$</p>
<p>Since $\... | Ángel Mario Gallegos | 67,622 | <p>If we let $(x,y)\to(0,0)$ on the $x$ axis we have
$$f(x,0)=\frac{2e^x(0)^2}{x^2+(0)^2}=0\quad\implies \quad f(x,y)\to0\quad\text{as } (x,y)\to(0,0)\text{ on the }x\text{ axis}$$
On the $y$ axis we have
$$f(0,y)=\frac{2e^0y^2}{0^2+y^2}=2\quad\implies \quad f(x,y)\to 2\quad\text{as } (x,y)\to(0,0)\text{ on the }y\text... |
777,691 | <p>I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$</p>
<p>I'm stuck.<br>
$n = a \cdot k_1$<br>
$n = b \cdot k_2$<br>
$\therefore a \cdot k_1 = b \cdot k_2$</p>
<p>EDIT: so for <a href="http://en.wikipedia.org/wiki/Fizz_buzz" rel="... | David | 119,775 | <p>You are possibly thinking of the following: if $a\mid n$ and $b\mid n$ and $a,b$ are relatively prime (have no common factor except 1), then $ab\mid n$.</p>
<p><strong>Proof</strong>. We have $n=ak$ and $n=bl$ for some integers $k,l$. Therefore $b\mid ak$; since $a,b$ are relatively prime this implies $b\mid k$, ... |
1,067,762 | <p>Is what I am doing below correct, please assist. </p>
<p>$$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ </p>
<p>Is this the rule on how to "invert" the limits, and does it matter if there are imaginary numbers in the sum; or not or is it all the same with both pure real an... | André Nicolas | 6,312 | <p>Recall that
$$\tan\left(\frac{\pi}{2}-x\right)=\cot x =\frac{1}{\tan x}.$$</p>
<p>Now
$$\frac{1}{\sqrt{2}+1}=\frac{1}{\sqrt{2}+1}\cdot\frac{\sqrt{2}-1}{\sqrt{2}-1}=\sqrt{2}-1.$$</p>
|
1,067,762 | <p>Is what I am doing below correct, please assist. </p>
<p>$$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ </p>
<p>Is this the rule on how to "invert" the limits, and does it matter if there are imaginary numbers in the sum; or not or is it all the same with both pure real an... | Dylan | 135,643 | <p>Here's an easy way:</p>
<p>$$\tan B \tan C = \big(\sqrt{2} - 1 \big) \big(\sqrt{2} + 1 \big) = 1$$</p>
<p>And we know that:</p>
<p>$$ \tan B \cot B = 1$$</p>
<p>So it follows that</p>
<p>$$\cot B = \tan C$$</p>
<p>Or</p>
<p>$$ \tan \left(\frac{\pi}{2} - B \right) = \tan C$$</p>
|
1,387,454 | <p>What is the sum of all <strong>non-real</strong>, <strong>complex roots</strong> of this equation -</p>
<p>$$x^5 = 1024$$</p>
<p>Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. Is there any way to determine number of real and non-real roots of an ... | user217285 | 217,285 | <p>By the fundamental theorem of algebra, the polynomial $a_0 + a_1x + \cdots + a_nx^n$ has $n$ (not necessarily distinct) roots, say $\alpha_1,\ldots,\alpha_n$. The polynomial $\frac{a_0}{a_n} + \frac{a_1}{a_n}x + \cdots + x^n$ also has roots $\alpha_1,\ldots,\alpha_n$. Factoring the latter polynomial and expanding,
\... |
2,425,916 | <p>What is this set describing?</p>
<p>{$n∈\mathbb{N}|n\ne 1$ and for all $a∈\mathbb{N}$ and $b∈\mathbb{N},ab=n$ implies $a= 1$ or $b= 1$}</p>
<p>Is it describing a subset of natural numbers, excluding 1, that is the product of two other natural numbers, of which one must be 1? Isn't that just every natural number ex... | Siong Thye Goh | 306,553 | <p>Prime numbers only have two positive divisors, of which one of them is $1$. Hence the set is describing the set of prime numbers.</p>
<p>It doesn't include composite numbers such as $6$. If $6 = ab$, we can't conclude that $a=1$ or $b=1$, since it is possible that $a=2$ and $b=3$.</p>
|
339,090 | <p>I would like to pose a question on a variation on the classical <a href="http://en.wikipedia.org/wiki/Coupon_collector%27s_problem#Extensions_and_generalizations" rel="nofollow">coupon collector's problem</a>: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number o... | Maya | 68,377 | <p>The method in <a href="http://www.jstor.org/stable/2308930" rel="nofollow">http://www.jstor.org/stable/2308930</a> looks like it can be adapted to this setting. It is not based on martingale arguments, but does give exact expressions. </p>
|
4,295,459 | <blockquote>
<p>Find the Taylor series of <span class="math-container">$$\frac{1}{(i+z)^2}$$</span> centered at <span class="math-container">$z_0 = i$</span>.</p>
</blockquote>
<p>Im thinking if I could find the Taylor series for <span class="math-container">$$\frac{1}{i+z}$$</span> I could use that <span class="math-c... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\frac 1 {(i+z)^{2}}=-\frac 1 4 (1+(z-i)/2i)^{-2}$</span>. Now use the expansion <span class="math-container">$(1+w)^{-2}=1+\frac {(-2)} {(1)} w+\frac {(-2)(-3)} {(1)(2)}w^{2}+\frac {(-2)(-3)(-4)} {(1)(2)(3)}w^{3}+\cdots$</span> for <span class="math-container">$|w| <1$</span>.</p>
|
2,294,969 | <p>I can't seem to find a path to show that:</p>
<p>$$\lim_{(x,y)\to(0,0)} \frac{x^2}{x^2 + y^2 -x}$$</p>
<p>does not exist.</p>
<p>I've already tried with $\alpha(t) = (t,0)$, $\beta(t) = (0,t)$, $\gamma(t) = (t,mt)$ and with some parabolas... they all led me to the limit being $0$ but this exercise says that there... | StackTD | 159,845 | <p>So you want to find a path that gives a non-zero limit. Take a path where:
$$\lim_{(x,y)\to(0,0)} \frac{x^2}{x^2 + \color{blue}{y^2 -x}}$$the blue part is $0$, since then the limit clearly simplifies to $1$; so take $x=y^2$.</p>
<p><em>Edit</em>: this comes down to s.harp's suggestion in the comment as well.</p>
|
3,077,808 | <p>I'm trying to find for which values <span class="math-container">$r$</span> the following improper integral converges.
<span class="math-container">$$\int_0^\infty x^re^{-x}\, dx$$</span>
What I have so far is that <span class="math-container">$x^r < e^{\frac{1}{2}x}$</span> for <span class="math-container">$x \g... | RRL | 148,510 | <p>Note that for all <span class="math-container">$r \in \mathbb{R}$</span>,</p>
<p><span class="math-container">$$\lim_{x \to \infty} \frac{x^r e^{-x}}{x^{-2}} = \lim_{x \to \infty} \frac{x^{r+2}}{e^x} = 0 $$</span></p>
<p>since the exponential function tends to infinity faster than any polynomial. Hence, by the li... |
1,969,934 | <p>Taken from <a href="https://en.wikipedia.org/wiki/E_(mathematical_constant)" rel="nofollow noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>The number $e$ is the limit $$e = \lim_{n \to \infty} \left (1 + \frac{1}{n} \right)^n$$</p>
</blockquote>
<p><strong>Graph of $f(x) = \left (1 + \dfrac{1}{x} \right)^x$</stro... | Dietrich Burde | 83,966 | <p>An algebraic way is Binomial Expansion, which is given by
$$\begin{eqnarray*}
\left(1+\frac{1}{n}\right)^{\!n} &=& 1+n\left(\frac{1}{n}\right)+\frac{n(n-1)}{2!}\left(\frac{1}{n}\right)^{\!2}+\frac{n(n-1)(n-2)}{3!}\left(\frac{1}{n}\right)^{\!3}+\cdots \\ \\
&=& 1+1+\frac{n(n-1)}{n^2}\cdot\frac{1}{2!}+... |
1,063,599 | <p>There is a deck made of $81$ different card. On each card there are $4$ seeds and each seeds can have $3$ different colors, hence generating the $ 3\cdot3\cdot3\cdot3 = 81 $ card in the deck.
