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 |
|---|---|---|---|---|
3,028,986 | <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... | Community | -1 | <p>You could split the integral: <span class="math-container">$$\int_{0}^{4 \pi} |\cos(\theta)| d\theta = \int_{0}^{2 \pi} |\cos(\theta)| d\theta + \int_{2 \pi}^{4 \pi} |\cos(\theta)| d\theta,$$</span> then consider the change of variable <span class="math-container">$\alpha = \theta - 2 \pi$</span>, so the RHS above b... |
1,081,383 | <p>Assume that $A$ is an $n\times n$ symmetric positive-definite matrix.</p>
<p>Prove that:</p>
<blockquote>
<p>the element of $A$ with maximum magnitude must lie on the diagonal. </p>
</blockquote>
| megas | 191,170 | <p>Hint:
Assume that this is not the case, <em>i.e.</em>, the entry of maximum magnitude is an off-diagonal entry $A_{ij}=A_{ji}$, with $j \neq i$.
To show that $A$ cannot be PSD, it suffices to construct an $x$ (a certificate)
such that
$$
x^{T}Ax < 0.
$$
How can we find one such $x$?
In order to answer that, yo... |
1,081,383 | <p>Assume that $A$ is an $n\times n$ symmetric positive-definite matrix.</p>
<p>Prove that:</p>
<blockquote>
<p>the element of $A$ with maximum magnitude must lie on the diagonal. </p>
</blockquote>
| Alex Silva | 172,564 | <p>Let $a_{max}$ be the element of $\mathbf{A}$ with maximum magnitude. Assume that it is not lie on the diagonal. Thus, there is exist a $2\times 2$ principal minor equal to $ab- a_{max}^2 \leq 0$ for some $a$ and $b$ in the main diagonal. Hence, as the matrix is positive definite, $a_{max}$ should be actually in the ... |
2,572,802 | <p>We were asked to find the number of five digit numbers $N=d_1d_2d_3d_4d_5$, where $d_i$ is the $i$th digit of the number and $d_1 < d_2 < d_3 < d_4 < d_5 $. The solution was trivial as for a given selection of five random distinct digits, there is only one way to arrange them in strict increasing order. ... | Especially Lime | 341,019 | <p>Here's an essentially equivalent approach that may be what you're looking for. Instead of looking for possible values $d_1,...,d_5\in\{1,...,9\}$ such that $d_1\leq d_2...\leq d_5$, try setting $c_i=d_i+i$ for each $i$. Now $c_1,...,c_5\in\{2,...,14\}$ and $c_1<c_2<...<c_5$, so there are ${}^{13}C_5$ ways t... |
1,730,352 | <p>I am really struggling to understand what modular forms are and how I should think of them. Unfortunately I often see others being in the same shoes as me when it comes to modular forms, I imagine because the amount of background knowledge needed to fully appreciate and grasp the constructions and methods is rather ... | Andrea Mori | 688 | <p>(a) ${\rm SL}_2(\Bbb Z)$ is a group in the sense that is an example of the algebraic structure called <a href="https://en.wikipedia.org/wiki/Group_%28mathematics%29">group</a>. :)</p>
<p>(b) That's not the group operation. The group operation in ${\rm SL}_2(\Bbb Z)$ (and in fact in any <a href="https://en.wikipedia... |
3,156,570 | <p>I need to evaluate the following limit:
<span class="math-container">$$
\lim_{x\downarrow 0} \dfrac{(1 - e^x)^{-1}}{x^c}
$$</span>
for different values of the constant <span class="math-container">$c$</span>.</p>
<p><em>What I've tried thus far:</em></p>
<p>We have that
<span class="math-container">$$
\lim_{x\down... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\frac x {1-e^{x}} \to -1$</span> as <span class="math-container">$x \to 0+$</span> by L'Hopital's Rule. Multiply numerator by <span class="math-container">$x$</span> and change <span class="math-container">$x^{c}$</span> in the denominator to <span class="math-container">$x^{c+1}$</span... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | alephzero | 223,485 | <p>Absolutely <strong><em>nothing</em></strong> in physics is completely described by a PDE, if you look at a sufficiently small resolution, because space and time are not continuous. (Since the OP has said in a comment that he doesn't know much physics, google for "Planck length" for more information.) </p>
<p>Howeve... |
3,854,785 | <p>Considering <span class="math-container">$$2\sin^2(x) = 1 - \cos(2x)$$</span>
to show <span class="math-container">$$8\sin^4(x) = 3 - 4\cos(2x) +\cos(4x)$$</span>
Assuming I did not how to initially do this proof properly, how would I be able to set up a proof that is still valid to show that <span class="math-conta... | Math Lover | 801,574 | <p><span class="math-container">$3 - 4 \cos(2x) + \cos(4x) = 3 - 4 (1 - 2 \sin^2 x) + (2 \cos^2 2x - 1)$</span></p>
<p><span class="math-container">$ = 8 \sin^2 x - 2 + 2 (1 - 2 \sin^2 x)^2 = 8 \sin^4x$</span></p>
|
2,398,215 | <p>If $f$ is continuous on $\mathbb{R}$ any of the following conditions are satisfied then $f$ must be a constant.</p>
<p>(1).$f(x)=f(mx),\forall x\in \mathbb{R},|m|≠1,m\in \mathbb{R}$</p>
<p>(2).$f(x)=f(2x+1),\forall x\in \mathbb{R}$</p>
<p>(3).$f(x)=f(x^2),\forall x\in \mathbb{R}$.</p>
<p>Suppose $f$ satisfy (1).... | trying | 309,917 | <p>Here the answer to your first question, where use is made of one of the main corollaries for continuous functions (see $(4)$ below).</p>
<p>Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $m\in\mathbb{R}$, $|m|\ne1$
$$f \text{ is continuous}\tag{1}$$
$$\forall x\in\mathbb{R},f(x)=f(mx)\tag{2}$$
<strong>Theorem.</strong... |
2,185,489 | <blockquote>
<p>Let $\xi \in \mathcal{L}^2(\Omega,P)$ be a random variable with finite variance. Show that $$(E(\xi))^2 \leq E(\xi^2)$$ </p>
</blockquote>
<p>Since $$\operatorname{Var}(\xi) := E(\xi^2) - (E(\xi))^2$$ this boils down to showing $$\operatorname{Var}(\xi) \geq 0$$ which is quite restrictive. Since the ... | Kenny Wong | 301,805 | <p>For an alternative proof, use Cauchy-Schwarz.</p>
<p>$$ E(\xi)^2 = \left( \int_\Omega \xi \right)^2 = \left(\int_\Omega 1.\xi \right)^2 \leq \int_\Omega 1^2 \times \int_\Omega \xi^2 = E(\xi^2).$$</p>
|
2,524,890 | <p>I know that if matrix $a$ is similar to matrix $b$ then $\operatorname{trace} a=\operatorname{trace} b$.</p>
<p>Does it go to the other side?</p>
<p>Thanks.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>we get
$$(b-4)^2=16(1-a)$$ i hope this solves your problem</p>
|
3,535,088 | <p>A function <span class="math-container">$\phi:X\rightarrow Y$</span> between two topological space <span class="math-container">$(X,\tau)$</span> and <span class="math-container">$(Y,\sigma)$</span> is continuous in <span class="math-container">$x\in X$</span> if and only if for any open set <span class="math-contai... | Lorenzo Cecchi | 523,193 | <p>Yes, it is true. Since <span class="math-container">$v_n\to v$</span> weakly <span class="math-container">$L^\infty$</span>, you have that <span class="math-container">$\langle u_n v_n,x\rangle=\langle u_n,v_n x\rangle$</span> (I'm assuming you are working with real valued functions) and <span class="math-container"... |
3,293,082 | <p>I know this question was asked on this site, but I didn't understand the answer. Could someone give me the simplest explanation of this? (High school level explanation)</p>
| fleablood | 280,126 | <p>They are completely different.</p>
<p><span class="math-container">$\sec x = \frac 1{\cos x} = 1\div \cos x$</span>. This is the multiplicative reciprocal, which is sometimes call the <em>multiplicative</em> inverse.</p>
<p><span class="math-container">$\arccos x$</span> is the function so that if <span class="ma... |
2,000,940 | <p>Suppose $A \subseteq \mathbb{R}$ is measurable and $f\colon A \to \mathbb{R}$ is Lipschitz on the set $A$, i.e there is some $K\ge 0$ such that $\lvert f(x)-f(y)\rvert \le K \lvert x-y\rvert$ for $x,y \in A$. </p>
<p>I'm trying to prove that
$$
m^\ast(f(E)) \le K\,m^\ast(E)\textrm{ for every set }E \subseteq A.
