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 |
|---|---|---|---|---|
3,911,297 | <p>Chapter 12 - Problem 26)</p>
<blockquote>
<p>Suppose that <span class="math-container">$f(x) > 0$</span> for all <span class="math-container">$x$</span>, and that <span class="math-container">$f$</span> is decreasing. Prove that there is a <em>continuous</em> decreasing function <span class="math-container">$g$</... | J. W. Tanner | 615,567 | <p>Using the Bezout relation <span class="math-container">$1=12\times3-7\times5$</span>, we have <span class="math-container">$(a^{12})^3/(a^7)^5=a\in\mathbb Q$</span>.</p>
|
1,276,177 | <p>Four fair 6-sided dice are rolled. The sum of the numbers shown on the dice is 8. What is the probability that 2's were rolled on all four dice?</p>
<hr>
<p>The answer should be 1/(# of ways 4 numbers sum to 8)
However, I can't find a way, other than listing all possibilities, to find the denominator. </p>
| WW1 | 88,679 | <p>I proved a relevant theorem <a href="https://math.stackexchange.com/questions/1259869/rocks-and-squares-balls-and-sticks/1260278#1260278">here</a> ( for some reason this was closed )</p>
<p>Theorem 1: The number of distinct $n$-tuples of whole numbers whose components sum to a whole number $m$ is given by </p>
<p>... |
2,473,132 | <p>Compute $$ \int_{0}^{ \infty }\frac{\ln xdx}{x^{2}+ax+b}$$
I tried to compute the indefinite integral by factorisation and partial fraction decomposition but it became nasty pretty soon. There must be another way to directly evaluate it without actually computing the indefinite integral which I don't know!</p>
| DXT | 372,201 | <p>Let $$I = \int^{\infty}_{0}\frac{\ln x}{x^2+ax+b}dx$$</p>
<p>Put $\displaystyle x = \frac{b}{t}$ and $\displaystyle dx = -\frac{b}{t^2}dt$</p>
<p>So $$I = \int^{\infty}_{0}\frac{\ln (b)-\ln(t)}{t^2+at+b}dt = \ln (b)\int^{\infty}_{0}\frac{1}{t^2+at+b}dt-I$$</p>
<p>So $$I = \frac{\ln (b)}{2}\int^{\infty}_{0}\frac{1... |
749,035 | <p>Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there infinitely many?</p>
| Robert Israel | 8,508 | <p>I hope I got the programming right: I get $679$ pairs less than $10000$, of which the first few are
$$
\eqalign{
[102,105], &[170,174], &[230,231], &[238,246], &[255,258], &[282,285], &[285,286],\cr
[366,370], &[399,402], &[429,430], &[430,434], &[434,435], &[438,442], &am... |
825,848 | <p>I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = {f_1}^{m_1} ... {f_n}^{m_n}$ then $\chi = {f_1}^{d_1} ... {f_n}^{d_n}$ with $f_i$ irreducible and $m_i \le d_i$.</p>
<p>Now I... | Marc van Leeuwen | 18,880 | <p>If you believe in invariant factors, this is easily seen as follows. The minimal polynomial is the last of the invariant factors (arranged so that each divides the next) and the characteristic polynomial is their product. It follows that the minimal polynomial divides the characteristic one (Cayley-Hamilton) and tha... |
2,340,606 | <p>$$x^2-2(3m-1)x+2m+3=0$$
Find the sum of solutions. It says that the sum equals to $-1$. I just can't wrap my head around this? Any help? Thx</p>
| Dr. Sonnhard Graubner | 175,066 | <p>$$x_{1,2}=3m-1\pm\sqrt{9m^2-8m-2}$$ therefore
$$x_1+x_2=2(3m-1)$$</p>
|
3,129,852 | <p>My question is pretty basic.</p>
<p>Here it goes:</p>
<blockquote>
<p>Is it always true that <span class="math-container">$$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$$</span>
where the <span class="math-container">$s_j$</span>'s are positive integers, and may be odd or even?</p>
</blockquote>
<p>We can pe... | Fabio Lucchini | 54,738 | <p>Since <span class="math-container">$2x+1\equiv(-1)^x\pmod 4$</span>, we have
<span class="math-container">$$\prod_{j=1}^{w}{(2s_j + 1)} \equiv (-1)^{\sum_{j}s_j} \pmod 4$$</span>
hence
<span class="math-container">$$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1\pmod 4\iff\sum_{j=1}^ws_j\equiv 0\pmod 2$$</span></p>
|
873,582 | <p>How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?</p>
| Martial | 99,769 | <p>use three equations:
$$a^3+b^3=(a+b)(a^2-ab+b^2)\quad (1)$$
$$a^3-b^3=(a-b)(a^2+ab+b^2)\quad (2)$$
$$(a+b)^3=a^3+3a^2b+3ab^2+b^3\quad (3)$$
for your problem:
$$left\\=(\sqrt[3]{\frac{1}{3}})^2-\sqrt[3]{\frac{1}{3}}\sqrt[3]{\frac{2}{3}}+(\sqrt[3]{\frac{2}{3}})^2\\=\frac{\frac{1}{3}+\frac{2}{3}}{\sqrt[3]{\frac{1}{3}}... |
317,981 | <p>Prove the following:</p>
<p>$$\lim_{n \to \infty} \displaystyle \int_0 ^{2\pi} \frac{\sin nx}{x^2 + n^2} dx = 0$$</p>
<p>How would I prove this? I know you have to show your steps, but I'm literally stuck on the first one, so I can't. </p>
| robjohn | 13,854 | <p>The simplest approach seems to be to note that
$$
\left|\frac{\sin(nx)}{x^2+n^2}\right|\le\frac1{n^2}
$$
so that
$$
\left|\int_0^{2\pi}\frac{\sin(nx)}{x^2+n^2}\,\mathrm{d}x\right|\le\frac{2\pi}{n^2}
$$</p>
|
602,973 | <p>I have a problem:</p>
<blockquote>
<p>For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$.</p>
</blockquote>
<p>=============================</p>
<p>Any help will be appreciated! Thanks!</p>
| user42070 | 113,916 | <p>First I would like to answer the first comment : It is only if you take the whole $\mathbb{R}^n$ that this two spaces coincide not at all for bounded domain. </p>
<p>For the problem : I will do it on $\mathbb{R}$ and $m=1$ and $\Omega=(0,1)$ (easy to adapt). To show that $u$ belongs to $W^{m,2}(\Omega)$ you only ne... |
3,455,967 | <p>Solve: (Hint: use <span class="math-container">$x\ln y=t$</span>)<span class="math-container">$$(xy+2xy\ln^2y+y\ln y)\text{d}x+(2x^2\ln y+x)\text{d}y=0$$</span></p>
<p>My Work:</p>
<p><span class="math-container">$$x\ln y=t, \text{ d}t=\ln y \text{ d}x+\frac{x}{y} \text{ d}y$$</span></p>
<p><span class="math-cont... | J.G. | 56,861 | <ol>
<li>You're right. Your final result implies <span class="math-container">$x=-2x\ln^2y-\frac{2x^2\ln y}{y}y^\prime-\ln y-\frac{x}{y}y^\prime$</span> and <span class="math-container">$y^\prime=-\frac{(2x\ln y+1)y\ln y+x}{x(1+2x\ln y)}$</span>, which is equivalent to the original equation.</li>
<li>The coefficients o... |
348,395 | <p>If <span class="math-container">$f$</span> a distribution with compact support then they exist <span class="math-container">$m$</span> and measures <span class="math-container">$f_\beta$</span>,<span class="math-container">$|\beta|\leq m$</span> such that
<span class="math-container">$$f=\sum_{|\beta|\leq m}\fra... | paul garrett | 15,629 | <p>In addition to the other two good answers, one can make a somewhat stronger assertion: let <span class="math-container">$u$</span> be a distribution on <span class="math-container">$\mathbb R^n$</span>, in the Sobolev space <span class="math-container">$H^{-\infty}$</span> (which contains all compactly-supported dis... |
4,093,406 | <p>Which of the equations have at least two real roots?