A tern is a winning one if,for every seed, the correspondent colors on the three card are or all the same or all different.</... | Empy2 | 81,790 | <p>For question one, ask a simpler question: Look at just the first seeds, how many ways can you color them?<br>
Two possibilities are: Card 1: Red, Card 2: Red, Card 3: Red<br>
and Card1:Red, Card2: Blue, Card3: Yellow.<br>
Now for four seeds, you make the choice of colors four times.<br>
You should check that you hav... |
1,521,739 | <p>The following is the Meyers-Serrin theorem and its proof in Evans's <em>Partial Differential Equations</em>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/XnzXY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XnzXY.png" alt="enter image description here"></a>
<a href="https://i.stack... | Lubin | 17,760 | <p>I’ll do the same thing as @heropup, but without the notation. To save myself typing, I’ll set $c=1/\sqrt2=\cos 45^\circ=\sin45^\circ$. Then you want a rotation of $45^\circ$, and to do this I’ll set
\begin{align}
x&=cX-cY\\y&=cX+cY\,.
\end{align}
Make these substitutions, and if I’m not mistaken, you get a n... |
873,803 | <p>I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about (or really self-studied) for that matter, is general polynomials beyond quadratics. In particular, I've been having ... | Anurag A | 68,092 | <p>Just use $P(x)=(x-r_1)(x-r_2)(x-r_3)$. Then compare the coefficients on each side.</p>
|
880,124 | <p>I'm trying to solve the question 1.36 from Fulton's algebraic curves book:</p>
<blockquote>
<p>Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and
$\dim_{\mathbb C}\mathbb C[X,Y]/I$.</p>
</blockquote>
<p>Obviously $V(I)=\{(0,0)\}$ and by a corolary in the same section we know that $\dim_{\mathbb C}... | Matemáticos Chibchas | 52,816 | <p><strong>Hint:</strong> Note that $Y^2=\bigl((Y^2-X^2)+(Y^2+X^2)\bigr)/2$ belongs to $I$, which in turn implies that $X^2=(Y^2+X^2)-Y^2\in I$. In other words $(X^2,Y^2)\subseteq I$, and a similar reasoning shows the reversed inclusion. Having this, what can you say about the higher order terms of an arbitrary polynom... |
880,124 | <p>I'm trying to solve the question 1.36 from Fulton's algebraic curves book:</p>
<blockquote>
<p>Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and
$\dim_{\mathbb C}\mathbb C[X,Y]/I$.</p>
</blockquote>
<p>Obviously $V(I)=\{(0,0)\}$ and by a corolary in the same section we know that $\dim_{\mathbb C}... | Aaron | 9,863 | <p>Given a ring $R$, every element of $R[x]$ is a sum of elements $a_0+a_1 x + a_2 x^2 + \ldots +a_n x^n$ where $a_i \in R$. Using this twice, we have that a basis for $\mathbb C[x,y]$ is the monomials $x^iy^j$ where $i,j\in \mathbb N$ Since $I=(x^2+y^2,x^2-y^2)=(x^2,y^2)$, we have that every monomial where either $i... |
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | Gil | 75,252 | <p>As a student who is suffering from the very same problem, I want to share my less professional solution. With this method, I feel like my studying became much more efficient.</p>
<p>When I read, I tend to be generous. I used to pick out every single detail and I gave an author or a lecturer a criticism about not wr... |
774,434 | <p><a href="https://www.wolframalpha.com/input/?i=Sum%5BBinomial%5B3n,n%5Dx%5En,%20%7Bn,%200,%20Infinity%7D%5D" rel="noreferrer">Wolfram alpha tells me</a> the ordinary generating function of the sequence $\{\binom{3n}{n}\}$ is given by
$$\sum_{n} \binom{3n}{n} x^n = \frac{2\cos[\frac{1}{3}\sin^{-1}(\frac{3\sqrt{3}\sqr... | Lucian | 93,448 | <p><strong>Too long for a comment:</strong> By investigating series of the form $S_a(x)=\displaystyle\sum_{n=0}^\infty{an\choose n}~x^n$, we notice that — in general — we get a $($ <a href="https://en.wikipedia.org/wiki/Generalized_hypergeometric_function" rel="nofollow">generalized</a> $)$ <a href="https:/... |
243,210 | <p>I have difficulty computing the $\rm mod$ for $a ={1,2,3\ldots50}$. Is there a quick way of doing this?</p>
| Felix Marin | 85,343 | <p>$$
\mbox{Set}\ {\rm y}\left(x\right) \equiv {\rm f}\left(x^{2}\right)\equiv{\rm F}\left(x\right)
$$</p>
<blockquote>
<p>$$
\mbox{You get}\quad {\rm F}'\left(x\right)
={{\rm F}^{2}\left(x\right) - {\rm F}\left(x\right)\over 2}
$$</p>
</blockquote>
|
4,294,714 | <p>The problem is as follows:</p>
<p>You have an unlimited number of marbles and each marble is one of 16 different colours. You have to choose 6 marbles and order is irrelevant. How many different combinations of 6 marbles are there?</p>
| user2661923 | 464,411 | <p>Ignore passenger D, as a non-factor.</p>
<p>Computation will be <span class="math-container">$$\frac{N\text{(umerator)}}{D\text{(enominator)}},$$</span></p>
<p>where <span class="math-container">$D = 7^3$</span>, which represents <span class="math-container">$7$</span> choices for each of <span class="math-container... |
2,363,390 | <p>This question arises from an unproved assumption made in a proof of L'Hôpital's Rule for Indeterminate Types of $\infty/\infty$ from a Real Analysis textbook I am using. The result is intuitively simple to understand, but I am having trouble formulating a rigorous proof based on limit properties of functions and/or ... | Stephen K. | 307,717 | <p>I believe I have a proof for the special case when the domain is an interval. It is interesting to me that the proof of this simple assertion is not so trivial (although if there is a simplier route to getting this result, I would be interested in finding out).</p>
<p><strong>Theorem</strong> Let $f$ be a continuou... |
1,834,756 | <p>The Taylor expansion of the function $f(x,y)$ is:</p>
<p>\begin{equation}
f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)}{\partial x \partial y}
\end{equation}</p>
<p>When $f=(x,y,z)$ is the following true?</p>
<p>$$\begin{alig... | Kamal Kant Misra | 672,693 | <p>It is not correct! We should have the expansion as
<span class="math-container">$f(x+u,y+v,z+w)\approx f(x,y,z) + u \frac{\partial f(x,y,z)}{\partial x} + v\frac{\partial f(x,y,z)}{\partial y} + w\frac{\partial f(x,y,z)}{\partial z} + \frac{1}{2!} \left[u^2 \frac{\partial^2 f(x,y,z)}{\partial x^2} + v^2 \frac{\parti... |
1,594,622 | <p>As an example, calculate the number of $5$ card hands possible from a standard $52$ card deck. </p>
<p>Using the combinations formula, </p>
<p>$$= \frac{n!}{r!(n-r)!}$$</p>
<p>$$= \frac{52!}{5!(52-5)!}$$</p>
<p>$$= \frac{52!}{5!47!}$$</p>
<p>$$= 2,598,960\text{ combinations}$$</p>
<p>I was wondering what the l... | kaiten | 301,758 | <p>Consider drawing $1$ card at a time.
The first card can be any of the $52$ cards.
The second can be any of the remaining $51$.