$$<... | ChristophorusX | 239,293 | <p>We do not know if the function can be extended to $\mathbb{R}$ by continuous extension theorem, so we handle it using the method we use to prove Growth Lemma.</p>
<p>For $\forall E \subseteq A$, we can cover $E$ with $\{I_k\}$, s.t.
$$
E \subseteq \cup_k I_k\quad \textrm{and}\quad \sum_k m(I_k) \le m^\ast (E) + \va... |
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... | joriki | 6,622 | <p>Funny enough, someone just posted a question on the <a href="https://math.stackexchange.com/questions/125539/power-reduction-formula">Power-reduction formula</a> two hours ago. Using that, you readily get the result </p>
<p>$$\frac{2\pi}{2^n}\binom{n}{n/2}\;.$$</p>
|
454,040 | <p>I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$</p>
<p>could any one give me any hint?</p>
| martini | 15,379 | <p><strong>Hint</strong>: $(0,1) = (0,\frac 12] \cup [\frac 12, 1)$. Can you map each part onto $(0,1]$?</p>
|
454,040 | <p>I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$</p>
<p>could any one give me any hint?</p>
| Balbichi | 24,690 | <p>From The Hint of Martini the Map $f(x)=2x; x\in (0,{1\over 2}]$ and $f(x)=1;x\in [{1\over 2},1)$ will work</p>
|
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... | Henno Brandsma | 4,280 | <p>Every ordinal is a well-ordered set, and the first one that is uncountable is by definition $\omega_1$. So this set which is uncountable because there are uncountably many different ways to well-order a countable set, is thus called "the set of all countable ordinals". By definition, all $\alpha < \omega_1$ are c... |
2,114,619 | <p>An intruder has a cluster of 64 machines, each of which can try 10^6 passwords per second. How long does it take him to try all legal passwords if the requirements for the password are as follows:</p>
<ul>
<li>passwords can be 6, 7, or 8 characters long</li>
<li>each character is either a lower-case letter or a dig... | Stella Biderman | 123,230 | <p>The methodology is correct, and if the calculator says that the end number is what you say it is, then it's right too :P</p>
|
259,308 | <p>The output of <code>ListPointPlot3D</code> is shown below:
<a href="https://i.stack.imgur.com/ypt73.png" rel="noreferrer"><img src="https://i.stack.imgur.com/ypt73.png" alt="enter image description here" /></a>
I only want to connect the dots in such a way that it forms a ring-like mesh. However, when I use <code>Li... | kglr | 125 | <p><strong>Update 2:</strong> Additional methods using <code>Partition[pts, 20]</code> with <code>BSplineSurface</code> and <code>BSplineFunction</code>:</p>
<pre><code>array = Partition[pts, 20];
</code></pre>
<p>We can get the surface (without lines connecting the points) using <code>array</code> with <a href="https:... |
1,854,823 | <p>How do I express the hyperplane $x+y=1$ as the span of two vectors or more?</p>
<p>P. S. We have a 3D space.</p>
| DonAntonio | 31,254 | <p>Using only basic analytic geometry: find three different non-collinear points on the plane, for example</p>
<p>$$A=(1,0,1)\;,\;\;B=(1,0,0)\;,\;\;C=(0,1,0)$$</p>
<p>and now construct the directed vectors</p>
<p>$$\vec{AB}=B-A=(0,0,-1)\;,\;\;\vec{AC}=C-A=(-1,1,-1)$$</p>
<p>and then the plane is</p>
<p>$$\pi:\;A+r... |
2,397,564 | <p>Question:</p>
<p>Prove that if $ \ A\cup B \subseteq C \cup D,\ A \cap B =$ ∅ $\land \ C \subseteq A \implies B \subseteq D$.</p>
<p>My attempt:</p>
<p>Let $ \ x\in B \implies x \in A \cup B \implies x \in C \cup D \because A\cup B \subseteq C \cup D$.</p>
<p>Now, $ x \in C \lor x\in D$. If $\ x \in C \implies... | Aryabhata | 1,102 | <p>It is correct. Well done.</p>
<p>Perhaps a bit of suggested modification in the "But that's not possible part" to make it clearer.</p>
<p>Since we started with $x \in B$, if $x \in A$, then $x \in A \cap B$ which is not possible, as $A \cap B = \phi$.</p>
|
138,520 | <p>I am attempting to show that the series $y(x)\sum_{n=0}^{\infty} a_{n}x^n$ is a solution to the differential equation $(1-x)^2y''-2y=0$ provided that $(n+2)a_{n+2}-2na_{n+1}+(n-2)a_n=0$</p>
<p>So i have:
$$y=\sum_{n=0}^{\infty} a_{n}x^n$$
$$y'=\sum_{n=0}^{\infty}na_{n}x^{n-1}$$
$$y''=\sum_{n=0}^{\infty}a_{n}n(n-1)x... | Community | -1 | <p>You are right till the last step.</p>
<p>You have $$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^{n}-2\sum_{n=0}^{\infty}n(n+1)a_{n+1}x^{n}+\sum_{n=0}^{\infty}n(n-1)a_{n}x^{n}-2\sum_{n=0}^{\infty} a_{n}x^n=0$$</p>
<p>which gives us
$$\sum_{n=0}^{\infty} \left((n+2)(n+1) a_{n+2} - 2 n (n+1) a_{n+1} + (n^2-n-2)a_n \right)x... |
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 ... | Randall | 464,495 | <p>Note that the statement is true if $n=2$. Now consider $f(x) = \ln x - \frac{1}{x}$. Since $f'(x) = \frac{1}{x} + \frac{1}{x^2}=\frac{x+1}{x^2}$ is clearly positive for $x \geq 2$, $f$ is increasing over the same span. Thus the gap between $\frac{1}{n}$ and $\ln n$ is only getting wider, so we have $\frac{1}{n} &... |
3,715,715 | <p>Let’s say I have a set <span class="math-container">$X$</span> and a set <span class="math-container">$Y$</span>, and <span class="math-container">$X \subseteq Y$</span>. Is it possible to state that <span class="math-container">$|X| \leq |Y|$</span> (<span class="math-container">$|X|$</span> cardinality of <span cl... | TransfiniteGuy | 474,380 | <p>If we talk about cardinality of two sets, say <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, the way to show that <span class="math-container">$|A|\leq |B|$</span> is giving an injective function <span class="math-container">$f: A \longrightarrow B$</span>. </p>
<p>Now, in your... |
2,391,453 | <p>Let $A\subseteq[0,1]^2$ be a Lebesgue-measurable set. For $x\in [0,1]$, we define $A_x$ as $\{y:(x,y)\in A\}$.<br><br> <strong>Prove that $A_x$ is measurable for almost all $x\in[0,1]$.</strong> <br><br> I know that the "almost every'' is indeed needed, since we could have $A=V\times\{0\}$ for $V$ the Vitali set -- ... | Matematleta | 138,929 | <p>I think the following argument works:</p>
<p>Let $(I\times I,\mathcal L, \lambda_2)$ be the completion of $(I\times I,\mathscr B([0,1])\times \mathscr B([0,1]), \lambda\times \lambda ),$ set $f=1_A$ and choose a $\mathscr B([0,1])\times \mathscr B([0,1])$- measurable function $g$ such that $g=f\ \text{a.e.}\ \lamb... |
2,391,453 | <p>Let $A\subseteq[0,1]^2$ be a Lebesgue-measurable set. For $x\in [0,1]$, we define $A_x$ as $\{y:(x,y)\in A\}$.<br><br> <strong>Prove that $A_x$ is measurable for almost all $x\in[0,1]$.</strong> <br><br> I know that the "almost every'' is indeed needed, since we could have $A=V\times\{0\}$ for $V$ the Vitali set -- ... | JS_ | 354,831 | <p>This is just a more detailed and extended version of the accepted answer. I just want to make sure everything is understood properly.</p>
<p>Let $\lambda_d$ denote the $d$-dimensional Lebesgue measure and $\mathcal{L}_d$ the set of Lebesgue-measurable subsets of $\mathbb{R}^d$. Moreover, let $\mathcal{L}_d\times\ma... |
1,163,033 | <p>I want to calculate $ 8^{-1} \bmod 77 $ </p>
<p>I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem.</p>
<p>But how to move further now. Should i calculate $ 8^{59} $ and then divide it by $ 77 $ or is there any other theorem i can use ? </p>
| Mark Bennet | 2,906 | <p>Your question is equivalent to solving $8x+77y=1$ for integers $x$ and $y$. This can be solved using the division algorithm, or alternatively (equivalently) solving modulo $8$ - which is small enough to do by trial and error or observation.</p>
<p>So noting that $77\equiv 5 \bmod 8$ and $5\times 5 \equiv 1 \bmod 8$... |
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>
| Doubt | 61,559 | <p>Consider the $2\times2$ case with
\begin{align}
X =
\begin{bmatrix}
x_1 & x_3\\ x_2 & x_4
\end{bmatrix},\quad A=
\begin{bmatrix}
a_1 & a_3\\ a_2 & a_4
\end{bmatrix},\quad Y=
\begin{bmatrix}
y_1 & y_3\\ y_2 & y_4
\end{bmatrix}.