<span class="math-container">\begin{aligned}
x^4-5x^2-36 & = 0 & (1) \\
x^4-13x^2+36 & = 0 & (2) \\
4x^4-10x^2+25 & = 0 & (3)
\end{aligned}</span>
I wasn't able to notice something clever, so I solved each of the equations. The first one h... | Will Jagy | 10,400 | <p><span class="math-container">$$ x^4 -13 x^2 + 36 = (x^2 + 6)^2 - 25 x^2 = (x^2 + 5x+6) (x^2 - 5x+6) $$</span>
and both quadratic factors have real roots(positive discriminants).</p>
<p><span class="math-container">$$ 4 x^4 -10 x^2 + 25 = (2x^2 + 5)^2 - 30 x^2 = (2x^2 + x \sqrt{30}+5) (2x^2 - x \sqrt{30}+5) $$</spa... |
2,549,834 | <p>Ive been looking at this problem and trying to use examples online to try to solve it but I get stuck. </p>
<p>It says to use mathematical induction to prove
(1/1*4)+(1/4*7)+(1/7*10)+ ... + (1/(3n-2)(3n+1)) = n/(3n+1)</p>
<p>I solve for n=1 and substitute k for n, but I don’t really know what to do after that step... | Community | -1 | <p>Not sure if you are allowed to use this in your answer, but note that $|f(x)|=g(f(x))$ where $g:\mathbb R\to \mathbb R$, $g(x)=|x|$. Thus, you can first prove that $g$ is continuous, and then invoke the theorem that the composition of two continuous functions is continuous.</p>
<p>Now, note that $g$ is continuous ... |
507,975 | <p>I'm new here and unsure if this is the right way to format a problem, but here goes nothing. I'm currently trying to solve an inequality proof to show that $n^3 > 2n+1$ for all $n \geq 2$.</p>
<p>I proved the first step $(P(2))$, which comes out to $8>5$, which is true.</p>
<p>In the next step we assume that... | njguliyev | 90,209 | <p>Hint: $$\lim_{t\to 0} \sin t \ln \frac{\cos t}{\sin^2t} = -2 \lim_{t\to 0} (\sin t \cdot \ln \sin t) = 0.$$</p>
|
507,975 | <p>I'm new here and unsure if this is the right way to format a problem, but here goes nothing. I'm currently trying to solve an inequality proof to show that $n^3 > 2n+1$ for all $n \geq 2$.</p>
<p>I proved the first step $(P(2))$, which comes out to $8>5$, which is true.</p>
<p>In the next step we assume that... | Felix Marin | 85,343 | <p>$\lim_{x \to \pi/2}\left[\sec\left(x\right)\tan\left(x\right)\right]^{\cos\left(x\right)} = \lim_{x \to 0}x^{-2x} = \lim_{x \to 0}{\rm e}^{-2x\ln\left(x\right)}$.</p>
<p>Since
$\lim_{x \to 0}\left[-2x\ln\left(x\right)\right] = -2\lim_{x \to 0}{\ln\left(x\right) \over 1/x} = -2\lim_{x \to 0}{1/x \over -1/x^{2}} = 0$... |
2,814,793 | <p>I don't know what I did wrong. Can anyone point out my mistake? The problem is:</p>
<blockquote>
<p>For $\lim\limits_{x\to1}(2-1/x)=1$, finding $\delta$, such that if $0<|x-1|<\delta$, then $|f(x)-1|<0.1$</p>
</blockquote>
<p>Here is what I did:</p>
<p>Since $|f(x)-1|=|2-1/x-1|=|1-1/x|=|x-1|/x<0.1$,... | ShyGuy | 563,183 | <p>The answer means:</p>
<p><strong>Claim.</strong> If $|x - 1| < 1/11$, then $|f(x) - 1| < 1/10$.</p>
<p>Now we prove this. If $|x - 1| < 1/11$, then $10/11 < x < 12/11$. It means</p>
<p>\begin{align}
f(x) - 1 &= 1 - 1/x < 1 - 11/12 = 1/12 < 1/ 10 \\
f(x) - 1 &= 1 - 1/x > 1 - 11/10 ... |
1,678,922 | <p>let B be a commutative unital real algebra and C its complexification
viewed as the cartesian product of B with itself.
If M is a maximal ideal in A, is the cartesian product of M with itself
a maximal ideal in C? </p>
| ray | 319,283 | <p>indeed, (z,iz) (z,-iz)=(0,0); so (z,iz) belongs to a max ideal which strictly contains the ideal $\{(0,0)\}$. What is then the relation between the
max ideals in B and its complexification C?</p>
|
1,920,776 | <p>As I know there is a theorem, which says the following: </p>
<blockquote>
<p>The prime ideals of $B[y]$, where $B$ is a PID, are $(0), (f)$, for irreducible $f \in B[y]$, and all maximal ideals. Moreover, each maximal ideal is of the form $m = (p,q)$, where $p$ is an irreducible element in $B$ and $q$ is an irred... | Rob | 151,459 | <p>The way we represent it is ambiguous. If you are explicit about all unstated bits, then simply flipping all bits negates the number. But you need to have a representation with a decimal point and bits stating what all unstated bits are. For example, say that we are explicit about what unstated bits are so that we... |
1,624,888 | <h2>Question</h2>
<p>In the following expression can $\epsilon$ be a matrix?</p>
<p>$$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \epsilon^2 E_2 |m_2\rangle + \dots) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) $... | user103093 | 303,143 | <p>Yes and No. When I say yes I mean it is possible in several ways. But it does not make sense.</p>
<p>As usual for a physicist you did not specify what the space is in which your states (kets) live and thus not what the $H$ and $H_1$ are. But of course they are meant to be operators which you can consider to be gene... |
142,939 | <p>I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a homotopy exists even if $M$ is not symplectic? how does the homotopy look like? is it differentiable, only continous ... ... | Igor Rivin | 11,142 | <p>This question is discussed at length in <a href="http://www.emis.de/journals/UIAM/actamath/PDF/38-105-128.pdf" rel="nofollow">the very nice survey by A. Tralle.</a> (Homotopy properties of closed symplectic manifolds).</p>
|
142,939 | <p>I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a homotopy exists even if $M$ is not symplectic? how does the homotopy look like? is it differentiable, only continous ... ... | Ian Agol | 1,345 | <p>A <a href="http://www.ams.org/mathscinet-getitem?mr=1625732" rel="nofollow">result of Szabo</a> implies that there are infinitely many homeomorphic but non-diffeomorphic 4-manifolds which do not admit a symplectic structure (the fact that they are homeomorphic is not explicitly stated, but follows from Freedman's cl... |
2,470,062 | <blockquote>
<p><span class="math-container">$$\sqrt{k-\sqrt{k+x}}-x = 0$$</span></p>
<p>Solve for <span class="math-container">$k$</span> in terms of <span class="math-container">$x$</span></p>
</blockquote>
<p>I got all the way to
<span class="math-container">$$x^{4}-2kx^{2}-x+k^{2}-x^{2}$$</span>
but could not facto... | zwim | 399,263 | <p>In this kind of problem, you have to be very careful about the domain of definition. Squaring the equation an find an equivalent polynomial equations is not enough, you need to verify if the solutions found are effective solutions of the original equation.</p>
<p>First two remarks : </p>
<ul>
<li>$x=\sqrt{\cdots}\... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | Andreas | 317,854 | <p>Set $\sin^2(x) = 1 - \cos^2(x) = y^2$ Then you have
$$
v=\sqrt{a^2(1-y^2)+b^2 y^2}+\sqrt{b^2 (1-y^2)+a^2 y^2}
$$
One extremum is obtained (differentiate w.r.t. y) at
$$
\sqrt{a^2(1-y^2)+b^2 y^2}= \sqrt{b^2 (1-y^2)+a^2 y^2}
$$
or
$$
y^2 = 1/2
$$</p>
<p>The other extremum is obtained at $y=0$. </p>
<p>So the diffe... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | Narasimham | 95,860 | <p>There is no need to do a lot of calculus. The ellipses are symmetric with respect to axes of symmetry which fact should be exploited. </p>
<p>It is periodic function, period = $ 2 \pi$. Maximum inter-distance at $ \theta= n \pi/2 $ between ends of major/minor axes as shown and minimum distance $ =0 $ at $ \theta= ... |
1,781,467 | <p>First, I know what the right answer is, and I know how to solve it. What I'm trying to figure out is why I can't get the following process to work.</p>
<p>The probability that we get 2 consecutive heads with one flip is 0. The probability that we get 2 consecutive heads with 2 flips = 1/4. The probability of gettin... | Med | 261,160 | <p>The problem is that your experiment is not well-defined. To solve the mentioned expectation problem, you would define the experiment as follows:</p>
<p>You toss a coin until you get two heads in a row and then you would stop. So there are no two consecutive heads before that.</p>
<p>each of the probabilities, that... |
388,815 | <p>I am trying to compute the integral: $$\int_{4}^{5} \frac{dx}{\sqrt{x^{2}-16}}$$