The third can be any of $50$... etc</p>
<p>So you have $52\times 51\times 50\times 49\times 48$ possibilities for $5$ cards. This is more conveniently written $\dfrac{52!}{47!}$.</p>
<p>Bu... |
309,234 | <p>I am looking for good, detailed references for "mod $p$ lower central series".</p>
<p>So far I only find papers such as (<a href="https://core.ac.uk/download/pdf/81193793.pdf" rel="nofollow noreferrer">https://core.ac.uk/download/pdf/81193793.pdf</a>, <a href="https://www.sciencedirect.com/science/article/pii/00409... | Community | -1 | <p>For the free group, it is called the Zassenahus filtration. Golod-Shafarevich groups are defined in terms of it. <a href="https://arxiv.org/abs/1206.0490" rel="nofollow noreferrer">Ershov's survey</a> on Golod-Shafarevich groups is excellent. Highly recommended. It is published as Ershov, Mikhail Golod-Shafarevich ... |
711,168 | <p>Let $V$ be an $n$-dimensional real inner product space and let $a=\lbrace v_1,v_2,\dots v_n \rbrace$ be an orthonormal basis for $V$. Let $W$ be a subspace of $V$ with orthonormal basis $B = \lbrace w_1, w_2,\dots w_k\rbrace$. Let $A = \lbrace [w_1]a, [w_2]a,\dots [w_k]a\rbrace$ and let $P_w$ be the orthogonal proje... | dani_s | 119,524 | <p>For a counterexample consider $G = \{0, 1\}$ and define $*$ by $$0 * 0 = 0 \\ 0 * 1 = 1 \\ 1 * 0 = 0 \\ 1 * 1 = 1$$</p>
<p>Show that your axioms hold; on the other hand $G$ is not a group because for example there is no identity.</p>
|
1,790,311 | <p>Show the following equalities $5 \mathbb{Z} +8= 5\mathbb{Z} +3= 5\mathbb{Z} +(-2)$.</p>
<p>$5 \mathbb{Z} +8=\{5z_{1}+8: z_{1} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +3=\{5z_{2}+3: z_{2} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +(-2)=\{5z_{3}+(-2): z_{3} \in \mathbb{Z}\}$.</p>
<p>So, how can prove to use these de... | MPW | 113,214 | <p><strong>Hint:</strong> You need to show that $8$, $3$, and $-2$ are all in the same coset of $5\mathbb Z$. Said differently, you need to show that
$8-3$, $8-(-2)$, and $3-(-2)$ are all in $5\mathbb Z$.</p>
|
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| DVD | 77,260 | <p>An eigenvector $v$ of a matrix $A$ is a directions unchanged by the linear transformation: $Av=\lambda v$.
An eigenvalue of a matrix is unchanged by a change of coordinates: $\lambda v =Av \Rightarrow \lambda (Bu) = A(Bu)$. These are important invariants of linear transformations.</p>
|
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| Ciro Santilli OurBigBook.com | 53,203 | <p><strong>Eigenvalues and eigenvectors are central to the definition of measurement in quantum mechanics</strong></p>
<p>Measurements are what you do during experiments, so this is obviously of central importance to a Physics subject.</p>
<p>The state of a system is a vector in <a href="https://en.wikipedia.org/wiki/H... |
507,827 | <p>Let $a_n$ be a positive sequence. Prove that
$$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$</p>
| Siméon | 51,594 | <p>Without loss of generality we can assume $a_1=1$.</p>
<p>Taking logarithms and <em>seeking for a contradiction</em>, suppose that there exists $0 < \alpha < 1$ such that for all $n$ large enough,
$$
\ln \left(\dfrac{1+a_{n+1}}{a_n}\right) = \ln a_{n+1} - \ln a_n + \ln\left(1+\frac{1}{a_{n+1}}\right)\leq \frac... |
142,677 | <p>Consider the following list of equations:</p>
<p>$$\begin{align*}
x \bmod 2 &= 1\\
x \bmod 3 &= 1\\
x \bmod 5 &= 3
\end{align*}$$</p>
<p>How many equations like this do you need to write in order to uniquely determine $x$?</p>
<p>Once you have the necessary number of equations, how would you actually ... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ It <em>can</em> be done simply without CRT. $\rm\:x\equiv -2\:\ (mod\ \rm3,5)\iff x\equiv -2\equiv 13\pmod{ 15}\:$ Now since $13\equiv 1\pmod 2\:$ we conclude $\rm\:x\equiv 13\:\ (mod\ 2,15)\iff x\equiv 13\pmod{30}\:$ </p>
<p>Hence your hunch was correct: it <em>is easy</em> (these are o... |
400,749 | <p>The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an absolute minimum for the function. </p>
<p>So, if we have a continuous function on $[a,b]$ we're guaranteed to have... | William Stagner | 49,220 | <p>This is a difference between <strong>necessary</strong> and <strong>sufficient</strong> conditions, see <a href="http://en.wikipedia.org/wiki/Necessary_and_sufficient" rel="nofollow">here</a>.</p>
<p>The extreme value theorem states that continuity on a closed interval is <strong>sufficient</strong> to ensure that ... |
400,749 | <p>The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an absolute minimum for the function. </p>
<p>So, if we have a continuous function on $[a,b]$ we're guaranteed to have... | user64494 | 64,494 | <p><a href="http://en.wikipedia.org/wiki/Semi-continuity" rel="nofollow">Semi-continuous</a> functions have this property.</p>
|
2,325,421 | <blockquote>
<p>If $f$ is a linear function such that $f(1, 2) = 0$ and $f(2, 3) = 1$, then what is $f(x, y)$?</p>
</blockquote>
<p>Any help is well received.</p>
| Sahiba Arora | 266,110 | <p><strong>Hint:</strong> $\{(1,2),(2,3)\}$ forms a basis of $\mathbb{R}^2$.</p>
|
507,454 | <p>I had a geometry class which was proctored using the Moore method, where the questions were given but not the answers, and the students were the source of all answers in the class. One of the early questions which we never solved is listed in the title.</p>
<p>In this case, use any reasonable definition of "betwee... | lab bhattacharjee | 33,337 | <p>Like Stefan,</p>
<p>$$(6a\pm1)^2=36a^2\pm12a+1=24a^2+24\frac{a(a\pm1)}2+1\equiv1\pmod{24}$$ as the product of two consecutive integers is always even</p>
<p>Observe that $6a\pm1$ is not necessarily prime, but $(6a\pm1,6)=1$ </p>
<p>So, any number $p$ relatively prime to $6$ will satisfy this</p>
|
2,592,282 | <p>If $f$and $g$ is one to one then ${f×g}$ and ${f+g}$ is one to one </p>
<p>Is this true? If not, I need some clarification to understand </p>
<p>Thanks</p>
| operatorerror | 210,391 | <p>No. For the first take
$$
f(x)=x,\;g(x)=-x
$$
making $f+g\equiv 0$. </p>
<p>For the second, take $f(x)=g(x)=x$.</p>
|
2,592,282 | <p>If $f$and $g$ is one to one then ${f×g}$ and ${f+g}$ is one to one </p>
<p>Is this true? If not, I need some clarification to understand </p>
<p>Thanks</p>
| Soumajit Das | 431,228 | <p>Well consider f(x) = x and g(x) = x^3
Both the functions are one-one but f × g is not an one- one function but f+g is one - one. </p>
|
2,929,887 | <p>Show that if a square matrix <span class="math-container">$A$</span> satisfies the equation <span class="math-container">$p(A)=0$</span>, where <span class="math-container">$p(x) = 2+a_1x+a_2x^2+...+a_kx^k$</span> where <span class="math-container">$a_1,a_2,...,a_k$</span> are constant scalars, then <span class="mat... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$Ax=0$</span>. Then <span class="math-container">$0=p(A)x=2x+0+\cdots+0$</span> so <span class="math-container">$x=0$</span>. So <span class="math-container">$A$</span> is injective, hence invertible. </p>