\end{align}
Then
\begin{align}
\begin{bmatrix}
y_1 & y_3\\ y_2... |
2,526,716 | <p>Define, by structural induction, a function $f : A^* \to A^*$ that removes all occurrences of the letter $a$. For instance, we should have
$f(abcbab) = bcbb$ and $f(bc) = bc$.</p>
<p>I came up with this:</p>
<p>$f(\lambda) = \lambda$ (empty word)</p>
<p>$f(xw) = xf(w)$ if $x$ is not $a$</p>
<p>$f(w)$, otherwise.... | Michael Rozenberg | 190,319 | <p>It's the distance between the tangent line $yy_1=p(x+x_1)$ to the parabola $y^2=x$ and $y=4x+4$, which parallel to the tangent line.</p>
<p>Since $p=\frac{1}{2}$, we obtain $\frac{1}{2y_1}=4$, which gives $y_1=\frac{1}{8}$ and $\left(\frac{1}{64},\frac{1}{8}\right)$ is a touching point. </p>
<p>Id est, for the di... |
1,591,311 | <p>I've been thinking about this problem which I think is interesting, but can't solve it.</p>
<p>There are $n$ distinguishable items, and $b$ distinguishable bins. Each bin has to include at least one item. But, once some set of items are placed in a bin, they become indistinguishable. How many ways are there to plac... | Eric Thoma | 35,667 | <p>First lets not have the restriction that all the bins are nonempty. There are $b$ choices of bins for each ball, and so there are $b^n$ different configurations.</p>
<p>How many configurations are we counting that we should not? Lets leave one bin empty. There are $b$ choices for the empty bin, and $(b-1)^n$ ways t... |
1,591,311 | <p>I've been thinking about this problem which I think is interesting, but can't solve it.</p>
<p>There are $n$ distinguishable items, and $b$ distinguishable bins. Each bin has to include at least one item. But, once some set of items are placed in a bin, they become indistinguishable. How many ways are there to plac... | Marc van Leeuwen | 18,880 | <p>Your second condition is implicit in the usual notion of bin in combinatorics: there is no imposed ordering among the items inside the bin (which is unrelated to the possibility to distinguish the items among each other). So you are just asking for the number of surjective maps from the set of your $n$ items to the ... |
26,152 | <p>In my textbook, they said:</p>
<p>$$2x^{3} + 7x - 4 \equiv 0 \pmod{5}$$</p>
<p>The solution of this equation are the integers with $x \equiv 1 \pmod{5}$, as can be seen by testing $x = 0, 1, 2, 3,$ and $4.$</p>
<p>And I have no clue how do they had $x \equiv 1 \pmod{5}$. I tested as they suggested:</p>
<p>Let $y... | Alex J. | 8,057 | <p>$$\begin{aligned}
2x^3+7x-4 \equiv 0 ( mod 5)\\
2x^3+2x-4 \equiv 0 ( mod 5)\\
2(x-1)(x^2+x+2) \equiv 0 ( mod 5)
\end{aligned}$$</p>
<p>Now you have two solutions $x-1 \equiv 0 (mod5)$ or $x^2+x+2 \equiv 0 (mod5)$.
You can continue and verify if $x \equiv 1 ( mod 5)$ is the only solution.</p>
|
750,417 | <p><strong>Question:</strong></p>
<blockquote>
<p>Initially we have a list of numbers $1,2,3,\cdots,2013$.an operation is defined that taking two numbers $a, b$ out from the list, but add $a+b$ into it instead, what is the minimum number of operations required that the sums of any number of numbers in the list can n... | achille hui | 59,379 | <p>The minimum number of operations is either $503$ or $504$. I cannot figure out which is the answer.</p>
<p>Among the list of $2013$ numbers $( 1, 2, \ldots, 2013 )$, there are $1006$ pairs that sum to $2014$: $$( 1, 2013), ( 2, 2012 ), \ldots, ( 1006, 1008 )\tag{*1}$$</p>
<p>In order for the final list and the sum... |
3,335,081 | <p>Is a temperature change in Celsius larger than a temperature change in Fahrenheit?</p>
<p><strong>The teacher offers this second way of thinking about the question.</strong></p>
<blockquote>
<p>If the temperature increases by 1 degree Celsius, does it also increase by 1 degree Fahrenheit? Or is one temperature c... | Tanner Swett | 13,524 | <p>What you have so far is correct, but in my opinion, you haven't <em>quite</em> answered the question that was asked.</p>
<p>Before I get to that, though, I have just a minor comment. You write that</p>
<p><span class="math-container">$$T_{\triangle F} = \frac95T_C + 32 - T_F + \frac95T_{\triangle C}.$$</span></p>
... |
3,365,361 | <p>Suppose that <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is analytic at <span class="math-container">$x=0$</span>, and <span class="math-container">$T(x)$</span> its Taylor series at <span class="math-container">$x=0$</span>, with radius of convergence <span class="math-container">$R>0$</span>... | DonAntonio | 31,254 | <p>Because <span class="math-container">$\;\operatorname{tr}(AB)=\operatorname{tr}(BA)\;$</span> , so</p>
<p><span class="math-container">$$\operatorname{tr}(Q\Lambda^kQ^{-1})=\operatorname{tr}\left(Q(\Lambda^kQ^{-1})\right)=\operatorname{tr}((\Lambda^kQ^{-1})Q)=\operatorname{tr}(\Lambda^k(Q^{-1}Q))=\operatorname{tr}(... |
1,622,779 | <p>I have been asked to calculate $\frac{1+i \tan \alpha}{1-i \tan \alpha}$, where $\alpha \in \mathbb{R}$.</p>
<p>So, I multiplied top and bottom by the complex conjugate of the denominator:</p>
<p>$\frac{(1+i \tan \alpha)(1+ i \tan \alpha)}{(1-i \tan \alpha)(1+ i \tan \alpha) } = \frac{1 + 2 i \tan \alpha + i^{2} \... | Community | -1 | <p>I'll continue from where you left to get a simpler form :</p>
<ul>
<li>The real part :</li>
</ul>
<p>$$\frac{1-\tan \alpha^2}{1+ \tan \alpha^2}=\frac{\cos^2 \alpha-\sin^2 \alpha}{\sin^2 \alpha+ \cos^2 \alpha}=\cos2\alpha$$</p>
<ul>
<li>The imaginary part :</li>
</ul>
<p>$$2\sin \alpha \cos \alpha=\sin 2\alpha$$<... |
792,356 | <p>There is a dark night and there is a very old bridge above a canyon. The bridge is very weak and only 2 men can stand on it at the same time. Also they need an oil lamp to see holes in the bridge to avoid falling into the canyon.</p>
<p>Six man try to go through that bridge. They need 1,3,4,6,8,9(first man, second ... | MJD | 25,554 | <p>A minimal solution is:</p>
<ul>
<li>1 and 6 cross the bridge; 1 comes back (7)</li>
<li>1 and 3 cross the bridge, 1 comes back (4)</li>
<li>8 and 9 cross the bridge; 3 comes back (12)</li>
<li>1 and 3 cross the bridge, 1 comes back (4)</li>
<li>1 and 4 cross the bridge (4)</li>
</ul>
<p>Total is $7+4+12+4+4 = 31$.... |
239,136 | <p>I was given this question and I'm not really sure how to approach this...</p>
<p>Assume $(r,s) = 1$. Prove that If $G = \langle x\rangle$ has order $rs$, then $x = yz$, where $y$ has order $r$, $z$ has order $s$, and $y$ and $z$ commute; also prove that the factors $y$ and $z$ are unique.</p>
| Hagen von Eitzen | 39,174 | <p>From $(r,s)=1$ we find integers $n,m$ with $nr+ms=1$.