The question is related to hyperbolic functions, so I let $x = 4\cosh(u)$ therefore the integral becomes: $$-\int_{0}^{\ln(2)}\frac{4\sinh(u)}{\sqrt{16-16\cosh^{2}(u)}}du = -\int_{0}^{\ln(2)}1du = -\ln(2)$$</p>
<p>The answer is $\ln(2)... | Community | -1 | <p>Setting $x = 4 \cosh(u)$, gives us $dx = 4 \sinh(u)du$. Hence the integral becomes
$$\int_0^{\ln(2)} \dfrac{4 \sinh(u) du}{\sqrt{16\cosh^2(u) - 16}} = \int_0^{\ln(2)} \dfrac{4 \sinh(u) du}{4 \sinh(u)} = \ln(2)$$</p>
|
1,770,508 | <p>I am confronted with the following definition:</p>
<blockquote>
<p>Let <span class="math-container">$K$</span> be a field and <span class="math-container">$e_1,e_2,\ldots,e_n$</span> the standard basis of the <span class="math-container">$K$</span> vector space <span class="math-container">$K^n$</span>.</p>
<p>For <... | Jack D'Aurizio | 44,121 | <p>I think it is best to use some elementary geometry. The volume of a cone is one third of the product between the base area and the height, by Cavalieri's principle. The distance of the vertex of the cone (i.e. the origin) from the plane $x+4z=a$ is $\frac{a}{\sqrt{17}}$ by <a href="http://mathworld.wolfram.com/Point... |
1,852,664 | <p>I'm working through Mumford's Red Book, and after introducing the definition of a sheaf, he says "Sheaves are almost standard nowadays, and we will not develop their properties in detail." So I guess I need another source to read about sheafs from. Does anybody know of any expository papers that cover them? I'd pref... | Babai | 36,789 | <p>I understand that you don't want to dig deep into a separate text book of Sheaf Theory. Still my suggestion would <strong>Sheaf Theory by B. R. Tennison</strong>. It is a 163 page book. But just for the introduction to sheaf you can just read the first two chapters of it, which is some 30 pages. The book is self con... |
3,070,127 | <p>Let <span class="math-container">$B := \{(x, y) \in \mathbb{R}^2
: x^2 + y^2 \le 1\}$</span>be the closed ball in <span class="math-container">$\mathbb{R^2}$</span> with center at the origin.
Let I denote the unit interval <span class="math-container">$[0, 1].$</span> Which of the following statements are true?</p>
... | jmerry | 619,637 | <p>It's correct, but we can do better - a closed form in which we're not summing an increasing number of terms.</p>
<p>Exactly half of the functions with nonzero sum have positive sum, by symmetry. There are <span class="math-container">$2^{2n}$</span> total functions. There are <span class="math-container">$\binom{2n... |
400,296 | <p>There are n persons.</p>
<p>Each person draws k interior-disjoint squares.</p>
<p>I want to give each person a single square out of his chosen k, so that the n squares I give are interior-disjoint.</p>
<p>What is the minimum k (as a function of n) for which I can do this?</p>
<p>NOTES:</p>
<ul>
<li>For n=1, obv... | Erel Segal-Halevi | 29,780 | <p>This answer: <a href="https://math.stackexchange.com/questions/412831/square-coloring">Square coloring</a> proves that, in the axis-aligned version, with $n=2$ people, $k=3$ squares are enough. This is a tight bound.</p>
<p>This answer: <a href="https://cs.stackexchange.com/questions/12275/team-construction-in-tri-... |
3,497,328 | <p>I'm stuck. Can I get a hint? I heard the answer is zero.</p>
<p>I'm guessing we use the SSA congruent triangle theorem. </p>
<p>If <span class="math-container">$m∠A = 50°$</span>, side <span class="math-container">$a = 6$</span> units, and side <span class="math-container">$b = 10$</span> units, what is the maximu... | Community | -1 | <p><a href="https://i.stack.imgur.com/xcF3rm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xcF3rm.png" alt="enter image description here"></a></p>
<p>The rumors were true: there are no points common to the ray and the circle.</p>
|
2,959,628 | <p>I'm struggling to find the density function of <span class="math-container">$Y=X^3$</span> where <span class="math-container">$X \sim \mathcal{N}(0,1)$</span> with density function <span class="math-container">$\phi(\cdot)$</span>. </p>
<p>Since <span class="math-container">$g(X) = X^3$</span> is monotonic I suppos... | DonAntonio | 31,254 | <p>Observe that</p>
<p><span class="math-container">$$\sqrt{x^3+4x}-\sqrt{x^3+x}=\frac{3}{\sqrt x\left(\sqrt{1+\frac4{x^2}}+\sqrt{1+\frac1{x^2}}\right)}\xrightarrow[x\to\infty]{}0$$</span></p>
|
2,811,908 | <p>I have this monster as the part of a longer calculation. My goal would be to somehow make it... nicer.</p>
<p>Intuitively, I would try to somehow utilize the derivate and the integral against each other, but I have no idea, exactly how.</p>
<p>I suspect, this might be a relative common problem, i.e. if we want to ... | Hashimoto | 425,635 | <p>If $f$ is continuous you can just use the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Leibniz integral rule</a> to assert: $\frac{d}{db}\int_0^1 e^{bx} f(x)dx = \int_0^1 x e^{bx} f(x)dx$</p>
<p>The Leibniz integral rule says that if $g$ and its partial derivatives are con... |
2,806,164 | <p>I recently came across a question that asked for the derivative of $e^x$ with respect to $y$. I answered $\frac{d}{dy}e^x$ but the answer was $e^x\frac{dx}{dy}$. How is that the answer? I am confused.</p>
| Aritra Chakraborty | 546,416 | <p>I feel the question needs to be clearer. It needs to be mentioned what exactly is $y$. Like one of the comment says, $y$ may not be a dependent variable. In that case, $\frac{\partial{e^{x}}}{\partial{y}}$ equates to zero. However, in case $x$ is a function of $y$ then, $\frac{de^{x}}{dy}$ can be written as $$\frac{... |
165,382 | <p>Is there any number $a+b\sqrt{5}$ with $a,b \in \mathbb{Z}$ with norm (defined by $|a^2−5b^2|$) equal 2?</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm\,\ 2\:|\:a^2\!-\!5b^2\! = (a\!-\!b)(a\!+\!b)\!-\!4b^2\Rightarrow\, 2\:|\:a\pm b\:\Rightarrow\:2\:|\:a\mp b\:\Rightarrow\:4\:|\:a^2\!-\!5b^2$</p>
|
4,220,751 | <p>In the example in the following slide, we follow the highlighted formula. With regard to the highlight, I'm confused why the number is greater <strong>or equal</strong> to <span class="math-container">$2^{n-1}$</span>, while only need to be less than <span class="math-container">$2^n$</span> (not less than <strong>o... | Asinomás | 33,907 | <p>You want to count the walks of length at most <span class="math-container">$20$</span> in the digraph.</p>
<p>Separate them by the different vertices that appear, since the graph without loops is just a directed path:</p>
<p>If there's only one vertex it's <span class="math-container">$20$</span>.</p>
<p>If there is... |
531,080 | <blockquote>
<p>Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane. Let $\Omega$ be a closed, bounded set in the $xy$-plane with $2$-volume $B$. Pick a point $Q$ in $P$ and make a pyramid by joining each point in $\Omega$ to $Q$ with a straight line segment. Find the $3$-volume of this pyramid.</p>
</blo... | Bill Kleinhans | 73,675 | <p>Euclid solves this by dividing a triangular prism into three pyramids of equal volume.</p>
|
4,480,276 | <p>[This question is rather a very easy one which I found to be a little bit tough for me to grasp. If there is any other question that has been asked earlier which addresses the same topic then kindly link this here as I am unable to find such questions by far.]</p>
<p>let <span class="math-container">$f(x)=x.$</span... | Tryst with Freedom | 688,539 | <p>You want an interval around a point in the domain such that difference of the value of function evaluated at that point from the function evaluated at other points in the interval is less than a number.</p>
<p>In this case, you want to find the size of <span class="math-container">$\delta$</span> such that for <span... |
1,648,354 | <p>We have been taught that linear functions, usually expressed in the form $y=mx+b$, when given a input of 0,1,2,3, etc..., you can get from one output to the next by adding some constant (in this case, 1).