|
3,117,459 | <p>I am interested in approximating the natural logarithm for implementation in an embedded system. I am aware of the Maclaurin series, but it has the issue of only covering numbers in the range (0; 2).</p>
<p>For my application, however, I need to be able to calculate relatively precise results for numbers in the ran... | Peter Foreman | 631,494 | <p>For the natural logarithm I recently wrote an algorithm for this myself. In my implementation, I found that the fastest method is to use the following iterative method. First take <span class="math-container">$x_0$</span> as an initial approximation to the logarithm, then use
<span class="math-container">$$x_n=x_{n-... |
3,857,071 | <p>I'm doing work with wind direction data, and will be coding a function that checks whether the a given wind direction is bewteen lower and upper limit or bound</p>
<p>e.g.:</p>
<ol>
<li>Is 5 degrees is between 315 degrees and 45 degrees? True</li>
<li>Is 310 degrees between 315 degrees and 45 degrees? False</li>
<li... | Math Lover | 801,574 | <p>The way I understand, your bounds are <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> where either of them can be bigger like in your first case <span class="math-container">$\alpha = 315^0, \beta = 45^0$</span> whereas in the third case, it is <span class="math-container... |
216,082 | <ol>
<li><p>How to prove that $\mathbb{R}^k$ is connected?</p></li>
<li><p>Let $C$ be an infinite connected set in $\mathbb{R}^k$. How can I show that $C\bigcap \mathbb{Q}^k$ is nonempty?</p></li>
</ol>
| DonAntonio | 31,254 | <p>1) For any $\,a,b\in\Bbb R^k\,$ , the straighline $\,\{(t-1)a+tb\;:\;t\in\Bbb R\}\,$ is completely contained in $\,R^k\,$, so it is path connected and thus connected.</p>
|
1,722,692 | <p>I am asked to find</p>
<p>$$\lim_{x \to 0} \frac{\sqrt{1+x \sin(5x)}-\cos(x)}{\sin^2(x)}$$</p>
<p>and I tried not to use L'Hôpital but it didn't seem to work. After using it, same thing: the fractions just gets bigger and bigger.</p>
<p>Am I missing something here?</p>
<p>The answer is $3$</p>
| Henricus V. | 239,207 | <p>Use the equivalent infinitesimal
$$ \lim_{x \to 0} \frac{(\sin x)^2}{x^2} = 1
$$
to change the denominator. Now l'Hopital only need to be applied twice.</p>
|
2,922,077 | <p>An urn contains $29$ red, $18$ green, and $12$ yellow balls. Draw two balls without replacement What is the probability that the number of red balls in the sample is exactly $1$ or the number of yellow balls in the sample is exactly $1$ (or both)? What about with replacement? </p>
<p>I can't seem to figure this out... | Mike Earnest | 177,399 | <p>Using the fact that the function $f(t)=|t|^p$ is convex, we have
\begin{align}
|b+a|^p+|b-a|^p
&=2\big(\tfrac12f(b+a)+\tfrac12f(b-a)\big)\\
&\ge 2f\big(\tfrac12(b+a)+\tfrac12(b-a)\big)\\
&=2f(b) = 2|b|^p.
\end{align}
To see $f$ is convex, note $f''(t)=p(p-1)|t|^{p-2}\ge 0$. This suffices for all $p\ge 2$... |
1,500,156 | <p>Call a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ nondecreasing if $x,y \in \mathbb{R}^n$ with $x \geq y$ implies $f(x) \geq f(y)$. Suppose $f$ is a nondecreasing, but not necessarily continuous, function on $\mathbb{R}^n$, and $D \subset \mathbb{R}^n$ is compact. Show that if $n = 1$, $f$ always has a maximu... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Consider the set
$$
A = \{(t,-t) \in \Bbb R^2: t \in [0,1]\}
$$
note that any function on $A$ is, by definition, non-decreasing.</p>
|
733,101 | <p>I've been stuck for a while on this question and haven't found applicable resources.</p>
<p>I have 10 choices and can select 3 at a time. I am allowed to repeat choices (combination), but the challenge is that ABA and AAB are not unique.</p>
<p>10 choose 3 is the question.</p>
<p>I have been working on a smaller ... | Calvin Lin | 54,563 | <p>This approach works because 3 is small.</p>
<p>Case 1: 3 distinct objects - ABC<br>
There are ${ 10 \choose 3}$ ways here.</p>
<p>Case 2: 2 distinct objects, 1 repeated twice - AAB<br>
There are $ 10 \times 9$ ways here.</p>
<p>Case 3: 1 distinct object, 1 repeated thrice - AAA<br>
There are $10$ ways here.</p>
... |
2,336,744 | <p>Any two co-prime number $a,b$ with $a>b+2$ we have $a^2+b^2$ is not divisible by $a-b$, $a,b \in \mathbb{N}$.
But how to prove this?</p>
| Paolo Leonetti | 45,736 | <p>If $a-b \mid a^2+b^2$ then $\gcd(a^2+b^2,a-b)=a-b \ge 3$. Now
$$
\gcd(a^2+b^2,a-b)=\gcd(a^2+b^2-(a-b)^2,a-b)=\gcd(2ab,a-b).
$$
Since $\gcd(a,b)=1$ then every prime dividing $a$ or $b$ cannot divide $a-b$. Hence
$$
\gcd(a^2+b^2,a-b) \in \{1,2\}.
$$
In particular, your conjecture is true.</p>
|
1,726,416 | <p>$\displaystyle \sum_{n=1}^∞ (-1)^n\dfrac{1}{n}.\dfrac{1}{2^n}$</p>
<p>Knowing that</p>
<ol>
<li>An alternating harmonic series is always convergent</li>
<li>Riemann series are always convergent when $p>1$</li>
</ol>
<p>Is it safe to say that the product of these two is convergent (as described above)?</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: Your series is absolutely convergent because $|a_n| \leq \left(\dfrac{1}{2}\right)^n$. And if you have $2$ converging series, say $\displaystyle \sum_{k=1}^\infty a_k, \displaystyle \sum_{k=1}^\infty b_k$, then their product as you define it yourself $\displaystyle \sum_{k=1}^\infty a_kb_k$ ma... |
210,401 | <p>In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that </p>
<p>$$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$</p>
<p>for every positive integer $n$?</p>
<p>A few thoughts: Graham and Lin (<em>Math. Mag.</em> 51 (1978) 174-176) give a test for $(s_n)... | Will Sawin | 18,060 | <p>Let's use the notation $\{ x\}$ for the fractional part of a number $x$.</p>
<p>Assume $u, v$, and $u/v$ are all irrational.</p>
<p>Then, $\{un\}$ and $\{vn\}$ behave as independent uniform random variables. (This is proved by Fourier analysis, vindicating James Cranch's suggestion) $s_{n+1}-s_n$ depends on $\{un... |
1,583,887 | <p>This problem is from an an Introduction to Abstract Algebra by Derek John that I am solving.</p>
<p>I am trying to prove that any group of order 1960 aren't simple, so I am doing it by contradiction, but I got stuck in the middle.</p>
<p>Suppose that $|G| = 1960 = 2^3 * 5 * 7^2$, by Sylow theory we have 2,5,7 sub... | 2'5 9'2 | 11,123 | <p>If the group is not simple, then there actually <em>must</em> be $8$ Sylow-$7$ subgroups. It's the only option give the Sylow theorems.</p>
