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 |
|---|---|---|---|---|
3,157,338 | <p>If we have a linear recurrence relation on a sequence <span class="math-container">$\{x_n\}$</span>, then I know how to find the worst case asymptotic growth. We consider the largest absolute value <span class="math-container">$\alpha$</span> of any root of the characteristic polynomial. Then, independent of the ini... | eyeballfrog | 395,748 | <p>Hint: express the operator as an infinite sum of derivatives, then work in Fourier space.</p>
|
3,028,986 | <p>How is this integral
<span class="math-container">$$\dfrac{1}{4} \int_{0}^{4\pi} \left| \cos \theta \right| \; d\theta$$</span>
equal to
<span class="math-container">$$\dfrac{1}{2} \int_{0}^{2\pi} \left| \cos \theta \right| \; d\theta$$</span> </p>
<p>On attempting to solve this integral, found this on a solution... | TurlocTheRed | 397,318 | <p>Remember the integral represents a limit of a sum of areas. <span class="math-container">$d\theta$</span> represents a change in <span class="math-container">$\theta$</span>, the width of a rectangle. The value of the function represents the height. If the change in <span class="math-container">$\theta$</span> is co... |
3,028,986 | <p>How is this integral
<span class="math-container">$$\dfrac{1}{4} \int_{0}^{4\pi} \left| \cos \theta \right| \; d\theta$$</span>
equal to
<span class="math-container">$$\dfrac{1}{2} \int_{0}^{2\pi} \left| \cos \theta \right| \; d\theta$$</span> </p>
<p>On attempting to solve this integral, found this on a solution... | Paramanand Singh | 72,031 | <p>Use the formula <span class="math-container">$$\int_{0}^{2a}f(x)\,dx=2\int_{0}^{a}f(x)\,dx$$</span> provided <span class="math-container">$f$</span> satisfies <span class="math-container">$f(2a-x)=f(x)$</span>. The formula above is proved by splitting the integral as sum of integrals over <span class="math-container... |
2,138,448 | <p>Survival game: Consider $3$ players, $A, B$ and $ C$, taking turns shooting at each other. Any player can shoot at only one opponent at a time (and each of them has to make a shot whenever it is his/her turn). </p>
<p>Each shot of $A$ is successful with probability $1/3$, each shot of $B$ is successful with probabi... | PMar | 415,956 | <p>This problem has appeared in 'the literature' before. If one assumes that each player is allowed to miss deliberately at any time, then A can do better by deliberately missing his first shot. Analysis of this action is just like analysis of A initially shooting at C, except the case where A kills C is avoided; th... |
221,017 | <p>If I have the following list:</p>
<pre><code>https://pastebin.com/nqyf4yY5
</code></pre>
<p>How can I find the closest value to "89" in the "T[C]" column and its corresponding value in the "DH,aged-DH,unaged (J/g)" column?.</p>
<p>Thank you in advanced,</p>
| Bob Hanlon | 9,362 | <p>Given your <code>data</code></p>
<pre><code>data[[5]] // InputForm
(* {"Time(s)", "T[C]", "K(T)=k^(1/n)",
"dx/dT", "x(t)",
"DH,aged-DH,unaged (J/g)",
"Check dx"} *)
values = data[[6 ;;]];
</code></pre>
<p>You are asking for data that corresponds to headers for columns {2, 6}.</p>
<p>The entry for the val... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | Jeff | 313,346 | <p>First, you should not believe in anything in mathematics, in particular weak solutions of PDEs. They are sometimes a useful tool, as others have pointed out, but they are often not unique. For example, one needs an additional entropy condition to obtain uniqueness of weak solutions for scalar conservation laws, like... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | user7530 | 7,530 | <p>To the excellent longer answers above I will add a short one: weak solutions in a conveniently-chosen (and in particular, <em>finite-dimensional</em>) function space can often be explicitly computed, whereas strong solutions often cannot (even if one can prove a solution must theoretically exist). Computability has ... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | ktoi | 149,608 | <p>The existing answers provide good reasons towards the question in the title, but from the perspective of a geometer I feel the applications in physics aren't quite as convincing. It's true that singular phenomena that arises in for example conservation laws requires a suitable notion of a generalised solution, but w... |
3,931,807 | <p>I need to find max and min of <span class="math-container">$f(x,y)=x^3 + y^3 -3x -3y$</span> with the following restriction: <span class="math-container">$x + 2y = 3$</span>.</p>
<p>I used the multiplier's Lagrange theorem and found <span class="math-container">$(1,1)$</span> is the minima of <span class="math-conta... | Math Lover | 801,574 | <p>This is from your working -</p>
<p><span class="math-container">$(3x^2 -3, 3y^2 -3) = \lambda (1,2)$</span></p>
<p><span class="math-container">$3x^2 - 3 = \lambda, 3y^2-3 = 2\lambda$</span></p>
<p>Equating <span class="math-container">$\lambda$</span> from both equations,</p>
<p><span class="math-container">$6x^2-6... |
255,827 | <p>I've had trouble coming up with one.</p>
<p>I know that if I could find </p>
<p>an irreducible poly $p(x)$ over $\mathbb{Q}$
which has roots $\alpha, \beta, \gamma\in Q(\alpha)$,</p>
<p>then $|\mathbb{Q}(\alpha) : \mathbb{Q}| $ = 3 and would be a normal extension,
as $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha,\beta,\g... | Gregor Botero | 31,955 | <p>Try to find a polynomial with discriminant $D$ that satisfies $\sqrt{D}\in\mathbb{Q}$.</p>
<p>Why does this help?</p>
<p>First, the only possibilities for the Galois group $G$ are $S_3$ and $A_3$, as Ben Millwood remarked.</p>
<p>Second, every element of $G$ must fix $\sqrt{D}\in\mathbb{Q}$. But you can check th... |
21,491 | <p>The question is prompted by change of basis problems -- the book keeps multiplying the bases by matrix $S$ from the left in order to keep subscripts nice and obviously matching, but in examples bases are multiplied by $S$ (the change of basis matrix) from whatever side. So is matrix multiplication commutative if at ... | Eric Naslund | 6,075 | <p>Definitely not. Yuan's comment is also not correct, diagonal matrices do not necessarily commute with non-diagonal matrices. Consider $$\left[\begin{array}{cc}
1 & 1\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
a & 0\\
0 & b\end{array}\right]=\left[\begin{array}{cc}
a & b\\
0 & b\... |
1,232,532 | <p>First, I'm not looking for an answer here, I'm just looking to understand the problem so that I can prove it. I'm trying to analyzing the worst case running time of an algorithm, and it must has summation notation. What keeping me back is that I don't understand how to express <code>doSomething(n-j)</code> in summat... | Tom | 230,703 | <p>when you squared both sides, you generated an extraneous solution (two solutions instead of one)..
