qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,623,521 | <p>Prove that: $\tan (2\tan^{-1} (x))=2\tan (\tan^{-1} (x)+\tan^{-1} (x^3))$</p>
<p>My Attempt'
Let $\tan^{-1} (x)=A$
$$x=\tan (A)$$
Now,
L.H.S.$=\tan (2\tan^{-1} (x))$
$$=\tan (2A)$$
$$=\dfrac {2\tan (A)}{1-\tan^2 (A)}$$
$$=\dfrac {2x}{1-x^2}$$</p>
| Intelligenti pauca | 255,730 | <p>Tension $\vec T_1$ is the easy one, because we have a static balance of forces:
$$
\vec T+\vec T_1+m\vec g=0.
$$
From the vertical components one then gets (I'm using the same letters without arrows to denote the magnitudes of vectors, as it is customary in kinematics):
$$
T_1\cos30°=mg,
\quad\hbox{that is:}\quad
T_... |
2,122,955 | <p><em>My question is from T.Y.Lam- A First Course in Noncommutative Rings.</em></p>
<blockquote>
<p>For a ring $R$, if all left $R$-modules are semisimple, then all short exact sequences of left $R$-modules split.</p>
</blockquote>
<p>A module $M$ is a semisimple module if every submodule $N$ of $M$ is a direct su... | rschwieb | 29,335 | <p>By the first isomorphism theorem, you always have that $B/A\cong C$.</p>
<p>Since $B$ is semisimple, there exists some submodule $C'$ such that $A\oplus C'=B$. By the second isomorphism theorem
$$\frac{B}{A}=\frac{A\oplus C'}{A}\cong\frac{C'}{A\cap C'} =\frac{C'}{\{0\}}\cong C'$$.</p>
<p>So $C\cong C'$, and $A\op... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| heropup | 118,193 | <p>I would start off with a much simpler example. Suppose we wanted to locate $1/3$ on a number line, but the number line is such that every interval is divided into tenths, not thirds (e.g., like a metric ruler). How would you locate such a number? Well, you would start off by moving $3/10$ units from $0$, so now y... |
174,676 | <p>When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:</p>
<p>"Notice" that adding $13$ on the right and subtracting $13x$ on the left gives:
$$-4x \equiv 20 \pmod{13}$$</p>
<p>so that $$x \eq... | Bill Dubuque | 242 | <p><span class="math-container">$bx\equiv a\pmod{\!m}$</span> has a <em>unique</em> solution <a href="https://math.stackexchange.com/a/2383809/242"><span class="math-container">$\!\iff\!b\,$</span> is coprime to the modulus <span class="math-container">$m$</span></a>. If so, <a href="https://math.stackexchange.com/a/32... |
174,676 | <p>When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:</p>
<p>"Notice" that adding $13$ on the right and subtracting $13x$ on the left gives:
$$-4x \equiv 20 \pmod{13}$$</p>
<p>so that $$x \eq... | CopyPasteIt | 432,081 | <p>Another offbeat process but with algorithmic potential.</p>
<p>Solve <span class="math-container">$9x \equiv 7 \pmod{13}$</span>.</p>
<p><span class="math-container">$\quad 9x = 7 + 13y \implies 0 \equiv 1 + y \pmod{3} \implies y \equiv 2 \pmod{3}$</span></p>
<p>and</p>
<p><span class="math-container">$\quad y : 2 \... |
1,056,397 | <p>I saw the following problem:
$$\lim_{x\to \infty} \sqrt{9x^2+x}-3x$$
My first thought was to say that the $x$ term is overpowered when $x$ becomes large enough, so the square root becomes just $\sqrt{9x^2} = 3x$, and the value of the limit is zero.</p>
<p>However, the solution given is $1\over6$, and starts with:
$... | Arthur | 99,272 | <p>$$\frac{x}{\sqrt{9x^2 + x} + 3x} = \frac{1}{\sqrt{9+\frac{1}{x}}+3}$$</p>
|
3,366,981 | <p>I have been looking for a concrete answer on this matter since leaving secondary school and thought now to ask online since YouTube and Wikipedia seem too convoluted. </p>
<p>I remember watching <a href="https://www.youtube.com/watch?v=Rzac520CESc" rel="nofollow noreferrer">this</a> video on the proof behind the nC... | Adrian Keister | 30,813 | <p>So everything looks good so far:
<span class="math-container">\begin{align*}
I_x&=\lambda\int_{0}^{2\pi}y^2r\, d\theta\\
&=\lambda\int_{0}^{2\pi}r^2\sin^2(\theta)\,r\, d\theta\\
&=\lambda r^3\int_{0}^{2\pi}\sin^2(\theta)\, d\theta\\
&=\frac{m}{2\pi r}\,r^3\,\pi\\
&=\frac{mr^2}{2},
\end{align*}</s... |
2,251,112 | <p>A small college has 1095 students. What is the approximate probability that more than five students were born on Christmas day? Assume that the birthrates are constant throughout the year and that each year has 365 days.</p>
<p>I tried doing<br>
$X \sim Pn(3) $ and calculating $P(X\gt5)$. My calculation turned out ... | robjohn | 13,854 | <p>The probability of having exactly $k$ people with birthdays on Christmas would be
$$
\binom{1095}{k}\frac1{365^k}\left(\frac{364}{365}\right)^{1095-k}
$$
Adding these probabilities for at most $5$ people, we get
$$
\sum_{k=0}^5\binom{1095}{k}\frac1{365^k}\left(\frac{364}{365}\right)^{1095-k}=0.916358737
$$
Thus, the... |
3,911,489 | <p>How to prove <span class="math-container">$$I=\int_{0}^{\infty} \frac{\sin x}{x(x^2+1)}dx=\frac{\pi}{2}(1-\frac{1}{e})\, ?$$</span></p>
<p>Let <span class="math-container">$$f(z)=\frac{\sin z}{z(z^2+1)}=\frac{e^{iz}-e^{-iz}}{2i\,z(z^2+1)}.$$</span></p>
<p>Then I have no idea how to deal with <span class="math-contai... | Mark Viola | 218,419 | <p>First note that the integrand is an even function so that</p>
<p><span class="math-container">$$\int_0^\infty \frac{\sin(x)}{x(x^2+1)}\,dx=\frac12 \int_{-\infty }^\infty \frac{\sin(x)}{x(x^2+1)}\,dx\tag1$$</span></p>
<p>Next, we can use Euler's Formula <span class="math-container">$e^{ix}=\cos(x)+i\sin(x)$</span> in... |
3,911,489 | <p>How to prove <span class="math-container">$$I=\int_{0}^{\infty} \frac{\sin x}{x(x^2+1)}dx=\frac{\pi}{2}(1-\frac{1}{e})\, ?$$</span></p>
<p>Let <span class="math-container">$$f(z)=\frac{\sin z}{z(z^2+1)}=\frac{e^{iz}-e^{-iz}}{2i\,z(z^2+1)}.$$</span></p>
<p>Then I have no idea how to deal with <span class="math-contai... | J.G. | 56,861 | <p>It helps to use <span class="math-container">$\tfrac{1}{x(x^2+1)}=\tfrac1x-\tfrac{x}{x^2+1}$</span> to reduce the problem to <span class="math-container">$I=\tfrac12\pi-J$</span> with<span class="math-container">$$J:=\int_0^\infty\tfrac{x\sin xdx}{x^2+1}=\tfrac12\Im\int_{\Bbb R}\tfrac{x\exp ixdx}{x^2+1}=\tfrac12\Im[... |
413,982 | <p>How can I divide this using long division?</p>
<p>$$\frac{ax^3-a^2x^2-bx^2+b^2}{ax-b}$$</p>