<p>So $G$ permutes these $8$ subgroups through conjugation. That means there is a map from $G$ to $S_8$. The kernel of this map is a normal subgroup, so the kernel is either al... |
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | José Carlos Santos | 446,262 | <p>No. Let $A$ be the set of all transpostions. Then $\langle A\rangle=S_n$, but not subset of $A$ generates $A_n$, since no transposition belongs to $A_n$.</p>
|
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | Andrea Mori | 688 | <p>Certainly not for <em>any</em> set of generators.</p>
<p>For instance, one knows that $S_n$ is generated by (some) cycles of lngth 2 and not all subgroups have the same property.</p>
|
68,817 | <p>I have two questions after reading the Hahn-Banach theorem from Conway's book ( I have googled to know the answer but I have not found any result yet. Also I am not sure that whether my questions have been asked here somewhere on this forum - so please feel free to delete them if they are not appropriate )</p>
<p>H... | Gerald Edgar | 454 | <p>$Y$ is called an <em>injective Banach space</em> if the extension exists for all $X$, $M$, and $f$. An example is $Y = l^\infty$. (Should be in Banach space text books. Here's a paper: <a href="http://www.jstor.org/pss/1998210" rel="nofollow">http://www.jstor.org/pss/1998210</a> )</p>
|
68,817 | <p>I have two questions after reading the Hahn-Banach theorem from Conway's book ( I have googled to know the answer but I have not found any result yet. Also I am not sure that whether my questions have been asked here somewhere on this forum - so please feel free to delete them if they are not appropriate )</p>
<p>H... | godelian | 12,976 | <p>Note quite what you asked, but related:</p>
<p>Continuous extensions of (continuous) functionals from $M$ are unique if and only if $M$ is a dense subspace of $X$. Otherwise its closure is a proper closed subspace and therefore there exists a nonzero bounded functional $\phi$ vanishing on the closure, which implies... |
623,796 | <p>What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ?</p>
<p>Thanks in advance.</p>
| lsp | 64,509 | <p>$f(x) = \sqrt{x^2 - 4x - 5}$</p>
<p>Since the square root of a negative number is imaginary, the condition is that :
$$x^2 - 4x - 5 \geq 0$$
$$ (x-2)^2 - 9 \geq 0$$
$$ (x-2) \geq 3$$ or
$$ (x-2) \leq -3$$
Therefore the domain of the function $f(x)$ would be:
$$(-\infty, -1] \cup [5, +\infty)$$</p>
<p>Hope the answ... |
623,796 | <p>What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ?</p>
<p>Thanks in advance.</p>
| WLOG | 21,024 | <p>The domain is $D =\{ x \in \mathbb{R} | x^{2}-4x-5 \geq 0 \}$.</p>
<p>The roots of $x^{2}-4x-5$ are $-1$ and $5$ so $D =(-\infty, -1] \cup [5, +\infty)$.</p>
|
182,091 | <p>3D graphics can be easily rotated interactively by clicking and dragging with the mouse.</p>
<p>Is there a simple way to achieve the same for animated 3D graphics? I would like to rotate them interactively (in real time) <em>while</em> the animation is running.</p>
<hr>
<p>Here's an example animation, mostly take... | Szabolcs | 12 | <p>One possible solution is to make a separate trackball control.</p>
<pre><code>{vp, vv} = {ViewPoint, ViewVertical} /. Options[Graphics3D];
Graphics3D[{Cuboid[]}, Boxed -> False, SphericalRegion -> True,
RotationAction -> "Clip",
Prolog -> {GrayLevel[.8], Disk[Scaled[{1/2, 1/2}], Scaled[1/2]]},
As... |
128,122 | <p>Original Question: Suppose that $X$ and $Y$ are metric spaces and that $f:X \rightarrow Y$. If $X$ is compact and connected, and if to every $x\in X$ there corresponds an open ball $B_{x}$ such that $x\in B_{x}$ and $f(y)=f(x)$ for all $y\in B_{x}$, prove that f is constant on $X$. </p>
<p>Here's my attempt:
Cover ... | Asaf Karagila | 622 | <p>Since $f(X)=\{a_1,a_2,\ldots,a_n\}$ we can take $A_k=\bigcup\{B_x\mid f(x)=a_k\}$ to be an open set in $X$ for every $k\leq n$. </p>
<p>These are disjoint open sets and their union is $X$. Use the fact the space is connected to deduce $k=1$. </p>
|
2,798,026 | <blockquote>
<p>There are three vector $a$, $b$, $c$ in three-dimensional real vector space, and the inner product between them $a\cdot a=b\cdot b = a\cdot c= 1, a\cdot b= 0, c\cdot c= 4$. When setting $x = b\cdot c$, answer the following question: when $a, b, c$ are linearly dependent, find all possible values of $... | Stefan | 453,800 | <p>For the first identity, $a \times b$ is orthogonal to both $a,b$, hence $c$ must be in the plane spanned by the orthonormal $a,b$.</p>
<p>So we can write $c$ as
$$
c = \alpha a + \beta b
$$</p>
<p>plugging this in we see that
$$
1 = a \cdot c = \alpha a \cdot a = \alpha
$$</p>
<p>and
$$
x = b \cdot c = \beta b ... |
2,328,567 | <p>I'm trying to search for any kind of development in the mathematics (science, astronomy, even astrology or other kind of early studies that envolve any kind of math) expecially in early england and in the carolingian empire. </p>
<p>The problem I have is that it seams that math died in there: every work seems to be... | Riju | 420,099 | <p>Homeomorphism means a continuous bijection whose inverse is continuos too. Now use the fact that f is continuous iff for every open set $U$ of Y , $f^{-1}(U)$ is open in X. The bijection is needed for the other direction, when you have to prove f is homeomorphism. $f^{-1}$ exists since it is a bijection and continu... |
287,043 | <p>Consider the problem of finding the limit of the following diagram:</p>
<p>$$ \require{AMScd} \begin{CD}
& & & & E
\\ & & & & @VVV
\\ && C @>>> D
\\ & & @VVV
\\A @>>> B
\end{CD} $$</p>
<p>The abstract definition of the limit involves an adjunction... | Dylan Wilson | 6,936 | <p>Let me try to turn my comments into an answer (I think it's also essentially what Vladimir was saying). Suppose you have some diagram $F: K \to \mathcal{C}$. To compute the limit of $F$ is the same as computing the right Kan extension $\epsilon_*F$ along the map $\epsilon: K \to \bullet$. The process you're describi... |
1,469,859 | <p>If I have a positive <code>x</code>, are there more integers below <code>x</code> or above <code>x</code>?</p>
<p>I was discussing this with some friends and we came up with two opposing ideas:</p>
<ol>
<li>No, since you can always count one more in either direction.</li>
<li>Yes, since the infinite amount of numb... | vadim123 | 73,324 | <p>$\mathbb{Z}$ is countable. Hence all subsets of $\mathbb{Z}$ are countably infinite, or finite. There aren't different sizes of infinity in the subsets of $\mathbb{Z}$ -- the only way you can get two subsets of different sizes is if at least one of them is finite.</p>
|
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | Siméon | 51,594 | <p>As $x$ tends to $0$, $\cos(2x)$ tends to $1$. Hence, using $\ln(1+u) \sim u$, $\sin u \sim u$ and $1 - \cos u\sim \frac{u^2}{2}$,