...I don't know where you received this problem, but is it possible this is a "trick" type problem? I mean, you could simply flip the sign on both sides...so
-(sqrt(2x-1)) = x
hope this helped a little...-</p>
|
3,394,050 | <p>I'm having trouble with this problem.</p>
<blockquote>
<p>Using logical equivalencies prove that <span class="math-container">$(p \land q)\implies (p \lor q)$</span> is a tautology.</p>
</blockquote>
| user0102 | 322,814 | <p>According to the equivalence <span class="math-container">$(a\rightarrow b) \Longleftrightarrow \neg a\vee b$</span>, the De Morgan's laws, the associativity and commutativity of the logical operator <span class="math-container">$\vee$</span>, one has</p>
<p><span class="math-container">\begin{align*}
(p\wedge q) \... |
3,007,443 | <p>I've heard the words "internal" and "external" generalization of concepts in category theory.</p>
<p>Specifically, i heard the idea that the concept of 'power set' has an internal and an external generalization in category theory.</p>
<p>What is the difference between these two?</p>
| Giorgio Mossa | 11,888 | <p>First of all one need to understand the concept of internalization.</p>
<p>Generally many classical constructions which can be given inside some specific category (usually <span class="math-container">$\mathbf{Set}$</span>) can be expressed in the language of category theory in terms of objects, arrows and more gen... |
2,170,382 | <p>I'm working on a question that asks to:</p>
<p>Find the area in the first quadrant bounded by the curves;
$\ xy = 1, xy=5, y=e^2x, y=e^5x $. </p>
<p>I would very much appreciate help solving this question (including the method of how to find the transformation expressions for $\ u$ and $v$ to use in the Jacobian... | Julián Aguirre | 4,791 | <p>A change of variable $u=x\,y$, $v=y/x$ will transform the domain of integration into a rectangle.</p>
|
2,600,776 | <blockquote>
<p>A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then</p>
<p>(<em>A</em>) $E(X) > 0$</p>
<p>(<em>B</em>) $E(X) < 0$</p>
<p>(<em>C</em>) $P(X\leq 0) > 0.5$</p>
<p>(<em>D</em>) $P(X\ge0) < 0.25$</p>
<p>\begin{eqnarray}... | StubbornAtom | 321,264 | <p>The required expectation is nothing but $2\mathbb E(X\Phi(X))$ where $X\sim\mathcal N(0,1)$.</p>
<p>Integrating by parts ( taking $\Phi(x)$ as 1st function and $x\phi(x)$ as 2nd function) and using the fact that $\phi'(x)=-x\phi(x)$, it can be shown that </p>
<p>$$\mathbb E(X\Phi(X))=\int_{\mathbb R}x\Phi(x)\phi(x... |
1,831,243 | <p>Kronecker "delta" function is generally defined as
$\delta(i,j)=1$ if $i$ is equal to $ j$, otherwise $0$.</p>
<p>How about if $j$ is not an integer? I mean let $j$ is a half open interval defined as $j=(0,1]$ and $ i$ has any value on interval $[0,1]$,
then can we use Kronecker delta to find if $i$ belongs to $j$... | parsiad | 64,601 | <p>What you are looking for is the indicator function:
$$
\mathbf{1}_{A}(x)=\begin{cases}
1 & \text{if }x\in A\\
0 & \text{otherwise}
\end{cases}
$$
In your case, $A=(0,1]$ (I have used $x$ instead of $i$ and $A$ instead of $j$ since it is more customary to use lower case letters at the end of the alphabet for ... |
478,517 | <blockquote>
<p>Construct a topological mapping of the open disk $|z|<1$ onto the whole plane.</p>
</blockquote>
<p>I represent $z=re^{i\theta}$. I thought about the bijection from $(0,1)$ to $(0,\infty)$, which is given by $x\rightarrow \dfrac1x-1$. Applying this to the norm, we will get the mapping $re^{i\theta... | njguliyev | 90,209 | <p>Hint: Try another bijection between $(0,1)$ and $(0,+\infty)$.</p>
<blockquote class="spoiler">
<p> $\tan \frac{\pi x}{2}$.</p>
</blockquote>
|
125,592 | <p>I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals:</p>
<p>$$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$</p>
<p>where $u(t)$ is Heavside function</p>
<p>well I applied the formula that says that the convolution of this two signal is... | example | 27,652 | <p>It is also possible with partial integration, though getting the closed formula from the other solution is not as easy to see.</p>
<p>$$ C(n):=\int_0^{2\pi}\!\!\!\cos^n(x)\,dx =\int_0^{2\pi}\!\!\!\cos^{n-1}(x)\cos(x)\,dx $$
partial integration gives</p>
<p>$$ = (n-1)\int_0^{2\pi}\!\!\!\cos^{n-2}(x)\sin^2(x)\,dx$$
... |
1,608,645 | <p>Is there supposed to be a fast way to compute recurrences like these?</p>
<p>$T(1) = 1$</p>
<p>$T(n) = 2T(n - 1) + n$</p>
<p>The solution is $T(n) = 2^{n+1} - n - 2$. </p>
<p>I can solve it with:</p>
<ol>
<li><p>Generating functions.</p></li>
<li><p>Subtracting successive terms until it becomes a pure linear re... | Robert Israel | 8,508 | <p>Just as in linear algebra, the general solution of a linear non-homogeneous equation is a particular solution + the general solution of the homogeneous equation.</p>
<p>The homogeneous equation $T(n) = 2 T(n-1)$ has the obvious solutions $c 2^n$.</p>
<p>For a particular solution of the non-homogeneous equation $T(... |
1,608,645 | <p>Is there supposed to be a fast way to compute recurrences like these?</p>
<p>$T(1) = 1$</p>
<p>$T(n) = 2T(n - 1) + n$</p>
<p>The solution is $T(n) = 2^{n+1} - n - 2$. </p>
<p>I can solve it with:</p>
<ol>
<li><p>Generating functions.</p></li>
<li><p>Subtracting successive terms until it becomes a pure linear re... | Claude Leibovici | 82,404 | <p>Let me make the problem slightly more complex with, for example, $$T_n=a \, T_{n-1}+b+c n+d n^2$$ ($a,b,c,d$ being given); set $$T_n=U_n+\alpha +\beta n+\gamma n^2$$ Now, replace in the original expression
$$U_n+\alpha+\beta n +\gamma n^2=a\left(U_{n-1}+\alpha +\beta (n-1)+\gamma (n-1)^2\right)+b+c n+d n^2$$ t... |
3,762,174 | <p>I have some confusion in integration . My confusion marked in red and green circle as given below<a href="https://i.stack.imgur.com/4fc1I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4fc1I.png" alt="enter image description here" /></a></p>
<p>Im not getting why <span class="math-container">$$\... | Alex | 38,873 | <p>In the integral in your problem, the bounds are <span class="math-container">$0<y<x$</span>, so, as @zkutch wrote, if you plot the graph <span class="math-container">$y=f(x)=x$</span>, the area will be the lower triangle if you split the unit square <span class="math-container">$([0,1]\times[0,1])$</span> with... |
14,612 | <p>For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ? </p>
| Lee Mosher | 7,258 | <p>Spaces that are homeomorpic to the Cantor set arise naturally in many mathematical settings, particularly in dynamical systems.</p>
<p>For one dynamical example, the Cantor set is homeomorphic to the phase space of any infinite <a href="https://en.wikipedia.org/wiki/Bernoulli_process" rel="nofollow noreferrer">Bern... |
2,867,042 | <blockquote>
<p>Find the value of
$$\tan\theta \tan(\theta+60^\circ)+\tan\theta \tan(\theta-60^\circ)+\tan(\theta + 60^\circ) \tan(\theta-60^\circ) + 3$$
(The answer is $0$.)</p>
</blockquote>
<p>My try: Let $\theta$ be $A$, $60^\circ -\theta$ be $B$, and $60^\circ + \theta$ be $C$. I simplified the result and ... | Batominovski | 72,152 | <p>I agree with Jamie Radcliffe. Let $[n]:=\{1,2,\ldots,n\}$ and $\mathcal{P}_n$ denote the power set $\mathcal{P}\big([n]\big)$. Suppose that $\mathcal{P}_n$ has a decomposition into pairwise disjoint symmetric chains