Let $y=x^{ms}$, $z=x^{nr}$.
Then $yz=x^{ms+nr}=x$.
The fact that $x$ and $y$ commute is trivial because the cyclic group $G$ is abelian.
Also, we have $y^r=(x^m)^{rs}=1$, $z^s=(x^n)^{rs}=1$, hence the orders are at least divisors of $r$ and $s$, respectively.
I... |
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... | Cookie | 111,793 | <p>We have $a_n = \frac{1}{(2n)!}$ and $a_{n+1} = \frac{1}{(2n+2)!}$</p>
<p>\begin{align}\lim_{n \rightarrow \infty} \Big|\frac{a_{n+1}}{a_n}\Big| &= \lim_{n \rightarrow \infty} \Bigg|\frac{(2n)!}{(2n+2)!}\Bigg| \\
&= \lim_{n \rightarrow \infty} \Bigg|\frac{(2n)(2n-1)(2n-2)\cdots}{(2n+2)(2n+1)(2n)(2n-1)(2n-2)\... |
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... | MoonKnight | 115,071 | <p>1) It is symmetry when $x>0$ and $x<0$, so we can only prove $x>0$. And the other half part of the proof is very similar</p>
<p>2) $\forall x_0>0$, it is easy to see that $x_1=(x_0+a/x_0)/2>\sqrt{a}$</p>
<p>3) If $x_n^2>a$, then follow your derivation until the second last line</p>
<p>$$
x_{n+1}... |
135,426 | <p>$$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw$$</p>
<p>(if it helps, in my setting $\varphi$ is the CDF of some arbitrary uniform distribution). So I want to get a nice expression for this integral and it seems to suggest FTC, but I tried a change of variable and ended up with a $q$ inside the integrand... | The Chaz 2.0 | 7,850 | <p>For a similar approach to Jyrki's, we use the result that the <a href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29#Alternating_harmonic_series" rel="nofollow">alternating harmonic series</a> converges to $\ln 2$. Let's subtract one from both sides of your inequality, then change all the signs to g... |
3,498,199 | <p>Suppose if a matrix is given as</p>
<p><span class="math-container">$$ \begin{bmatrix}
4 & 6\\
2 & 9
\end{bmatrix}$$</span></p>
<p>We have to find its eigenvalues and eigenvectors.</p>
<p>Can we first apply elementary row operation . Then find eigenvalues.</p>
<p>Is their any relation on the matrix if ... | Aryaman Maithani | 427,810 | <p>No, elementary row operations need not preserve eigenvalues and/or eigenvectors.</p>
<p>Examples.</p>
<p><span class="math-container">$$\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} \overset{R_1 \mapsto R_1 + R_2}{\longrightarrow} \begin{bmatrix}1 & 1 \\0 & 1\end{bmatrix}$$</span></p>
<p>In this case,... |
221,351 | <p>I asked the following question (<a href="https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con">https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con</a>) on math.stackexchange.com and ... | David E Speyer | 297 | <p>This argument is fairly standard, but it is quicker to repeat it than to find a reference: Let $G$ be a finite subgroup of $GL_n(\mathbb{Q})$. Set $\Lambda = \sum_{g \in G} g \cdot \mathbb{Z}^n \subset \mathbb{Q}^n$. Then $\Lambda$ is a finitely generated torsion free abelian group, hence isomorphic to $\mathbb{Z}^r... |
4,337,820 | <p>We know that during projection 3D space points <span class="math-container">$(x, y, z)$</span> projects to projection plane which has 2D points <span class="math-container">$(x, y).$</span> But during matrix calculation we use homogenous coordinates is of the form <span class="math-container">$(x, y, 1).$</span>
And... | bubba | 31,744 | <p>Short answer: the two concepts “projective plane” and “projection plane” are different things, though they are loosely related.</p>
<p>Longer answer …</p>
<p>The “projective plane”, often denoted by <span class="math-container">$P^2$</span>, is an abstract mathematical concept. It’s used in a field of mathematics ca... |
2,697,069 | <p>Two series of functions are given in which I cannot figure out how to find $M_n$ of the second problem. $$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)$$.. </p>
<p>I have determined the $M_n$ for problem no. $1.$ [$\space|\sum_{n=1}^{\infty} \f... | Botond | 281,471 | <p>Hint:
From $2x=5y$ you have that $y=\frac{2}{5}x$, and from $\frac{y}{3}=\frac{z}{4}$ you have that $\frac{y}{z}=\frac{3}{4}$</p>
|
2,697,069 | <p>Two series of functions are given in which I cannot figure out how to find $M_n$ of the second problem. $$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)$$.. </p>
<p>I have determined the $M_n$ for problem no. $1.$ [$\space|\sum_{n=1}^{\infty} \f... | Peter Szilas | 408,605 | <p>An option:</p>
<p>1)$2x=5y,$ divide both sides by by $2z:$</p>
<p>1') $x/z = (5/2)(y/z).$</p>
<p>2)$y/3=z/4$, then</p>
<p>2')$ y/z = 3/4$ (why?)</p>
<p>Substuting 2') into the RHS of 1'):</p>
<p>$x/z= (5/2)(3/4)=15/8.$</p>
|
2,416,424 | <p>It is known that the collection of finite mixtures of Gaussian Distributions over $\mathbb{R}$ is dense in $\mathcal{P}(\mathbb{R})$ (the space of probability distributions) under convergence in distribution metric.</p>
<p>I'm interested to know the following:</p>
<p>Let $P_X$ be a random variable with finite $p$ ... | kingW3 | 130,953 | <p>The equations $2^x=16$ and $-2^x=-16$ are equivalent however the functions $f(x)=-2^x+16$ and $f(x)=2^x-16$ are not. </p>
<p>You can take a look at the <a href="https://www.desmos.com/calculator/iesqpyxuqg" rel="nofollow noreferrer">graph</a> here, they are symmetric around the $x$ axis which means that when one is... |
3,180,914 | <p>Let <span class="math-container">$G$</span> be a cyclic group of order <span class="math-container">$n$</span>. Let <span class="math-container">$G_k$</span> the subgroup
<span class="math-container">$$G_k=\left\{x^k: x\in G\right\}.$$</span>
Is it true that <span class="math-container">$[G:G_k]\in\{1,k\}$</span>?</... | ΑΘΩ | 623,462 | <p>If <span class="math-container">$G$</span> is cyclic of order <span class="math-container">$n$</span>, say <span class="math-container">$G=\langle a \rangle$</span> then the subgroup of <span class="math-container">$m$</span>-powers is also cyclic generated by <span class="math-container">$a^m$</span> and it can be ... |
54,496 | <p>If there a group G acting on a variety V.