$$
\frac{\ln(\cos(2x))}{x\sin x} \sim \frac{\cos(2x)-1}{x\times x} \sim \frac{-\frac{(2x)^2}{2}}{x^2} \sim - 2
$$</p>
|
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Peter Humphries | 3,803 | <p>It depends very much on what areas of harmonic analysis you're interested in, of course. Grafakos' books are excellent and really quite advanced, and if you wish to continue in that style of harmonic analysis, then there's not much else you can do other than start reading many of the articles that he cites. On the o... |
1,025,117 | <p>Let $V$ be finite dim $K-$vector space. If w.r.t. any basis of $V$, the matrix of $f$ is a diagonal matrix, then I need to show that $f=\lambda Id$ for some $\lambda\in K$. </p>
<p>I am trying a simple approach: to show that $(f-\lambda Id)(e_i)=0$ where $(e_1,...,e_2)$ is a basis of $V$. Let the diagonal matrix b... | UserX | 148,432 | <p>Hint; Let $t=\frac1x$${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$</p>
|
3,261,334 | <blockquote>
<p>Prove isomorphism of groups <span class="math-container">$\langle G, + , {}^{-1}\rangle$</span> and <span class="math-container">$\langle G, *,{}^{-1}\rangle$</span>, where <span class="math-container">$a*b=b+a$</span><br>
<span class="math-container">$\forall a,b \in G$</span></p>
</blockquote>
<... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> Define <span class="math-container">$f(a)=a^{-1}$</span> and prove that it is an isomorphism.</p>
|
2,505,171 | <p>How many numbers are there if you do not allow leading $0$'s? </p>
<p>In how many of the numbers in each case is no digit $j$ in the $j$th place?</p>
<p>If leading $0$'s are allowed? </p>
<p>If they are not allowed? </p>
<p>I know how to answer this if the numbers $0$ through $9$ can be repeated, but I am gett... | JMoravitz | 179,297 | <p>Approach via the <a href="https://en.wikipedia.org/wiki/Rule_of_product" rel="nofollow noreferrer">rule of product</a> (<em>also called the multiplication principle</em>) which can be paraphrased as the following:</p>
<blockquote>
<p>If you wish to count how many outcomes there are to a particular scenario and you ... |
4,496,913 | <blockquote>
<p>If <span class="math-container">$x=\frac12(\sqrt[3]{2009}-\frac{1}{\sqrt[3]{2009}})$</span>, what is the value of <span class="math-container">$(x+\sqrt{1+x^2})^3$</span>?</p>
</blockquote>
<p>I solved this problem as follow,</p>
<p>Assuming <span class="math-container">$\sqrt[3]{2009}=\alpha$</span> , ... | dxiv | 291,201 | <blockquote>
<p>Assuming <span class="math-container">$\sqrt[3]{2009}=\alpha$</span> , we have <span class="math-container">$x=\dfrac12\left(\alpha-\dfrac1{\alpha}\right)$</span></p>
</blockquote>
<p>We also have:</p>
<p><span class="math-container">$$
\begin{align}
x = \frac{1}{2}\left(x+\sqrt{1+x^2} \;+\; x-\sqrt{1+x... |
3,420,459 | <p>I have this <a href="https://i.stack.imgur.com/ew8Id.png" rel="nofollow noreferrer">question:</a></p>
<blockquote>
<p>If <span class="math-container">$a\otimes b=a^b-b^a$</span>, what is <span class="math-container">$(3\otimes 2)\otimes (4\otimes 1)$</span>?</p>
</blockquote>
<p>The answer in the solution set I ... | J. W. Tanner | 615,567 | <p>Just plug it in:</p>
<p><span class="math-container">$(3\otimes 2)\otimes (4\otimes 1)=(3^2-2^3)\otimes(4^1-1^4)=1\otimes3=1^3-3^1=\;?$</span></p>
|
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| GEdgar | 442 | <p>Write <span class="math-container">$c(x) = f(x) + \frac{1}{f(x)}$</span>. Solve a quadratic equation to see that
<span class="math-container">$f(x)$</span> is either <span class="math-container">$(c(x)+\sqrt{c(x)^2-4}\;)/2$</span> or <span class="math-container">$(c(x)-\sqrt{c(x)^2-4}\;)/2$</span> . So, for all <s... |
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| Fabio Lucchini | 54,738 | <p>Let
<span class="math-container">\begin{align}
&a=\liminf_{x\to 0}f(x)&&b=\limsup_{x\to 0}f(x)
\end{align}</span>
<a href="https://math.stackexchange.com/q/205346">Since</a>
<span class="math-container">$$\liminf_{x\to 0}\left(f(x)+\frac 1{f(x)}\right)\leq\liminf_{x\to 0}f(x)+\limsup_{x\to 0}\frac 1{f(x)... |
2,056,499 | <p>I am trying to prove that if
$$
\lim_{x \to c} (f(x)) = L_1
\\ \lim_{x \to c} (g(x)) = L_2
\\ L_1, L_2 \geq 0
$$
Then
$$
\lim_{x \to c} f(x)^{g(x)} = (L_1)^{L_2}
$$</p>
<p>I am doing this for fun, and my prof said that it shouldn't be too hard, but all I got so far is
$$
\forall \epsilon >0 \ \exists \delta >... | DanielWainfleet | 254,665 | <p>It is often convenient to write $0^0=1,$ for example, in "Let $p(x)=\sum_{j=0}^n a_jx^j$ " it is assumed that $a_0x^0=a_0$ when $x=0.$</p>
<p>But if $L_1=L_2=0$ then $f(x)/g(x)$ can converge to any non-negative value, or fail to converge. Examples: Let $c=0:$ </p>
<p>(1). Let $f_1(x)=1/e^{1/|x|}$ for $x\ne 0$ a... |
35,964 | <p>This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong:</p>
<p>Question:
Show that a ball dropped from height of <em>h</em> feet and bounces in such a way that each bounce is $\frac34$ of the height of... | André Nicolas | 6,312 | <p>Before jumping to a formula, let us calculate a little. The distance travelled until the first contact with the ground is $h$. </p>
<p>The distance travelled between the first contact and the second is $(h)(2)(3/4)$
(up and then down). The distance travelled from second contact to third is $(h)(2)(3/4)^2$, and s... |
1,893,609 | <p>I am trying to show that $A=\{(x,y) \in \Bbb{R} \mid -1 < x < 1, -1< y < 1 \}$ is an open set algebraically. </p>
<p>Let $a_0 = (x_o,y_o) \in A$. Suppose that $r = \min\{1-|x_o|, 1-|y_o|\}$ then choose $a = (x,y) \in D_r(a_0)$. Then</p>
<p>Edit: I am looking for the proof of the algebraic implication t... | Andres Mejia | 297,998 | <p>Fun fact: The type of quadrilateral (for ease of argument, draw the convex one defined by the four points in question) is called <a href="https://en.wikipedia.org/wiki/Cyclic_quadrilateral" rel="nofollow">cyclic</a>. In Euclid's Elements, Book 3, Proposition 22, it is proven that a quadrilateral is cyclic if and onl... |
4,133,760 | <p>Dr Strang in his book linear algebra and it's applications, pg 108 says ,when talking about the left inverse of a matrix( <span class="math-container">$m$</span> by <span class="math-container">$n$</span>)</p>
<blockquote>
<p><strong>UNIQUENESS:</strong> For a full column rank <span class="math-container">$r=n . A x... | Community | -1 | <p>If <span class="math-container">$A$</span> is <span class="math-container">$m \times n$</span>, then the following are equivalent:</p>
<ol>
<li><span class="math-container">$A$</span> has full column rank <span class="math-container">$n$</span></li>
<li>The columns of <span class="math-container">$A$</span> are line... |
1,221,586 | <p>How would one compute?