$$
\begin{array}{c|l}
\text{Input} & \text{Output} \\
\hline
0 & 1\\
1 & 2\\
2 & 3
\end{array}
... | Sri-Amirthan Theivendran | 302,692 | <p>The map satisfies the recurrence relation $x_{n+1}={x_n}^2$ with base case $x_1=2$. By backtracking or induction one can show that $x_n=2^{2^{n-1}}$ for all $n\in\mathbb{N}$.</p>
|
2,447,677 | <p>Consider the parametric equations given by \begin{align*} x(t)&=\sin{t}-t,\\ y(t) & = 1-\cos{t}.\end{align*}</p>
<p>I want to write these parametric equations in Cartesian form. </p>
<p>In order to eliminate the sine and cosine terms I think I probably need to consider some combination of $x(t),y(t), x(t)... | velut luna | 139,981 | <p>$$t=\arccos(1-y)$$
$$x=\sin(\arccos(1-y))-\arccos(1-y)$$
$$=\sqrt{1-(1-y)^2}-\arccos(1-y)$$</p>
|
1,570,983 | <p>The volume of a sphere with radius $r$ is given by the formula $V(r) = \frac{4 \pi}{3} r^3$.</p>
<p>a) If $a$ is a given fixed value for $r$, write the formula for the linearization of the volume function $V(r)$ at $a$.</p>
<p>b) Use this linearization to calculate the thickness $\Delta r$ (in $cm$) of a layer of ... | Christian Blatter | 1,303 | <p>The volume $\Delta V$ of paint is approximatively given by
$$\Delta V=V(a+\Delta r)-V(a)\doteq V'(a)\>\Delta r=4\pi a^2\>\Delta r\ .\tag{1}$$
In your problem the unknown is the thickness $\Delta r$ of the paint layer. From $(1)$ we immediately get
$$\Delta r\doteq{\Delta V\over 4\pi\>a^2}\ .$$</p>
|
1,885,434 | <p>I'm currently reading through <em>Introductory Discrete Mathematics</em> by V.K. Balakrishnan and came across the following theorem:</p>
<p>If $X$ is a set of cardinality $n$, then the number of $r$-collections from $X$ is $\binom{r + n - 1}{n - 1}$, where $r$ is any positive integer.</p>
<p>To me, it seems like t... | Community | -1 | <p>I know your point is not to get another solution.</p>
<p>Anyway, I would prefer to rewrite the terms with $b-r,b,b+r$, giving
$$(b-r)^2-b(b+r),b^2-(b-r)(b+r),(b+r)^2-(b-r)b$$
or
$$-3br+r^2,r^2,3br+r^2.$$</p>
|
165,154 | <p>a) Let $\,f\,$ be an analytic function in the punctured disk $\,\{z\;\;;\;\;0<|z-a|<r\,\,,\,r\in\mathbb R^+\}\,$ . Prove that if the limit $\displaystyle{\lim_{z\to a}f'(z)}\,$ exists finitely, then $\,a\,$ is a removable singularity of $\,f\,$</p>
<p><strong>My solution and doubt:</strong> If we develop $\,f... | tatterdemalion | 34,528 | <p>The baseband bandwidth is defined to be the highest frequency of the signal.</p>
<p>In our case, the baseband would be $f_{base} = (2k+1)f_{square}$, the largest frequency that is an odd multiple of the fundamental square wave frequency smaller than $4 kHz$.</p>
|
1,377,927 | <p>Prove using mathematical induction that
$(x^{2n} - y^{2n})$ is divisible by $(x+y)$.</p>
<p><strong>Step 1:</strong> Proving that the equation is true for $n=1 $</p>
<p>$(x^{2\cdot 1} - y^{2\cdot 1})$ is divisible by $(x+y)$ </p>
<p><strong>Step 2:</strong> Taking $n=k$</p>
<p>$(x^{2k} - y^{2k})$ is divisible ... | Mythomorphic | 152,277 | <p>For $n=k$, assume $P(k)$ is true, we have</p>
<p>$$x^{2k}-y^{2k}=A(x-y)$$, where A is a polynomial.</p>
<p>For $n=k+1$, </p>
<p>\begin{align}
x^{2k+2}-y^{k+2}&=x^2[A(x-y)+y^{2k}]-y^{2k+2}\\&=A(x-y)x^2+x^2y^{2k}-y^{2k+2}\\&=A(x-y)x^2+y^{2k}(x^2-y^2)\\&=A(x-y)x^2+y^{2k}(x-y)(x+y)\\&=(x-y)[Ax^2+y... |
1,363,074 | <p>Q- If roots of quad. Equation $x^2-2ax+a^2+a-3=0$ are real and less than $3$ then,</p>
<p>a) $a<2$ </p>
<p>b)$2<a<3$ </p>
<p>c)$a>4$</p>
<p>In this ques., i used $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and then if $a$ will be $1,2$ only then the root will be defined but if we use $3$ then there will be only... | egreg | 62,967 | <p>The discriminant of the polynomial is
$$
4a^2-4(a^2+a-3)=12-4a
$$
so you know that $12-4a\ge0$ and so $a\le 3$, which excludes (c).</p>
<p>For $a=0$, the equation is
$$
x^2-3=0
$$
and the roots are less than $3$, which excludes (b).</p>
<p>Now, how can you completely verify the assert (a)? The largest root of the ... |
91,766 | <p>Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler char... | Liviu Nicolaescu | 20,302 | <p>For a complete proof of the Gauss-Bonnet-Chern for <em>arbitrary</em> vector bundles (not just tangent bundles) see Section 8.3.2 of <a href="http://www.nd.edu/~lnicolae/Lectures.pdf" rel="noreferrer">these notes</a>. The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern... |
533,628 | <p>I'm learning how to take indefinite integrals with U-substitutions on <a href="https://www.khanacademy.org/math/calculus/integral-calculus/u_substitution/v/u-substitution" rel="nofollow">khanacademy.org</a>, and in one of the videos he says that: $$\int e^{x^3+x^2}(3x^2+2x) \, dx = e^{x^3+x^2} + \text{constant}$$
I ... | Arash | 92,185 | <p>Hint: Use <a href="http://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="noreferrer">Stolz-Cesaro Lemma</a>.</p>
<hr>
<p>To see the direct proof, consider that if :
$$
\displaystyle\left|\frac{\displaystyle\sum_{i=1}^na_i}{n}-\frac{\displaystyle\sum_{i=1}^{n-1}a_i}{n-1}\right|\\
=\displaystyle\left|... |
3,766,146 | <p>Show that:</p>
<p><span class="math-container">$$\lim\limits_{N\rightarrow\infty}\sum\limits_{n=1}^N\frac{1}{N+n}=\int\limits_1^2 \frac{dx}{x}=\ln(2)$$</span></p>
<hr />
<p><strong>My attempt:</strong></p>
<p>We build a Riemann sum with:</p>
<p><span class="math-container">$1=x_0<x_1<...<x_{N-1}<x_N=2$</... | Oliver Díaz | 121,671 | <p>Your solution using a Riemann sum approximation to the integral <span class="math-container">$\int^2_1\frac{dx}{x}$</span> looks fine to me. Yves Daoust is much more direct.