$$\mathcal{P}_n=\bigcup_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor}\,\bigcup_{r=1}^{t_k}\,\mathca... |
2,396,073 | <p>Let $\omega_1$ be the first uncountable ordinal. In some book, the set $\Omega_0:=[1,\omega_1)=[1,\omega_1]\backslash\{\omega_1\}$ is called the set of countable ordinals. Why? It is obvious that it is an uncountable set, because $[1,\omega_1]$ is uncountable. The most possible reason I think is that for any $x\pr... | Ross Millikan | 1,827 | <p>It is just like $\omega$ being the set of all finite ordinals. Every member of $\omega$ is finite but $\omega$ itself is infinite. Similarly, $\Omega_0$ is uncountable but all its members are (finite or) countable.</p>
|
2,706,165 | <p>So if $y=\log(3-x) = \log(-x+3)$ then you reflect $\log(x)$ in the $y$ axis to get $\log(-x)$.</p>
<p>Then because it is $+3$ inside brackets you then shift to the left by $3$ giving an asymptote of $x=-3$ and the graph crossing the $x$ axis at $(-4,0)$. </p>
<p>However this does not work. The answer shows the $+3... | pjs36 | 120,540 | <p>There is a small set of algebraic operations that correspond to geometric transformations:</p>
<p>When we have the graph of a function $y = f(x)$...</p>
<p><strong>Shifting:</strong></p>
<ul>
<li><p>The substitution $x \mapsto x - h$ shifts a graph $h$ units to the right (that'd be left, if $h$ is negative)</p></... |
35,151 | <p>Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever solving NP-Complete problems in Quantum Polynomial time?</p>
<p><a href="http://en.wikipedia.org/wiki/QMA" rel="nofollow... | Greg Kuperberg | 1,450 | <p>David is right about one thing. Scott had a discussion about this on his blog and I was also involved.</p>
<p>On the one hand, many complexity theorists simply also assume that BQP does not contain NP, just as they assume that P does not contain NP. The evidence for the former is not as dramatic as that for the l... |
347,171 | <p>Let <span class="math-container">$\text{ppTop}$</span> denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor <span class="math-container">$FG: \text{ppTop}\to \text{Gp}$</span> where GP is the category of groups.</p>
... | NWMT | 38,698 | <p>There is a more topological way. If you assume that <span class="math-container">$X$</span> admits a universal covering <span class="math-container">$\tilde X$</span> (so <span class="math-container">$X$</span> is path connected and semilocally simply connected, I believe) then the <span class="math-container">$G=\p... |
2,448,696 | <p>Show that $\frac{1}{n}<\ln n$, for all $n>1$ where n is a positive integer</p>
<p>I've tried using induction by multiplying both sides by $\ln k+1$ and $\frac{1}{k+1}$ but but all it does is makes it more complicated, I've tried using the fact that $k>1$ and $k+1>2$ during the inductive $k+1$ step, but ... | Peter Szilas | 408,605 | <p>Consider:</p>
<p>$e \lt n^n$ for $n\gt 1.$</p>
<p>Take $\log$ of both sides:</p>
<p>$1 \lt n\log(n)$ $ $ $\rightarrow:$</p>
<p>$1/n \lt \log(n)$ for $n \gt 1$.</p>
|
623,428 | <blockquote>
<p>Suppose $$
Y = X^TAX,
$$ where $Y$ and $A$ are both known $n\times n$, real, symmetric matrices. The unknown matrix $X$ is restricted to $n\times n$.</p>
</blockquote>
<p>I think there should be at least one real valued solution for $X$. How do I solve for $X$? </p>
| rschwieb | 29,335 | <p>A solution is not possible for all $Y$ and $A$.</p>
<p>For example, suppose $\operatorname{rank}(Y)>\operatorname{rank}(A)$. Then $\operatorname{rank}X^\top AX\leq\operatorname{rank}(A)<\operatorname{rank}(Y)$, so we can't hope for equality.</p>
<p>Another restriction is that $\det(Y)=\det(X)^2\det(A)$. So f... |
1,561,370 | <p>Is there any graphical interface in <a href="http://gap-system.org" rel="noreferrer">GAP</a>? Something like <a href="https://www.rstudio.com/" rel="noreferrer">RStudio</a> for <a href="https://www.r-project.org/" rel="noreferrer">R</a> or <a href="http://andrejv.github.io/wxmaxima/" rel="noreferrer">WxMaxima</a> fo... | Russ Woodroofe | 562,386 | <p>I want to follow up on Alexander's answer. <a href="https://cocoagap.sourceforge.io/" rel="nofollow noreferrer">Gap.app</a>, which is one of the <a href="http://www.gap-system.org/Packages/undep.html" rel="nofollow noreferrer">Undeposited Implementations for GAP</a> that Alexander mentions briefly, is back in activ... |
3,365,361 | <p>Suppose that <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is analytic at <span class="math-container">$x=0$</span>, and <span class="math-container">$T(x)$</span> its Taylor series at <span class="math-container">$x=0$</span>, with radius of convergence <span class="math-container">$R>0$</span>... | Bernard | 202,857 | <p>Another reason is that the trace is the sum of the eigenvalues, and two similar matrices have the same eigenvalues.</p>
|
118,540 | <p>Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.</p>
<p>Assuming that $X$ is rational over $k$, is the quotient $X/G$ always rational?</p>
<p>If $k=\mathbb{C}$, we can use Castelnuovo's theorem and see that $X/G$ is unirational... | Jason Starr | 13,265 | <p>This is not always true, and cubic threefolds give a counterexample over the field $k=\mathbb{C}(t)$. Let $\mathcal{Y}$ be a smooth cubic hypersurface in $\mathbb{P}^4_{\mathbb{C}}$. Let $L\subset \mathcal{Y}$ be a line. Denote by $\mathcal{X}$ the (locally closed) subvariety of $\mathcal{Y}\times L$ parameterizi... |
800,363 | <p>What is </p>
<blockquote>
<p>$$\lim_{x\to 0}\left(\frac{x}{e^{-x}+x-1}\right)^x$$</p>
</blockquote>
<p>Using the expansion of <a href="http://en.wikipedia.org/wiki/Exponential_function" rel="nofollow">$e^x$</a>, I get that the function</p>
<blockquote>
<p>$$y=\left(\frac{x}{e^{-x}+x-1}\right)^x$$</p>
</blockq... | Did | 6,179 | <p>WA interprets the number
$$
u(x)=\left(\frac{x}{\mathrm e^{-x}+x-1}\right)^x
$$
when $x\gt0$ as
$$
u(x)=\exp\left(x\log\left(\frac{x}{\mathrm e^{-x}+x-1}\right)\right),
$$
and when $x\lt0$ as
$$
u(x)=\exp\left(x\log\left(\frac{-x}{\mathrm e^{-x}+x-1}\right)+\mathrm i\pi x\right).
$$
Then both limits are indeed $1$ (... |
119,636 | <p>I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.</p>
<p>For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
according to <a href="http://www.wolframalpha.com/input/?i=partial+sum+of+n+2%5En" rel="noreferrer">http://www.wolfr... | Community | -1 | <p>Hint: </p>
<ul>
<li><p>The Geometric series...</p></li>
<li><p>Differentiation.</p></li>
</ul>
|
119,636 | <p>I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.</p>
<p>For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
according to <a href="http://www.wolframalpha.com/input/?i=partial+sum+of+n+2%5En" rel="noreferrer">http://www.wolfr... | Marc van Leeuwen | 18,880 | <p>Observe that your formula $\sum_{n=0}^{m}nr^n$ can be obtained from $\sum_{n=0}^{m}x^n$ by applying $x\frac d{dx}$ (deriving and then multiplying by $x$) and then substituting $r$ for $x$. Now for geometric series one has the well known formula
$$
\sum_{n=0}^mx^n=\frac{x^0-x^{m+1}}{1-x}
$$
and applying $x\frac d... |
1,158,642 | <p>Let $E$ be a vector bundle of rank $r$ and let $\phi:E\rightarrow \mathbb C_p$ non vanishing map to the skyscraper sheaf.
consider the kernel $F$ of this sheaf which is a sub-bundle of $E$, every fiber of $F$ has a rank $r$, just that over $p$ which has rank $r-1$.