<p><strong>Edit</strong></p>
<p>Sorry guys I wrote it wrong... Fixed it now.</p>
| user49685 | 49,685 | <p>Here's one way to approach it. You can try <strong>grouping</strong> the elements in the numerator. Note that, you should try grouping in the sense that $ax - b$ must appear. So that they can calcel themselves out. So here it goes:</p>
<p>$$\dfrac{\color{red}{ax^3} \color{blue}{- a^2 x^2} \color{red}{- bx^2} \color... |
848,631 | <p>The function</p>
<p>$$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$</p>
<p>is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ it is the same as $\pi \leq x < 2 \pi$.</p>
<p>So can we state that the period of $f(y)$ with $N$ od... | Joshua Mundinger | 106,317 | <p>Writing sine in exponential form we have
$$f(x+\pi) = \frac{e^{iN(x+\pi)}-e^{-iN(x+\pi)}}{e^{i(x+\pi)}-e^{-i(x+\pi)}}$$
Now for $N$ odd, we have $e^{-iN\pi} = e^{iN\pi} = e^{i\pi}=e^{-i\pi}=-1$, so
$$f(x+\pi) = \frac{-e^{iNx}+e^{-iNx}}{-e^{ix}+e^{-ix}} = f(x)$$</p>
|
1,789,414 | <blockquote>
<p><strong>Question :</strong> Prove that the equation $x + \cos(x) + e^{x} = 0$ has
<em>exactly</em> one root</p>
</blockquote>
<hr>
<p><em>This is what I thought of doing:</em></p>
<p>$$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$
By using the Intermediate Value Theorem on the open interval $(-\... | aarbee | 87,430 | <p>The given equation can be written as <span class="math-container">$$x+\cos x=-e^x$$</span></p>
<p><span class="math-container">$x+\cos x$</span> is a familiar increasing <a href="https://www.desmos.com/calculator" rel="nofollow noreferrer">graph</a>.</p>
<p><span class="math-container">$-e^x$</span> is also easy to ... |
1,789,414 | <blockquote>
<p><strong>Question :</strong> Prove that the equation $x + \cos(x) + e^{x} = 0$ has
<em>exactly</em> one root</p>
</blockquote>
<hr>
<p><em>This is what I thought of doing:</em></p>
<p>$$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$
By using the Intermediate Value Theorem on the open interval $(-\... | MosikasikaC19 | 1,008,452 | <p>The Intermediate Value Theorem give you existence of one <span class="math-container">$x$</span> satisfying their statements, for the unicity you have to show the bijection of <span class="math-container">$f(x)=1+\cos x+ e^x$</span><br />
Consider this fonction <span class="math-container">$$f(x)=x+\cos x+e^x$$</spa... |
23,818 | <p>I'm trying to define some notation so that Mathematica code would be more functional, similar to Haskell (just for fun): currying, lambdas, infix operator to function conversion, etc.. And I have some questions about it:</p>
<ul>
<li>Is it possible to make all Mathematica <code>h_[x1_,x2_,...]</code> functions to w... | Mr.Wizard | 121 | <h2>Currying</h2>
<p>I don't know if it is possible to make <em>all</em> functions work in the Currying form (<code>h[x1][x2][..]</code>) but it is at least possible to extend Hold behavior to all arguments which natively that pattern will not have. I will copy my favorite method which I learned from <a href="https:/... |
618,763 | <p>Let $\mathbb{F}$ be a field, $ \mathbb{R} = \mathbb{F}[X,X^{-1}]$ the ring of Laurent polynomials over $\mathbb{F}$</p>
<blockquote>
<blockquote>
<p>a) find the group of units in R.</p>
<p>b) find an Euclidean norm in R.</p>
</blockquote>
</blockquote>
<p>so for a, I understand the units are $uX^{... | Pete L. Clark | 299 | <p>This may well be overkill, but it is a result of Motzkin and Samuel that any localization of a Euclidean domain is Euclidean. A proof is given in Theorem 2.33 of <a href="http://alpha.math.uga.edu/%7Epete/OrdinalInvariant.pdf" rel="nofollow noreferrer">this paper of mine</a>.</p>
<p>Since you say you want a <em>hin... |
3,570,943 | <p>From the comments I got, my question amonts to : can the " inverse of inverse" law be derived from the definition of division. </p>
<hr>
<p>Division is defined as : <span class="math-container">$\dfrac AB = A.\dfrac1B$</span>, that is, </p>
<p>" dividisng by A by B is, by definition, mutiplicating A by the inver... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">\begin{align}
\dfrac{1}{\left\{\dfrac CD \right\}}
&= \dfrac{1}{\left\{\dfrac CD \right\}} \cdot 1 \\
&= \dfrac{1}{\left\{\dfrac CD \right\}} \cdot \dfrac DD \\
&= \dfrac{1 \cdot D}{\dfrac CD\cdot D } \\
&= \dfrac DC
\end{align}</span></p>
<p>Therefore</p... |
2,022,529 | <p>$$\Large
\lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt
$$</p>
<p>I am trying to calculate this limit. Using L’Hôpital’s rule, I am getting it as $1/4$, but the book says it's $3/8$. I don't know where I am doing the mistake.</p>
| Dhanvi Sreenivasan | 332,720 | <p>We divide the 7 people into 2 groups, of 5 people with no problem, and 2 people who don't want to work with each other.</p>
<p>To make a committee that will fail, we need to take both of the people from the 2-group, i.e, $2 \choose 2$</p>
<p>Now, there is only one space left in the committee, and we have 5 possibl... |
1,039,326 | <p>What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$?</p>
<p>I know how to do this using complex algebra. but that takes a long time.
Can someone show me a faster way?</p>
| Community | -1 | <p>Taking the modulus, $|z|^3=2\cdot2\cdot|z|$, and $|z|=0$ or $|z|=2$ ($|z|=-2$ cannot hold).</p>
<p>Taking the argument, $3\arg z=\pi+\pi/3-\arg z+2k\pi$, and $\arg z=\pi/3+k\pi/2$.</p>
<p>$$z=0,1+i\sqrt3,-\sqrt3+i,-1-i\sqrt3,\sqrt3-i.$$</p>
|
1,460,704 | <p>Intuitively we know that $n^2$ grows faster than $n$, thus the difference tends to negative infinity. But I have trouble proving it symbolically because of the indeterminate form $\infty - \infty$. Is there anyway to do this without resorting to the Epsilon-Delta definition ?</p>
| A.Γ. | 253,273 | <p>Another way: complete the square $n-n^2=\frac14-\bigl(n-\frac12\bigr)^2\to -\infty$.</p>
|
391,569 | <p>I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.</p>
<p>$\eqalign{
& \int {{{\sin }^3}x} dx \cr
& = \int {\sin x({{\sin }^2}x} )dx \cr
& = \int {\sin x({1 \over 2... | Caran-d'Ache | 66,418 | <p>Isn't it faster to use subsitution like $\cos(x)=t$?