A similar method was develop <a href="https://www.youtube.com/watch?v=TyA8kQrYzNE" rel="nofollow noreferrer">here</a> to estimate another nice ... |
4,396,765 | <p>What would be an efficient algorithm to determine if <span class="math-container">$n \in \mathbb{N}$</span> can be written as <span class="math-container">$n = a^b$</span> for some <span class="math-container">$a,b \in \mathbb{N}, b>1$</span>?</p>
<p>So far, I've tried:</p>
<pre><code>def ispower(n):
if n<... | Gareth Ma | 948,125 | <p>If you restrict <span class="math-container">$b\geq 2$</span>, then the most efficient way is <em>probably</em> simply testing each root <span class="math-container">$b = 2, 3, \ldots, \lfloor\log_2 n\rfloor$</span>. This is of runtime <span class="math-container">$O(\log n)$</span> and can be implemented using the ... |
2,889,651 | <p>The given points are $M(3,-1,2)$ and $M1(0,1,2)$ , plane A is passing though this 2 points. Plane B $2x-y+2z-1=0$ and its normal to A. How to we find the equation of plane A? I have tried with cross product of vector $MM1$ and vector $b=(2,-1,2)$, but it didn't work.</p>
| P Vanchinathan | 28,915 | <p>Take an odd permutation on $n$ symbols for large $n$, for example three parallel transpositions for $n=6$ such as (12)(34)(56). The corresponding permutation matrix will be orthogonal and have determinant $-1$.</p>
<p>Its eigen space corresponding to $-1$ eigenvalue is 3-dimensional and hence not a reflection ma... |
72,943 | <p>I want find the minima of a (multivariable) function under a constraint which has to be fulfilled on a whole interval, let's say
$$
\nabla f (\underline x) = 0 \ \\ \ c(\underline x,s)\geq0\ \forall s\in [0,1].
$$
How do I implement such a condition into the <code>Minimize[{f[x1,x2,...,xn],c[x1,...,xn,s]>=0 ?},{x... | Elena Fortina | 53,568 | <p>This belongs to the class of Semi-Infinite Programming problems for which ad-hoc algorithms must be used. The most intuitive one (discretization) involves building a finite grid on the interval and imposing the constraint on the grid points only, thus obtaining a classical constrained optimization problem with n con... |
2,409,377 | <p>I'm trying to prove that if $F \simeq h_C(X)$ or "$X$ represents the functor $F$", then $X$ is unique up to unique isomorphism. I already know that if $h_C(X) \simeq F \simeq h_C(Y)$ that $s: X \simeq Y$ since Yoneda says that $h_C(X)$ is fully faithful, so reflects isomorphisms (in either direction). If $h_C(X) ... | Community | -1 | <p>You're asking the wrong question — you're considering the wrong kind of isomorphism. Natural isomorphisms between functors are irrelevant here — the subject is about natural isomorphisms between <em>fuctors equipped with a natural transformation from $F$</em></p>
<p>Put differently, the relevant notions... |
9,540 | <p>I'm following <a href="http://reference.wolfram.com/mathematica/ref/FinancialData.html">http://reference.wolfram.com/mathematica/ref/FinancialData.html</a></p>
<p>I get the following:</p>
<pre><code>In[6]:= DateListLogPlot[FinancialData["^DJI", All]]
</code></pre>
<blockquote>
<p>During evaluation of In[6]:= Da... | user2047 | 2,047 | <p>The problem here is with the data provider Yahoo!. There has been intermittent problems with Yahoo! over the past few weeks with the DJI. The workaround is <code>WolframAlpha[]</code> as described in one of the answers above.</p>
<hr>
<p>(from Searke's comment)</p>
<p>Yahoo! is <a href="http://help.yahoo.com/kb/i... |
1,419,185 | <p>I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't take a picture of the problem so I attempted to trace it.</p>
<p><a href="https://i.stack.imgur.com/FzMSf.png" rel="... | lhf | 589 | <p>If this is a quartic then you're given the three roots of its derivative. You can integrate this cubic to recover the quartic and use the known point to find the leading coefficient and the constant of integration. Note that $x=0$ is a zero of both the quartic and the cubic.</p>
<p>Solution:</p>
<blockquote class=... |
698,702 | <p>Let $V$ be a vector space with finite dimension $n$ and $T:V\longrightarrow V$ is a linear transformation such that $T^{2}=0$. Then</p>
<ol>
<li><p>$rank(T)\leq\frac{n}{2}$</p></li>
<li><p>$n(T)\leq\frac{n}{2}$</p></li>
<li><p>$rank(T)\geq n(T)$</p></li>
<li><p>$rank(T)\geq \frac{n}{2}$</p></li>
</ol>
| J.R. | 44,389 | <p>I assume that you mean $n(T)=\dim \ker T$.</p>
<ol>
<li><strong>True</strong>. This follows from <a href="http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29#Properties" rel="nofollow">Sylvester's rank inequality</a>:
$$\operatorname{rank}(A)+\operatorname{rank}(B)-n\le \operatorname{rank}(AB)$$</li>
</ol>
<p>2... |
788,814 | <p>I need to generate binomial random numbers:</p>
<blockquote>
<p>For example, consider binomial random numbers. A binomial random
number is the number of heads in N tosses of a coin with probability p
of a heads on any single toss. If you generate N uniform random
numbers on the interval $(0,1)$ and count th... | Greg Martin | 16,078 | <p>Here's an algorithm that gives a good approximation precisely when $np$ is small. The probability that there are exactly $k$ heads is $\binom nk p^k(1-p)^{n-k}$. Note that
$$
\frac{\binom n{k+1} p^{k+1}(1-p)^{n-(k+1)}}{\binom nk p^k(1-p)^{n-k}} = \frac p{1-p} \frac{n-k}{k+1},
$$
which is small when $np$ is small; he... |
788,814 | <p>I need to generate binomial random numbers:</p>
<blockquote>
<p>For example, consider binomial random numbers. A binomial random
number is the number of heads in N tosses of a coin with probability p
of a heads on any single toss. If you generate N uniform random
numbers on the interval $(0,1)$ and count th... | Hagen von Eitzen | 39,174 | <p>The following method is theoretically exact, provided we have a "good" random generator <code>urand()</code>$\in[0,1)$. It uses the geometric distribution, i.e. the number of trials until a probability $p$ event occurs for the first time:</p>
<pre><code>int getGeometric(double p) {
return ceil( log( urand() ) /... |
306,337 | <p>I have a question.
Is this integral improper?
$$\int_0^\infty \frac{5x}{e^x-e^{-x}} \, dx = \int_0^a \frac{5x}{e^{x}-e^{-x}} \, dx+ \int_a^\infty \frac{5x}{e^x-e^{-x}} \, dx$$</p>
<p>Why is $\displaystyle\int_0^a \frac{5x}{e^{x}-e^{-x}}dx$ an improper integral at point $x=0$? And $\displaystyle\int_a^\infty \frac{5... | Michael Hardy | 11,667 | <p>An integral
$$
\int_a^b f(x) \, dx
$$
is improper at $a$ if it must be defined as
$$
\lim_{c\downarrow a} \int_c^b f(x)\,dx.