$$ \oint_{|z|=1} \frac{dz}{\sin \frac{1}{z}} $$</p>
<p>Can one "generalize" the contour theorem and take the infinite series of the residues at each singularity? </p>
| Karanko | 227,830 | <p>If you want to generalize the residue theorem for infinitely many singularities, understand how the residue theorem is proven first. One replaces the loop by a circuit that encloses the singularities plus some arcs that are integrated in both directions. </p>
<p>In the case of infinitely many singularities you can ... |
1,179,981 | <p>As the title suggests. Let $G$ be a group, and suppose the function $\phi: G \to G$ with $\phi(g)=g^3$ for $g \in G$ is a homomorphism. Show that if $3 \nmid |G|$, $G$ must be abelian.</p>
<p>By considering $\ker(\phi)$ and Lagrange's Theorem, we have $\phi$ must be an isomorphism (right?), but I'm not really sure ... | Ben West | 37,097 | <p>Note that $(gh)^3=\varphi(gh)=\varphi(g)\varphi(h)=g^3h^3$. This implies $ghghgh=ggghhh$, and hence after cancelling, $hghg=gghh$, or $(hg)^2=g^2h^2$. </p>
<p>I claim that every element commutes with every square in $G$. Let $x\in G$ be arbitrary, and let $a^2\in G$ be an arbitrary square. Since $\varphi$ is an aut... |
195,006 | <p>I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.</p>
<p>Since I may even be using incorrect terminology, I'll try to explain what the terms I'm using mean in my mind. Please correct my term... | hmakholm left over Monica | 14,366 | <p>Given your objections to the other answers, here is how I understand your concept of "sequentially ordered":</p>
<ul>
<li>The set is totally ordered (<em>i.e.,</em> for any $a$, $b$ we must have $a\le b$ or $b\le a$).</li>
<li>Any element that has a successor has a <em>first</em> successor.</li>
<li>Any element tha... |
230,126 | <p>A family of functions is known as <span class="math-container">$\left(\varphi_{0}, \varphi_{1}, \cdots, \varphi_{n}\right)$</span>.</p>
<p>I'd like to know how to express their inner product conveniently as follows:</p>
<p><span class="math-container">$$\left(\begin{array}{cccc}
\left(\varphi_{0}, \varphi_{0}\right)... | NonDairyNeutrino | 46,490 | <p>Since you're dealing with inner products, you can take advantage of the symmetry that <span class="math-container">$\left<f(x), g(x)\right> = \left<g(x), f(x)\right>$</span> to only compute the <span class="math-container">$n(n+1)/2$</span> upper triangular elements instead of the total <span class="math... |
1,147,373 | <p>I need to prove that
<span class="math-container">$$I = \int^{\infty}_{-\infty}u(x,y) \,dy$$</span>
is independent of <span class="math-container">$x$</span> and find its value, where
<span class="math-container">$$u(x,y) = \frac{1}{2\pi}\exp\left(+x^2/2-y^2/2\right)K_0\left(\sqrt{(x-y)^2+(-x^2/2+y^2/2)^2}\right)$$<... | KStarGamer | 885,224 | <p>Although I'm about 7 years late, here is an answer anyway for anyone interested:</p>
<blockquote>
<p><strong>Claim</strong>
<span class="math-container">$$I = \frac{e}{2} \sqrt{\pi} \, \text{erfc} (1)$$</span>
and is thus independent of <span class="math-container">$x$</span>.</p>
</blockquote>
<p><em>Proof.</em></p... |
3,201,996 | <p>I have an 8-digit number and you have an 8-digit number - I want to see if our numbers are the same without either of us passing the other our actual number. Hashing the numbers is the obvious solution. However, if you send me your hashed number and I do not have it - it is very easy to hash all the permutations of ... | user1952500 | 64,332 | <p>You will likely need to use a scheme with a trusted third party. Let the two parties be A and B, and the trusted third party be X. </p>
<ol>
<li>A sends X its key K1 and gets back some sort of token T1. This token needs to have some small lifetime guarantees etc. </li>
<li>B sends X it’s key K2 and gets back anothe... |
1,225,122 | <p>So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when working with these ideals is when we reduce the generating set to something much simpler. For e.g.</p>
<p>Consider the I... | Thomas Poguntke | 222,154 | <p>The fact that $x-1$ divides both of the generators of $I$ only means that $I \subseteq (x-1)$. For the other inclusion, try subtracting one of the generators from the other.</p>
<p>Subsequently, it is also true that $(x^3-x^2+x) \subseteq (x)$, but the other inclusion is false (the LHS does not contain polynomials ... |
2,284,178 | <p>Let the roots of the equation:
$2x^3-5x^2+4x+6$ be $\alpha,\beta,\gamma$</p>
<ol>
<li>State the values of $\alpha+\beta+\gamma,\alpha\gamma+\alpha\beta+\beta\gamma,\alpha\beta\gamma$</li>
<li>Hence, or otherwise, determine an equation with integer coefficients which has $\frac{1}{\alpha^2}\frac{1}{\beta^2}\frac{1}{... | John Lou | 404,782 | <p>I think you're asking how to find the numerator of the fraction giving the other values. </p>
<p>Notice that:</p>
<p>$$(\alpha \beta + \alpha \gamma + \beta \gamma)^2 = (\alpha \beta)^2 + (\alpha \gamma)^2+(\beta \gamma)^2 + (\alpha + \beta + \gamma)(2)(\alpha \beta \gamma)$$
which you can find by simply expanding... |
2,284,178 | <p>Let the roots of the equation:
$2x^3-5x^2+4x+6$ be $\alpha,\beta,\gamma$</p>
<ol>
<li>State the values of $\alpha+\beta+\gamma,\alpha\gamma+\alpha\beta+\beta\gamma,\alpha\beta\gamma$</li>
<li>Hence, or otherwise, determine an equation with integer coefficients which has $\frac{1}{\alpha^2}\frac{1}{\beta^2}\frac{1}{... | Bernard | 202,857 | <p>All this is about <em>Vieta's relations</em> between the elementary symmetric funnctions of the roots of a polynomial and its coefficients. Let's denote
$$s=\alpha+\beta+\gamma,\quad q=\alpha\beta+\beta\gamma+\gamma\alpha, \quad p=\alpha\beta\gamma.$$
Thus you have to find the values of</p>
<ul>
<li>$S=\dfrac1{\alp... |
1,977,577 | <p>if a function is lebesgue integrable, does it imply that it is measurable?
(without any other assumption)</p>
<p>The reason why I ask this is because royden, in his book, kind of imply about a measurable function when assuming the function to be lebesgue integrable</p>
| Anindya Biswas | 357,212 | <p>In order to define integrability, the measurability is a necessary condition. For an example, let $A$ be the Vitali set in $[0,1]$ and $B=[0,1]-A$. Define $f=\chi_A -\chi_B$. Notice that $|f|=\chi_{[0.1]}$ and hence integrable. Now range of $f$ is $\{-1,1,0\}$ and $A=f^{-1}\{1\}$, $B=f^{-1}\{-1\}$ and $[0,1]^c=f^{-1... |
258,215 | <p>How can we show that if $f:V\to V$
Then for each $m\in \mathbb {N}$ $$\operatorname{im}(f^{m+1})\subset \operatorname{im}(f^m)$$
Please help,I am stuck on this.</p>
| Harald Hanche-Olsen | 23,290 | <p>This is an answer to the bonus (?) question in the comments. If $f$ is not injective, say $f(x)=f(y)$ for some $x$, $y$ with $x\ne y$, then for any set $B\subset Y$, either $x$ and $y$ are both in $f^{-1}(B)$, or none of them are. So any set $A$ that contains one but not the other is not an inverse image.</p>
<p>Co... |
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | Jika | 143,836 | <p><strong>HINT______________$1$:</strong> $$\sqrt{24}=2\sqrt{6}.$$</p>
<p><strong>HINT______________$2$:</strong> $$a^2=b^2\Leftrightarrow a=b\,\vee a=-b.$$</p>
|
1,547,972 | <p>Why is $\sin(x^2)$ similar of $\ x \sin(x)$? </p>
<p>I graphed it using desmos and when I look at it, the behavior as x approaches zero seems to be to oscillate less. </p>
<p>Yet as x approaches infinity and negative infinity $\sin(x^2)$ oscillates between y=1 and y=-1 while $\ x *sin(x)$ oscillates between y=x a... | Akiva Weinberger | 166,353 | <p>In calculus, you learn that many functions can be written as "infinite polynomials." For example, there's this:
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dotsb$$
Yes, those are factorials. A consequence of this is that, since $\sin\pi=0$, we have:
$$0=\pi-\frac{\pi^3}{3!}+\frac{\pi^5}{5!}-\frac{\pi^7}... |
1,547,972 | <p>Why is $\sin(x^2)$ similar of $\ x \sin(x)$? </p>
<p>I graphed it using desmos and when I look at it, the behavior as x approaches zero seems to be to oscillate less. </p>
<p>Yet as x approaches infinity and negative infinity $\sin(x^2)$ oscillates between y=1 and y=-1 while $\ x *sin(x)$ oscillates between y=x a... | Claude Leibovici | 82,404 | <p><em>This is not an answer but it is too long for a comment.</em></p>
<p>You received the explanation of what you observed.</p>
<p>I give you another one which is also amazing : function $y=\sin^x(x)$ in the range $(2\pi,3\pi)$ almost looks like a gaussian. Doing things similar as in the answers, consider Taylor ex... |
3,684,331 | <p>I am working on a problem from my Qual</p>
<p>"Let <span class="math-container">$T:V\to V$</span> be a bounded linear map where <span class="math-container">$V$</span> is a Banach space. Assume for each <span class="math-container">$v\in V$</span>, there exists <span class="math-container">$n$</span> s.t. <span cla... | Integrand | 207,050 | <p>Put <span class="math-container">$x=z-\pi$</span>; it suffices to show the inequality for for <span class="math-container">$z\geq \pi$</span>.