The action is algebraic.
What is the definition of algebro-geometric quotient of this action?</p>
<p>I hope you can give a very basic explanation.</p>
<p>Thanks.</p>
| JBorger | 1,114 | <p>The are several possible meanings. Which one it is would surely depend on the context. </p>
<p>The straight-up meaning is the one that works in any category. If $G$ acts on $X$ then a quotient is a universal object $X/G$ with a $G$-equivariant map $X\to X/G$, where $X/G$ has the trivial $G$ action. Here, 'universal... |
1,406,796 | <p>Can someone please show me how they would work it out as I have never come across this before.</p>
<p>$$(x^2-5x+5)^{x^2-36} =1$$</p>
| Hirshy | 247,843 | <p>We have that $a^b=1$ if and only if $a=1$ and arbitrary $b$ or $b=0$ and arbitrary $a$ (one might have to talk about the case $a=b=0$). Having this you should be able to get two seperate equations from $$(x^2-5x+5)^{x^2-36}=1$$ which can be easily solvedfor $x$.</p>
<p>Edit: I should drink a cup of coffee first aft... |
1,808,258 | <p>I was reading about orthogonal matricies and noticed that the $2 \times 2$ matrix
$$\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} $$
is orthogonal for every value of $\theta$ and that every $2\times 2$ orthogonal matrix can be expressed in this form. I then wonder... | ajotatxe | 132,456 | <p>The third point implies that the parametrization goes over the circle at a constant speed $1$, which is, of course, false in general.</p>
<p>Take for example $v(t)=\langle \sin t^2,\cos t^2\rangle $ for $t\in[0,\sqrt{2\pi})$.</p>
|
3,182,802 | <p>Show that if <span class="math-container">$ \sigma $</span> is a solution to the equation <span class="math-container">$ x^2 + x + 1 = 0 $</span> then the following equality occurs:</p>
<p><span class="math-container">$$ (a +b\sigma + c\sigma^2)(a + b\sigma^2 + c\sigma) \geq 0 $$</span></p>
<p>I looked at the solu... | Travis Willse | 155,629 | <p><strong>Hint</strong> Rearranging the defining equation gives <span class="math-container">$\require{cancel} \sigma^2 = -\sigma - 1$</span>. So, for example, the second term, <span class="math-container">$a b \sigma^2$</span> becomes <span class="math-container">$-a b \sigma - a b$</span>.</p>
<p>Alternatively, we ... |
1,304,344 | <p>How do I find the following:</p>
<p>$$(0.5)!(-0.5)!$$</p>
<p>Can someone help me step by step here?</p>
| Zelos Malum | 197,853 | <p>Gamma function the rescue! It is generalized of factorial to all non-negative even values
$$(0.5)!=\Gamma(-0.5)=-2\sqrt{\pi}$$
$$(-0.5)!=\Gamma(-1.5)=\frac{4}{3}\sqrt{\pi}$$</p>
|
382,293 | <p>Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set $A.$ Once we've got that, it'd be a matter of showing that a representation $($as a sum, $q+a)$ exists for each real number $($which should be the case by constru... | Brian M. Scott | 12,042 | <p>HINT: You need $A$ to have the property that if $a,b\in A$ and $a\ne b$, then $(a+\Bbb Q)\cap(b+\Bbb Q)=\varnothing$. Suppose that $x\in(a+\Bbb Q)\cap(b+\Bbb Q)$; then there are $p,q\in\Bbb Q$ such that $a+p=x=b+q$ and hence $b=a+(p-q)\in a+\Bbb Q$ Conversely, if $b\in a+\Bbb Q$, then it’s clear that $(a+\Bbb Q)\cap... |
158,720 | <p>By induction I can prove :
$$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} = \frac{(D+M)!}{D!M!} $$</p>
<p>However, I couldn't derive the right hand side directly.</p>
<p>It would be of great help if anyone can solve it!!</p>
| Norbert | 19,538 | <p>Here is a combinatorial proof.</p>
<p>Assume you have $D+1$ types of cakes, and you allowed to choose $M$ cakes. Of course you can take several cakes of the same type. The amount of possible choices is <a href="http://en.wikipedia.org/wiki/Combination" rel="nofollow">combination with repetitions</a>:
$$
{(D+1)+M - ... |
205,080 | <p>I have a problem that I cannot figure out how to do. The problem is:<br>
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?<br><br>
I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have tried to find the inverse of function s but I got stuck ... | Community | -1 | <p><strong>Hint:</strong> one way would be to sketch it, notice the minimum and maximum (call them $m$ and $M$), and show that $y>M$ and $y<m$ lead to contradictions (show also that $y=M$ and $y=m$ for certain values of $x$)</p>
|
205,080 | <p>I have a problem that I cannot figure out how to do. The problem is:<br>
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?<br><br>
I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have tried to find the inverse of function s but I got stuck ... | Matthew Conroy | 2,937 | <p>To find the range, we want to find all $y$ for which there exists an $x$ such that
$$ y = \frac{x+2}{x^2+5}.$$
We can solve this equation for $x$:
$$ y x^2 + 5y = x+2$$
$$ 0 = y x^2 -x + 5y-2$$
If $y \neq 0$, this is a quadratic equation in $x$, so we can solve it with the quadratic formula:
$$
x = \frac{ 1 \pm \sqr... |
2,473,780 | <p>So I have the limit $$\lim_{x\rightarrow \infty}\left(\frac{1}{2-\frac{3\ln{x}}{\sqrt{x}}}\right)=\frac{1}2,$$ I now want to motivate why $(3\ln{x}/\sqrt{x})\rightarrow0$ as $x\rightarrow\infty.$ I cam up with two possibilites:</p>
<ol>
<li><p>Algebraically it follows that $$\frac{3\ln{x}}{\sqrt{x}}=\frac{3\ln{x}}{... | Bernard | 202,857 | <ol>
<li>This is a standard result from high school</li>
<li>If you nevertheless want to deduce it from the limit of $\dfrac{\ln x}x$, use the properties of logarithm:
$$\frac{\ln x}{\sqrt x}=\frac{2\ln(\sqrt x)}{\sqrt x}\xrightarrow[\sqrt x\to\infty]{}2\cdot 0=0$$</li>
</ol>
|
148,257 | <p>Let $f:\mathbb R\to\mathbb R$ be continuous. Suppose $(x_n)_n$ and $(y_n)_n$ are sequences in $\mathbb R$ such that the sequence $(x_n-y_n)_n$ converges to $0$. Does this mean that the sequence $(f(x_n)-f(y_n))_n$ converges to $0$?</p>
<p>I feel like it is true, since the definition of continuity states that $f$ pr... | copper.hat | 27,978 | <p>Try $f(x) = e^{x^2}$, with $y_n = n$, $x_n = n+\frac{1}{n}$. Then $(x_n-y_n) \to 0$, but $f(x_n)-f(y_n) = e^{n^2}(e^{2+\frac{1}{n^2}}-1)$, which is unbounded.</p>
|
2,375,736 | <p>If the Ratio of the roots of $ax^2+bx+c=0$ be equal to the ratio of the roots of $a_1x^2+b_1x+c_1=0$, then how one prove that $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?</p>
| Donald Splutterwit | 404,247 | <p>Hint :
let $\alpha$ and $\beta$ be the roots of $ax^2+bx+c=0$ & let $\gamma$ and $\delta$ be the roots of $a_1 x^2+b_1 x+c_1 =0$.