$$
In Lebesgue's theory of integration, that wuold be necessary only if for every $c>a$, the integral $\displaystyle\int_a^c f(x)\,dx$ cannot be defined because the integrals of the positiv... |
120,808 | <pre><code>Limit[Sum[k/(n^2 - k + 1), {k, 1, n}], n -> Infinity]
</code></pre>
<p>This should converge to <code>1/2</code>, but <code>Mathematica</code> simply returns <code>Indeterminate</code> without calculating (or so it would appear). Any specific reason why it can't handle this? Did I make a mistake somewhere... | Jens | 245 | <p>In cases where you can't get a symbolic result, it's also possible to use a completely numerical approach:</p>
<pre><code>Needs["NumericalCalculus`"]
sum[n_?NumberQ] := NSum[k/(n^2 - k + 1), {k, 1, n}]
NLimit[sum[n], n -> Infinity]
(* ==> 0.499999 *)
</code></pre>
|
2,229,244 | <p>combinatorial proof of $2^{n+1}\nmid (n+1)(n+2)\dots (2n)$.</p>
<p>I don't have any ideas about proving $sth \nmid sthx$ using combinatorics for showing $sth \mid sthx$ we show that there is a problem that gives $\frac{sthx}{sth*k}$ but what about proving sth doesn't divide sth using combinatorics?</p>
| Ethan Bolker | 72,858 | <p>You have the logic backwards. The argument shows directly that "evaluation at $i$" is a surjection, without any mention or discussion of cardinality. There's no "because" needed in that part of the proof.</p>
<p>Now <em>because</em>
you have found a surjection you know the cardinality of $\mathbb{R}[x]$ is at leas... |
3,162,294 | <p>I've been trying to prove this statement by opening up things on the left hand side using the chain rule but am really getting nowhere. Any tips/hints would be very helpful and appreciated!</p>
| Peter Foreman | 631,494 | <p>I assume throughout that the die used is <span class="math-container">$6$</span> sided.</p>
<p>Taking each value of <span class="math-container">$X$</span> separately there are <span class="math-container">$11$</span> possible values the 'larger sum' can take - <span class="math-container">$\{2,3,4,5,6,7,8,9,10,11,... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | Barry Cipra | 86,747 | <p>Let <span class="math-container">$u=1-|r|$</span> and note that <span class="math-container">$|r|\lt1$</span> implies <span class="math-container">$0\lt u\le1$</span>. This implies <span class="math-container">$0\le1-u^2\lt1$</span>, which in turn implies <span class="math-container">$0\le1-u\lt1/(1+u)$</span>, whic... |
2,940,649 | <p>solve <span class="math-container">$$\frac{|x|-1}{|x|-3} \ge 0$$</span></p>
<p>here <span class="math-container">$x$</span> is not equal to <span class="math-container">$3$</span> and <span class="math-container">$-3$</span>.</p>
| Phil H | 554,494 | <p>You don't have enough information to determine the intersection of fish with red and blue stripes. Knowing the total number of fish in the tank would help. Let's call that <span class="math-container">$N$</span>.</p>
<p>Then <span class="math-container">$.45N = 70 + 50 - 2RB$</span> where <span class="math-contain... |
1,912,628 | <p>Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!</p>
| user326210 | 326,210 | <p>What about if we define $H = \{ \frac{1}{n} : n\in \mathbb{N}\} \cup \{0\}$, a sort of standard countable compact set with 0 at its sole limit point, then define your countable compact set to be:</p>
<p>$$S = \{ x + y \mid x, y \in H\}.$$</p>
<p>To unpack the thought behind this definition:</p>
<ol>
<li>This set ... |
4,075,667 | <p>[this is question. I want to know about iv.] I want to know that without being defined everywhere ,can a mapping be onto and one-to-one?
In iv - D has four elements and B has three elements, while in question only three elements are used. So it cannot be function as per definition. Then how it will onto or one to o... | Ykh | 897,215 | <p>One- one and onto is merely a type of function . So yes the function needs to be defined everywhere in it's domain for it to be one - one or onto. The question you are referring to will therefore will be neither defined in domain nor one- one or onto</p>
|
1,715,232 | <p>Suppose $f_n$ converges uniformly to $f$ and $f_n$ are differentiable. Is it true that f will be differentiable?</p>
<p>My initial guess is no because $f_n= \frac{\sin(nx)}{\sqrt n}.$ Is this right? And more examples would be greatly appreciated.</p>
| Plutoro | 108,709 | <p>Your sequence converges uniformly to 0, which is differentiable. Consider $f_n(x)=\sqrt{x^2+1/n}$, which converges uniformly to $|x|$. Here is an outline of how you would prove that:</p>
<ol>
<li>Prove that the largest difference between $f_n(x)$ and $|x|$ occurs when $x=0$.</li>
<li>Prove that if $n$ is large enou... |
535,226 | <p>I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq N$ the sequence is monotonically increasing or decreasing). However, I'm having a lot of trouble finding a solution th... | Brian M. Scott | 12,042 | <p>HINT for the last part: Note that</p>
<p>$$\frac{a_{n+1}}{a_n}=\frac{\frac{(n+1)^{n+1}}{(n+1)!}}{\frac{n^n}{n!}}=\frac{(n+1)^{n+1}n!}{n^n(n+1)!}=\frac{(n+1)^{n+1}}{n^n(n+1)}=\left(\frac{n+1}n\right)^n\;.$$</p>
|
323,781 | <p>In the <a href="https://arxiv.org/abs/1902.07321" rel="noreferrer">paper</a> by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in <a href="https://www.pnas.org/content/early/2019/05/20/1902572116" rel="noreferrer">PNAS</a>) the abstract includes</p>
<blockquote>
<p>In the... | Will Sawin | 18,060 | <p>The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, because it's not clear what that would even mean unless the zeroes are all contained in some bounded interval. (In f... |
1,535,914 | <p>When I plot the following function, the graph behaves strangely:</p>
<p><span class="math-container">$$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$</span></p>
<p>While <span class="math-container">$\lim_{x\to +\infty} f(x) = e$</span> the graph starts to fade at <span class="math-container">$x \approx 6$</span>... | hmakholm left over Monica | 14,366 | <p>Your problem is the finite precision of floating-point arithmetic. There are only so many numbers near $1$ that can be represented by the computer's floating-point format, and the larger your $x$ is, the more of the <em>difference</em> between $1$ and $1+\frac{1}{x^{16}}$ (which is what really matters when raising t... |
3,217,130 | <p>How can I prove this by induction? I am stuck when there is a <span class="math-container">$\Sigma$</span> and two variables, how would I do it? I understand the first step but have problems when i get to the inductive step.</p>
<p><span class="math-container">$$\sum_{j=1}^n(4j-1)=n(2n+1)$$</span></p>
| Hongyi Huang | 619,069 | <p>Argue by contradiction: suppose there exists <span class="math-container">$a,b$</span> such that <span class="math-container">$ab\ne ba$</span>, then <span class="math-container">$b\ne a^{-1}ba$</span>. Let <span class="math-container">$c = a^{-1}ba$</span>, then <span class="math-container">$ba = ac$</span> and so ... |
40,463 | <p>If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it ... | Sergei Ivanov | 4,354 | <p>Q3: Laman's theorem is the same on the sphere.</p>
<p>Indeed, a configuration with $n$ vertices and $m$ edges is defined by a system of $m$ equations in $2n-3$ variables (there are $2n$ coordinates of points, but we may assume that the first point is fixed and the direction of one of the edges from the first point ... |
40,463 | <p>If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it ... | j.c. | 353 | <p>The first paragraph of <a href="https://arxiv.org/abs/0709.3354" rel="nofollow noreferrer">Some notes on the equivalence of first-order rigidity in various geometries</a> by Franco V. Saliola and Walter Whiteley states:</p>
<blockquote>
<p>In this paper, we explore the connections among the theories of first-order r... |