So why we say that $F$ has a rank $r$??
thanks <... | Georges Elencwajg | 3,217 | <p>Let $X$ be a smooth curve, $E$ a vector bundle on $X$ of rank $r$ and $\mathcal E$ the locally free sheaf of sections of $E$.<br>
Suppose you have an exact sequence of sheaves $ 0\to \mathcal F\to \mathcal E \to \mathbb C_p \to 0$ with $\mathbb C_p$ the skyscraper sheaf concentrated at $p$ with stalk $\mathbb C$... |
732,996 | <p><img src="https://i.stack.imgur.com/kXJEt.png" alt="enter image description here"></p>
<p>Hi! I am working on some ratio and root test online homework problems for my calc2 class and I am not sure how to completely solve this problem. I guessed on the second part that it converges, but Im not sure how to solve of t... | Ellya | 135,305 | <p>$\rho=\lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|=\lim_{n\rightarrow\infty}|\frac{\frac{1}{(2n+2)!}}{\frac{1}{(2n)!}}|=\lim_{n\rightarrow\infty}|\frac{(2n)!}{(2n+2)!}|=\lim_{n\rightarrow\infty}\frac{1}{(2n+1)(2n+2)}=\lim_{n\rightarrow\infty}\frac{1}{4n^2+5n+2}=0$</p>
<p>I think you may have overlooked the fact t... |
605,772 | <p>Solving $x^2-a=0$ with Newton's method, you can derive the sequence $x_{n+1}=(x_n + a/x_n)/2$ by taking the first order approximation of the polynomial equation, and then use that as the update. I can successfully prove that the error of this method converges quadratically. However, I can't seem to prove this for th... | Ross Millikan | 1,827 | <p>After the second line, you can go to $ |\frac 14(x_n-\frac a{x_n})^2|$</p>
|
3,498,199 | <p>Suppose if a matrix is given as</p>
<p><span class="math-container">$$ \begin{bmatrix}
4 & 6\\
2 & 9
\end{bmatrix}$$</span></p>
<p>We have to find its eigenvalues and eigenvectors.</p>
<p>Can we first apply elementary row operation . Then find eigenvalues.</p>
<p>Is their any relation on the matrix if ... | Luca Citi | 197,925 | <p>As others have noted you can't apply arbitrary elementary row operations to a matrix and expect the eigenvalues/vectors be preserved. The closest you can do is to apply them to both rows and columns in a specific way as follows.</p>
<p>Consider the matrix
<span class="math-container">$$
T =
\begin{bmatrix}
1 &... |
221,351 | <p>I asked the following question (<a href="https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con">https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con</a>) on math.stackexchange.com and ... | Geoff Robinson | 14,450 | <p>David Speyer has answered the question, but let me add some background. The general result is that if $R$ is a principal ideal domain with field of fractions $K$, and $G$ is a finite group, then every finite dimensional representation of $G$ over $K$ may be realised over $R$ ( ie, is equivalent to a representation o... |
465,255 | <p>Does there exists any form of Algebra where operators can be assumed as variables?</p>
<p>For example:
$$
1+2\times3=7
$$
can be considered as:
$$
1\:(\mathrm{\,X})\:2\:(\mathrm{Y})\:3=7
$$
?</p>
| vishva8kumara | 614,215 | <p>I don't think you have to do any calculation to get the answer. This can be just estimated.</p>
<ul>
<li><p>4 Questions with 25 points each is max 100 points.</p>
</li>
<li><p>400 students answering 4 questions - 200 ~ 350 gets each answer right.</p>
</li>
</ul>
<p>Average must be between 50 and 100.</p>
<p>Only <st... |
2,697,069 | <p>Two series of functions are given in which I cannot figure out how to find $M_n$ of the second problem. $$1.\space \sum_{n=1}^{\infty} \frac{1}{1+x^n}, x\in[k,\infty)\\ 2. \space \sum_{n=1}^{\infty} (\cos x)^n, x\in(0,\pi)$$.. </p>
<p>I have determined the $M_n$ for problem no. $1.$ [$\space|\sum_{n=1}^{\infty} \f... | Rohan Shinde | 463,895 | <p>$$2x=5y$$
$$\Rightarrow y=\frac {2x}{5}$$
$$\frac y3=\frac z4\Rightarrow \frac {x}{15}=\frac z8$$
$$\Rightarrow \frac xz=\frac {15}{8}$$</p>
|
2,697,069 | <p>Two series of functions are given in which I cannot figure out how to find $M_n$ of the second problem. $$1.\space \sum_{n=1}^{\infty} \frac{1}{1+x^n}, x\in[k,\infty)\\ 2. \space \sum_{n=1}^{\infty} (\cos x)^n, x\in(0,\pi)$$.. </p>
<p>I have determined the $M_n$ for problem no. $1.$ [$\space|\sum_{n=1}^{\infty} \f... | TheSimpliFire | 471,884 | <p>We have $$2x=5y\implies \color{red}{x=\frac52}\color{blue}y$$ and $$\frac{y}{3} = \frac{z}{4}\implies \color{blue}{y=\frac34z}$$ so $$\color{red}{x=\frac52}\cdot\color{blue}{\frac34z}\implies \boxed{\frac xz=\frac{15}8}$$</p>
|
10,427 | <p>I like Mathematica, but it's syntax baffles me.</p>
<p>I am trying to figure out how to minimize the whitespace around a graphic.</p>
<p>For example,</p>
<pre><code>ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]},
Boxed -> True, Axes -> False]
</code></pre>
<p><img src="https://i.s... | Yves Klett | 131 | <p><code>ImageCrop</code> seems to be a bit buggy (at least right here in Version 8.04, Win 64). It tends to crop lightly coloured areas rather agressively. You could try the following work-around, which works more reliably:</p>
<pre><code>imcrop[img_] := ImagePad[img, -BorderDimensions[img, 0]]
g = Graphics3D[{Specu... |
2,571,909 | <p>$$\left|\frac{-10}{x-3}\right|>\:5$$</p>
<ul>
<li>Find the values that $x$ can take. </li>
</ul>
<p>I know that</p>
<p>$$\left|\frac{-10}{x-3}\right|>\:5$$
and
$$\left|\frac{-10}{x-3}\right|<\:-5$$</p>
| nonuser | 463,553 | <p>$$\left|\frac{-10}{x-3}\right|>\:5$$
so $$10> 5|x-3| \Longrightarrow -2<x-3<2 \Longrightarrow 1<x<5; x\ne 3$$</p>
|
590,205 | <p>I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times [0,1]/((0,1)\times\{1\}).$$ But how do I show that it can not be embedded in an Euclidean space? Cause for me it looks like it is possible... | Stefan Hamcke | 41,672 | <p>The cone $C(J)$ where $J=\mathrm{int}(I)$ is not first countable. Consider the subspace $$B:=S\times I:=\left\{\frac1n\middle|n\in\Bbb N\right\}\times I$$ of $J\times I$. It is closed, and each closed and saturated subset $A$ of $B$ is either disjoint from $J\times\{1\}$, in this case it is saturated in $J\times I$,... |
1,200,358 | <blockquote>
<p>Assume the $n$-th partial sum of a series $\sum_{n=1}^\infty a_n$ is the following:
$$S_n=\frac{8n-6}{4n+6}.$$
Find $a_n$ for $n > 1$.</p>
</blockquote>
<p>I'm really stuck on what to do here.</p>
| ASB | 111,607 | <p>Observe that for each $n\in \mathbb{N}$, </p>
<p>$a_{n+1}=S_{n+1}-S_n= \dfrac{8(n+1)-6}{4(n+1)+6}-\dfrac{8n-6}{4n+6}$ and $a_1=S_1=\dfrac{8.1-6}{4.1+6}=\dfrac{1}{5}$.</p>
|
2,672,097 | <p>What are the must-know concepts and best resources for preparing the <strong>mathematical background for advanced machine learning studies</strong>?</p>
<p>Currently, looking into the book <strong>What is Mathematics? by Richard Courant</strong> to strengthen my fundamentals. Are there any better references that ca... | user3658307 | 346,641 | <p>It definitely depends on what you want to do, since ML is a relatively large and diverse field now. A quick summary might be something like this:</p>
<p><strong>Basics</strong> (i.e. needed for the more advanced ones below)</p>
<ul>
<li>Linear algebra (e.g. matrix operations and decompositions, vector spaces)</li>... |
1,088,734 | <p>It's possible the integral bellow. What way I must to use for solve it.</p>
<p>$$\int \sin(x)x^2dx$$</p>
| kmbrgandhi | 132,855 | <p>Here's a hint: every cubic can be factored in the following way:
$$p(x) = (x-r_1)(x-r_2)(x-r_3)$$
You know, from the given, that $r_1$, $r_2$, $r_3$ are <em>distinct</em> positive integers, and you know that $r_1r_2r_3 = 26$. It turns out that there is only one possible unordered triple $(r_1, r_2, r_3)$ that sati... |
4,000,576 | <blockquote>
<p>What is the value of the following integral:
<span class="math-container">$$\int_0^{2\pi}\frac{1}{4\cos^2(t)+9\sin^2(t)}\mathrm{d}t$$</span>