$$\eqalign{
& \int {{{\sin }^3}x} dx \cr
& = \int {\sin x({{\sin }^2}x} )dx \cr
& =-\int \sin^2(x)d\cos(x) \cr
& = -\int (1-\cos^2(x)) d\cos(x) \cr
& = -\int (1-t^2) dt \cr }
$$</p>
|
391,569 | <p>I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.</p>
<p>$\eqalign{
& \int {{{\sin }^3}x} dx \cr
& = \int {\sin x({{\sin }^2}x} )dx \cr
& = \int {\sin x({1 \over 2... | pritam | 33,736 | <p>Using the formula $$\sin (3x)=3\sin x-4\sin^3 x$$ you can calculate the integral in just one step.</p>
|
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Quinn Culver | 11,030 | <p>After subtracting $1$ this is equivalent to $\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \cdots = \frac{1}{3}$. Behold:</p>
<p><img src="https://i.stack.imgur.com/qkD4g.jpg" alt="enter image description here"></p>
<p>Image taken from <a href="http://mrhonner.files.wordpress.com/2010/10/infinite-series-triangle.jpg... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| user47693 | 47,693 | <p>I'm not really sure if it is possible to prove this using induction but we can do something else.</p>
<p>The sum $\displaystyle 1 + \frac{1}{4} + \frac{1}{4^2} + \cdots \frac{1}{4^k}$ can be written as $\displaystyle\sum_{n=0}^k \frac{1}{4^n}$. Now $\displaystyle\sum_{n=0}^k \displaystyle\frac{1}{4^n} = \frac{1-\fr... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Pete L. Clark | 299 | <p>The statement of your question is quite confused. Clearly $1 + \frac{1}{4} + \ldots + \frac{1}{4^n}$ is a quantity that gets larger with $n$ -- you're adding on more positive numbers -- so it cannot be equal to any fixed number.</p>
<p>I think what you want to show by induction is:</p>
<blockquote>
<p>For every... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Hagen von Eitzen | 39,174 | <p>If you multiply $s_n:=1+q+q^2+\ldots + q^n$ with $q$ you get $q+q^2+\ldots +q^n+q^{n+1}$, which differs from $a_n$ by only dropping the first and adding a new last term. More procisely, we find $qs_n=s_n-1+q^{n+1}$, hence (if $q\ne 1$)
$$s_n=\frac{q^{n+1}-1}{q-1}=\frac1{1-q}+\frac1{q-1}\cdot q^{n+1}.$$
In your case,... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| JTP - Apologise to Monica | 60,518 | <p>If you represent 4/3 using binary notation, you get:</p>
<p>1.0101.... (I don't know how to do a 'bar' to show repeating.)</p>
<p>The first 1 after the decimal is read as 1/4 the zero is in the 1/8 place, the next 1 is 1/16, etc. This reads as matching your infinite series. </p>
|
2,325,650 | <p>This question is based on the fourth question in the 2003 edition of the <a href="https://www.vwo.be/vwo/files/finale03.pdf" rel="nofollow noreferrer">Flemish Mathematics Olympiad</a>.</p>
<blockquote>
<p>Consider a grid of points with integer coordinates. If one chooses the number $R$ appropriately, the circle w... | Daniel Schepler | 337,888 | <p>Let $a_1 + b_1 i, \ldots, a_n + b_n i$ be distinct irreducibles of the Gaussian integers $\mathbb{Z}[i]$, with $a_k > b_k > 0$ for each $k$. Then $a_1 + b_1 i, a_1 - b_1 i, \ldots, a_n + b_n i, a_n - b_n i$ are also distinct irreducibles such that no pair has ratio equal to a unit of $\mathbb{Z}[i]$. Therefo... |
390,757 | <p>suppose <strong>we don't know that Halting problem is not recursive</strong>.</p>
<p>I want to prove that <strong>complement of halting set is not r.e.</strong> then we can find <strong>halting problem is not recursive</strong>.</p>
<p><strong>Can you direct prove that complement of halting set is not r.e.??</stro... | Brian M. Scott | 12,042 | <p>HINT: If a set and its complement are both recursively enumerable, the set is ... ?</p>
<p><strong>Added:</strong> This hint was for the original version of the question, which assumed that the halting problem was undecidable and did not ask for a direct proof.</p>
|
390,757 | <p>suppose <strong>we don't know that Halting problem is not recursive</strong>.</p>
<p>I want to prove that <strong>complement of halting set is not r.e.</strong> then we can find <strong>halting problem is not recursive</strong>.</p>
<p><strong>Can you direct prove that complement of halting set is not r.e.??</stro... | Christopher Rose | 296,508 | <p>For notational convention we will call the halting set $K$ and the complement of the halting set $\overline{K}$. </p>
<p>Definition 1: A set $S$ is recursively enumerable if and only if it is the domain of some computable partial function $f$.</p>
<p>Theorem: $\overline{K}$ is not the domain of some computable pa... |
1,856,071 | <p>If I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is wrong if they do not intersect (when not extending them).</p>
<p>Is there a possibility to express a 90 degree angle bet... | Narasimham | 95,860 | <p>We can recognize by what changes occur in the equation of the line.The product of slopes is -1. </p>
<p>If the lines are in slope intercept form</p>
<p>$$ y= m x + a \,\rightarrow y= -x/m + const $$</p>
<p>If in full intercept form </p>
<p>$$ \frac{x}{a}+ \frac{y}{b} =1 \rightarrow \frac{x}{b} - \frac{y}{a}... |
7,725 | <p>Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: <a href="http://www.stitz-zeager.com/szprecalculus07042013.pdf">http://www.stitz-zeager.com/szprecalculus07042013.pdf</a></p>
<p>However, it does not cover convergence,limits etc. and those topics we... | schremmer | 4,815 | <p>OK, here is a directly relevant answer: <strong>asymptotic expansions</strong>, which are <em>finite</em> expansions by definition, are used to approximate $f(x_{0}+h)$ and can be used with divergent series (That is what Poincaré invented them for) as well as convergent series. </p>
<p>The second part of my answer ... |
2,314,369 | <p>My objective is to represent a unit square using a single matrix.<br>The four corners of the unit square are represented by column vectors as follows,</p>
<p>$\left[\begin{matrix} 0 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 1 \end{matrix}\right],\left[\begin{... | Peter Melech | 264,821 | <p>$\begin{pmatrix}1&0\\0&1\end{pmatrix}$ might be understood to represent</p>
<p>the set $\{x\in\mathbb{R}^2:x=t\begin{pmatrix}1\\0\end{pmatrix}+s\begin{pmatrix}0\\1\end{pmatrix},0\leq t,s\leq 1\}$, the square, and more genearlly</p>
<p>$\begin{pmatrix}a_{11} & a_{12}\\a_{21}&a_{22}\end{pmatrix}$ mig... |
2,888,464 | <p>Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying </p>
<p>$f(x) \equiv 1$ mod $(x-1)$</p>
<p>$f(x) \equiv 0$ mod $(x-3)$</p>
<p>Which of the following statements are true?</p>
<p>a) $S$ is empty .</p>
<p>b) $S$ is a singleton.</p>
<p>c)$S$ is a finite non-empty set.</p>
<p>d) $S$... | Mostafa Ayaz | 518,023 | <p>Let we have a general polynomial form as following$$f(x)=a_0+a_1x+\cdots +a_dx^d$$We know that for some polynomials $a(x)$ and $b(x)$ $$f(x)=1+(x-1)a(x)=b(x)(x-3)$$therefore $$f(1)=1\\f(3)=0$$ which leads to $$(1)\quad a_0+a_1+\cdots +a_d=1\\(2)\quad a_0+3a_1+\cdots +3^da_d=0$$by subtracting $(1)$ from $(2)$ we obta... |
2,888,464 | <p>Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying </p>
<p>$f(x) \equiv 1$ mod $(x-1)$</p>
<p>$f(x) \equiv 0$ mod $(x-3)$</p>
<p>Which of the following statements are true?</p>
<p>a) $S$ is empty .</p>
<p>b) $S$ is a singleton.</p>
<p>c)$S$ is a finite non-empty set.</p>
<p>d) $S$... | farruhota | 425,072 | <p>Your solution is short and correct. Indeed:
$$\begin{align}\begin{cases}f(x)\equiv 1\bmod{(x-1)}\\ f(x)\equiv 0\bmod{(x-3)}\end{cases} &\iff
\begin{cases}f(x)=(x-1)a(x)+1\\ f(x)=(x-3)b(x) \end{cases} \Rightarrow \\
\begin{cases}f(5)\equiv 1\bmod{4}\\f(5)\equiv 0\bmod{2}\end{cases} &\iff\begin{cases} f(5)=4a(... |
2,572,564 | <p>So I have a straight line, in the classic <span class="math-container">$y = mx + b$</span>, and I'm just trying to translate the formula for the line a certain distance along its normal.</p>
<p>For example, with <a href="https://i.stack.imgur.com/r8vct.jpg" rel="nofollow noreferrer">this graph</a>, how would I tran... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Use the basic formulae:
$$\lim_{h\to0}\dfrac{\ln(1+h)}h=1$$
$$\lim_{u\to0}\dfrac{\sin u}u=1$$</p>
|
1,368,447 | <p><img src="https://i.stack.imgur.com/TL80w.jpg" alt="enter image description here"></p>
<p>So I tried the good old Calculus 1 approach and turned this into an optimization problem. The equations got REALLY hairy, but it was okay since this was the graphing calculator section of the exam. I called the longer part of ... | DeepSea | 101,504 | <p>Let $a, b, c$ be the length of the parts of the longer diagonal, and the half the shorter diagonal. Then: $S = ac + bc \leq \dfrac{a^2+c^2}{2}+\dfrac{b^2+c^2}{2}=\dfrac{2^2}{2}+\dfrac{5^2}{2}=\dfrac{29}{2}$</p>