The ratio of their roots are equal if
\begin{eqnarray*}
\frac{\alpha}{\beta} = \frac{\gamma}{\delta}.
\end{eqnarray*}</p>
<p>Further hint : $\color{red}{\alpha+\beta=-\frac{b}{a}}$ ... |
2,375,736 | <p>If the Ratio of the roots of $ax^2+bx+c=0$ be equal to the ratio of the roots of $a_1x^2+b_1x+c_1=0$, then how one prove that $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?</p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>Note that , if the roots of the equation $ax^2+bx+c=0$ are $x_1$ and $x_2$, than
$$
\frac{b}{ac}=\frac{\left(\frac{x_1}{x_2}+1\right)^2}{\frac{x_1}{x_2}}
$$</p>
|
3,784,471 | <p>To solve this exercise,</p>
<p><span class="math-container">$$|\arccos(\cos(x))|<\pi/4$$</span></p>
<p>I have thought to apply this condition,
<span class="math-container">$$|f(x)|<k, \quad k\in \Bbb R^+, \iff -k<f(x)<k$$</span></p>
<p>Hence,</p>
<p><span class="math-container">$$-\frac \pi4<\arccos(\... | tkf | 117,974 | <p>Consider the algebra over <span class="math-container">$\mathbb{F}_3$</span> with basis (as a vector space over <span class="math-container">$\mathbb{F}_3$</span>) the set <span class="math-container">$\{1,x,x^2\}$</span> and multiplication given by:
<span class="math-container">\begin{eqnarray*}
x(x^2)&=&x... |
3,784,471 | <p>To solve this exercise,</p>
<p><span class="math-container">$$|\arccos(\cos(x))|<\pi/4$$</span></p>
<p>I have thought to apply this condition,
<span class="math-container">$$|f(x)|<k, \quad k\in \Bbb R^+, \iff -k<f(x)<k$$</span></p>
<p>Hence,</p>
<p><span class="math-container">$$-\frac \pi4<\arccos(\... | tkf | 117,974 | <p>We provide a family of examples of finite <a href="https://en.wikipedia.org/wiki/Division_algebra" rel="nofollow noreferrer">division algebras</a>, which additionally have a right identity. This does not answer the revised version of the question which asks for a finite division algebra (other than a finite field) ... |
775,265 | <p>Please help me get the answer to this question.</p>
<p>Prove $f(x)=\sqrt{2x-6}$ is continuous at $x=4$ by using precise definition. ($\epsilon-\delta$ definition of limits.)</p>
| Mr.Fry | 68,477 | <p>There is no problem with "reo's" work, he/she gave you a good sketch of the background work. Now we will proceed in the forward argument which somehow where we "magically" know what $\delta$ has to be. </p>
<p>Proof: Given $\epsilon >0$ choose $\delta $= min{$1,\frac{\epsilon}{\sqrt{2}}$}. Then, $|f(x)-f(4)| = \... |
1,022,523 | <blockquote>
<p>Make a series expansion of $f(z)=\dfrac{1}{z^2+z-6}$ valid in the region $2<|z|<3$.</p>
</blockquote>
<p>By partial fractions,</p>
<p>$$f(z) = \frac{1}{(z-2)(z+3)} = \frac{1}{5(z-2)}-\frac{1}{5(z+3)}.$$</p>
<p>From here, how are these fractions expanded into a geometric series?</p>
| Jo Wehler | 169,961 | <p>The correspondence between Grothendieck and Serre during the period 1955-1987 illustrates how Grothendieck's mathematical work shifted from functional analysis to algebraic geometry. The correspondence edited by the American Mathematical Society has the special flavour of being bilingual.</p>
<p><a href="http://rad... |
2,747,753 | <p>Let $x\in\mathbb{R}$. Demonstrate that if the numbers $a = x^3–x$ and $b = x^2 +1$ are rational, then $x$ is rational.</p>
| lulu | 252,071 | <p>$x^2+1\in \mathbb Q \implies x^2\in \mathbb Q$</p>
<p>From this, we deduce that $x^3=rx$ for some $r\in \mathbb Q$. It is easy to see that $r=1\implies x=0,\pm 1$ which are all rational. Assume, then, that $r\neq 1$.</p>
<p>But this implies that $x^3-x=x(r-1)\in \mathbb Q\implies x\in \mathbb Q$ as desired.</p>
|
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | rm -rf | 5 | <p>You can use <code>Subtract</code> or <code>Differences</code> to get the differences in the index and use <code>FreeQ</code> to test if every element is <code>True</code>. This removes the need for the ugly <code>#2[[1]]-#2[[2]]</code> and also the <code>Flatten</code> and <code>And@@</code> at the end.</p>
<pre><c... |
8,382 | <h3>Context</h3>
<p>I'm writing a function that look something like:</p>
<pre><code>triDiagonalQ[mat_] := MapIndexed[ #1 == 0 || Abs[#2[[1]]-#2[[2]]] <= 1 &, mat, {2}] //
Flatten // And @@ # &
</code></pre>
<p>Now, things like <code>#2[[1]]</code> and <code>#2[[2]]</code> are somewhat hard to read. I... | Leonid Shifrin | 81 | <p>You can use macros / code generation to be able to use the syntax you like. Here is one possibility:</p>
<pre><code>ClearAll[withLiteralIndices];
SetAttributes[withLiteralIndices, HoldAll];
withLiteralIndices[code_, inds : {__Symbol}] :=
Block[inds,
Unevaluated[code] /.
MapIndexed[
Function[... |
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"... | user121955 | 121,955 | <p>Also, the definition of a derivative that I learned was $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$...basically rise over run as run ($\mathrm{d}x$) approaches $0$ (thus the tangent line concept).</p>
|
881,282 | <p>Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.</p>
| 2'5 9'2 | 11,123 | <p>One solution is "clear" at $\frac{2}{\sqrt{3}}$. I was motivated to look for something like this by trying to write $1$ as the sum of two simple fractions, and the presence of $3$ and $4$. </p>
<p>It's also halfway between the two vertical asymptotes of $\frac1{x^2}+\frac{1}{\left(4-\sqrt{3}x\right)^2}$, and in a s... |
3,596,566 | <p>I've been going through <a href="https://math.stackexchange.com/a/27177/288450">this proof</a>.</p>
<p>And I'm wondering what allows me to change the order of the integral and the infinite sum.</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \left( \sum_{n \ge 0} \frac{2^n t^n x^n}{n!} \right) e^{-x^... | AlanD | 356,933 | <p>The series can be written as
<span class="math-container">$$
\sum_{n=0}^\infty \frac{(2tx)^n}{n!}e^{-x^2}\leq \sum_{n=0}^\infty \frac{(2|t||x|)^n}{n!}e^{-x^2}=e^{-(x^2-2|t||x|)}=e^{-[(|x|-|t|)^2-|t|^2]}=e^{t^2}e^{-(|x|-|t|)^2}
$$</span>
Obviously, the right hand term is integrable (it is essentially a Gaussian dist... |
3,970,641 | <p>I have 927 unique sequences of the numbers 1, 2 and 3, all of which sum to 12 and represent every possible one-octave scale on the piano, with the numbers representing the intervals between notes in half-steps (i.e., adjacent keys). For example, the <a href="https://en.wikipedia.org/wiki/Major_scale" rel="nofollow n... | Linguosaurus Rex | 1,149,751 | <p>Very interesting question! I've also been thinking about entropy in sequences, but with regards to language. I'm surprised I haven't been able to find much about the entropy of sequences (still looking), but I think I've been able to work some parts of it out.</p>
<p>As the first commenter mentioned, Shannon's entro... |
2,339,101 | <blockquote>
<p>There are six socks in a drawer. The socks are of two colors: black and white. If you draw two socks randomly, the probability that you get white socks is $\frac{2}{3}$. What is the probability of getting black socks, when two socks are drawn at a time?</p>
</blockquote>
<p>There is no detail about t... | jvdhooft | 437,988 | <p>If there are $k$ white socks, the probability of getting two white socks equals:</p>
<p>$$\frac{k}{6} \cdot \frac{k-1}{5} = \frac{k(k-1)}{30} = \frac{2}{3} \iff k(k-1) = 20 \iff k=5$$</p>
<p>Since there are five white socks, it is impossible to draw more than one black sock. As such, the probability of getting two... |
3,014,766 | <p>I am supposed to find the derivative of <span class="math-container">$ 2^{\frac{x}{\ln x}} $</span>. My answer is <span class="math-container">$$ 2^{\frac{x}{\ln x}} \cdot \ln 2 \cdot \frac{\ln x-x\cdot \frac{1}{x}}{\ln^{2}x}\cdot \frac{1}{x} .$$</span> Is it correct? Thanks. </p>