<span class="math-container">$\frac\pi9$</span> ; <span class="math-container">$\frac\pi6$</span> ; <span class="math-container">$\frac\pi3$</span> ; <span class="... | Quanto | 686,284 | <p>Integrate with Fourier series as follows</p>
<p><span class="math-container">$$\int_0^{2\pi}\frac{1}{4\cos^2 t+9\sin^2 t}{d}t
=\int_0^{2\pi}\frac{2}{13-5\cos 2t} {d}t\\
=\int_0^{2\pi}\left( \frac16 + \frac13\sum_{n=1}^\infty \frac{1}{5^n} \cos (2n t) \right)dt
=\frac\pi3
$$</span></p>
|
582,283 | <p>$H$ is subgroup of $G$ with $H$ not equal $G$.</p>
<p>Be $S=G-H$. I am being asked to prove that $\langle S \rangle=G$.</p>
<p>Some tip to solve this? I think in $S_3$ is possible but I can´t prove.</p>
| Marc van Leeuwen | 18,880 | <p>Hint. The complement $S$ contains at least one element. You can fix any one $s\in S$, and produce any $h\in H$ by a <em>single</em> multiplication involving $s$.</p>
|
1,304,344 | <p>How do I find the following:</p>
<p>$$(0.5)!(-0.5)!$$</p>
<p>Can someone help me step by step here?</p>
| Tim Raczkowski | 192,581 | <p>Use the gamma function:</p>
<p>$$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\,dt.$$</p>
|
4,468,112 | <p>Let <span class="math-container">$a,b\in\mathbb R$</span> with <span class="math-container">$a<b$</span>, <span class="math-container">$$\mathcal D_{[a,\:b]}:=\{(t_0,\ldots,t_k):k\in\mathbb N\text{ and }a=t_0<\cdots<t_k\}$$</span> and <span class="math-container">$$\mathcal T_\varsigma:=\{(\tau_1,\ldots,\ta... | Z Ahmed | 671,540 | <p><span class="math-container">$\int_{a}^{b} f(x) dg= \int f(x)\frac{dg}{dx} dx =\int_{a}^t f(x) g'(x) dx+\int_{t}^{b} f(x) g'(x) dx,$</span></p>
<p>where <span class="math-container">$g'(x)$</span> has a finite jump discontinuity at <span class="math-container">$x=t$</span>.</p>
<p><strong>Edit:</strong></p>
<p>For ... |
962,691 | <p>I'm trying to integrate $ \int_0^1\frac {u^2 + 1}{u - 2}du$</p>
<p>I've calculated that this equates to $ [\frac{u^2}{2}+2u +5ln(u-2)]_0^1 $</p>
<p>But then I have to evaluate $ln(-1)$ and $ln(-2)$ which are obviously not defined in the real plane. I have drawn the graph and I know for certain that this integral e... | Dr. Sonnhard Graubner | 175,066 | <p>rewrite your integrand in the form $\frac{u^2-4}{u-2}+\frac{5}{u-2}$</p>
|
1,649,907 | <p>Please kindly forgive me if my question is too naive, i'm just a <em>prospective</em> undergraduate who is simply and deeply fascinated by the world of numbers.</p>
<p>My question is: Suppose we want to prove that $f(x) > \frac{1}{a}$, and we <em>know</em> that $g(x) > a$, where $f,g$ and $a$ are all positive... | Ross Millikan | 1,827 | <p>No, you could have $g$ huge and $f$ tiny. Let $a=10, g=1000, f=1/50$</p>
|
23,953 | <p>I cited the diagonal proof of the uncountability of the reals as an example of a <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23708#23708">`common false belief'</a> in mathematics, not because there is anything wrong with the proof but because it is commonly belie... | Jason Dyer | 441 | <p>From <a href="http://books.google.com/books?id=vrQLbbxGNMsC" rel="nofollow">Labyrinth of thought: a history of set theory and its role in modern mathematics</a> by José Ferreirós and José Ferreirós Domínguez:</p>
<blockquote>
<p>page 184 (quoting a margin note of
Cantor's) </p>
<p>Besides, the theorem of p... |
23,953 | <p>I cited the diagonal proof of the uncountability of the reals as an example of a <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23708#23708">`common false belief'</a> in mathematics, not because there is anything wrong with the proof but because it is commonly belie... | John Stillwell | 1,587 | <p>Cantor's diagonal argument first appears in his 1891 paper
"Über eine elementare Frage der Mannigfaltigkeitslehre", <em>Jahresbericht der Deutschen
Mathematiker-Vereinigung</em> 1: 75–78, in which he generalizes the argument to prove that
any set has more subsets than elements. The 1891 paper has the diagonal argum... |
1,445,913 | <p>Given 2 lines r and s. </p>
<ul>
<li>r and s don't have an intersection point</li>
<li>none of them touch the origin (0,0,0)
What approach should I use to find the equation of the line that cross the origin and also cross r and s?</li>
</ul>
<p>if necessary, we can consider r and s as:</p>
<pre><code> x = at +... | Yes | 155,328 | <p>If $n \geq 3$, then $n-1$ and $n+1$ are $> 1$, so $n^{2}-1 = (n-1)(n+1) > 1$. Every prime is by definition $\neq 1$ and can and only can be divided by $1$ and itself. But $n-1$ and $n+1$ divide $n^{2}-1$. Thus $n^{2}-1$ is composite by definition.</p>
|
362,881 | <p>I am going to try to explain this as easily as possible. I am working on a computer program that takes input from a joystick and controls a servo direction and speed. I have the direction working just fine now I am working on speed. To control the speed of rotation on the servo I need to send it so many pulses per s... | Michael Hardy | 11,667 | <p>$$
\int_{-\infty}^\infty f(x)\delta(x)\,dx = f(0).
$$
$$
\int_{-\infty}^\infty f(x)\Big(e^x \delta(x)\Big)\,dx = \text{what?}
$$
But look at that last integral this way:
$$
\int_{-\infty}^\infty \Big(f(x)e^x\Big) \delta(x)\,dx.
$$
This is equal to the value of the function $x\mapsto f(x)e^x$ at $x=0$, because the de... |
248,267 | <p>It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is:</p>
<p>$$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^\nu}{\partial x^\gamma \over \partial \ti... | Yuri Vyatkin | 2,002 | <p>This is very straightforward, just substitute the transformation rules and collect the terms.</p>
<p>Here are some details.</p>
<p>The inverse metric transforms, as we know, by the rule:
$$
g^{\mu \lambda} = \frac{\partial{\bar{x}}^\mu}{\partial{x}^\alpha} \frac{\partial{\bar{x}}^\lambda}{\partial{x}^\delta} g^{\a... |
3,807,900 | <p>I have just met this exercise in functional analysis, asking us to determine if these two subspaces of the Hilbert space <span class="math-container">$\ell^2$</span> of square-summable complex sequences are closed:</p>
<blockquote>
<ol>
<li>The set of all sequences <span class="math-container">$\{x_n\}_{n=1}^{\infty... | Matematleta | 138,929 | <p>Hints:</p>
<p>For <span class="math-container">$1).\ $</span> define the linear functional <span class="math-container">$x\mapsto \sum_{n=1}^{\infty} \frac{1}{n} x_n$</span>, and show it is bounded, hence continuous.</p>
<p>For <span class="math-container">$2).\ $</span> Consider the sequence of sequences</p>
<p><sp... |
3,154,212 | <p>I'm working a lot with series these days, and I would like to know if there are any texts, papers, articles that might suggest a general outline for finding <span class="math-container">$n$</span>th partial sums of convergent series. Most of my searching turns up methods for finding the sums of geometric/telescopin... | gandalf61 | 424,513 | <p>If <span class="math-container">$u=v$</span> then we can take <span class="math-container">$w=u=v$</span>.</p>
<p>So let's assume that <span class="math-container">$u \ne v$</span>. Then <span class="math-container">$u$</span> and <span class="math-container">$v$</span> span a <span class="math-container">$2$</span... |
10,880 | <p>I am posting to formally register my disapproval of <a href="https://math.stackexchange.com/users/93658/anti-gay">this user's</a> name.</p>
<p>I believe it constitutes hate speech. If you look at the comments on this user's answers, you will see that many others do too. The name is already causing a lot of trouble, ... | Douglas S. Stones | 139 | <p>Reminds me of this SMBC comic:</p>
<p><a href="http://www.smbc-comics.com/?id=1904" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ACzEL.gif" alt="enter image description here"></a></p>
|
69,050 | <p>The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group,
I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question.