|
2,331,255 | <p>Let $K$ be a set of all Killing vector fields on $\mathbf R^n$ (with the Euclidean metric $\bar g$) which vanish at the origin.</p>
<p>(A vector field $V$ on a Riemannian manifold $(M, g)$ is said to be a <strong><em>Killing vector field</em></strong> if the flow of $V$ acts by isometries of $M$. This is equivalent... | Willie Wong | 1,543 | <p>Here is a slightly simpler, non-geometric, proof:</p>
<p>Consider the coordinate derivatives
$$ a_{ijk} = \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j} V^k $$
from calculus we know that this expression is symmetric in $i$ and $j$. Killing's equation implies that this expression is antisymmetric in $j$... |
3,340,723 | <p>How can we find the total number of ways in which we can divide <span class="math-container">$n$</span> elements into two subsets such that none of them are empty and the union of both sets should be equal to the whole set?</p>
<p>Eg. If <span class="math-container">$S=\{1,2,3\}$</span>, the answer can be <span cla... | Wuestenfux | 417,848 | <p>Well, the number you are searching is the Stirling number of the 2nd kind.</p>
<p>Let <span class="math-container">$n,k\geq 1$</span>. Define <span class="math-container">$S(n,k)$</span> as the number of partitions of an <span class="math-container">$n$</span>-element set such that each partition consists of <span ... |
2,450,838 | <p>So I am hoping to show that $f(x,y)=\dfrac{x}{y}$ is continuous when $x>0$ and $y>0$.</p>
<p>I am not sure how to approach this problem. My idea was that taking $$\dfrac{\partial f}{\partial x}=\dfrac{1}{y}$$ and $$\dfrac{\partial f}{\partial y}=\dfrac{-x}{y^2}$$</p>
<p>My logic was that since both partials ... | Community | -1 | <p>First show by <em>definition</em> that $\pi_1(x,y)=x$ and $\pi_2(x,y)=y$ are both continuous on ${\bf R}^2$ and in particular on $S:=\{(x,y)\in{\bf R}^2\mid x>0,y>0\}$. </p>
<p>Now note that $f$ is the product of two continuous functions:
$$
f(x,y)=\pi_1(x,y)\cdot g(\pi_2(x,y))
$$
where $g(z):=\frac{1}{z}$ is... |
2,918,453 | <p>After reading <a href="https://math.stackexchange.com/questions/208317/show-sum-n-0-infty-frac1a2n2-frac1a-pi-coth-a-pi2a2/208407#208407">Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$</a> and noodling around on wolfram alpha, I discovered </p>
<p>$$ \begin{align}
&\coth(x \pi)=\frac{x... | Mason | 552,184 | <p>There is some nice things to see here: </p>
<p>$$\tanh(x \pi)=\frac{4x}{\pi}\sum_{n=-\infty}^\infty{\frac{1}{(2n+1)^2+4x^2}}$$</p>
<p>$$\Big(\tanh(x \pi) \Big)'= \Big( \frac{1}{\pi}\sum_{n=-\infty}^\infty{\frac{4x}{(2n+1)^2+4x^2}} \Big)'$$</p>
<p>$$ \pi\operatorname{sech}^2(\pi x)=\frac{1}{\pi}\sum_{n=-\infty}^{... |
4,386,333 | <p>This is Exercise 1.6(iii) from Intro. to K-Theory for <span class="math-container">$C^*$</span>-Algebras by Rørdam et al.</p>
<p>If <span class="math-container">$A$</span> is a unital <span class="math-container">$C^*$</span>-algebra, and <span class="math-container">$a \in A$</span> is invertible, then I want to sh... | Martin Argerami | 22,857 | <p>You can use the Polar Decomposition. In principle you have to do this in a von Neumann algebra, but you always have this if you represent <span class="math-container">$A\subset B(H)$</span>.</p>
<p>You have <span class="math-container">$a=v|a|$</span>, with <span class="math-container">$|a|=(a^*a)^{1/2}\in C^*(a)$<... |
2,675,758 | <p>For example:</p>
<p>There are two green marbles and two red marbles in a bag. Two marbles are chosen at random, what is the probability that the two marbles which have been chosen are of the same colour?</p>
<p>Ordered + distinct: <br>
Marbles = $G1, G2, R1, R2$ <br>
Event space = $\{|G1~G2|, |G2~G1|, |R1~R2|, |R2... | Mathew Mahindaratne | 525,941 | <p>I assume this is base $10$ logarithmic we are talking here. If so, $\log_{10}(x+2)^2 = 2$ means $(x+2)^2 = 10^2$. Similarly, $\ln(x+2)^2=\log_{e}(x+2)^2 = 2$ means $(x+2)^2 = e^2$. The rest is just algebra.</p>
|
115,630 | <p>Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.</p>
<p>Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)</p>
<p>... | Tryst with Freedom | 688,539 | <p>For a totally different angle to the subject, we can get a taste of what the tensor product means by considering Dehn's invariant. I'll try to shortly explain what it is, if we have a polyhedron with edge lengths <span class="math-container">$l_i$</span> and <a href="https://en.wikipedia.org/wiki/Dihedral_angle" rel... |
2,600,062 | <p>I'm stuck at doing this problem. It's actually a problem related to circuit analysis. I'm given this equation:
$$\frac{(R_3+\frac{1}{\omega C}I)*(\omega Li)}{(R_3+\frac{1}{\omega Ci})+(\omega Li)}$$
Now to rationalize it I will multiply by
$$\frac{R_3+\frac{1}{\omega C}i-\omega Li}{R_3+\frac{1}{\omega C}i-\omega Li}... | Jan Eerland | 226,665 | <p>Well, as an electrical engineer you'll write (where $\text{j}^2=-1$):</p>
<p>$$\underline{\text{Z}}_{\space\text{in}}=\frac{\left(\underline{\text{Z}}_{\space\text{R}}+\underline{\text{Z}}_{\space\text{C}}\right)\cdot\underline{\text{Z}}_{\space\text{L}}}{\underline{\text{Z}}_{\space\text{R}}+\underline{\text{Z}}_{... |
3,081,918 | <p>It was claimed <a href="https://math.stackexchange.com/a/2387782/446262">here</a> that the convergence of the series<span class="math-container">$$\sum_{n=2}^\infty \frac{\Lambda(n)-1}{n^{1/2}\log^3 n}\tag1$$</span>(where <span class="math-container">$\Lambda$</span> is the <a href="https://en.wikipedia.org/wiki/Von... | reuns | 276,986 | <p>Claymath official <a href="http://www.claymath.org/sites/default/files/official_problem_description.pdf" rel="nofollow noreferrer">description</a> of the Riemann hypothesis claims the RH is true iff <span class="math-container">$\pi(x) = Li(x)+O(x^{1/2}\log x)$</span> so <span class="math-container">$\psi(x) = x+O(x... |
3,682,514 | <blockquote>
<p>The hyperbola is given with the following equation: <span class="math-container">$$3x^2+2xy-y^2+8x+10y+14=0$$</span>
Find the asymptotes of this hyperbola. (<span class="math-container">$\textit{Answer: }$</span> <span class="math-container">$6x-2y+5=0$</span> and <span class="math-container">$2x+2y... | Saket Gurjar | 769,080 | <p>Here is a view I have on hyperbola's asymptotes :</p>
<p>The asymptotes and the hyperbola behave the same way as we go to a point <span class="math-container">$(x,y)$</span> at an infinite distance from the centre of hyperbola, along the asymptotes.</p>
<p>So for a hyperbola: <span class="math-container">$H:ax^2+b... |
3,888,805 | <p>"Six men and three women are standing in a supermarket queue. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?"</p>
<p>I went about solving this question by considering 3 cases, first with a single... | N. F. Taussig | 173,070 | <p>You are counting each selection with two women twice and each selection with three women three times, once for each way you could have designated one of the women who is selected as the woman you have selected and the remaining woman or women as additional people.</p>
<p>Suppose you select two women, Abigail and Bea... |
3,556,490 | <p>Can someone explain me how to solve the equation <span class="math-container">$\frac{x-n}{x+n} = e^{-x}$</span>, where <span class="math-container">$n$</span> is a non-zero natural number ? Unfortunately, I have not even an idea how to start. Any hint is much appreciated. </p>
<p>Many thanks in advance. </p>
| Piquito | 219,998 | <p>HINT.-No way to calculate in closed form by elementary means. You have to try with numerical methodes. You have in particular for a possible approximation
<span class="math-container">$$1+x+\frac{x^2}{2!}+\cdots+\frac{x^k}{k!}+\cdots =e^x=\frac{x+n}{x-n}$$</span>
<span class="math-container">$$\frac{1}{x-n}=-\left(\... |
1,457,797 | <p>I'm told that a circle intersects the y-axis at $(0,0)$ and $(0, -4)$.</p>
<p>I have a tangent that starts at $(0, -6)$. I want to find the point of intersection.</p>
<p>I am taking the midpoint of the circle to be $(0, -2)$ and the radius to be $2$ from the points above.</p>
<p>The equation of the circle then is... | R.N | 253,742 | <p>see this, it shows that your answer is not correct <a href="https://i.stack.imgur.com/4XB35.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4XB35.png" alt="enter image description here"></a></p>
|
45,800 | <p>Let's say I want to calculate the following scalar-by-matrix derivative</p>
<p>$$\frac{\partial}{\partial A} \text{tr} \left[(\vec X^T A)^T (\vec X^T A)\right],$$</p>
<p>with $\vec X$ and $A$ being a $n \times 1$ and a $n \times m$ matrix, respectively.