<span class="math-container">$$
z-\pi -\sin(z-\pi)\geq \frac{(z-\pi)^3}{z^2}
$$</span>
<span class="math-container">$$
z-\pi +\sin(z)\geq z -3\pi+\frac{3\pi^2}{z} -\frac{\pi^... |
44,306 | <p>This morning I realized I have never understood a technical issue about Cauchy's theorem (homotopy form) of complex analysis. To illustrate, let me first give a definition.</p>
<p>(In what follows $\Omega$ will always denote an open subset of the complex plane.)</p>
<p><em>Definition</em> Let $\gamma, \eta\colon [... | Malik Younsi | 1,197 | <p>If I understand your question correctly, the problem that $H$ may be non-smooth can be solved by approximating with polygonal smooth paths, see for example Rudin's real and complex analysis (3rd edition), thm 10.40 and the remark after it. </p>
<p>As an interesting note, Rudin adds that another way to circumvent th... |
452,292 | <p>Let $E$ be a Lebesgue measurable set in $\mathbb{R}$. Prove that
$$\lim_{x\rightarrow 0} m(E\cap (E+x))=m(E).$$ </p>
| Etienne | 80,469 | <p>First assume that $m(E)<\infty$. In this case you can write
$$m(E\cap (E+x))=\int_{\mathbb R} \mathbf 1_{E}(u)\mathbf 1_E(u-x)=\mathbf 1_{E}*\mathbf 1_{-E} (x)\, .$$
Since $f=\mathbf 1_E\in L^1$ (because $m(E)<\infty$) and $g=\mathbf 1_{-E}\in L^\infty$, the convolution $f*g$ is an everyywhere defined $contin... |
3,775,554 | <p><strong>Identity for a set X:</strong></p>
<p><em>The set X has an identity under the operation if there is
an element <strong>j</strong> in set X such that <strong>j * a = a * j = a</strong> for all elements <strong>a</strong> in set X.</em></p>
<p>According to my college book the counting numbers don' t have an ... | Barry Cipra | 86,747 | <p>In comments the OP clarifies the question:</p>
<blockquote>
<p>"Which properties are true for the counting numbers, whole numbers,
integers, rational numbers, irrational numbers, and real numbers under
the operation of addition?" The properties are: closed, identity and
inverse.</p>
</blockquote>
<p>The ve... |
4,170,150 | <p>I have the following two definitions:</p>
<p>Let <span class="math-container">$S$</span> be a subset of a Banach space <span class="math-container">$X$</span>.</p>
<ol>
<li><p>We say that <span class="math-container">$S$</span> is <em>weakly bounded</em> if <span class="math-container">$l\in X^*$</span>, the dual sp... | Speripro | 885,263 | <p>The fact that X is Banach it's not necessary.
Suppose that X is a normed space.
Recall that the map</p>
<p><span class="math-container">$j: X\to X^{**}$</span>, <span class="math-container">$s\mapsto \hat{s}$</span></p>
<p>is an <strong>isometry</strong>.</p>
<hr />
<p>Per your definition of weakly boundedness the s... |
536,068 | <p>If X and Y are iid random variables with distribution $F(x)=e^{-e^{-x}}$ and we let $Z=X-Y$ find the distribution function of $F_Z(x)$. I get $F_Z(x)=\frac{e^x}{1+e^x}$ but that doesn't match the answer that the professor gave us. Is what I have correct? I used the standard approach where I integrate over a region t... | mathemagician | 49,176 | <p>Here's what I did </p>
<p>$F_Z(z)=\mathbb{P}\{Z\leq z\}=\mathbb{P}\{X-Y\leq z\}=\mathbb{P}\{X\leq Y+z\}$. Now if $A\subseteq \mathbb{R}^2$ is defined as $A=\{(x,y)\in\mathbb{}R^2\;|\;x\leq y+z\}$ we get that the above equals $\int_A f_{X,Y}(x,y) dxdy$. Since $X,Y$ are independent we have $f_{X,Y}(x,y)=f_X(x)\cdot f... |
2,459,493 | <p>Here's the <span class="math-container">$C^1$</span> norm : </p>
<p><span class="math-container">$|| f || = \sup | f | + \sup | f '|$</span></p>
<p>where the supremum is taken on <span class="math-container">$[a, b]$</span>. </p>
<p>Please, justify your answer (proofs or counterexamples are needed). </p>
| Wildcard | 276,406 | <p>Ignoring the restriction, there are five choices for each of three positions.</p>
<p>The restriction excludes ordered sequences drawn from the space wherein there are only three choices for each of three positions.</p>
<p>Final answer is:</p>
<p>$$5^3-3^3=98$$</p>
|
1,302,891 | <p>I would like to ask you a question about the following question.</p>
<p>Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow \ c-}{f(x)}$ and $\lim_{x \ \rightarrow \ c+}{f(x)}$ both exist.</p>
<p>If $f$ is continuous at $c$ then obv... | Disintegrating By Parts | 112,478 | <p>Note that $f(x) \le f(c)$ for $a < x < c$. Therefore the following exists
$$
\sup_{x < c} f(x)
$$
and you can show that the following limit exists, too:
$$
\lim_{x\uparrow c}f(x) = \sup_{x < c} f(x).
$$</p>
|
4,132,439 | <p>I know that <span class="math-container">$\int_1^\infty \frac1xdx$</span> diverges. I can probe that <span class="math-container">$\int_1^\infty \frac1{\ln(x)}dx$</span> diverges, as <span class="math-container">$\forall x>1:\frac1x<\frac1{\ln(x)}$</span></p>
<p>I also know that <span class="math-container">$\... | RRL | 148,510 | <p>Since <span class="math-container">$\ln x = 7 \ln x^{1/7}$</span> we have <span class="math-container">$\ln ^7 x = (7 \ln x^{1/7})^7 < 7^7 x$</span> and</p>
<p><span class="math-container">$$\int _2^\infty \frac{dx}{\ln^7 x} > \int _2^\infty \frac{dx}{7^7x}= +\infty$$</span></p>
<p>The integral over <span cla... |
4,132,439 | <p>I know that <span class="math-container">$\int_1^\infty \frac1xdx$</span> diverges. I can probe that <span class="math-container">$\int_1^\infty \frac1{\ln(x)}dx$</span> diverges, as <span class="math-container">$\forall x>1:\frac1x<\frac1{\ln(x)}$</span></p>
<p>I also know that <span class="math-container">$\... | hamam_Abdallah | 369,188 | <p><strong>hint</strong>
Near infinity
<span class="math-container">$$\lim_{x\to+\infty}x.\frac{1}{(\ln(x))^7}=+\infty\implies$$</span>
for <span class="math-container">$ x $</span> large enough</p>
<p><span class="math-container">$$x.\frac{1}{(\ln(x))^7}>1\implies$$</span></p>
<p><span class="math-container">$$\fra... |
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