My Question is :</p>
<ol>
<li>Why is the name Qu... | Jyrki Lahtonen | 11,619 | <p>Often I think of a quotient group in terms of (loss of) information. When we move from a group to its quotient group we lose some information about the identity of the elements. For example, when we map an element of the additive group of integers <span class="math-container">$\mathbf{Z}$</span> to the quotient grou... |
876,310 | <p>So I <em>think</em> I understand what differentials are, but let me know if I'm wrong.</p>
<p>So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in terms of the differentials $\text{dy}$ and $\text{dx}$, we take the derivative $f'(x)$ as our "primitive"... | Community | -1 | <p>$\mathrm{d}y$ depends only on $y$: it doesn't depend on any choice of $x$ or anything else: that's one of the big advantages to differentials (as opposed to, say, partial derivatives).</p>
<p>A differential is a gadget that expresses <em>how</em> something varies. There are three main things you can do with such a ... |
105,750 | <p>Given a <code>ContourPlot</code> with a set of contours, say, this:</p>
<p><a href="https://i.stack.imgur.com/cKoyo.jpg"><img src="https://i.stack.imgur.com/cKoyo.jpg" alt="enter image description here"></a></p>
<p>is it possible to get the contours separating domains with the different colors in the form of lists... | Michael E2 | 4,999 | <p>To answer the last question, the contour domains (since V8) are enclosed separately in <a href="http://reference.wolfram.com/language/ref/GraphicsGroup.html"><code>GraphicsGroup</code></a>, each which you can cull and turn into a region:</p>
<pre><code>plot = ContourPlot[x*Exp[-x^2 - y^2], {x, 0, 3}, {y, -3, 3},
... |
909,734 | <p>I have answered this question to the best of my knowledge but somehow I feel as if I am missing something? Can I further prove this statement or add anything to it? </p>
<p>Question: </p>
<p>Let $m \in \mathbb N$. Prove that the congruence modulo $m$ relation on $\mathbb Z$ is transitive. </p>
<p>My attempt:</p>
... | G Tony Jacobs | 92,129 | <p>I've often seen the congruence relation modulo $m$ defined as "$a\equiv b \pmod{m}$ means m|(a-b)". If that's the definition that you're working with, then the fact you need to use in this proof is: m|p and m|q imply m|(p+q).</p>
<p>Then, using $a-b$ for $p$, and $b-c$ for $q$, you have your result.</p>
|
619,477 | <blockquote>
<p>Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob.
"That's funny, i was in it too, and i don't remember ever seeing you there."
"Well," admitted Alice sheepishly, "I guess i did skip class a lot." "Yeah, ... | Community | -1 | <p>I'm not sure if this is what you're looking for, but why not use the pigeonhole principle?</p>
<p>Let $L_n$ represent the $n$th lecture. During $L_1$, either Bob or Alice attended, or neither attended. During $L_2$, either Bob or Alice attended, or neither attended. This is true for every lecture up to $L_n$. </p>
... |
197,730 | <blockquote>
<p>Prove that the states of the 8-puzzle are divided into two disjoint sets such that any
state in one of the sets is reachable from any other state in that set, but not from any state in the other set. To do so, you can use the following fact: think of the board as a one-dimensional array, arranged i... | user21820 | 21,820 | <p>Don't know why this wasn't answered, but the basic idea is just to prove that you can move any 3 pieces into the top-left 2 times 2 square and cycle them before moving them back, hence proving that you can perform any 3-cycle. Then prove that the set of 3-cycles generates the alternating group. To do so prove that w... |
3,014,766 | <p>I am supposed to find the derivative of <span class="math-container">$ 2^{\frac{x}{\ln x}} $</span>. My answer is <span class="math-container">$$ 2^{\frac{x}{\ln x}} \cdot \ln 2 \cdot \frac{\ln x-x\cdot \frac{1}{x}}{\ln^{2}x}\cdot \frac{1}{x} .$$</span> Is it correct? Thanks. </p>
| David G. Stork | 210,401 | <p><em>Mathematica</em> gives:</p>
<p><span class="math-container">$$\frac{\log (2) 2^{\frac{x}{\log (x)}} (\log (x)-1)}{\log ^2(x)}$$</span></p>
|
268,360 | <p>Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?</p>
| Community | -1 | <p>Let $x^a=y$, $z^b=x$ and $z^c=y$. Then $z^{ab}=(z^b)^a=x^a=y=z^c$ so that $ab=c$.</p>
|
268,360 | <p>Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?</p>
| Michael Hardy | 11,667 | <p>I will presume that what was meant was $\displaystyle\log_x y = \frac{\log_z y}{\log_z x}$.</p>
<p>Notice that this is true if and only if $(\log_x y)(\log_z x) = \log_z y$, and that holds if and only if $\displaystyle z^{(\log_x y)(\log_z x)}=y$.</p>
<p>So
$$
z^{\Big((\log_x y)(\log_z x)\Big)} = \Big(z^{\log_z x}... |
268,360 | <p>Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?</p>
| Alan | 54,910 | <p>Demonstrate: $ \log_xy=\frac{\log_ay}{\log_ax}; x,y,a \in \mathbb{R} $</p>
<p>We initially have:$$ f(x,y)= \log_xy$$
We transform it to the exponential form: $$ x^{f(x,y)}=y$$
We apply logarithm of base $a$ for $a \in \mathbb{R}$ on both sides of the equation:$$\log_a{x^{f(x,y)}}=\log_ay$$
Applying the exponential... |
3,802,806 | <p>I and a friend are trying to find all endomorphisms <span class="math-container">$f$</span> of <span class="math-container">$\mathcal{M}_n(\mathbb{R})$</span> such that <span class="math-container">$f({}^t M)={}^t f(M)$</span> for all <span class="math-container">$M$</span>. We believe they are of the form <span cla... | user1551 | 1,551 | <p>Let <span class="math-container">$f$</span> be a linear endomorphism on <span class="math-container">$M_n(\mathbb R)$</span>. Then
<span class="math-container">$$
f(M^T)=f(M)^T\quad\forall M\tag{1}
$$</span>
if and only if
<span class="math-container">$$
f(M)=\frac12\left(g(M)+g(M^T)^T\right)\quad\forall M\tag{2}
$$... |
130,465 | <p>i just started university so im pretty new to all this new math. My problem is to solve this <code>recursive sequence</code>: $a_{n+1} = a_{n}^3$ with: $a_{0} = \frac{1}{2}$ and: $n \in N$</p>
<p>I've to analyse convergence and if its convergent i've to get the limit of this sequence.</p>
<p>I dont know how to sta... | MathematicalPhysicist | 13,374 | <p>When the limit exists we have $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} a_{n+1}=a$, so to find the limit you need to solve the equation: $a^3=a$.</p>
<p>But that's not enougth, we can see from the definition of $a_n$ that it's non-negative (why?), and its decreasing to zero, so we can infer fro... |
130,465 | <p>i just started university so im pretty new to all this new math. My problem is to solve this <code>recursive sequence</code>: $a_{n+1} = a_{n}^3$ with: $a_{0} = \frac{1}{2}$ and: $n \in N$</p>
<p>I've to analyse convergence and if its convergent i've to get the limit of this sequence.</p>
<p>I dont know how to sta... | Georgy | 139,717 | <p>Another idea is to define a new sequence $b_n=\log a_n$. Then your recursive equation
$$a_{n+1}=a_n^3$$
becomes
$$b_{n+1}=3b_n$$
which is pretty straight forward.</p>
|
2,820,796 | <p>In How many ways can a 25 Identical books can be placed in 5 identical boxes. </p>
<p>I know the process by counting but that is too lengthy .