Is there a way in Mathematica to get the result</p>
<p>$$2 \... | userrandrand | 86,543 | <h2>Outline</h2>
<ul>
<li><strong>Discussion</strong></li>
<li><strong>Using Differentials</strong></li>
<li><strong>Using the NonCommutativeMultiply package NCAlgebra</strong></li>
<li><strong>Using MatrixD</strong></li>
<li><strong>Using xAct</strong></li>
</ul>
<h3>Discussion (code sections below)</h3>
<p>There is a... |
3,879,201 | <p>Let <span class="math-container">$E$</span> be a normed <span class="math-container">$\mathbb R$</span>-vector space, <span class="math-container">$\mu$</span> be a finite signed measure on <span class="math-container">$(E,\mathcal B(E))$</span> and <span class="math-container">$$\hat\mu:E'\to\mathbb C\;,\;\;\;\varp... | Michael Rozenberg | 190,319 | <p>Your answer is right.</p>
<p>I like the following way: <span class="math-container">$$x^5-x^4-x^3+x^2+x=x(x^4-x^3-x^2+x+1)=$$</span>
<span class="math-container">$$=x(x^4+2x^3-x^2-2x+1)=x(x^2+x-1)^2.$$</span></p>
|
4,135,473 | <p>Can you explain how the lower integral is the supremum of a sum? Isn't <span class="math-container">$L(f,P)$</span> just a sum and therefore a single number? Definition 5.13 says from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>, not from each <span class="math-container">$x_j$<... | N. S. | 9,176 | <p>My favourite, and the cleanest way in my oppinion is the following classical solution:</p>
<p>Let <span class="math-container">$f(x)= \lfloor x\rfloor+\lfloor x+\frac{1}{2}\rfloor-\lfloor 2x \rfloor$</span>.</p>
<p>Then, it is trivial to see that</p>
<ul>
<li><span class="math-container">$f(x+\frac{1}{2})=f(x)$</spa... |
864,816 | <p>I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc.</p>
<p>Of course, there are the constant functions $f(x)=0$ and $f(x)=2$. I also showed that $1+b^x$, where $b>0$, is another solution. Are there any other ni... | Community | -1 | <p>From your solution, I thought to consider the change of variable $f(x) = 1 + g(x)$: your functional equation then becomes</p>
<p>$$ g(x) g(-x) = 1 $$</p>
<p>and now the entire solution space becomes obvious: you can pick $g(x)$ to be <em>any</em> function on the positive reals that is nowhere zero, and then the va... |
3,219,260 | <p>Trying to find the easiest and most general method in finding last <span class="math-container">$n$</span> digits of a number.
I know the trick lies in finding the remainder when the number is divided by <span class="math-container">$10^n$</span> but still not able to perform the rest of the steps to reach the answe... | Bill Dubuque | 242 | <p>Below I show that the method in JWT's answer is also essentially Binomial Theorem based.</p>
<p><span class="math-container">$\begin{align}
7^{\large 2}\ &= -1 + 50 \\[.3em]
\Rightarrow\ \ \ \ \ \ \ 7^{\large 4}\ &=\ \ \ 1 + 24\cdot100\ \, (=\, 1 -2(50)+ 2500\ \ \rm by\ squaring\ above)\\[.3em]
\Rightarr... |
2,596,347 | <p>Suppose $X$ is compact and Hausdorff and that $f:X \to Y$ is continuous, closed, and surjective. How can I show that $Y$ is Hausdorff?</p>
| Not Mike | 480,631 | <p>Since $X$ is compact Hausdorff, $X$ it is normal.
Next, since $f$ is closed, the single-element subsets of $Y$ are closed (as $\{y\} = f''\{x\}$ for some $x \in X$.) It follows that for each pair of distinct $y_0, y_1 \in Y$, the sets $X_0 = f^{-1}(y_0)$ and $X_1 = f^{-1}(y_1)$ are disjoint closed subsets of $X$. N... |
4,502,175 | <p>I want to create a simple demo of moving an eye (black circle) inside a bigger circle with a black stroke when moving a mouse. I have cursor positions mouseX and mouseY on a canvas and I need to map the value of mouse position into a circle so the eye is moving inside the circle.</p>
<p>This should be trivial but I ... | user2661923 | 464,411 | <p>I may not be understanding the problem correctly. If not, please advise:</p>
<p>It seems as if you are facing two mathematical challenges:</p>
<ul>
<li><p>You are given a circle with center at the Cartesian coordinate <span class="math-container">$(0,0)$</span>, of radius <span class="math-container">$(r)$</span>. ... |
4,651,215 | <p>Three Cowboys have a shoot out. A shoots with 1/3 accuracy, B 1/2, C one (never misses). A goes first. The cowboys attempt to shoot the cowboy with the best shooting ability. What’s the probability that A is still alive?</p>
<p>Edit: We assume that after A, B goes first and then C</p>
<p>My approach:</p>
<p>Since it... | P. Lawrence | 545,558 | <p>There are exactly 4 possibilities, for any <span class="math-container">$a$</span>. (i) If <span class="math-container">$a\equiv 0 \pmod 4,$</span> then <span class="math-container">$a^2 \equiv 0 \pmod 4$</span> (ii) If <span class="math-container">$a\equiv 1 \pmod 4,$</span> then <span class="math-container">$a^2... |
1,241,586 | <p>I want to prove that for all positive integers
$n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$.
By mathematical induction:</p>
<p>1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$</p>
<p>2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, then prove that $2+2^2+2^3+...+2^n+2^{n+1}=2^{n+2}-2$ holds.</p>
<p>I have no idea how to proceed wit... | Mark Joshi | 106,024 | <p>start with zero, the question is then the same as
$$2^0+2^1+2^2+...+2^n=2^{n+1}-1$$
The binary for the LHS is
$$
11111\dots 1
$$
with $n$ ones.