| Robert Z | 299,698 | <p>You are almost correct. You have just an <strong>extra</strong> factor <span class="math-container">$1/x$</span> at the end. The correct derivative is
<span class="math-container">$$D(2^{\frac{x}{\ln x}})=2^{\frac{x}{\ln x}} \cdot \ln 2 \cdot D\left(\frac{x}{\ln x}\right)=2^{\frac{x}{\ln x}} \cdot \ln 2 \cdot \fr... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | orangeskid | 168,051 | <p>It is enough to assume
<span class="math-container">$$|f'(x)|\le f^2(x)$$</span></p>
<p>To show that if <span class="math-container">$\{x \ | \ f(x) = 0\}$</span> is nonvoid then it equals <span class="math-container">$\mathbb{R}$</span>. </p>
<p>Clearly it is a closed set. Let us show that it is also open.</p>... |
352,983 | <p>How to find this expression $(1000!\mod 3^{300})$?</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="https://math.stackexchange.com/questions/141196/highest-power-of-a-prime-p-dividing-n">this</a>, the highest power of $3$ in $1000!>300$</p>
<p>So, the remainder $=0$</p>
|
1,195,625 | <p>We have $X = R^n$ and the discrete metric:</p>
<p>$d(x,y) = 0$, if $x=y$ and $d(x,y) = 1$ in all other cases.</p>
<p>Is this space separable or not? I tried to prove, that the answer for that is no.</p>
<p>Let us have a random $x=(x_1, x_2, ..., x_n)$ vector from $R^n$. If $X$ is separable, then such $q$ exists, ... | Casteels | 92,730 | <p>Let $n=2k$. Then $T_2(n)$ is just the complete bipartite graph $K_{k,k}$. </p>
<p>Let $G=(V,E)$ be a triangle-free graph on $n$ vertices with $\delta(G)=k$. Let $v\in V$ have degree $\delta(G)$. Let $S$ be the neighbours of $v$, and let $T=V\setminus(S\cup v)$. So $|S|=k$ and $|T|=k-1$</p>
<p>Since $G$ is triangle... |
3,753,060 | <blockquote>
<p>If <span class="math-container">$\int f(x)dx =g(x)$</span> then <span class="math-container">$\int f^{-1}(x)dx $</span> is equal to</p>
<p>(1) <span class="math-container">$g^{-1}(x)$</span></p>
<p>(2) <span class="math-container">$xf^{-1}(x)-g(f^{-1}(x))$</span></p>
<p>(3) <span class="math-container">... | Kavi Rama Murthy | 142,385 | <p>Ignoring the constant of integration the answer is (2):<span class="math-container">$$\int f^{-1}(x)dx=\int yf'(y)dy=yf(y)-\int f(y)dy$$</span> (where I have used integration by parts). Hence <span class="math-container">$$\int f^{-1}(x)dx=f^{-1}(x)x-g(y)=xf^{-1}(x)-g(f^{-1}(x))$$</span>.</p>
|
265,189 | <p>Integrate, $$\int_{0}^{\frac{\pi}{2}}\sin (\tan\theta) \mathrm{d\theta}$$</p>
| Mhenni Benghorbal | 35,472 | <p>First, make the change of variables $ x = \arctan(t) $ to transform the integral t0</p>
<p>$$\int_{0}^{\frac{\pi}{2}}\sin (\tan\theta) \mathrm{d\theta} = \int_{0}^{\infty}\frac{\sin(t)}{t^2+1} {dt}
\\
= -\frac{1}{2}\,{{\rm e}^{-1}}{ \operatorname {E_1} } \left( -1 \right) +\frac{1}{2}\,{{\rm e}}\,{\operatorname{E_1... |
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... | Boyku | 567,523 | <p>Number of partitions of m into b positive parts (blocks) is described here: <a href="https://oeis.org/A008284" rel="nofollow noreferrer">https://oeis.org/A008284</a> in the OEIS article. </p>
<p>$P(m,b) = P(m-1, b-1) + P(m-k,b)$</p>
<p>The recurrence formula means that given a $P(m,b)$ partition of m into b blocks... |
11,266 | <p>I have a list of time durations, which are strings of the form: <code>"hh:mm:ss"</code>. Here's a sample for you to play with:</p>
<pre><code>durations = {"00:09:54", "00:31:24", "00:40:07", "00:11:58", "00:13:51", "01:02:32"}
</code></pre>
<p>I want to convert all of these into numbers in seconds, so that I can a... | J. M.'s persistent exhaustion | 50 | <p>One method:</p>
<pre><code>3600 FromDMS[ToExpression[StringSplit[#, ":"]]] & /@ durations
{594, 1884, 2407, 718, 831, 3752}
</code></pre>
|
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... | Keivan Karai | 3,635 | <p>The quotients of an abelian group are in bijection with the subgroups of its Pontryagin dual. Now, every finite abelian group is isomorphic to its dual.</p>
|
2,555,399 | <p>The question is to find out the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{1/2}$</p>
<p>I tried using multinomial theorem but here the exponent is a fraction and I couldn't get how to proceed.Any ideas?</p>
| Bernard | 202,857 | <p>Expand $(1+u)^{\tfrac12}$ up to order $3$:
$$(1+u)^{\tfrac12}=1+\frac12 u-\frac18u^2+\frac1{16}u^3+o(u^3),$$
and compose with $u=-2x+3x^2-4x^3$:</p>
<ul>
<li><p>$u^2=4x^2-12x^3+o(x^3)$,</p></li>
<li><p>$u^3=u^2\cdot u=-8x^3+o(x^3)$.</p></li>
</ul>
<p>One finally obtains$$1-x+x^2-x^3+o(x^3).$$</p>
|
1,136,192 | <p>I need to solve this integral but I have no idea about how to procede, this is the integral:</p>
<p>$$\int \frac{x-1}{x+4x^3}\mathrm dx$$</p>
<p>This is how I solve the first part:</p>
<p>$$\int \frac{x}{x+4x^3}\mathrm dx - \int \frac{1}{x+4x^3}\mathrm dx$$</p>
<p>$$\int \frac{1}{1+4x^2}\mathrm dx - \int \frac{1... | idm | 167,226 | <p><strong>Hint:</strong></p>
<p>$$\ln(n+1)\leq n+1\implies \frac{1}{\ln(n+1)}\geq\frac{1}{n+1}$$</p>
|
244,241 | <p>How can I find minimum distance between cone and a point ?</p>
<p><strong>Cone properties :</strong><br/>
position - $(0,0,z)$<br/>
radius - $R$<br/>
height - $h$</p>
<p><strong>Point properties:</strong><br/>
position - $(0,0,z_1)$</p>
| WimC | 25,313 | <p>This was already mentioned by Rahul but I think it deserves an answer in its own right. Digital signal processing of 1d (sound) and 2d (images) real data would take incredible amounts of time and would be much harder to understand if it weren't for the discrete Fourier transform and its fast implementations. This f... |
1,344,464 | <p>Consider <span class="math-container">$$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$</span></p>
<p>It's a power series with a radius, <span class="math-container">$R=1$</span>. at <span class="math-container">$x=1$</span> it converges. Hence, by Abel's thorem:</p>
<p><span class="math-container">$$\lim_... | Ian | 83,396 | <p>The fundamental theorem tells you that</p>
<p>$$f(x)=f(a)+\int_a^x f'(t) dt.$$</p>
<p>It's convenient to choose $a$ such that $f(a)=0$, because then</p>
<p>$$f(x)=f(a)+\int_a^x f'(t) dt = \int_a^x f'(t) dt.$$</p>
<p>Since your function is a power series with no constant term, it's not hard to see that you can us... |
4,034,709 | <p>What will be the operator norm of the matrix <span class="math-container">$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$</span> where <span class="math-container">$a,b,c,d \in \Bbb C\ $</span>?</p>
<p>According to the definition of the operator norm it turns out that <span class="math-container">$$\|A\|... | Yalikesifulei | 521,468 | <p>Let <span class="math-container">$A$</span> be <span class="math-container">$n\times n$</span> matrix, <span class="math-container">$\|{A}\| = \sup\left\{ \|Ax\| : \|x\| = 1\right\}$</span>. For any <span class="math-container">$x$</span> with <span class="math-container">$\| x\| = 1$</span> we have
<span class="mat... |
45,911 | <p>I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{... | Zhen Lin | 5,191 | <p>It's a good habit to distinguish the coordinates of a point from vectors. As everyone else has pointed out, Euclidean space is special — but I'll add that on top of that, cartesian coordinates on Euclidean space is special. If you use, for example, polar coordinates on Euclidean space, you'll find that you can't sub... |
2,054,676 | <p>I know that m is even and m/2 is odd, but I don't know where/how I can use this. Also, 3y^2 is odd and the sum is odd when x^2 is even. I'm trying to prove that its always odd, but I'm stuck.