I want different approach by which I can easily calculate required number in Exam hall in few minutes. </p>
<p>Process of Counting :
This problem can be taken partitions of... | Andrew Woods | 153,896 | <p>The answer of Foobaz John defined $p_k$ and $p_{\le k}$.</p>
<p>Notice first of all that $p_{\le k}(n)=p_k(n+k)$. (That's because we can add one object to each part to ensure that there are no parts of size zero.) Thus, while we must be careful to distinguish them, the tables for these two functions are very simila... |
34,874 | <p>If you visit this <a href="http://www.springerlink.com/content/ug8h1563j3484211/" rel="nofollow">link</a>, you'll see at the top of the PDF view. Basic properties of finite abelian groups:</p>
<p>Every quotient group of a finite abelian group is isomorphic to a subgroup.</p>
<p>If the above statement true, it wou... | Pete L. Clark | 1,149 | <p>The result you are interested in is Theorem 19 on page 8 of</p>
<p><a href="http://alpha.math.uga.edu/%7Epete/4400algebra2point5.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/4400algebra2point5.pdf</a></p>
<p>As I explain there, this fact is a kind of duality statement, but it lies deeper than the f... |
12,114 | <p>I retired after 25 years of teaching and moved to Israel a year ago. My Hebrew is okay, but before moving here, I had no experience talking about math in Hebrew. I have been learning Hebrew math vocabulary by reading math textbooks and taking an online math course in Hebrew. </p>
<p>I recently started volunteering... | Morten Engelsmann | 3,502 | <p>If your students are willing to take the time, I would say you can add a lot of value to their understanding and skills by aproaching the challenge from a "socratic" point of view.</p>
<ul>
<li><p>Facilitate conceptualization through "concept cards":
I have my students make a mindmap with the math concept in the c... |
244,241 | <p>How can I find minimum distance between cone and a point ?</p>
<p><strong>Cone properties :</strong><br/>
position - $(0,0,z)$<br/>
radius - $R$<br/>
height - $h$</p>
<p><strong>Point properties:</strong><br/>
position - $(0,0,z_1)$</p>
| Tom Oldfield | 45,760 | <p>One basic example is with eigenvalues and eigenvectors of matrices. Often real matrices are not diagonalisable over $\mathbb{R}$ because they have imaginary eigenvalues, wnad knowing things about these eigenvalues can tell us a lot about the transformation that the matrix represents. The obvious example is the $2D$ ... |
4,539,167 | <p><span class="math-container">$$
g(x) =\min_y f(x, y) =\min_y x^TAx + 2x^TBy + y^TCy
$$</span>
where <span class="math-container">$x\in \mathbb R^{n\times 1}$</span>, <span class="math-container">$y\in \mathbb R^{m\times 1}$</span>, <span class="math-container">$A\in \mathbb R^{n\times n}$</span>, <span class="math-c... | greg | 357,854 | <p><span class="math-container">$
\def\a{\lambda}
\def\B{BC^{-1}B^T}
\def\o{{\tt1}}\def\p{\partial}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\qiq{\quad\implies\quad}
\def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
\def\CLR#1{\c{\LR{#1}}}
\def\fracLR#1#2{\LR{\... |
1,986,249 | <blockquote>
<p>Let q be a positive integer such that $q \geq 2$ and such that for any
integers a and b, if $q|ab$, then $q|a$ or $q|b$. Show that $\sqrt{q}$
is irrational.</p>
</blockquote>
<p>Proof;</p>
<p>Let assume $\sqrt{q}$ is a rational number, where $n \neq 0$ and $\gcd (m,n)=1$, meaning $\sqrt{q} = \fr... | Ojas | 382,895 | <p><strong>Proof using <a href="http://mathworld.wolfram.com/BezoutsIdentity.html" rel="nofollow">Bézout's Identity</a></strong></p>
<p>For $\sqrt{q}$ to be irrational, $q$ must not be a perfect square. Thus, we only concern us with non-perfect square $q$.</p>
<p>Assume that $\sqrt{q}$ is rational. Therefore $\sqrt{q... |
2,802,959 | <p>If I write
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?</p>
<p>Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$? </p>
<p>Q2:</p>
<p>If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?</p>
| Eff | 112,061 | <blockquote>
<p>If I write $x\in[0,1]$ does it mean that $x$ can be ANY number between $0$ and $1$?</p>
</blockquote>
<p><strong>Yes.</strong></p>
<p>If $x\in [0,1]$ then $x$ can be any number between $0$ and $1$ (inclusive). Another way to write this is $0 \leq x \leq 1$.</p>
<p>A related notation is $(0,1)$, or ... |
2,802,959 | <p>If I write
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?</p>
<p>Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$? </p>
<p>Q2:</p>
<p>If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?</p>
| Chappers | 221,811 | <p>$[0,1]$ is (defined as) the set $\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$, i.e. it is a set that contains every real number between $0$ and $1$ (inclusive). It contains an uncountable number of elements.</p>
<p>$\{0,1\}$ is a set containing 2 elements: $0$, and $1$.</p>
|
3,522,752 | <p>Solve the following equation:
<span class="math-container">$$y=x+a\tan^{-1}p$$</span>
<span class="math-container">$$\text{where p}=\frac{dy}{dx}$$</span>
Differentiating both side w.r.t. x,
<span class="math-container">$$\frac{dy}{dx}=1+\frac{a}{1+p^2}\frac{dp}{dx}\\
\implies p=1+\frac{a}{1+p^2}\frac{dp}{dx}$$</spa... | user577215664 | 475,762 | <p><span class="math-container">$$\frac{dy}{dx}=1+\frac{a}{1+p^2}\frac{dp}{dx}$$</span>
<span class="math-container">$$\implies p=1+\frac{a}{1+p^2}\frac{dp}{dx}$$</span>
It's separable
<span class="math-container">$$ \frac {dp}{(p-1)({1+p^2})}=\frac {dx} a$$</span>
Use fraction decomposition. And integrate.</p>
<hr>
... |
118,406 | <p>I have a single flat directory with over a million files. I just wanted to take a sample of the first few files but <code>FileNames</code> doesn't include a "only the first n" option, and so it took over a minute:</p>
<p><a href="https://i.stack.imgur.com/s5cBS.png" rel="nofollow noreferrer"><img src="https://i.sta... | Alexey Golyshev | 23,402 | <p>New function in Mathematica 11 <code>FileSystemMap</code> with option <code>MaxItems</code> (<a href="https://reference.wolfram.com/language/ref/FileSystemMap.html">documentation</a>) can be useful here.</p>
<pre>
dir = "C:\\Users\\Alexey\\Documents";
n = 10;
f = FileSystemMap[#&, dir, MaxItems -> n] // Keys;
</pre... |
1,070,008 | <p>Is being $T_1$ is a topological invariant?
Is being a first-countable space is a topological invariant?