add $1$ to get
$$
10000\dots 0,
$$
with $n$ zeros in binary. This is the binary for $2^{n+1}$ and are done.</p>
|
2,437,466 | <p>A point, $p$, is defined as a cluster point of a set $S$ if $\forall \epsilon > 0,$ there exists an open ball of radius $\epsilon$ centered at $p$ that contains infinitely many points in $S$.</p>
<p>We want to prove that a subset $S$ of a metric space $(E,d)$ is closed iff $S$ contains all of its cluster points.... | Community | -1 | <p>For $k+1$ it is equivalent to</p>
<p>$$
\frac{4^k}{k+1}<\frac{(2k+1)(k+2)(2k)!}{2(k+1)^2(k!)^2}$$</p>
<p>It is enough to prove that</p>
<p>$$\frac{(2k+1)(k+2)}{2(k+1)^2}>1$$</p>
<p>Notice that the above fraction is monotonically decreasing and its limit at $k\to\infty$ is $1$, therefore it is always greate... |
429,178 | <p>Consider the following function $$ \Pi(t) = \begin{cases} 1 \ \ \text{if} \ |t| < 1/2 \\ 0 \ \ \text{if} \ |t| \geq 1/2 \end{cases} $$</p>
<p>This is not a periodic function. Why can we define the fourier coefficients of $\Pi(t)$ as $$c_n = \frac{1}{T} \int_{-T/2}^{T/2} e^{-2 \pi int/T} \Pi(t) \ dt $$</p>
<p>$$... | John Gowers | 26,267 | <p>You've used the variable $T$ in both expressions without telling us what it represents. If we interpret $T$ as the period in the Fourier expansion, and compute</p>
<p>$$
\sum_{n=-\infty}^\infty c_ne^{2\pi int/T}
$$</p>
<p>with $c_n$ defined as in your question, we get the periodic function which agrees with your ... |
1,902,939 | <p>I was going through the problem attached herewith. the fact is I could not understand 2 things. </p>
<p>Firstly, how did the author came up with the idea of choosing the event sets the way he has described? How is this intuitive?</p>
<p>Secondly, I tried to find the $P(A_1)$ directly, and I came up with the follow... | Brian M. Scott | 12,042 | <p>An alternative approach is to observe that since $ab\in(aH)(bH)$, we must have $(aH)(bH)=abH$. We can now define an operation $*$ on the set $\mathscr{L}$ of left cosets of $H$ by $aH*bH=abH$. It's straightforward to check that this operation is a well-defined group operations on $\mathscr{L}$ with identity $H=1_GH$... |
1,504,888 | <p>Let $R$ be an integral domain. The following is a well known fact:</p>
<blockquote>
<p>Let $a,b \in R$. Then $(a)=(b)$ if and only if there exist a unit $u \in R$ such that $a=ub$.</p>
</blockquote>
<p>I would like to generalize this result for ideals which are generated by two elements. So the question is:</p>
... | Remy | 284,272 | <p>We consider the following condition.</p>
<blockquote>
<p><strong>Condition.</strong> There exists $A \in \operatorname{GL}_2(R)$ such that
$$\begin{pmatrix} a \\ b\end{pmatrix} = A \begin{pmatrix} a' \\ b'\end{pmatrix}.$$</p>
</blockquote>
<p>The sufficiency of the condition has already been noted, so I will o... |
1,851 | <p>Is it possible to modify closing reasons menu? If so I would love to see a small addition, maybe others have other wishes too.</p>
<p>So, the last <a href="https://mathematica.meta.stackexchange.com/q/1010/5478">Granular off-topic close reasons</a> lead to creating</p>
<blockquote>
<p>This question arises due to... | Mr.Wizard | 121 | <p>We (community moderators) can change the custom close reasons but I believe we are limited by the software to a total of three reasons, therefore to enact what you propose would require eliminating one of the existing and commonly used reasons. Any proposed change would need to clearly specify exactly how the other... |
1,426,891 | <blockquote>
<p>Why does $$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy ?$$</p>
</blockquote>
<p>This came up while studying Fourier analysis. What's the underlying theorem?</p>
| idm | 167,226 | <p>$$\left(\int_a^b f(x)\mathrm dx\right)^2=\left(\int_a^b f(x)\mathrm dx\right)\left(\int_a^b f(x)\mathrm dx\right)\underset{x\ is\ a \ mute\ variable}{=}\left(\int_a^b f(x)\mathrm dx\right)\left(\int_a^b f(y)\mathrm dy\right)$$$$\underset{Fubini}{=}\int_a^b\int_a^bf(x)f(y)\mathrm dx\mathrm dy$$</p>
|
2,985,962 | <p>I was asking my friends a riddle about identifying hats. Each person has to correctly identify the colour of their own hat that was put on their head randomlly.
There is no defined number of either colour. So they could be all white or all black or any combination in between.</p>
<p>They gave an answer that gets 50... | T_M | 562,248 | <p>You are correct, 50% is what you would expect, on average, for straight guesses.</p>
<p>Your friend is missing the point of saying "expect, on average" if they claim that what happens "on average" won't happen some of the time.</p>
<p>The 50% rule does apply even if everyone is wearing black. It'... |
2,495,789 | <p>\begin{pmatrix}4&-6&2&-6\\ \:-2&3&-1&3\end{pmatrix}</p>
<p>Am I supposed to move the $0 $ to the other side of the equation and find the inverse of the matrix above or is there another way?</p>
| matteociccozzi | 496,782 | <p>Simply row reduce the coefficient matrix and let your independent variables (columns without leading ones) be your parameters. Being a homogenous system with 2 equations and 4 variables you will have infinite solutions (since every homogenous system has at least the trivial solution)</p>
|
2,536,877 | <p>I have to evaluate this integral:
$$\int \frac{\tan^5 x}{\cos^3{x}} dx.$$</p>
<p>I guess I should use substitution, but I don't know what to substitute =(</p>
| José Carlos Santos | 446,262 | <p>Observe that$$\frac{\tan^5 x}{\cos^3x}=\frac{\sin^5x}{\cos^8x}=\frac{\sin x(1-\cos^2x)^2}{\cos^8x}.$$So, do the substitution $\cos x=t$ and $\sin x\,\mathrm dx=-\mathrm dt$.</p>
|
125,645 | <p>I stumbled upon the following integral:</p>
<p>$$\int e^{-\left(\frac{\gamma}{2} + ip \right)^2} H_n \left( q-\frac{\gamma}{2} \right) H_n \left( q+\frac{\gamma}{2} \right) \, d\gamma \, ,$$</p>
<p>and I really need a closed form of it. I'm pretty sure it exists, and probably in terms of $\Gamma$ functions, since<... | Jens | 245 | <p>This is an integral I happen to have worked with before. The answer is a Laguerre polynomial, as can be verified like this for the first three indices as in the question:</p>
<pre><code>ll =
2 Table[LaguerreL[i, 2 (p^2 + q^2)] (-2)^i Sqrt[Pi] i!, {i, 0, 2}];
l1 = Table[
Integrate[
HermiteH[i, q + 1/2 γ] ... |
3,247,923 | <p>I’ve tried splitting this into the sum of two Chinese Dumbass triangles:</p>
<p>0</p>
<p>-5 0</p>
<p>0 15 -5</p>
<p>0 -5 0 0</p>
<p>And</p>
<p>1</p>
<p>2 2</p>
<p>2 -15 2</p>
<p>1 2 2 1</p>
<p>And that fails. Any hint or help would be appreciated!</p>
| Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$a=\min\{a,b,c\}$</span>, <span class="math-container">$b=a+u$</span> and <span class="math-container">$c=a+v$</span>.</p>
<p>Thus, by AM-GM we obtain:
<span class="math-container">$$\sum_{cyc}a(a-b)(a-2b)=2a(u^2-uv+v^2)+u^3-3u^2v+2uv^2+v^3\geq$$</span>
<span class="math-container">... |
764,973 | <p>here is the question: </p>
<p>$$
{\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right)
=
0;
\qquad\qquad
{\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1.
$$</p>
<ul>
<li>$\mathcal{L} (y''(t)) = s^2y(s) -s y(0) -y'(0)$</li>
<li>$\mathcal{L} (+2y'(t)) = 2(sy(s) -y(0))$</li>
<li... | Amzoti | 38,839 | <p>If the DE is written correctly, we should have arrived at:</p>
<p>$$y(s)=\dfrac{2s+3}{s^{2}+2s+5}$$</p>
<p>This yields:</p>
<p>$$y(t) = \dfrac{1}{2} e^{-t} (\sin(2 t)+4 \cos(2 t))$$</p>
|
3,997,110 | <p>I am not very experienced with math and need help with a proof. I need to show that the pair of functions <span class="math-container">$t$</span> and <span class="math-container">$1/t$</span> defined for <span class="math-container">$t>0$</span> are linearly independent. Is the following a sufficient proof? If so... | Nikola Ubavić | 588,411 | <p>Your proof is fine and straightforward.</p>
<p>Maybe little shorter proof would be to see that <span class="math-container">$\lim_{t\rightarrow\infty} at+b\frac{1}t$</span> can be zero if and only if <span class="math-container">$a=0$</span>. It follows immediately that <span class="math-container">$b=0$</span> too.... |
3,997,110 | <p>I am not very experienced with math and need help with a proof. I need to show that the pair of functions <span class="math-container">$t$</span> and <span class="math-container">$1/t$</span> defined for <span class="math-container">$t>0$</span> are linearly independent. Is the following a sufficient proof? If so... | Maths | 740,715 | <p>Well they are actually not linear independent. Take <span class="math-container">$t=1$</span> and <span class="math-container">$a=1;b=-1$</span></p>
<p>and You have <span class="math-container">$1+(-1*1/1$</span>)</p>
<p>@billiam Thanks for your answer, if it is only positive, then it is quite trivial, here is my pr... |
1,001,913 | <p>If we want to graph a horizontal line, we will do the following:</p>
<pre><code>y = 0x + 3
</code></pre>
<p>No matter the domain for x, the range for y will always be 3. Therefore, we have a horizontal line.</p>
<pre><code>y = 0(0) + 3 = (0,3)
y = 0(1) + 3 = (1,3)
y = 0(2) + 3 = (2,3)
</code></pre>
<p>Now the fo... | Marko Riedel | 44,883 | <p>By way of enrichment here is another algebraic proof using basic
complex variables.