730,018 | <p>Assume ZFC (and AC in particular) as the background theory.</p>
<p>If $(M,\in^M)$ is a model of ZFC (not necessarily transitive or standard), must there exist a bijection between $M$ and $$\{x \in M \mid (M,\in^M) \models x \mbox{ is an ordinal number}\}?$$</p>
<p>I am also interested in the cases where $M$ is ass... | Andrés E. Caicedo | 462 | <p>The answer is yes. However, the answer is no if we require the model itself to know the bijection. More specifically, the existence of a class bijection between $V$ and its ordinals is equivalent to the axiom of global choice, and it is consistent that $\mathsf{ZFC}$ holds but global choice fails. </p>
<p>Now, give... |
603,291 | <p>Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. </p>
<p>I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function... | Brian M. Scott | 12,042 | <p>HINT: Your idea is a good one. What happens when you divide the inequality by $|x-y|$?</p>
|
3,690,076 | <p>Is there a rigorous proof that <span class="math-container">$|G|=|\text{Ker}(f)||\text{Im}(f)|$</span>, for some homomorphism <span class="math-container">$f\,:\,G\rightarrow G'$</span>? Can anyone provide such a proof with explanations?</p>
| user792277 | 792,277 | <p><span class="math-container">$G$</span> acts on <span class="math-container">$Im(f)$</span> by <span class="math-container">$\theta_g(f(h))=f(gh)$</span>. Then we can use the formula <span class="math-container">$|G|=|Stab(e)||Orbit(e)|$</span>.</p>
|
3,690,076 | <p>Is there a rigorous proof that <span class="math-container">$|G|=|\text{Ker}(f)||\text{Im}(f)|$</span>, for some homomorphism <span class="math-container">$f\,:\,G\rightarrow G'$</span>? Can anyone provide such a proof with explanations?</p>
| diracdeltafunk | 19,006 | <p>Here is a proof in full detail.</p>
<p>You said in the comments that you know the first isomorphism theorem, which will make the proof quite simple. Let <span class="math-container">$f : G \to G'$</span> be the group homomorphism. The first isomorphism theorem tells us that <span class="math-container">$G / \ker(f)... |
1,017,738 | <p>So I know that the derivative of arccos is:
$-dx/\sqrt{1-x^2}$</p>
<p>So how would I find the derivative of $\arccos(x^2)$? What does the $-dx$ mean in the above formula?</p>
<p>Would it just be $-2x/\sqrt{1-x^2}$ ?</p>
| mfl | 148,513 | <p>If you have a composite function $h(x)=(g\circ f)(x)$ then its derivative is given by $$h'(x)=g'(f(x))f'(x).$$ (This is known as the Chain rule. See <a href="http://en.wikipedia.org/wiki/Chain_rule" rel="nofollow">http://en.wikipedia.org/wiki/Chain_rule</a> for more information.)</p>
<p>In your case, $f(x)=x^2$ and... |
1,017,738 | <p>So I know that the derivative of arccos is:
$-dx/\sqrt{1-x^2}$</p>
<p>So how would I find the derivative of $\arccos(x^2)$? What does the $-dx$ mean in the above formula?</p>
<p>Would it just be $-2x/\sqrt{1-x^2}$ ?</p>
| Sujaan Kunalan | 77,862 | <p>$$\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}$$</p>
<p>Using the chain rule, we can evaluate $\frac{d}{dx}\arccos (x^2)$.</p>
<p>$$\frac{d}{dx}\arccos(x^2)=-\frac{1}{\sqrt{1-(x^2)^2}}\frac{d}{dx}(x^2)=-\frac{1}{\sqrt{1-x^4}}\frac{d}{dx}(x^2)=-\frac{1}{\sqrt{1-x^4}}\cdot 2x$$</p>
|
169,919 | <blockquote>
<p>If $p$ is a prime, show that the product of the $\phi(p-1)$ primitive roots of $p$ is congruent modulo $p$ to $(-1)^{\phi(p-1)}$.</p>
</blockquote>
<p>I know that if $a^k$ is a primitive root of $p$ if gcd$(k,p-1)=1$.And sum of all those $k's$ is $\frac{1}{2}p\phi(p-1)$,but then I don't know how use... | lab bhattacharjee | 33,337 | <p>We know $ ord_m(a^k) =\frac{d}{(d, k)} $ where d=$ord_ma$ => $ord_ma=ord_m(a^{-1})$ where m is a natural number.</p>
<p>So, $a,a^{-1}$ must belong to the same order(d).</p>
<p>Now by the previous solution, $a≢a^{-1}(mod\ m)$ if d>2.</p>
<p>So, the product of all number belonging to the same order(d)≡1(mod m) if... |
749,097 | <p>In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this <em>dream product rule</em> true? There are of course trivial examples, and also many instances where the equality is true at a handful of points. Less o... | pxc3110 | 139,697 | <p>Let $u$ and $v$ be functions of $t$.
<br/>then $(uv)'=u'v'$
<br/>$\iff u'v+uv'=u'v'$
<br/>$\iff u'(v-v')+v'u=0$
<br/>$\iff u'+\frac {v'}{v-v'}u=0$
<br/>Let u be the unkown function, then multiply both sides by $e^{\int \frac{v'}{v-v'}}dt$:
$u'(e^{\int \frac{v'}{v-v'}dt})+(e^{\int \frac{v'}{v-v'}dt}\frac {v'}{v-v'})u... |
2,319,341 | <blockquote>
<p>If a rubber ball is dropped from a height of $1\,\mathrm{m}$ and continues to rebound to a height that is nine tenth of its previous fall, find the total distance in meter that it travels on falls only.</p>
</blockquote>
<h3>My Attempt:</h3>
<p>I tried if it could be solved using arithmetic progress... | REVOLUTION | 454,435 | <p>During the first drop it covers a distance of $1$m and then rises to a distance of $1\bullet\frac{9}{10}$ and then falls to that distance again to rise by $1\bullet\frac{9}{10}\bullet\frac{9}{10}$ and so on. You will notice that this is forming a geometric progression than a arithmetic one.</p>
<p>The summation of ... |
694,090 | <p>Let $V$ and $W$ be vector spaces over $\Bbb{F}$ and $T:V \to W$ a linear map. If $U \subset V$ is a subspaec we can consider the map $T$ for elements of $U$ and call this the restriction of $T$ to $U$, $T|_{U}: U \to W$ which is a map from $U$ to $W$. Show that</p>
<p>$$\ker T|_{U} = \ker T\cap U.$$</p>
<p>I know ... | 5xum | 112,884 | <p>First, take a vector $v$ from $\ker T|_U$ and show that it lies both in $\ker T$ and in $U$. It is simple to do both, since $\ker T|_U$ is a subspace of $U$ and $0=T|_U(v)=T(v)$.</p>
<p>Then, show that if a vector $v$ lies both in $U$ and $\ker T$, it also lies in $\ker T|_U$. Again, a simple task, since $T_U(v)=T(... |
694,090 | <p>Let $V$ and $W$ be vector spaces over $\Bbb{F}$ and $T:V \to W$ a linear map. If $U \subset V$ is a subspaec we can consider the map $T$ for elements of $U$ and call this the restriction of $T$ to $U$, $T|_{U}: U \to W$ which is a map from $U$ to $W$. Show that</p>
<p>$$\ker T|_{U} = \ker T\cap U.$$</p>
<p>I know ... | Riccardo | 74,013 | <p>Try to prove the double inclusion: Let $T'$ the restriction, let $x \in ker T'$, obviously $x \in U$, by definition of Kernel of a linear operator defined over $U$. But by the fact that $U \subset V$ then if $u \in U \Rightarrow u \in V$ and $0=T'(u) = T(u)$</p>
|
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | David White | 11,540 | <p>Such a paper might be appropriate for the <a href="http://www.gradmath.org/" rel="nofollow noreferrer">Graduate Journal of Mathematics</a>, since a readership of grad students might enjoy a bunch of interesting conjectures. I published a <a href="https://www.gradmath.org/article/an-overview-of-schema-theory/" rel="n... |
19,848 | <p><strong>{Xn}</strong> is a sequence of independent random variables each with the same
<strong>Sample Space {0,1}</strong> and <strong>Probability {1-1/$n^2$ ,1/$n^2$}</strong>
<br> <em>Does this sequence converge with probability one (Almost Sure) to the <strong>constant 0</strong>?</em>
<br><br>Essentially th... | mercio | 17,445 | <p>Yes, but how fast does the frequency drop ? The faster it does the more it is probable that (Xn) converges to 0.</p>
<p>Lemma : If (Xn) is a sequence of elements in {0,1}, Xn converges to 0 if and only if Xn has finitely many 1s :</p>
<p>If a sequence (Xn) converges to 0, then by definition of the limit, there exi... |
423,479 | <p>I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of </p>
<p>$$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$$</p>
<p>particularily compared to $$b_n = \frac{1}{4n}$$</p>
<p>I haven't studied asympto... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Use <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling approximation</a></p>