I need a little hint as to whether or not these sets are topological invariants.</p>
| Matthew Leingang | 2,785 | <p>A <em>toplogical invariant</em> is a property that is preserved under homeomorphism. So your first question is equivalent to:</p>
<blockquote>
<p>If $X$ and $Y$ are homeomorphic, and $X$ is $T_1$, is $Y$ also $T_1$?</p>
</blockquote>
<p>Let $f\colon Y \to X$ be a homeomorphism. Given that $X$ is $T_1$, and $f$... |
1,456,407 | <p><a href="https://i.stack.imgur.com/oy6T7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oy6T7.jpg" alt="enter image description here"></a></p>
<p>We need to find the area of the shaded region , where curves are in polar forms as $r = 2 \sin\theta$ and $r=1$.</p>
<p>I formulated the double integ... | mathlove | 78,967 | <p>Note that
$$42^{2k}-1=(42^k)^2-1=(42^k-1)(42^k+1)$$
where $1\lt 42^k-1\lt 42^k+1$.</p>
|
467,609 | <blockquote>
<p>Find the value of
$$\int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx $$</p>
</blockquote>
<p>I have tried using $\int_a ^bf(x) dx=\int_a^b f(a+b-x)dx$</p>
<p>$\displaystyle \int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx=\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=I$</p>
<p>I couldn't go any further with that!</p... | Ahaan S. Rungta | 85,039 | <p>Now, note that $$ \left( \displaystyle\int_0^\pi \dfrac {\pi}{1+\sin^2(x)} \, \mathrm{d}x \right) - I = I \implies I = \dfrac {\displaystyle\int_0^\pi \dfrac {\pi}{1+\sin^2(x)} \, \mathrm{d}x}{2}. $$</p>
<p>Try to find $ \displaystyle\int_0^\pi \dfrac {1}{1+\sin^2(x)} \mathrm{d}x $. </p>
|
467,609 | <blockquote>
<p>Find the value of
$$\int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx $$</p>
</blockquote>
<p>I have tried using $\int_a ^bf(x) dx=\int_a^b f(a+b-x)dx$</p>
<p>$\displaystyle \int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx=\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=I$</p>
<p>I couldn't go any further with that!</p... | N. S. | 9,176 | <p>You are almost there</p>
<p>$$2I=I+I= \displaystyle \int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx+\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=\pi\int _0 ^ \pi \dfrac{1}{1+\sin^2(x)} dx$$</p>
<p>The last integral can be calculated with the substitution $t =\tan(\frac{x}{2})$ or by writing $\sin(x)=\frac{1}{\csc(x)}$ (but... |
1,269,738 | <p>I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with. </p>
<p>Some examples that come to min... | Eric Stucky | 31,888 | <p>This sum-of-squares theorem of Fermat may qualify as an example:</p>
<blockquote>
<p>An odd prime $p$ is expressible as the sum of squares $x^2+y^2$ if and only if $p\equiv 1 \text{ mod } 4$.</p>
</blockquote>
<p>You can read <a href="http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squar... |
4,504,080 | <p>If it is given that <span class="math-container">$$\displaystyle \frac{1}{(20-x)(40-x)}+\displaystyle \frac{1}{(40-x)(60-x)}+....+\displaystyle \frac{1}{(180-x)(200-x)}= \frac{1}{256}$$</span> then how to find the maximum value of <span class="math-container">$x$</span> ? I tried solving it with <span class="math-co... | Alex Youcis | 16,497 | <p>I will post this answer just as a complement to <strong>Thomas Preu</strong>'s nice computation above.</p>
<p>Let us write <span class="math-container">$\Gamma:=\mathrm{Gal}(\mathbf{C}/\mathbf{R})$</span>. Also, let me model <span class="math-container">$\mathbf{Z}/2\mathbf{Z}$</span> as <span class="math-container"... |
576,553 | <p>Please, forgive me if this is an elementary question, as well as my the sloppy phrasing and notation.</p>
<p>Suppose we have two discrete probability distributions $p = {\lbrace p_i \rbrace}$ and $q={\lbrace q_i \rbrace}$, $i=1,\dots,n$, where $p_i=P(p=p_i)$ and $q_i=P(q=q_i)$. Let's represent them as vectors $\bol... | suvrit | 18,934 | <p>Neither of the two versions holds...</p>
<p>Here are counterexamples. Let $a=1.5$, $b=2$. Let
\begin{equation*}
p=[3,1,4]/8\quad q = [2,2,5]/9;
\end{equation*}
Then, we have</p>
<p>\begin{equation*}
\begin{split}
\|p\|_a &= 0.7329,\quad \|q\|_a = 0.7299\\
\|p\|_b &= 0.6374,\quad \|q\|_b = 0.6383.
\... |
2,312,913 | <p>In the triangle $ABC$, let $E$ be a point on $BC$ such that $BE : EC =
3: 2$. Pick points $D$ and $F$ on the sides $AB$ and $AC$ , correspondingly, so that
$3AD = 2AF$ . Let $G$ be the point of intersection of $AE$ and $DF$ . Given that $AB = 7$
and $AC = 9$, find the ratio $DG: GF$.</p>
<p>I have been working on t... | fonfonx | 247,205 | <p>Let us count the number of groups without any girls: there are $25 \choose 20$ possibilities (just pick 20 boys out of 25).</p>
<p>Let us count the total number of possible groups: here are $50 \choose 20$ possibilities (just pick 20 kids out of 50).</p>
<p>Consequently you have ${50 \choose 20} - {25 \choose 20}$... |
3,620,767 | <p><a href="https://imgur.com/a/i24lMmS" rel="nofollow noreferrer">https://imgur.com/a/i24lMmS</a></p>
<p>I tried solving this problem, but couldn't find an answer. Any suggestions? Thanks!</p>
| sammy gerbil | 203,175 | <p><a href="https://i.stack.imgur.com/xXuI8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xXuI8.png" alt="enter image description here"></a></p>
<p>The required area <span class="math-container">$A$</span> of the large square is the area of the middle square AFGC plus 4 x the area of the triangles... |
2,011,754 | <p>Can somebody help me to solve this equation?</p>
<p>$$(\frac{iz}{2+i})^3=-8$$ ?
I'm translating this into</p>
<p>$(\frac{iz}{2+i})=-2$</p>
<p>But i recon it's wrong ...</p>
| haqnatural | 247,767 | <p>You are in the right way
$$\frac { iz }{ 2+i } =\sqrt [ 3 ]{ 8\cdot \left( -1 \right) } =-2\left( \cos { \frac { k\pi }{ 3 } +i\sin { \frac { k\pi }{ 3 } } } \right) ,k=0,1,2$$ check for instance $k=0$ </p>
<p>$$\frac { iz }{ 2+i } =-2\\ iz=-4-2i\\ z=\frac { -4-2i }{ i } =\frac { -4i+2 }{ { i }^{ 2 } } =-2... |
1,506,532 | <p>How do I prove that the connected undirected graph having 10 nodes and 10 edges
contains a cycle.</p>
| happymath | 129,901 | <p><strong>Hint</strong>: Any tree with $n$ vertices can have atmost $n-1$ edges.</p>
|
1,506,532 | <p>How do I prove that the connected undirected graph having 10 nodes and 10 edges
contains a cycle.</p>
| TechJ | 281,154 | <p>Assume that it is not cyclic, so then it means it is a tree.</p>
<p>But for tree we have number of edges 1 less than number of vertices i.e. $n=n-1$</p>
<p>$n=10-1=9$ which contradicts the given statement.</p>
<p>Hence our assumption is wrong, so there exists a cycle.</p>
|
1,508,863 | <p>I have this homework problem assigned but I'm confused as to how to solve it:</p>
<p>For $n>2$ and $a\in\mathbb{Z}$ with $\gcd(a,n)=1$, show that $o_n(a)=m$ is odd $\implies o_n(-a)=2m$.</p>
<p>(where $o_n(a)=m$ means that $a$ has order $m$ modulo $n$).</p>
<p>We were also given this hint: Helpful to consider ... | Brian M. Scott | 12,042 | <p>The number of <a href="https://en.wikipedia.org/wiki/Derangement">derangements</a> of $[n]=\{1,\ldots,n\}$ is </p>
<p>$$d_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}\;,$$</p>
<p>so</p>
<p>$$\frac{n!}e-d_n=n!\sum_{k>n}\frac{(-1)^k}{k!}\;,$$</p>
<p>which is less than $\frac1{n+1}$ in absolute value. Thus for $n\ge 1$, $d... |
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