<P>
We seek to compute
$$\sum_{t=0}^{n-m} {t+k\choose t} {n-k-t\choose n-m-t}.$$</p>
<p>Introduce the integral representation
$${n-k-t\choose n-m-t}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon} \frac{(1+z)^{n-k-t}}{z^{n-m-t+1}} \; d... |
1,816,783 | <p>When can one conclude that a character table has non-real entries?</p>
<p>In particular, by constructing the character table for $\mathbb{Z}/3\mathbb{Z}$ or $A_4$ how does one determine that some of the entries will be nonreal?
Is the reason that the same complex values in the table for $\mathbb{Z}/3\mathbb{Z}$ als... | P Vanchinathan | 28,915 | <p>If an element of the group and its inverse are in the same conjugacy of of a group then the value of any character at that element will be a real number.</p>
<p>In the symmetric groups $S_n$ any element and its inverse have the same cycle decomposition structure and hence they are conjugates. </p>
|
234,435 | <p>This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded linear operator, which is also injective. If $T(A)$ is dense in $X$, does it follow that $A$ is dense in $X$? </p>
| Bill Johnson | 2,554 | <p>No, not even if $X=\ell_2$. Le $A$ be the closed span of $(e_n)_{n=2}^\infty$ and map $A$ to a proper dense subspace and $e_1$ to a vector not in that subspace.</p>
|
234,435 | <p>This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded linear operator, which is also injective. If $T(A)$ is dense in $X$, does it follow that $A$ is dense in $X$? </p>
| Nate Eldredge | 4,832 | <p>Here's an explicit example for what Bill's answer proposes.</p>
<p>Let $T_1$ be the operator which maps $e_n$ to $4^{-n} e_{n-1}$ for $n \ge 2$ and maps $e_1$ to $0$. That is, $T_1 \left(\sum_{n \ge 1} a_n e_n\right) = \sum_{n \ge 1} a_{n+1} 4^{-(n+1)} e_n$. Note $T_1$ is injective on $A$, and $T_1 A$ is dense (i... |
1,136,347 | <p>I'm trying to compute</p>
<p>$$
\omega^{3n/2 + 1} + \omega
$$</p>
<p>where $\omega$ is one of the $n^{th}$ roots of unity where $n$ is a multiple of $4$. Could anyone demonstrate how to do this?</p>
| Sai | 212,299 | <p>Since $n=4k$ we can write the expression as $ \omega ^{2k+1}+\omega=\omega(1+\omega^{2k})$. But now $\omega ^{2k}$ is a square root of $1 $ not equal to $1$. So it is definitely $-1$. Hence the expression is 0.</p>
|
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | Ross Millikan | 1,827 | <p>A simple answer that almost works is $f(x)=g(x)=-x,h(x)=x$ with the only problem being $f=g$ but we can patch that up by making each one the identity on different parts of the real line. Split the reals into $|x| \gt 2,1 \le |x| \le 2, |x| \lt 1$. Make each the identity on one part and $-x$ on the other two. </p>
... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | orion2112 | 518,984 | <p>(This really should be a comment but I needed more space, apologies).</p>
<p>You seem to think (reading the previous comments) that the ellipse's equation should "instruct" us, in a step-by-step manner, how to draw the ellipse. That doesn't have to be the case.</p>
<p>In fact, let us invent a new relation between ... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Blue | 409 | <p>There is no single equation for an ellipse, just as there is no single equation for a line. We choose a form to highlight information of interest in the current context.</p>
<p>Consider this sampling of ways to write the equation of a line:</p>
<blockquote>
<p>$$\begin{array}{rcccl}
\text{slope-intercept} &\... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | user5402 | 81,595 | <p>Draw two concentric circles of centre $O$ and radii $a$ and $b$ ($a<b$) in an orthonormal system of axes $(O,\textbf{i},\textbf{j})$. Every half-line from $O$ cuts the circles at points $M_a$ and $M_b$. Let $N_a$ and $N_b$ be the orthogonal projections of $M_a$ and $M_b$ over the $x$ axis and the $y$ axis respect... |
66,634 | <p>Here is a quote from <em>Lectures on Ergodic Theory</em> by Halmos:</p>
<blockquote>
<p>I cannot resist the temptation of
concluding these comments with an
alternative "proof" of the ergodic
theorem. If $f$ is a complex valued
function on the nonnegative integers,
write $\int f(n)dn=\lim
> \frac{1}{n... | Terry Tao | 766 | <p>I feel the answer is "no", at least while staying true to the spirit of Halmos's text. Halmos's "proof", if valid, would imply something far stronger (and false), namely that $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(T_n x)$ converged for almost every x, where $T_1, T_2, \ldots$ are an arbitrary sequence of m... |
1,027,249 | <p>Prove that $2$ is the only even prime:</p>
<p>I have tried, this is what I have done. Is this is suffitient? </p>
<p>Proof: To prove that $2$ is the only even prime
number we need to prove following:
(1) $2$ is an even prime number
(2) If there existed another even prime
number say $m$, then $m=2$</p>
<p>Proof of... | Henry | 6,460 | <p>Saying (1) is obvious may not be enough to be a proof: it is not a proof and (2) is also obvious.</p>
<p>I think it might be clearer if you were trying to prove (2) that any even positive integer other than $2$ is not prime. In any case for (2) you could to state that $k\gt 1$. </p>
|
438,744 | <p>Let $A$ and $B$ be real square matrices of the same size. Is it true that
$$\det(A^2+B^2)\geq0\,?$$</p>
<p>If $AB=BA$ then the answer is positive:
$$\det(A^2+B^2)=\det(A+iB)\det(A-iB)=\det(A+iB)\overline{\det(A+iB)}\geq0.$$</p>
| user1551 | 1,551 | <p>The ideas in <strong>Seirios</strong>' answer and <strong>Jyrki Lahtonen</strong>'s (now deleted) answer can be generalised to the following result:</p>
<blockquote>
<p>Let $n\ge2$ and $A\in M_n(\mathbb{C})$. Then $\det(A^2+B^2)\ge0$ for all $B\in M_n(\mathbb{R})$ if and only if $A^2$ is a nonnegative scalar mul... |
76,493 | <p>i'm interested in geometric interpretations of many linear algebra notions (check also related <a href="https://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors">geometric interpretation of matrix minors</a>). it came to me recently that geometric description of adjugate matrix (for example... | Igor Rivin | 11,142 | <p>Well, if your matrix happens to be invertible, the adjugate is the inverse. Otherwise, the adjugate gives the action on the $n-1$st exterior power (you can use the $*$ operator to map it back to the space itself. @Qiaochu has alluded to this in his answer to the question you cite. You can also check out Section 9 of... |
190,604 | <p><strong>Background</strong></p>
<p>I am currently trying to solve exercise 1.1.18 in Hatcher's Algebraic Topology. The part of the exercise I am interested in is the following:</p>
<blockquote>
<p>Using the technique in the proof of Proposition 1.14, show that if a space $X$ is obtained from a path-connected sub... | Math | 225,252 | <p>Since $\pi_1(S^n) =0 $ for $n>1$ so all elements of $\pi_1(X)$ can be realized as an element os $\pi_1(A)$.</p>
|
5,848 | <p>This is the question which i am attempting to solve and it seems to be difficult to get rid of the exponents.</p>
<blockquote>
<p>Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect each other at nine points.</p>
</blockquote>