<p>$$ n!=\Gamma(n+1) \sim \left(\frac{n}{e}\right)^n\sqrt{2 \pi n}. $$</p>
<p><strong>Added:</strong> If you use the above approximation, you will get</p>
<p>$$ a_n\sim 1-{\f... |
423,479 | <p>I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of </p>
<p>$$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$$</p>
<p>particularily compared to $$b_n = \frac{1}{4n}$$</p>
<p>I haven't studied asympto... | Julián Aguirre | 4,791 | <p>The following code in Mathematica</p>
<pre><code>Series[1 - (Gamma[x + 1/2]/(Sqrt[x] Gamma[x]))^2, {x, Infinity, 6}]
</code></pre>
<p>gives
$$
\frac{1}{4 x}-\frac{1}{32 x^2}-\frac{1}{128 x^3}+\frac{5}{2048 x^4}+\frac{23}{8192 x^5}-\frac{53}{65536 x^6}+O\left[\frac{1}{x}\right]^7
$$</p>
|
2,780,832 | <p>Let's define: </p>
<p>$$\sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i}$$
$$\cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}$$</p>
<blockquote>
<p>We are to prove that
$$\sin(z+w)=\sin(w) \cos(z) + \sin(z)\cos(w), \forall_{z,w \in \mathbb{C}}$$
using only the following statement: $\exp(z+w) = \exp(w)\exp(z)$.</p>
</bloc... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Since OP starts from left-hand-side and asks "where can I go from here", I'll give a solution staring from the left-hand-side.</p>
<p>\begin{align}
& \sin(z+w) \\
&= \frac{\exp(i(z+w)) - \exp(-i(z+w))}{2i} \\
&= \frac{\exp(iz)\exp(iw) - \exp(-iz)\exp(-iw)}{2i} \\
&= \frac{\exp(iz)\exp(iw) \color{blu... |
357,138 | <blockquote>
<p>If $a_1,a_2,\dotsc,a_n $ are positive real numbers, then prove that</p>
</blockquote>
<p>$$\lim_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+.....+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n.$$ </p>
<p>My Attempt: </p>
<p>Let $P=\lim_{x \to \infty} \left[\dfrac {a_1^{\frac{1}{x}}+a_2^{\frac {1... | Community | -1 | <p>Another method to resolve this problem is, take the inequality $GM \leqslant AM \leqslant x \text {th root mean}$, then use squeeze theorem.</p>
|
117,836 | <p>I'm not the best at math(but eager to learn) so please excuse me if I'm not explaining this problem correctly, I will try to add as much info to make it clear. I basically receive 2 pieces of data, one is a list of integers and the other is a target_sum, and I want to figure out all the ways I can use the list to ... | Stephen | 146,439 | <p>This is not an answer, since (as indicated in the comments) there is not going to be a fast way to write down the set you are looking at just because it can be so huge! There are reasonable algorithms for computing exactly how huge; here is one simple idea (it is not the most efficient way known to humans).</p>
<p>... |
106,887 | <p>Statements like</p>
<pre><code>A) A is false.
</code></pre>
<p>or</p>
<pre><code>B1) B2 is true.
B2) B1 is false.
</code></pre>
<p>cannot be assigned a truth-value due to their paradoxical use of self-reference. Are <em>all</em> statements lacking a truth-value self-referential, or are there non-self-referential... | JDH | 413 | <p>One of the main discoveries of set-theoretic research over the past fifty years is the widespread independence phenomenon, the phenomenon by which numerous fundamental statements of set theory are independent of the principal axioms of set theory. Many instances of this ubiquitous phenomenon are described in <a href... |
106,887 | <p>Statements like</p>
<pre><code>A) A is false.
</code></pre>
<p>or</p>
<pre><code>B1) B2 is true.
B2) B1 is false.
</code></pre>
<p>cannot be assigned a truth-value due to their paradoxical use of self-reference. Are <em>all</em> statements lacking a truth-value self-referential, or are there non-self-referential... | Community | -1 | <p>JDH has given a deep, interesting answer -- but it's deep and interesting in part because it relates to ZFC, which is a deep and interesting theory. Formal mathematical theories don't have to be deep and interesting, and it is possible to answer this question using very simple and straightforward examples.</p>
<p>H... |
373,313 | <p>Given a manifold <span class="math-container">$M$</span>, we can always embed it in some Euclidian space (general position theorem). Hence we can define the minimal embedding space of <span class="math-container">$M$</span> to be the smallest euclidean space that we can embed <span class="math-container">$M$</span> ... | Ryan Budney | 1,465 | <p>Similar to Sander's example, the Poincare Dodecahedral space does not smoothly embed in <span class="math-container">$\mathbb R^4$</span>, but it does embed topologically.</p>
|
147,361 | <p>i'm new in MATHEMATICA. I want to create an operator $D^{(f)}=\partial_x+f'-\partial^2_x$ and $D^{(g)}=\partial_x+g'-\partial^2_x$ and put it into a matrix element, then multiplied by a vector whose components are functions on $x$, say $u(x), v(x)$. For example $$\begin{pmatrix}D^{(f)} & D^{(g)}\\ D^{(g)} & ... | Carl Woll | 45,431 | <p>You can make use of my <a href="https://mathematica.stackexchange.com/a/162590/45431"><code>DifferentialOperator</code></a> paclet to do this. Install with:</p>
<pre><code>PacletInstall["https://github.com/carlwoll/DifferentialOperator/releases/download/0.1/DifferentialOperator-0.0.2.paclet"]
</code></pre>
<p>and ... |
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | Mnifldz | 210,719 | <p>When you take the derivative of $\frac{y}{x}$ with respect to $y$ you are computing $\frac{\partial }{\partial y} \frac{y}{x} = \frac{1}{x}$ because here you are holding $x$ constant. If you take the derivative of the same expression with respect to $x$ then you compute $\frac{\partial}{\partial x} \frac{y}{x} = - ... |
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | leftaroundabout | 11,107 | <p>There is <em>nothing</em> special about the symbol $x$. Unfortunately, there's a certain bias to calling $f$ <strong>the</strong> function and $x$ <strong>the</strong> variable, but really it should never<sup>1</sup> matter how variables are labelled as long as it's done consistently.</p>
<p>So in particular, if yo... |
84,982 | <p>I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by other professors, and I would like to use them for guided reading. <strong>I am wondering whether it is customary to as... | Community | -1 | <p>The world has changed rapidly with the advent of the web and desktop publishing, so it's going to be difficult to say what is customary. You will probably find that there are generational differences as well as differences between fields and institutions. For example, what is customary at MIT is for professors to pu... |
2,390,215 | <p>There exist a bijection from $\mathbb N$ to $\mathbb Q$, so $\mathbb Q$ is countable. And by well ordering principle $\mathbb N$ has a least member say $n_1$ which is mapped to something in $\mathbb Q$, In this way $\mathbb Q$ can be well arranged? Is my argument good? </p>
| John Griffin | 466,397 | <p>Yes! If you have a bijection $f:W\to S$, where $(W,\leq)$ is well ordered, then one can show that $(S,\preceq)$ is a well ordering where $\preceq$ is defined as follows.
Given $s_1,s_2\in S$, there must be unique $w_1,w_2\in W$ such that $s_1=f(w_1)$ and $s_2=f(w_2)$. We will say that
$$
s_1 \prec s_2\ \text{if}\ w_... |
188,139 | <p>Let $f$ be entire and non-constant. Assuming $f$ satisfies the functional equation $f(1-z)=1-f(z)$, can one show that the image of $f$ is $\mathbb{C}$?</p>
<p>The values $f$ takes on the unit disc seems to determine $f$...</p>
<p>Any ideas?</p>
| Arkady | 23,522 | <p>Since the function is entire, it misses at most one point by Picard's little theorem. If the point is say, $y$, then $1-y$ must be in the range, unless $y=1-y$. But, in this case, $y=\frac{1}{2}$ and we can deduce that $f(\frac{1}{2})=\frac{1}{2}$. So, $y\neq 1-y$. So, if there is a $z$ such that $f(z)=1-y$, then, $... |
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