Can someone please help?
Thanks</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ {\rm mod}\,\ 2\!:\ x\equiv y\ \Rightarrow\ {\rm mod}\,\ 4\!:\ x^2\equiv y^2\equiv -3y^2$</p>
|
617,389 | <p>How to sketch $y = \frac1{\sqrt{x-1}}$</p>
<p>My way:(which does not work here)</p>
<p>I normally solve these problems by squaring and converting them to equations of 2 degree curves(such as parabola, hyperbola, etc.) which I can easily plot. But this seems to go 3 degree as $xy^2$ term is coming.</p>
<p>Please h... | Clive Newstead | 19,542 | <p>Consider some basic properties of the function, which you can work out either by inspection or by considering derivatives:</p>
<ul>
<li>It is only defined for $x \ge 1$;</li>
<li>It has no roots, stationary points, inflection points, etc.;</li>
<li>It is always decreasing and convex;</li>
<li>It tends to $0$ as $x ... |
3,358,592 | <p>In the following quote, what does the notation <span class="math-container">$\{a_n\}$</span> mean?</p>
<blockquote>
<p>Дана последовательность Фибоначчи <span class="math-container">$\{a_n\}$</span>.</p>
</blockquote>
<p><strong>Translation:</strong> "You are given the Fibonacci sequence <span class="math-contai... | Bill | 735,723 | <p>I apologize for the late response. I recently came across this post and as it so happens, I have been experimenting with such functions recently and figured I would add my two-cents.</p>
<p>For my current research, I needed a smooth(ish) approximation to the Heaviside step function defined on a compact interval. I r... |
306,848 | <p>The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\ast}(map_{\ast}(X,Y);k) $$
where $k$ is a fixed field. Could we say something interesting in the case when $H_{\ast}(X;... | Gregory Arone | 6,668 | <p>An alternative spectral sequence for $H_∗(map_∗(X,Y);k)$ can be constructed using the approach in <a href="https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf" rel="nofollow noreferrer">this paper</a> of mine (apols for self-promotion). Actually, this spectral sequence is di... |
285,114 | <blockquote>
<p>Find the solution of the differential equation
$$\frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)}, y(0)=1$$</p>
</blockquote>
<p>Trial: $$\begin{align} \frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)} \\ \implies \frac{dy}{dx}=-\frac{1+(y/x)^2-10/x^2}{(y/x)(1+(y/x)^2+5/x^2)} \\ \implies v+x\... | Kns | 27,579 | <p>Your differential equation can be written as
$$ (x^{3}+x y^{2}-10 x)dx+(x^{2}y+y^{3}+5y)dy=0$$
So it is of the form $M(x,y)dx+N(x,y)dy=0$
Now, $$M_{y}=2xy$$ and $$N_{x}=2xy.$$
$\therefore$ The given differential equation is exact.
So its general solution is
$$\int M dx+\int (\text{Terms in $N$ which does not contai... |
3,897,361 | <p>Find the GS of the following system of DE's where the independent variable is <span class="math-container">$t$</span> and <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are the dependent variables</p>
<p><span class="math-container">\begin{cases}
x' = x-3y\\
y'=3x+7y
\end{cases}<... | user577215664 | 475,762 | <p><span class="math-container">$$\begin{cases}
x = -C_1e^{4t}-C_2te^{4t}+C_3e^{4t}\\
y = C_1e^{4t}+C_2te^{4t}
\end{cases}$$</span>
You should only end with two constants not three.
<span class="math-container">$$y=C_1e^{4t}+C_2te^{4t}$$</span>
<span class="math-container">$$y'=4C_1e^{4t}+C_2e^{4t}+4tC_2e^{4t}$$</span>... |
392,442 | <p>What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or disprove) the primality of every N-th number. I know this is a very large and subjective answer, however, I would li... | Caleb Stanford | 68,107 | <p>There are in fact many 'formula's which always generate prime numbers. Among the simplest ones listed by <a href="http://en.wikipedia.org/wiki/Formula_for_primes" rel="noreferrer">Wikipedia</a> are:</p>
<ul>
<li>There is some real number $A$ such that $\left\lfloor A^{3^n}\right\rfloor$ is always prime.</li>
<li>T... |
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Unknown | 5,627 | <p>Wiki also says: </p>
<blockquote>
<p>A famous network of conditional proofs is the NP-complete class of complexity theory</p>
</blockquote>
|
3,522,752 | <p>Solve the following equation:
<span class="math-container">$$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}$$</spa... | Oliver Jones | 55,622 | <p>The equation can be written as</p>
<p><span class="math-container">$$\frac{dy}{dx}=\tan \frac{1}{a}(y-x).$$</span></p>
<p>Now make the substitution <span class="math-container">$u=y-x$</span> to get </p>
<p><span class="math-container">$$
\frac{du}{dx}+1=\tan\frac{u}{a}.
$$</span></p>
<p>This equation is now sep... |
1,032,714 | <p>'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that $k_0 = i, k_n = j$ and $$U_{k_0} \cap U_{k_1} \neq \emptyset, U_{k_1} \cap U_{k_2} \neq \emptyset, ..., U_{k_{n-1}} \ca... | Jimmy R. | 128,037 | <p>No, the first formula is a special case of the second. Let $A=\{ω\}\in B$ and $B=B$, then if you substitute in the second formula, you obtain that $$Pr(ω\mid B)=Pr(A\mid B)=\frac{Pr(A\cap B)}{Pr(B)}=\frac{P(\{ω\}\cap B)}{Pr(B)}\overset{ω\in B}=\frac{Pr(\{ω\})}{Pr(B)}$$ as the first formula states.</p>
|
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... | lhf | 589 | <p>By the binomial theorem, $42^n = (43-1)^n=43a+(-1)^n$.</p>
<p>If $n$ is even, then $42^n-1$ is a multiple of $43$.</p>
<p>On the other hand, $42^n = (41+1)^n=41b+1$, and so $42^n-1$ is always a multiple of $41$. Thus, $42^n-1$ is not prime if $n>1$, regardless of the parity of $n$.</p>
|
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