<p>Any help would be appreciated. By the way i would al... | Bill Dubuque | 242 | <p>Like your earlier question about how to prove the associativity of the group law on elliptic curves, this too is a consequence of general results on intersection theory, esp. (variants of) <a href="http://en.wikipedia.org/wiki/Bezouts_theorem" rel="nofollow">Bezout's theorem</a>. For computational purposes you can e... |
2,661,221 | <p><strong>$\frac{1}{n}=O(\frac{1}{\ln n})$ or $\frac{1}{n}=o(\frac{1}{\ln n})$?</strong></p>
<p>I know that $n\geq\ln n, (n> 0)$, with which $\frac{1}{n}\leq \frac{1}{\ln n}$ and so $\frac{1}{n}=O(\frac{1}{\ln n})$.</p>
<p>To know if $\frac{1}{n}=o(\frac{1}{\ln n})$, I need to know if $\lim_{n\to \infty}\frac{\ln... | mucciolo | 222,084 | <p>$\DeclareMathOperator{\spn}{span}\DeclareMathOperator{\img}{Im}$There is really some linear algebra hidden there. Note that $\sum_{i=1}^{n}w_{i}a_{i}$ is a linear functional on $\mathbb{R}^n$ with respect to $w$. That is, $\alpha: \mathbb{R}^n \to \mathbb{R} : (w_1, \cdots, w_n) \mapsto \sum_{i=1}^{n}w_{i}a_{i}$ is ... |
2,041,849 | <p>I'm stuck on the following questions "In the alternating group $A_4$, let $H$ be the cyclic subgroup generated by $(123)$. Find all the right cosets of $H$ and all the left cosets of $H$. Is every right coset also a left coset."</p>
<p>The answer is as follows "$H = [idx_4 , (123), (132)]$ so each coset will have o... | Matt Samuel | 187,867 | <p>The choice was to an extent arbitrary. The fact is any single element from each coset can be chosen to represent it. </p>
<p>If your question is as to how to find these elements, you can work recursively. The easiest coset to find is $H$ itself. To find the next coset, pick any element $a_1$ not in $H$. This will g... |
2,041,849 | <p>I'm stuck on the following questions "In the alternating group $A_4$, let $H$ be the cyclic subgroup generated by $(123)$. Find all the right cosets of $H$ and all the left cosets of $H$. Is every right coset also a left coset."</p>
<p>The answer is as follows "$H = [idx_4 , (123), (132)]$ so each coset will have o... | Andreas Caranti | 58,401 | <p>I do not know how much group theory you have covered until now, but if you consider the subgroup of $A_{4}$
$$
V = \{ 1, (12)(34), (13)(24), (14)(23) \},
$$
you will have $$\{ v h : v \in V, h \in H\}
=
V H = A_{4} = H V = \{ h v : v \in V, h \in H\},$$ with $V \cap H = \{ 1 \}$.</p>
<p>So, although as noted in o... |
337,806 | <p>I'm stuck on this problem for few days and can't find the solution.Hope some one here can help me solve this. I'm so grateful for any any help:</p>
<blockquote>
<blockquote>
<p>Let $V$ be a finite-dimensional vector space and $f$ a non-degenerate symmetric bilinear form on $V$. With every basis $\beta = \{\al... | le duc quang | 38,024 | <p>In reality, I found another solution using 2 ways to calculate $\alpha_{i}'$.</p>
<p>Denote $f_{\beta} = A$. </p>
<p>For every $\alpha$ in $V$, we can express:
$\alpha = \sum_{i=1}^{n}b_{i}\alpha_{i}$</p>
<p>Then we have $f(\alpha, \alpha_{j}') = f(\sum_{i=1}^{n}b_{i}\alpha_{i}, \alpha_{j}') = b_{j}$. So we have... |
158,630 | <p>Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an element of $\mathbb{C}$ and thus would be $z \ \stackrel{\mathrm{def}}{=}\ Ae^{s T}$ (but then it would be different to t... | Noir | 324,626 | <p>There is a good reason to use <span class="math-container">$z$</span> instead of <span class="math-container">$e^{sT}$</span>. Before starting with any analysis, let me remind you that in analysis of signals and systems we are interested in analyzing the frequency spectrum of the signal, i.e. the Laplace transform o... |
194,985 | <p>How does one show that $$\lim_{n \rightarrow \infty}\int_{0}^{1}\frac{x^{n}}{1 + x^{n}}\, dx = 0?$$ My idea is to evaluate the inner integral, but I can't seem to be able to do that.</p>
| André Nicolas | 6,312 | <p>If you really want to evaluate an integral, note that
$$\frac{x^n}{1+x^n}\le \frac{x^{n-1}}{1+x^n}$$
on $[0,1]$ and let $u=1+x^n$. </p>
|
211,052 | <p>I want to modify a list of table row by adding an extra column. For that I <a href="https://reference.wolfram.com/language/ref/Map.html" rel="nofollow noreferrer"><code>Map</code></a> the data with a pure function that evluate the new column value from exisiting one and reconstruct a new list from that and the init... | Sylvain Leroux | 68,791 | <p>Somewhat a variation around the <a href="https://reference.wolfram.com/language/ref/Sequence.html" rel="nofollow noreferrer"><code>Sequence</code></a> symbol that was suggeste by @kglr in <a href="https://mathematica.stackexchange.com/a/211055/68791">another answer</a>:</p>
<pre><code>{#[[1;;3]] /. List -> Seque... |
1,960,793 | <p>Considering the (not most common) definition:</p>
<blockquote>
<p>A set is <em>infinite</em> if it is equipotent to a proper subset of itself. A set is <em>finite</em> if it is not infinite.</p>
</blockquote>
<p>How can I prove that a set $A$ is finite iif it is equipotent to $J_n=\{1,…,n\}$ for some $n{\in}\mat... | Bargabbiati | 352,078 | <p>Suppose that $A$ is not equipotent to $J_n$ $\forall n$. So you can find a surjective function $f:A\rightarrow\mathbb N$. The idea is that if the function is not surjective than you can find a bijection between $A$ and $J_n$ where $n$ is the maximum of $Im(f)$. So $A$ has at least the cardinality of $\mathbb N$, tha... |
245,853 | <p>The graphs of my provided code look approximately like the one in the picture below. How is it possible to plot such graphs? Please help how to break the axis scale as shown. Any suggestion appreciated. Thanks a lot.
<a href="https://i.stack.imgur.com/8hNeO.jpg" rel="nofollow noreferrer"><img src="https://i.stack.im... | Bob Hanlon | 9,362 | <p>First, I recommend that you avoid using subscripts except for display.</p>
<p>Since the range of your plots is only <code>{t, 0, 10^-14}</code> there are no regions where the functions are essentially constant and where a gap in the axis would be appropriate.</p>
<pre><code>Clear["Global`*"]
Format[Ti] = ... |
2,471,217 | <p>Contradiction or contra-positive? Or is direct easier? </p>
<p>∀n∈N, (n>3∧ n is prime)→∃q∈N,(n=6q+1∨n=6q+5)</p>
| Tom Miller | 487,910 | <p>Well, since primes greater than 3 are odd, the only other possible form is 6k + 3. But this is divisible by 3 so is not prime. </p>
|
3,378,655 | <p>I have an assignment about epsilon-delta proofs and I'm having trouble with this one. I have worked it through using some of the methods I've picked up for similar proofs but it's just something about this particular expression that doesn't sit right with me. Any feedback would be very helpful. This is how far I've ... | Bernard | 202,857 | <p><strong>Hint</strong>:
You may suppose, w.l.o.g. that <span class="math-container">$|x-2|<\frac12$</span>, in which case <span class="math-container">$|x-1|>\frac12$</span>, so that
<span class="math-container">$$\left|\frac{x-2}{x - 1}\right| <2|x-2|.$$</span></p>
|
3,187,628 | <p>On a keyboard 1-9 and A-D , how many possible way to have a 5 digit code with 3 different numbers and two identical letters (order isn't a problem)?</p>
<p>My try : 9C3 x 4C1 = 84 x 4 = 336</p>
| José Carlos Santos | 446,262 | <p><strong>Hint:</strong> What is the Taylor series of the sine function centered at <span class="math-container">$0$</span>?</p>
|
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